\(\int \frac {1}{x^{5/2} (a+b x^2)^2 (c+d x^2)^2} \, dx\) [819]

Optimal result

Integrand size = 24, antiderivative size = 532 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {-7 b^2 c^2+8 a b c d-7 a^2 d^2}{6 a^2 c^2 (b c-a d)^2 x^{3/2}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {b^{11/4} (7 b c-15 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} (b c-a d)^3}-\frac {b^{11/4} (7 b c-15 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} (b c-a d)^3}+\frac {d^{11/4} (15 b c-7 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{11/4} (b c-a d)^3}-\frac {d^{11/4} (15 b c-7 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{11/4} (b c-a d)^3}-\frac {b^{11/4} (7 b c-15 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{4 \sqrt {2} a^{11/4} (b c-a d)^3}-\frac {d^{11/4} (15 b c-7 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{4 \sqrt {2} c^{11/4} (b c-a d)^3} \] Output:

1/6*(-7*a^2*d^2+8*a*b*c*d-7*b^2*c^2)/a^2/c^2/(-a*d+b*c)^2/x^(3/2)+1/2*d*(a 
*d+b*c)/a/c/(-a*d+b*c)^2/x^(3/2)/(d*x^2+c)+1/2*b/a/(-a*d+b*c)/x^(3/2)/(b*x 
^2+a)/(d*x^2+c)+1/8*b^(11/4)*(-15*a*d+7*b*c)*arctan(1-2^(1/2)*b^(1/4)*x^(1 
/2)/a^(1/4))*2^(1/2)/a^(11/4)/(-a*d+b*c)^3-1/8*b^(11/4)*(-15*a*d+7*b*c)*ar 
ctan(1+2^(1/2)*b^(1/4)*x^(1/2)/a^(1/4))*2^(1/2)/a^(11/4)/(-a*d+b*c)^3+1/8* 
d^(11/4)*(-7*a*d+15*b*c)*arctan(1-2^(1/2)*d^(1/4)*x^(1/2)/c^(1/4))*2^(1/2) 
/c^(11/4)/(-a*d+b*c)^3-1/8*d^(11/4)*(-7*a*d+15*b*c)*arctan(1+2^(1/2)*d^(1/ 
4)*x^(1/2)/c^(1/4))*2^(1/2)/c^(11/4)/(-a*d+b*c)^3-1/8*b^(11/4)*(-15*a*d+7* 
b*c)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*x^(1/2)/(a^(1/2)+b^(1/2)*x))*2^(1/2)/ 
a^(11/4)/(-a*d+b*c)^3-1/8*d^(11/4)*(-7*a*d+15*b*c)*arctanh(2^(1/2)*c^(1/4) 
*d^(1/4)*x^(1/2)/(c^(1/2)+d^(1/2)*x))*2^(1/2)/c^(11/4)/(-a*d+b*c)^3
 

Mathematica [A] (verified)

Time = 1.04 (sec) , antiderivative size = 425, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {1}{24} \left (-\frac {4 \left (7 b^3 c^2 x^2 \left (c+d x^2\right )+a^3 d^2 \left (4 c+7 d x^2\right )+4 a b^2 c \left (c^2-c d x^2-2 d^2 x^4\right )+a^2 b d \left (-8 c^2-4 c d x^2+7 d^2 x^4\right )\right )}{a^2 c^2 (b c-a d)^2 x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {3 \sqrt {2} b^{11/4} (-7 b c+15 a d) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{11/4} (-b c+a d)^3}+\frac {3 \sqrt {2} d^{11/4} (15 b c-7 a d) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{11/4} (b c-a d)^3}+\frac {3 \sqrt {2} b^{11/4} (-7 b c+15 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{11/4} (b c-a d)^3}+\frac {3 \sqrt {2} d^{11/4} (-15 b c+7 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{11/4} (b c-a d)^3}\right ) \] Input:

Integrate[1/(x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^2),x]
 

Output:

((-4*(7*b^3*c^2*x^2*(c + d*x^2) + a^3*d^2*(4*c + 7*d*x^2) + 4*a*b^2*c*(c^2 
 - c*d*x^2 - 2*d^2*x^4) + a^2*b*d*(-8*c^2 - 4*c*d*x^2 + 7*d^2*x^4)))/(a^2* 
c^2*(b*c - a*d)^2*x^(3/2)*(a + b*x^2)*(c + d*x^2)) + (3*Sqrt[2]*b^(11/4)*( 
-7*b*c + 15*a*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqr 
t[x])])/(a^(11/4)*(-(b*c) + a*d)^3) + (3*Sqrt[2]*d^(11/4)*(15*b*c - 7*a*d) 
*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(c^(11/4 
)*(b*c - a*d)^3) + (3*Sqrt[2]*b^(11/4)*(-7*b*c + 15*a*d)*ArcTanh[(Sqrt[2]* 
a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(a^(11/4)*(b*c - a*d)^3) 
+ (3*Sqrt[2]*d^(11/4)*(-15*b*c + 7*a*d)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*S 
qrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(c^(11/4)*(b*c - a*d)^3))/24
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 630, normalized size of antiderivative = 1.18, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {368, 972, 25, 1049, 27, 1053, 27, 1020, 755, 1476, 1082, 217, 1479, 25, 27, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx\)

\(\Big \downarrow \) 368

\(\displaystyle 2 \int \frac {1}{x^2 \left (b x^2+a\right )^2 \left (d x^2+c\right )^2}d\sqrt {x}\)

\(\Big \downarrow \) 972

\(\displaystyle 2 \left (\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {\int -\frac {11 b d x^2+7 b c-4 a d}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )^2}d\sqrt {x}}{4 a (b c-a d)}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 \left (\frac {\int \frac {11 b d x^2+7 b c-4 a d}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )^2}d\sqrt {x}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 1049

\(\displaystyle 2 \left (\frac {\frac {\int \frac {4 \left (7 b^2 c^2-8 a b d c+7 a^2 d^2+7 b d (b c+a d) x^2\right )}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{4 c (b c-a d)}+\frac {d (a d+b c)}{c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {\frac {\int \frac {7 b^2 c^2-8 a b d c+7 a^2 d^2+7 b d (b c+a d) x^2}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{c (b c-a d)}+\frac {d (a d+b c)}{c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 1053

\(\displaystyle 2 \left (\frac {\frac {-\frac {\int \frac {3 \left (b d \left (7 b^2 c^2-8 a b d c+7 a^2 d^2\right ) x^2+(b c+a d) \left (7 b^2 c^2-15 a b d c+7 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{3 a c}-\frac {\frac {7 b^2 c}{a}+\frac {7 a d^2}{c}-8 b d}{3 x^{3/2}}}{c (b c-a d)}+\frac {d (a d+b c)}{c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {\frac {-\frac {\int \frac {b d \left (7 b^2 c^2-8 a b d c+7 a^2 d^2\right ) x^2+(b c+a d) \left (7 b^2 c^2-15 a b d c+7 a^2 d^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{a c}-\frac {\frac {7 b^2 c}{a}+\frac {7 a d^2}{c}-8 b d}{3 x^{3/2}}}{c (b c-a d)}+\frac {d (a d+b c)}{c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 1020

\(\displaystyle 2 \left (\frac {\frac {-\frac {\frac {a^2 d^3 (15 b c-7 a d) \int \frac {1}{d x^2+c}d\sqrt {x}}{b c-a d}+\frac {b^3 c^2 (7 b c-15 a d) \int \frac {1}{b x^2+a}d\sqrt {x}}{b c-a d}}{a c}-\frac {\frac {7 b^2 c}{a}+\frac {7 a d^2}{c}-8 b d}{3 x^{3/2}}}{c (b c-a d)}+\frac {d (a d+b c)}{c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 755

\(\displaystyle 2 \left (\frac {\frac {-\frac {\frac {a^2 d^3 (15 b c-7 a d) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {\sqrt {d} x+\sqrt {c}}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}\right )}{b c-a d}+\frac {b^3 c^2 (7 b c-15 a d) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}\right )}{b c-a d}}{a c}-\frac {\frac {7 b^2 c}{a}+\frac {7 a d^2}{c}-8 b d}{3 x^{3/2}}}{c (b c-a d)}+\frac {d (a d+b c)}{c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 1476

\(\displaystyle 2 \left (\frac {\frac {-\frac {\frac {a^2 d^3 (15 b c-7 a d) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}}{2 \sqrt {c}}\right )}{b c-a d}+\frac {b^3 c^2 (7 b c-15 a d) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {a}}\right )}{b c-a d}}{a c}-\frac {\frac {7 b^2 c}{a}+\frac {7 a d^2}{c}-8 b d}{3 x^{3/2}}}{c (b c-a d)}+\frac {d (a d+b c)}{c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle 2 \left (\frac {\frac {-\frac {\frac {a^2 d^3 (15 b c-7 a d) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}+\frac {b^3 c^2 (7 b c-15 a d) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}}{a c}-\frac {\frac {7 b^2 c}{a}+\frac {7 a d^2}{c}-8 b d}{3 x^{3/2}}}{c (b c-a d)}+\frac {d (a d+b c)}{c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle 2 \left (\frac {\frac {-\frac {\frac {a^2 d^3 (15 b c-7 a d) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}+\frac {b^3 c^2 (7 b c-15 a d) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}}{a c}-\frac {\frac {7 b^2 c}{a}+\frac {7 a d^2}{c}-8 b d}{3 x^{3/2}}}{c (b c-a d)}+\frac {d (a d+b c)}{c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 1479

\(\displaystyle 2 \left (\frac {\frac {-\frac {\frac {a^2 d^3 (15 b c-7 a d) \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}+\frac {b^3 c^2 (7 b c-15 a d) \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}}{a c}-\frac {\frac {7 b^2 c}{a}+\frac {7 a d^2}{c}-8 b d}{3 x^{3/2}}}{c (b c-a d)}+\frac {d (a d+b c)}{c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 \left (\frac {\frac {-\frac {\frac {a^2 d^3 (15 b c-7 a d) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}+\frac {b^3 c^2 (7 b c-15 a d) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}}{a c}-\frac {\frac {7 b^2 c}{a}+\frac {7 a d^2}{c}-8 b d}{3 x^{3/2}}}{c (b c-a d)}+\frac {d (a d+b c)}{c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {\frac {-\frac {\frac {a^2 d^3 (15 b c-7 a d) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}+\frac {b^3 c^2 (7 b c-15 a d) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}}{a c}-\frac {\frac {7 b^2 c}{a}+\frac {7 a d^2}{c}-8 b d}{3 x^{3/2}}}{c (b c-a d)}+\frac {d (a d+b c)}{c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle 2 \left (\frac {\frac {-\frac {\frac {a^2 d^3 (15 b c-7 a d) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}+\frac {b^3 c^2 (7 b c-15 a d) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}}{a c}-\frac {\frac {7 b^2 c}{a}+\frac {7 a d^2}{c}-8 b d}{3 x^{3/2}}}{c (b c-a d)}+\frac {d (a d+b c)}{c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}\right )\)

Input:

Int[1/(x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^2),x]
 

Output:

2*(b/(4*a*(b*c - a*d)*x^(3/2)*(a + b*x^2)*(c + d*x^2)) + ((d*(b*c + a*d))/ 
(c*(b*c - a*d)*x^(3/2)*(c + d*x^2)) + (-1/3*((7*b^2*c)/a - 8*b*d + (7*a*d^ 
2)/c)/x^(3/2) - ((b^3*c^2*(7*b*c - 15*a*d)*((-(ArcTan[1 - (Sqrt[2]*b^(1/4) 
*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4))) + ArcTan[1 + (Sqrt[2]*b^(1/4 
)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a]) + (-1/2*Log[Sqr 
t[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(Sqrt[2]*a^(1/4)*b^(1/ 
4)) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(2*Sqrt[2 
]*a^(1/4)*b^(1/4)))/(2*Sqrt[a])))/(b*c - a*d) + (a^2*d^3*(15*b*c - 7*a*d)* 
((-(ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)]/(Sqrt[2]*c^(1/4)*d^(1/4) 
)) + ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)]/(Sqrt[2]*c^(1/4)*d^(1/4 
)))/(2*Sqrt[c]) + (-1/2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sq 
rt[d]*x]/(Sqrt[2]*c^(1/4)*d^(1/4)) + Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4) 
*Sqrt[x] + Sqrt[d]*x]/(2*Sqrt[2]*c^(1/4)*d^(1/4)))/(2*Sqrt[c])))/(b*c - a* 
d))/(a*c))/(c*(b*c - a*d)))/(4*a*(b*c - a*d)))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 368

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) 
, x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e   Subst[Int[x^(k*(m + 1) 
 - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], 
 x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m 
] && IntegerQ[p]
 

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 972

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x 
^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*(p + 
 1))   Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*( 
b*c - a*d)*(p + 1) + d*b*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{ 
a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] & 
& IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
 

rule 1020

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^( 
n_))), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^n), x], x 
] - Simp[(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^n), x], x] /; FreeQ[{a, b 
, c, d, e, f, n}, x]
 

rule 1049

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m 
 + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p + 1))) 
, x] + Simp[1/(a*n*(b*c - a*d)*(p + 1))   Int[(g*x)^m*(a + b*x^n)^(p + 1)*( 
c + d*x^n)^q*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e 
- a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, 
g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]
 

rule 1053

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ 
))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b 
*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( 
m + 1))   Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) 
- e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 
) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 
0] && LtQ[m, -1]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.61

method	result	size
derivativedivides	\(-\frac {2 b^{3} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (15 a d -7 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{a^{2} \left (a d -b c \right )^{3}}-\frac {2 d^{3} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) \sqrt {x}}{x^{2} d +c}+\frac {\left (7 a d -15 b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c}\right )}{c^{2} \left (a d -b c \right )^{3}}-\frac {2}{3 a^{2} c^{2} x^{\frac {3}{2}}}\)	\(323\)
default	\(-\frac {2 b^{3} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (15 a d -7 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{a^{2} \left (a d -b c \right )^{3}}-\frac {2 d^{3} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) \sqrt {x}}{x^{2} d +c}+\frac {\left (7 a d -15 b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c}\right )}{c^{2} \left (a d -b c \right )^{3}}-\frac {2}{3 a^{2} c^{2} x^{\frac {3}{2}}}\)	\(323\)
risch	\(-\frac {2}{3 a^{2} c^{2} x^{\frac {3}{2}}}-\frac {\frac {2 a^{2} d^{3} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) \sqrt {x}}{x^{2} d +c}+\frac {\left (7 a d -15 b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c}\right )}{\left (a d -b c \right )^{3}}+\frac {2 c^{2} b^{3} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (15 a d -7 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{\left (a d -b c \right )^{3}}}{a^{2} c^{2}}\)	\(332\)

Input:

int(1/x^(5/2)/(b*x^2+a)^2/(d*x^2+c)^2,x,method=_RETURNVERBOSE)
 

Output:

-2*b^3/a^2/(a*d-b*c)^3*((1/4*a*d-1/4*b*c)*x^(1/2)/(b*x^2+a)+1/32*(15*a*d-7 
*b*c)*(a/b)^(1/4)/a*2^(1/2)*(ln((x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2) 
)/(x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4 
)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)))-2*d^3/c^2/(a*d-b*c) 
^3*((1/4*a*d-1/4*b*c)*x^(1/2)/(d*x^2+c)+1/32*(7*a*d-15*b*c)*(c/d)^(1/4)/c* 
2^(1/2)*(ln((x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*x^( 
1/2)*2^(1/2)+(c/d)^(1/2)))+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+2*arcta 
n(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)))-2/3/a^2/c^2/x^(3/2)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate(1/x^(5/2)/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate(1/x**(5/2)/(b*x**2+a)**2/(d*x**2+c)**2,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 761, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx =\text {Too large to display} \] Input:

integrate(1/x^(5/2)/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="maxima")
 

Output:

-1/16*(2*sqrt(2)*(7*b*c - 15*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1 
/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt 
(b))) + 2*sqrt(2)*(7*b*c - 15*a*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^ 
(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sq 
rt(b))) + sqrt(2)*(7*b*c - 15*a*d)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + s 
qrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(7*b*c - 15*a*d)*log(-sqrt 
(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))*b^3/ 
(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3) - 1/6*(4*a*b^2*c 
^3 - 8*a^2*b*c^2*d + 4*a^3*c*d^2 + (7*b^3*c^2*d - 8*a*b^2*c*d^2 + 7*a^2*b* 
d^3)*x^4 + (7*b^3*c^3 - 4*a*b^2*c^2*d - 4*a^2*b*c*d^2 + 7*a^3*d^3)*x^2)/(( 
a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*x^(11/2) + (a^2*b^3*c^5 
 - a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3)*x^(7/2) + (a^3*b^2*c^5 - 2 
*a^4*b*c^4*d + a^5*c^3*d^2)*x^(3/2)) - 1/16*(2*sqrt(2)*(15*b*c*d^3 - 7*a*d 
^4)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt( 
sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(15*b*c*d^3 
- 7*a*d^4)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x 
))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(15*b* 
c*d^3 - 7*a*d^4)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c) 
)/(c^(3/4)*d^(1/4)) - sqrt(2)*(15*b*c*d^3 - 7*a*d^4)*log(-sqrt(2)*c^(1/4)* 
d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)))/(b^3*c^5 - 3*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1012 vs. \(2 (416) = 832\).

Time = 0.30 (sec) , antiderivative size = 1012, normalized size of antiderivative = 1.90 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(1/x^(5/2)/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="giac")
 

Output:

-1/4*(7*(a*b^3)^(1/4)*b^3*c - 15*(a*b^3)^(1/4)*a*b^2*d)*arctan(1/2*sqrt(2) 
*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^3*b^3*c^3 - 3*s 
qrt(2)*a^4*b^2*c^2*d + 3*sqrt(2)*a^5*b*c*d^2 - sqrt(2)*a^6*d^3) - 1/4*(7*( 
a*b^3)^(1/4)*b^3*c - 15*(a*b^3)^(1/4)*a*b^2*d)*arctan(-1/2*sqrt(2)*(sqrt(2 
)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^3*b^3*c^3 - 3*sqrt(2)*a 
^4*b^2*c^2*d + 3*sqrt(2)*a^5*b*c*d^2 - sqrt(2)*a^6*d^3) - 1/4*(15*(c*d^3)^ 
(1/4)*b*c*d^2 - 7*(c*d^3)^(1/4)*a*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^( 
1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^3*c^6 - 3*sqrt(2)*a*b^2*c^5*d + 
3*sqrt(2)*a^2*b*c^4*d^2 - sqrt(2)*a^3*c^3*d^3) - 1/4*(15*(c*d^3)^(1/4)*b*c 
*d^2 - 7*(c*d^3)^(1/4)*a*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2 
*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^3*c^6 - 3*sqrt(2)*a*b^2*c^5*d + 3*sqrt(2 
)*a^2*b*c^4*d^2 - sqrt(2)*a^3*c^3*d^3) - 1/8*(7*(a*b^3)^(1/4)*b^3*c - 15*( 
a*b^3)^(1/4)*a*b^2*d)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sq 
rt(2)*a^3*b^3*c^3 - 3*sqrt(2)*a^4*b^2*c^2*d + 3*sqrt(2)*a^5*b*c*d^2 - sqrt 
(2)*a^6*d^3) + 1/8*(7*(a*b^3)^(1/4)*b^3*c - 15*(a*b^3)^(1/4)*a*b^2*d)*log( 
-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^3*b^3*c^3 - 3*sqr 
t(2)*a^4*b^2*c^2*d + 3*sqrt(2)*a^5*b*c*d^2 - sqrt(2)*a^6*d^3) - 1/8*(15*(c 
*d^3)^(1/4)*b*c*d^2 - 7*(c*d^3)^(1/4)*a*d^3)*log(sqrt(2)*sqrt(x)*(c/d)^(1/ 
4) + x + sqrt(c/d))/(sqrt(2)*b^3*c^6 - 3*sqrt(2)*a*b^2*c^5*d + 3*sqrt(2)*a 
^2*b*c^4*d^2 - sqrt(2)*a^3*c^3*d^3) + 1/8*(15*(c*d^3)^(1/4)*b*c*d^2 - 7...
 

Mupad [B] (verification not implemented)

Time = 10.50 (sec) , antiderivative size = 44436, normalized size of antiderivative = 83.53 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

int(1/(x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^2),x)
 

Output:

atan((((-(2401*b^15*c^4 + 50625*a^4*b^11*d^4 - 94500*a^3*b^12*c*d^3 + 6615 
0*a^2*b^13*c^2*d^2 - 20580*a*b^14*c^3*d)/(4096*a^23*d^12 + 4096*a^11*b^12* 
c^12 - 49152*a^12*b^11*c^11*d + 270336*a^13*b^10*c^10*d^2 - 901120*a^14*b^ 
9*c^9*d^3 + 2027520*a^15*b^8*c^8*d^4 - 3244032*a^16*b^7*c^7*d^5 + 3784704* 
a^17*b^6*c^6*d^6 - 3244032*a^18*b^5*c^5*d^7 + 2027520*a^19*b^4*c^4*d^8 - 9 
01120*a^20*b^3*c^3*d^9 + 270336*a^21*b^2*c^2*d^10 - 49152*a^22*b*c*d^11))^ 
(1/4)*(((-(2401*b^15*c^4 + 50625*a^4*b^11*d^4 - 94500*a^3*b^12*c*d^3 + 661 
50*a^2*b^13*c^2*d^2 - 20580*a*b^14*c^3*d)/(4096*a^23*d^12 + 4096*a^11*b^12 
*c^12 - 49152*a^12*b^11*c^11*d + 270336*a^13*b^10*c^10*d^2 - 901120*a^14*b 
^9*c^9*d^3 + 2027520*a^15*b^8*c^8*d^4 - 3244032*a^16*b^7*c^7*d^5 + 3784704 
*a^17*b^6*c^6*d^6 - 3244032*a^18*b^5*c^5*d^7 + 2027520*a^19*b^4*c^4*d^8 - 
901120*a^20*b^3*c^3*d^9 + 270336*a^21*b^2*c^2*d^10 - 49152*a^22*b*c*d^11)) 
^(1/4)*(117440512*a^25*b^38*c^59*d^4 - 3657433088*a^26*b^37*c^58*d^5 + 549 
78936832*a^27*b^36*c^57*d^6 - 531300876288*a^28*b^35*c^56*d^7 + 3709140467 
712*a^29*b^34*c^55*d^8 - 19931198390272*a^30*b^33*c^54*d^9 + 8577784532172 
8*a^31*b^32*c^53*d^10 - 303808739540992*a^32*b^31*c^52*d^11 + 903261116694 
528*a^33*b^30*c^51*d^12 - 2288995975299072*a^34*b^29*c^50*d^13 + 500618250 
6823680*a^35*b^28*c^49*d^14 - 9552410255032320*a^36*b^27*c^48*d^15 + 16064 
830746132480*a^37*b^26*c^47*d^16 - 24054442827448320*a^38*b^25*c^46*d^17 + 
 32403938271559680*a^39*b^24*c^45*d^18 - 39685869262602240*a^40*b^23*c^...
 

Reduce [B] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 2698, normalized size of antiderivative = 5.07 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int(1/x^(5/2)/(b*x^2+a)^2/(d*x^2+c)^2,x)
 

Output:

(90*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2* 
sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b**2*c**4*d*x + 90*sqrt 
(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)* 
sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b**2*c**3*d**2*x**3 - 42*sqrt(x 
)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sq 
rt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b**3*c**5*x + 48*sqrt(x)*b**(3/4)*a* 
*(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1 
/4)*a**(1/4)*sqrt(2)))*a*b**3*c**4*d*x**3 + 90*sqrt(x)*b**(3/4)*a**(1/4)*s 
qrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**( 
1/4)*sqrt(2)))*a*b**3*c**3*d**2*x**5 - 42*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2 
)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)* 
sqrt(2)))*b**4*c**5*x**3 - 42*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**( 
1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b* 
*4*c**4*d*x**5 - 90*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1 
/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b**2*c* 
*4*d*x - 90*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt 
(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b**2*c**3*d**2* 
x**3 + 42*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2 
) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b**3*c**5*x - 48*sqr 
t(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt...