\(\int \frac {x^{7/2}}{(a+b x^2)^2 (c+d x^2)^2} \, dx\) [813]

Optimal result

Integrand size = 24, antiderivative size = 480 \[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {(b c+a d) \sqrt {x}}{2 b (b c-a d)^2 \left (c+d x^2\right )}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\sqrt [4]{a} (5 b c+3 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{a} (5 b c+3 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{c} (3 b c+5 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{c} (3 b c+5 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}-\frac {\sqrt [4]{a} (5 b c+3 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{4 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}+\frac {\sqrt [4]{c} (3 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{4 \sqrt {2} \sqrt [4]{d} (b c-a d)^3} \] Output:

1/2*(a*d+b*c)*x^(1/2)/b/(-a*d+b*c)^2/(d*x^2+c)+1/2*a*x^(1/2)/b/(-a*d+b*c)/ 
(b*x^2+a)/(d*x^2+c)+1/8*a^(1/4)*(3*a*d+5*b*c)*arctan(1-2^(1/2)*b^(1/4)*x^( 
1/2)/a^(1/4))*2^(1/2)/b^(1/4)/(-a*d+b*c)^3-1/8*a^(1/4)*(3*a*d+5*b*c)*arcta 
n(1+2^(1/2)*b^(1/4)*x^(1/2)/a^(1/4))*2^(1/2)/b^(1/4)/(-a*d+b*c)^3-1/8*c^(1 
/4)*(5*a*d+3*b*c)*arctan(1-2^(1/2)*d^(1/4)*x^(1/2)/c^(1/4))*2^(1/2)/d^(1/4 
)/(-a*d+b*c)^3+1/8*c^(1/4)*(5*a*d+3*b*c)*arctan(1+2^(1/2)*d^(1/4)*x^(1/2)/ 
c^(1/4))*2^(1/2)/d^(1/4)/(-a*d+b*c)^3-1/8*a^(1/4)*(3*a*d+5*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)/b^(1/4)/(-a*d+b 
*c)^3+1/8*c^(1/4)*(5*a*d+3*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)/d^(1/4)/(-a*d+b*c)^3
 

Mathematica [A] (verified)

Time = 1.21 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.72 \[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {1}{8} \left (\frac {4 \sqrt {x} \left (2 a c+b c x^2+a d x^2\right )}{(b c-a d)^2 \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\sqrt {2} \sqrt [4]{a} (5 b c+3 a d) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{b} (b c-a d)^3}+\frac {\sqrt {2} \sqrt [4]{c} (3 b c+5 a d) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{\sqrt [4]{d} (-b c+a d)^3}-\frac {\sqrt {2} \sqrt [4]{a} (5 b c+3 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{b} (b c-a d)^3}+\frac {\sqrt {2} \sqrt [4]{c} (3 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{\sqrt [4]{d} (b c-a d)^3}\right ) \] Input:

Integrate[x^(7/2)/((a + b*x^2)^2*(c + d*x^2)^2),x]
 

Output:

((4*Sqrt[x]*(2*a*c + b*c*x^2 + a*d*x^2))/((b*c - a*d)^2*(a + b*x^2)*(c + d 
*x^2)) + (Sqrt[2]*a^(1/4)*(5*b*c + 3*a*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sq 
rt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(b^(1/4)*(b*c - a*d)^3) + (Sqrt[2]*c^(1/4 
)*(3*b*c + 5*a*d)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sq 
rt[x])])/(d^(1/4)*(-(b*c) + a*d)^3) - (Sqrt[2]*a^(1/4)*(5*b*c + 3*a*d)*Arc 
Tanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(b^(1/4)*(b 
*c - a*d)^3) + (Sqrt[2]*c^(1/4)*(3*b*c + 5*a*d)*ArcTanh[(Sqrt[2]*c^(1/4)*d 
^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(d^(1/4)*(b*c - a*d)^3))/8
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.20, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {368, 970, 1024, 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.

\(\displaystyle \int \frac {x^{7/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx\)

\(\Big \downarrow \) 368

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

\(\Big \downarrow \) 970

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

\(\Big \downarrow \) 1024

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1020

\(\displaystyle 2 \left (\frac {a \sqrt {x}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {b \left (\frac {a (3 a d+5 b c) \int \frac {1}{b x^2+a}d\sqrt {x}}{b c-a d}-\frac {c (5 a d+3 b c) \int \frac {1}{d x^2+c}d\sqrt {x}}{b c-a d}\right )}{b c-a d}-\frac {\sqrt {x} (a d+b c)}{\left (c+d x^2\right ) (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 755

\(\displaystyle 2 \left (\frac {a \sqrt {x}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {b \left (\frac {a (3 a d+5 b c) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {c (5 a d+3 b c) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {\sqrt {d} x+\sqrt {c}}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}\right )}{b c-a d}\right )}{b c-a d}-\frac {\sqrt {x} (a d+b c)}{\left (c+d x^2\right ) (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 1476

\(\displaystyle 2 \left (\frac {a \sqrt {x}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {b \left (\frac {a (3 a d+5 b c) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {c (5 a d+3 b c) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}}{2 \sqrt {c}}\right )}{b c-a d}\right )}{b c-a d}-\frac {\sqrt {x} (a d+b c)}{\left (c+d x^2\right ) (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle 2 \left (\frac {a \sqrt {x}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {b \left (\frac {a (3 a d+5 b c) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {c (5 a d+3 b c) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}\right )}{b c-a d}-\frac {\sqrt {x} (a d+b c)}{\left (c+d x^2\right ) (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle 2 \left (\frac {a \sqrt {x}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {b \left (\frac {a (3 a d+5 b c) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {c (5 a d+3 b c) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}\right )}{b c-a d}-\frac {\sqrt {x} (a d+b c)}{\left (c+d x^2\right ) (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 1479

\(\displaystyle 2 \left (\frac {a \sqrt {x}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {b \left (\frac {a (3 a d+5 b c) \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {c (5 a d+3 b c) \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}\right )}{b c-a d}-\frac {\sqrt {x} (a d+b c)}{\left (c+d x^2\right ) (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 \left (\frac {a \sqrt {x}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {b \left (\frac {a (3 a d+5 b c) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {c (5 a d+3 b c) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}\right )}{b c-a d}-\frac {\sqrt {x} (a d+b c)}{\left (c+d x^2\right ) (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {a \sqrt {x}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {b \left (\frac {a (3 a d+5 b c) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {c (5 a d+3 b c) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}\right )}{b c-a d}-\frac {\sqrt {x} (a d+b c)}{\left (c+d x^2\right ) (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle 2 \left (\frac {a \sqrt {x}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {b \left (\frac {a (3 a d+5 b c) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {c (5 a d+3 b c) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}\right )}{b c-a d}-\frac {\sqrt {x} (a d+b c)}{\left (c+d x^2\right ) (b c-a d)}}{4 b (b c-a d)}\right )\)

Input:

Int[x^(7/2)/((a + b*x^2)^2*(c + d*x^2)^2),x]
 

Output:

2*((a*Sqrt[x])/(4*b*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)) - (-(((b*c + a*d) 
*Sqrt[x])/((b*c - a*d)*(c + d*x^2))) + (b*((a*(5*b*c + 3*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[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(Sq 
rt[2]*a^(1/4)*b^(1/4)) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + S 
qrt[b]*x]/(2*Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a])))/(b*c - a*d) - (c*(3*b 
*c + 5*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] + Sqrt[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)))/(b*c - a*d))/(4*b*(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 970

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[(-a)*e^(2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n) 
^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Simp[e^(2*n) 
/(b*n*(b*c - a*d)*(p + 1))   Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d 
*x^n)^q*Simp[a*c*(m - 2*n + 1) + (a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^ 
n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[ 
n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && 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 1024

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f 
_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c 
+ d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*( 
p + 1))   Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b 
*c - a*d)*(p + 1) + d*(b*e - a*f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; Fr 
eeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -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 = 1.51 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.63

method	result	size
derivativedivides	\(\frac {2 a \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (3 a d +5 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{\left (a d -b c \right )^{3}}-\frac {2 c \left (\frac {\left (-\frac {a d}{4}+\frac {b c}{4}\right ) \sqrt {x}}{x^{2} d +c}+\frac {\left (5 a d +3 b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c}\right )}{\left (a d -b c \right )^{3}}\)	\(302\)
default	\(\frac {2 a \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (3 a d +5 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{\left (a d -b c \right )^{3}}-\frac {2 c \left (\frac {\left (-\frac {a d}{4}+\frac {b c}{4}\right ) \sqrt {x}}{x^{2} d +c}+\frac {\left (5 a d +3 b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c}\right )}{\left (a d -b c \right )^{3}}\)	\(302\)

Input:

int(x^(7/2)/(b*x^2+a)^2/(d*x^2+c)^2,x,method=_RETURNVERBOSE)
 

Output:

2*a/(a*d-b*c)^3*((1/4*a*d-1/4*b*c)*x^(1/2)/(b*x^2+a)+1/32*(3*a*d+5*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*c/(a*d-b*c)^3*((-1/4*a*d+ 
1/4*b*c)*x^(1/2)/(d*x^2+c)+1/32*(5*a*d+3*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*arctan(2^(1/2)/(c/d 
)^(1/4)*x^(1/2)-1)))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 15.14 (sec) , antiderivative size = 4935, normalized size of antiderivative = 10.28 \[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(x^(7/2)/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x**(7/2)/(b*x**2+a)**2/(d*x**2+c)**2,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 617, normalized size of antiderivative = 1.29 \[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx =\text {Too large to display} \] Input:

integrate(x^(7/2)/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="maxima")
 

Output:

-1/16*(2*sqrt(2)*(5*b*c + 3*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)*(5*b*c + 3*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))) + sqrt(2)*(5*b*c + 3*a*d)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt 
(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(5*b*c + 3*a*d)*log(-sqrt(2)* 
a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))*a/(b^3*c 
^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) + 1/16*(2*sqrt(2)*(3*b*c + 5 
*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqr 
t(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(3*b*c + 5 
*a*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/sq 
rt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(3*b*c + 5* 
a*d)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d 
^(1/4)) - sqrt(2)*(3*b*c + 5*a*d)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + s 
qrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)))*c/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2* 
b*c*d^2 - a^3*d^3) + 1/2*((b*c + a*d)*x^(5/2) + 2*a*c*sqrt(x))/(a*b^2*c^3 
- 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^4 
+ (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 912 vs. \(2 (368) = 736\).

Time = 0.27 (sec) , antiderivative size = 912, normalized size of antiderivative = 1.90 \[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(x^(7/2)/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="giac")
 

Output:

-1/4*(5*(a*b^3)^(1/4)*b*c + 3*(a*b^3)^(1/4)*a*d)*arctan(1/2*sqrt(2)*(sqrt( 
2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*b^4*c^3 - 3*sqrt(2)*a*b^ 
3*c^2*d + 3*sqrt(2)*a^2*b^2*c*d^2 - sqrt(2)*a^3*b*d^3) - 1/4*(5*(a*b^3)^(1 
/4)*b*c + 3*(a*b^3)^(1/4)*a*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 
2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*b^4*c^3 - 3*sqrt(2)*a*b^3*c^2*d + 3*sqrt( 
2)*a^2*b^2*c*d^2 - sqrt(2)*a^3*b*d^3) + 1/4*(3*(c*d^3)^(1/4)*b*c + 5*(c*d^ 
3)^(1/4)*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^( 
1/4))/(sqrt(2)*b^3*c^3*d - 3*sqrt(2)*a*b^2*c^2*d^2 + 3*sqrt(2)*a^2*b*c*d^3 
 - sqrt(2)*a^3*d^4) + 1/4*(3*(c*d^3)^(1/4)*b*c + 5*(c*d^3)^(1/4)*a*d)*arct 
an(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^ 
3*c^3*d - 3*sqrt(2)*a*b^2*c^2*d^2 + 3*sqrt(2)*a^2*b*c*d^3 - sqrt(2)*a^3*d^ 
4) - 1/8*(5*(a*b^3)^(1/4)*b*c + 3*(a*b^3)^(1/4)*a*d)*log(sqrt(2)*sqrt(x)*( 
a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*b^4*c^3 - 3*sqrt(2)*a*b^3*c^2*d + 3*s 
qrt(2)*a^2*b^2*c*d^2 - sqrt(2)*a^3*b*d^3) + 1/8*(5*(a*b^3)^(1/4)*b*c + 3*( 
a*b^3)^(1/4)*a*d)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt( 
2)*b^4*c^3 - 3*sqrt(2)*a*b^3*c^2*d + 3*sqrt(2)*a^2*b^2*c*d^2 - sqrt(2)*a^3 
*b*d^3) + 1/8*(3*(c*d^3)^(1/4)*b*c + 5*(c*d^3)^(1/4)*a*d)*log(sqrt(2)*sqrt 
(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^3*c^3*d - 3*sqrt(2)*a*b^2*c^2* 
d^2 + 3*sqrt(2)*a^2*b*c*d^3 - sqrt(2)*a^3*d^4) - 1/8*(3*(c*d^3)^(1/4)*b*c 
+ 5*(c*d^3)^(1/4)*a*d)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d)...
 

Mupad [B] (verification not implemented)

Time = 4.36 (sec) , antiderivative size = 34921, normalized size of antiderivative = 72.75 \[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

int(x^(7/2)/((a + b*x^2)^2*(c + d*x^2)^2),x)
 

Output:

atan(((-(81*b^4*c^5 + 625*a^4*c*d^4 + 1500*a^3*b*c^2*d^3 + 1350*a^2*b^2*c^ 
3*d^2 + 540*a*b^3*c^4*d)/(4096*a^12*d^13 + 4096*b^12*c^12*d - 49152*a*b^11 
*c^11*d^2 + 270336*a^2*b^10*c^10*d^3 - 901120*a^3*b^9*c^9*d^4 + 2027520*a^ 
4*b^8*c^8*d^5 - 3244032*a^5*b^7*c^7*d^6 + 3784704*a^6*b^6*c^6*d^7 - 324403 
2*a^7*b^5*c^5*d^8 + 2027520*a^8*b^4*c^4*d^9 - 901120*a^9*b^3*c^3*d^10 + 27 
0336*a^10*b^2*c^2*d^11 - 49152*a^11*b*c*d^12))^(1/4)*((-(81*b^4*c^5 + 625* 
a^4*c*d^4 + 1500*a^3*b*c^2*d^3 + 1350*a^2*b^2*c^3*d^2 + 540*a*b^3*c^4*d)/( 
4096*a^12*d^13 + 4096*b^12*c^12*d - 49152*a*b^11*c^11*d^2 + 270336*a^2*b^1 
0*c^10*d^3 - 901120*a^3*b^9*c^9*d^4 + 2027520*a^4*b^8*c^8*d^5 - 3244032*a^ 
5*b^7*c^7*d^6 + 3784704*a^6*b^6*c^6*d^7 - 3244032*a^7*b^5*c^5*d^8 + 202752 
0*a^8*b^4*c^4*d^9 - 901120*a^9*b^3*c^3*d^10 + 270336*a^10*b^2*c^2*d^11 - 4 
9152*a^11*b*c*d^12))^(1/4)*(((405*a^2*b^9*c^8*d^3)/2 + 1674*a^3*b^8*c^7*d^ 
4 + (9843*a^4*b^7*c^6*d^5)/2 + 6884*a^5*b^6*c^5*d^6 + (9843*a^6*b^5*c^4*d^ 
7)/2 + 1674*a^7*b^4*c^3*d^8 + (405*a^8*b^3*c^2*d^9)/2)/(a^8*d^8 + b^8*c^8 
+ 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^ 
3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7) + (-(81*b^ 
4*c^5 + 625*a^4*c*d^4 + 1500*a^3*b*c^2*d^3 + 1350*a^2*b^2*c^3*d^2 + 540*a* 
b^3*c^4*d)/(4096*a^12*d^13 + 4096*b^12*c^12*d - 49152*a*b^11*c^11*d^2 + 27 
0336*a^2*b^10*c^10*d^3 - 901120*a^3*b^9*c^9*d^4 + 2027520*a^4*b^8*c^8*d^5 
- 3244032*a^5*b^7*c^7*d^6 + 3784704*a^6*b^6*c^6*d^7 - 3244032*a^7*b^5*c...
 

Reduce [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 2423, normalized size of antiderivative = 5.05 \[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int(x^(7/2)/(b*x^2+a)^2/(d*x^2+c)^2,x)
 

Output:

( - 6*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*c*d**2 - 6*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*d**3*x**2 - 10*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* 
c**2*d - 16*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*c*d**2*x**2 - 6*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*d**3*x**4 - 10*b**(3/4)*a**(1/4)*sqrt(2)*ata 
n((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt( 
2)))*b**2*c**2*d*x**2 - 10*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**2*c*d**2*x 
**4 + 6*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*c*d**2 + 6*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*d**3*x**2 + 10*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*c**2*d + 16*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*c*d**2*x**2 + 6*b**(3/ 
4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b)...