\(\int \frac {x}{\sqrt {c+d x} (a+b x^2)^3} \, dx\) [741]

Optimal result

Integrand size = 20, antiderivative size = 672 \[ \int \frac {x}{\sqrt {c+d x} \left (a+b x^2\right )^3} \, dx=-\frac {(c-d x) \sqrt {c+d x}}{4 \left (b c^2+a d^2\right ) \left (a+b x^2\right )^2}-\frac {d \sqrt {c+d x} \left (b c^2 x+a d (6 c-5 d x)\right )}{16 a \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )}+\frac {d^2 \left (b^{3/2} c^3+13 a \sqrt {b} c d^2+\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}-\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}\right )}{32 \sqrt {2} a b^{3/4} \left (b c^2+a d^2\right )^{5/2} \sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}-\frac {d^2 \left (b^{3/2} c^3+13 a \sqrt {b} c d^2+\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}+\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}\right )}{32 \sqrt {2} a b^{3/4} \left (b c^2+a d^2\right )^{5/2} \sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}-\frac {d^2 \left (b^{3/2} c^3+13 a \sqrt {b} c d^2-\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt {b c^2+a d^2}+\sqrt {b} (c+d x)}\right )}{32 \sqrt {2} a b^{3/4} \left (b c^2+a d^2\right )^{5/2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}} \] Output:

-1/4*(-d*x+c)*(d*x+c)^(1/2)/(a*d^2+b*c^2)/(b*x^2+a)^2-1/16*d*(d*x+c)^(1/2) 
*(b*c^2*x+a*d*(-5*d*x+6*c))/a/(a*d^2+b*c^2)^2/(b*x^2+a)+1/64*d^2*(b^(3/2)* 
c^3+13*a*b^(1/2)*c*d^2+(-5*a*d^2+b*c^2)*(a*d^2+b*c^2)^(1/2))*arctan(((b^(1 
/2)*c+(a*d^2+b*c^2)^(1/2))^(1/2)-2^(1/2)*b^(1/4)*(d*x+c)^(1/2))/(-b^(1/2)* 
c+(a*d^2+b*c^2)^(1/2))^(1/2))*2^(1/2)/a/b^(3/4)/(a*d^2+b*c^2)^(5/2)/(-b^(1 
/2)*c+(a*d^2+b*c^2)^(1/2))^(1/2)-1/64*d^2*(b^(3/2)*c^3+13*a*b^(1/2)*c*d^2+ 
(-5*a*d^2+b*c^2)*(a*d^2+b*c^2)^(1/2))*arctan(((b^(1/2)*c+(a*d^2+b*c^2)^(1/ 
2))^(1/2)+2^(1/2)*b^(1/4)*(d*x+c)^(1/2))/(-b^(1/2)*c+(a*d^2+b*c^2)^(1/2))^ 
(1/2))*2^(1/2)/a/b^(3/4)/(a*d^2+b*c^2)^(5/2)/(-b^(1/2)*c+(a*d^2+b*c^2)^(1/ 
2))^(1/2)-1/64*d^2*(b^(3/2)*c^3+13*a*b^(1/2)*c*d^2-(-5*a*d^2+b*c^2)*(a*d^2 
+b*c^2)^(1/2))*arctanh(2^(1/2)*b^(1/4)*(b^(1/2)*c+(a*d^2+b*c^2)^(1/2))^(1/ 
2)*(d*x+c)^(1/2)/((a*d^2+b*c^2)^(1/2)+b^(1/2)*(d*x+c)))*2^(1/2)/a/b^(3/4)/ 
(a*d^2+b*c^2)^(5/2)/(b^(1/2)*c+(a*d^2+b*c^2)^(1/2))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.32 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.54 \[ \int \frac {x}{\sqrt {c+d x} \left (a+b x^2\right )^3} \, dx=\frac {d \left (-\frac {2 \sqrt {a} \sqrt {c+d x} \left (b^2 c^2 d x^3+a^2 d^2 (10 c-9 d x)+a b \left (4 c^3-3 c^2 d x+6 c d^2 x^2-5 d^3 x^3\right )\right )}{d \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )^2}+\frac {\left (-2 i \sqrt {b} c+5 \sqrt {a} d\right ) \arctan \left (\frac {\sqrt {-b c-i \sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+i \sqrt {a} d}\right )}{\sqrt {b} \left (\sqrt {b} c+i \sqrt {a} d\right )^2 \sqrt {-b c-i \sqrt {a} \sqrt {b} d}}+\frac {\left (2 i \sqrt {b} c+5 \sqrt {a} d\right ) \arctan \left (\frac {\sqrt {-b c+i \sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-i \sqrt {a} d}\right )}{\sqrt {b} \left (\sqrt {b} c-i \sqrt {a} d\right )^2 \sqrt {-b c+i \sqrt {a} \sqrt {b} d}}\right )}{32 a^{3/2}} \] Input:

Integrate[x/(Sqrt[c + d*x]*(a + b*x^2)^3),x]
 

Output:

(d*((-2*Sqrt[a]*Sqrt[c + d*x]*(b^2*c^2*d*x^3 + a^2*d^2*(10*c - 9*d*x) + a* 
b*(4*c^3 - 3*c^2*d*x + 6*c*d^2*x^2 - 5*d^3*x^3)))/(d*(b*c^2 + a*d^2)^2*(a 
+ b*x^2)^2) + (((-2*I)*Sqrt[b]*c + 5*Sqrt[a]*d)*ArcTan[(Sqrt[-(b*c) - I*Sq 
rt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c + I*Sqrt[a]*d)])/(Sqrt[b]*(Sqrt 
[b]*c + I*Sqrt[a]*d)^2*Sqrt[-(b*c) - I*Sqrt[a]*Sqrt[b]*d]) + (((2*I)*Sqrt[ 
b]*c + 5*Sqrt[a]*d)*ArcTan[(Sqrt[-(b*c) + I*Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d* 
x])/(Sqrt[b]*c - I*Sqrt[a]*d)])/(Sqrt[b]*(Sqrt[b]*c - I*Sqrt[a]*d)^2*Sqrt[ 
-(b*c) + I*Sqrt[a]*Sqrt[b]*d])))/(32*a^(3/2))
 

Rubi [A] (verified)

Time = 3.22 (sec) , antiderivative size = 1012, normalized size of antiderivative = 1.51, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {561, 25, 27, 1492, 27, 1492, 27, 1483, 27, 1142, 25, 27, 1083, 219, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1492

\(\displaystyle -\frac {2 \left (\frac {d^4 \int \frac {2 a b (6 c-5 (c+d x))}{d^2 \left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{16 a b \left (a d^2+b c^2\right )}+\frac {d^2 (c-d x) \sqrt {c+d x}}{8 \left (a d^2+b c^2\right ) \left (a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {d^2 \int \frac {6 c-5 (c+d x)}{\left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{8 \left (a d^2+b c^2\right )}+\frac {d^2 (c-d x) \sqrt {c+d x}}{8 \left (a d^2+b c^2\right ) \left (a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^2}\)

\(\Big \downarrow \) 1492

\(\displaystyle -\frac {2 \left (\frac {d^2 \left (\frac {d^4 \int \frac {2 b \left (c \left (b c^2+13 a d^2\right )+\left (b c^2-5 a d^2\right ) (c+d x)\right )}{d^4 \left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{8 a b \left (a d^2+b c^2\right )}-\frac {\sqrt {c+d x} \left (c \left (b c^2-11 a d^2\right )-(c+d x) \left (b c^2-5 a d^2\right )\right )}{4 a \left (a d^2+b c^2\right ) \left (a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 \left (a d^2+b c^2\right )}+\frac {d^2 (c-d x) \sqrt {c+d x}}{8 \left (a d^2+b c^2\right ) \left (a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {d^2 \left (\frac {\int \frac {c \left (b c^2+13 a d^2\right )+\left (b c^2-5 a d^2\right ) (c+d x)}{\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}d\sqrt {c+d x}}{4 a \left (a d^2+b c^2\right )}-\frac {\sqrt {c+d x} \left (c \left (b c^2-11 a d^2\right )-(c+d x) \left (b c^2-5 a d^2\right )\right )}{4 a \left (a d^2+b c^2\right ) \left (a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 \left (a d^2+b c^2\right )}+\frac {d^2 (c-d x) \sqrt {c+d x}}{8 \left (a d^2+b c^2\right ) \left (a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^2}\)

\(\Big \downarrow \) 1483

\(\displaystyle -\frac {2 \left (\frac {d^2 \left (\frac {\frac {d^2 \int \frac {\sqrt {2} c \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \left (b c^2+13 a d^2\right )+\sqrt [4]{b} \left (\frac {\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}}{\sqrt {b}}-c \left (b c^2+13 a d^2\right )\right ) \sqrt {c+d x}}{\sqrt [4]{b} \left (c+d x+\frac {\sqrt {b c^2+a d^2}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt [4]{b}}\right )}d\sqrt {c+d x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {a d^2+b c^2} \sqrt {\sqrt {a d^2+b c^2}+\sqrt {b} c}}+\frac {d^2 \int \frac {\sqrt {2} c \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \left (b c^2+13 a d^2\right )+\sqrt [4]{b} \left (b c^3+13 a d^2 c-\frac {\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}}{\sqrt {b}}\right ) \sqrt {c+d x}}{\sqrt [4]{b} \left (c+d x+\frac {\sqrt {b c^2+a d^2}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt [4]{b}}\right )}d\sqrt {c+d x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {a d^2+b c^2} \sqrt {\sqrt {a d^2+b c^2}+\sqrt {b} c}}}{4 a \left (a d^2+b c^2\right )}-\frac {\sqrt {c+d x} \left (c \left (b c^2-11 a d^2\right )-(c+d x) \left (b c^2-5 a d^2\right )\right )}{4 a \left (a d^2+b c^2\right ) \left (a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 \left (a d^2+b c^2\right )}+\frac {d^2 (c-d x) \sqrt {c+d x}}{8 \left (a d^2+b c^2\right ) \left (a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {d^2 \left (\frac {\frac {d^2 \int \frac {\sqrt {2} c \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \left (b c^2+13 a d^2\right )+\sqrt [4]{b} \left (\frac {\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}}{\sqrt {b}}-c \left (b c^2+13 a d^2\right )\right ) \sqrt {c+d x}}{c+d x+\frac {\sqrt {b c^2+a d^2}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt [4]{b}}}d\sqrt {c+d x}}{2 \sqrt {2} \sqrt {b} \sqrt {a d^2+b c^2} \sqrt {\sqrt {a d^2+b c^2}+\sqrt {b} c}}+\frac {d^2 \int \frac {\sqrt {2} c \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \left (b c^2+13 a d^2\right )+\sqrt [4]{b} \left (b c^3+13 a d^2 c-\frac {\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}}{\sqrt {b}}\right ) \sqrt {c+d x}}{c+d x+\frac {\sqrt {b c^2+a d^2}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt [4]{b}}}d\sqrt {c+d x}}{2 \sqrt {2} \sqrt {b} \sqrt {a d^2+b c^2} \sqrt {\sqrt {a d^2+b c^2}+\sqrt {b} c}}}{4 a \left (a d^2+b c^2\right )}-\frac {\sqrt {c+d x} \left (c \left (b c^2-11 a d^2\right )-(c+d x) \left (b c^2-5 a d^2\right )\right )}{4 a \left (a d^2+b c^2\right ) \left (a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 \left (a d^2+b c^2\right )}+\frac {d^2 (c-d x) \sqrt {c+d x}}{8 \left (a d^2+b c^2\right ) \left (a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^2}\)

\(\Big \downarrow \) 1142

\(\displaystyle -\frac {2 \left (\frac {\left (\frac {\frac {\left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \left (b^{3/2} c^3+13 a \sqrt {b} d^2 c+\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}\right ) \int \frac {1}{c+d x+\frac {\sqrt {b c^2+a d^2}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt [4]{b}}}d\sqrt {c+d x}}{\sqrt {2} \sqrt {b}}+\frac {1}{2} \sqrt [4]{b} \left (\frac {\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}}{\sqrt {b}}-c \left (b c^2+13 a d^2\right )\right ) \int -\frac {\sqrt {2} \left (\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}-\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}\right )}{\sqrt [4]{b} \left (c+d x+\frac {\sqrt {b c^2+a d^2}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt [4]{b}}\right )}d\sqrt {c+d x}\right ) d^2}{2 \sqrt {2} \sqrt {b} \sqrt {b c^2+a d^2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}}+\frac {\left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \left (b^{3/2} c^3+13 a \sqrt {b} d^2 c+\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}\right ) \int \frac {1}{c+d x+\frac {\sqrt {b c^2+a d^2}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt [4]{b}}}d\sqrt {c+d x}}{\sqrt {2} \sqrt {b}}+\frac {1}{2} \sqrt [4]{b} \left (b c^3+13 a d^2 c-\frac {\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}}{\sqrt {b}}\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}+\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}\right )}{\sqrt [4]{b} \left (c+d x+\frac {\sqrt {b c^2+a d^2}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt [4]{b}}\right )}d\sqrt {c+d x}\right ) d^2}{2 \sqrt {2} \sqrt {b} \sqrt {b c^2+a d^2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}}}{4 a \left (b c^2+a d^2\right )}-\frac {\sqrt {c+d x} \left (c \left (b c^2-11 a d^2\right )-\left (b c^2-5 a d^2\right ) (c+d x)\right )}{4 a \left (b c^2+a d^2\right ) \left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )}\right ) d^2}{8 \left (b c^2+a d^2\right )}+\frac {(c-d x) \sqrt {c+d x} d^2}{8 \left (b c^2+a d^2\right ) \left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )^2}\right )}{d^2}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2 \left (\frac {\left (\frac {\frac {\left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \left (b^{3/2} c^3+13 a \sqrt {b} d^2 c+\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}\right ) \int \frac {1}{c+d x+\frac {\sqrt {b c^2+a d^2}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt [4]{b}}}d\sqrt {c+d x}}{\sqrt {2} \sqrt {b}}-\frac {1}{2} \sqrt [4]{b} \left (\frac {\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}}{\sqrt {b}}-c \left (b c^2+13 a d^2\right )\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}-\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}\right )}{\sqrt [4]{b} \left (c+d x+\frac {\sqrt {b c^2+a d^2}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt [4]{b}}\right )}d\sqrt {c+d x}\right ) d^2}{2 \sqrt {2} \sqrt {b} \sqrt {b c^2+a d^2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}}+\frac {\left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \left (b^{3/2} c^3+13 a \sqrt {b} d^2 c+\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}\right ) \int \frac {1}{c+d x+\frac {\sqrt {b c^2+a d^2}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt [4]{b}}}d\sqrt {c+d x}}{\sqrt {2} \sqrt {b}}+\frac {1}{2} \sqrt [4]{b} \left (b c^3+13 a d^2 c-\frac {\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}}{\sqrt {b}}\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}+\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}\right )}{\sqrt [4]{b} \left (c+d x+\frac {\sqrt {b c^2+a d^2}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt [4]{b}}\right )}d\sqrt {c+d x}\right ) d^2}{2 \sqrt {2} \sqrt {b} \sqrt {b c^2+a d^2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}}}{4 a \left (b c^2+a d^2\right )}-\frac {\sqrt {c+d x} \left (c \left (b c^2-11 a d^2\right )-\left (b c^2-5 a d^2\right ) (c+d x)\right )}{4 a \left (b c^2+a d^2\right ) \left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )}\right ) d^2}{8 \left (b c^2+a d^2\right )}+\frac {(c-d x) \sqrt {c+d x} d^2}{8 \left (b c^2+a d^2\right ) \left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )^2}\right )}{d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {\left (\frac {\frac {\left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \left (b^{3/2} c^3+13 a \sqrt {b} d^2 c+\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}\right ) \int \frac {1}{c+d x+\frac {\sqrt {b c^2+a d^2}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt [4]{b}}}d\sqrt {c+d x}}{\sqrt {2} \sqrt {b}}-\frac {\left (\frac {\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}}{\sqrt {b}}-c \left (b c^2+13 a d^2\right )\right ) \int \frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}-\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{c+d x+\frac {\sqrt {b c^2+a d^2}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt [4]{b}}}d\sqrt {c+d x}}{\sqrt {2}}\right ) d^2}{2 \sqrt {2} \sqrt {b} \sqrt {b c^2+a d^2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}}+\frac {\left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \left (b^{3/2} c^3+13 a \sqrt {b} d^2 c+\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}\right ) \int \frac {1}{c+d x+\frac {\sqrt {b c^2+a d^2}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt [4]{b}}}d\sqrt {c+d x}}{\sqrt {2} \sqrt {b}}+\frac {\left (b c^3+13 a d^2 c-\frac {\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}}{\sqrt {b}}\right ) \int \frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}+\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{c+d x+\frac {\sqrt {b c^2+a d^2}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt [4]{b}}}d\sqrt {c+d x}}{\sqrt {2}}\right ) d^2}{2 \sqrt {2} \sqrt {b} \sqrt {b c^2+a d^2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}}}{4 a \left (b c^2+a d^2\right )}-\frac {\sqrt {c+d x} \left (c \left (b c^2-11 a d^2\right )-\left (b c^2-5 a d^2\right ) (c+d x)\right )}{4 a \left (b c^2+a d^2\right ) \left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )}\right ) d^2}{8 \left (b c^2+a d^2\right )}+\frac {(c-d x) \sqrt {c+d x} d^2}{8 \left (b c^2+a d^2\right ) \left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )^2}\right )}{d^2}\)

\(\Big \downarrow \) 1083

\(\displaystyle -\frac {2 \left (\frac {\left (\frac {\frac {\left (-\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \left (b^{3/2} c^3+13 a \sqrt {b} d^2 c+\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}\right ) \int \frac {1}{-c+2 \left (c-\frac {\sqrt {b c^2+a d^2}}{\sqrt {b}}\right )-d x}d\left (2 \sqrt {c+d x}-\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}}{\sqrt [4]{b}}\right )}{\sqrt {b}}-\frac {\left (\frac {\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}}{\sqrt {b}}-c \left (b c^2+13 a d^2\right )\right ) \int \frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}-\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{c+d x+\frac {\sqrt {b c^2+a d^2}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt [4]{b}}}d\sqrt {c+d x}}{\sqrt {2}}\right ) d^2}{2 \sqrt {2} \sqrt {b} \sqrt {b c^2+a d^2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}}+\frac {\left (\frac {\left (b c^3+13 a d^2 c-\frac {\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}}{\sqrt {b}}\right ) \int \frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}+\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{c+d x+\frac {\sqrt {b c^2+a d^2}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt [4]{b}}}d\sqrt {c+d x}}{\sqrt {2}}-\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \left (b^{3/2} c^3+13 a \sqrt {b} d^2 c+\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}\right ) \int \frac {1}{-c+2 \left (c-\frac {\sqrt {b c^2+a d^2}}{\sqrt {b}}\right )-d x}d\left (\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}}{\sqrt [4]{b}}+2 \sqrt {c+d x}\right )}{\sqrt {b}}\right ) d^2}{2 \sqrt {2} \sqrt {b} \sqrt {b c^2+a d^2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}}}{4 a \left (b c^2+a d^2\right )}-\frac {\sqrt {c+d x} \left (c \left (b c^2-11 a d^2\right )-\left (b c^2-5 a d^2\right ) (c+d x)\right )}{4 a \left (b c^2+a d^2\right ) \left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )}\right ) d^2}{8 \left (b c^2+a d^2\right )}+\frac {(c-d x) \sqrt {c+d x} d^2}{8 \left (b c^2+a d^2\right ) \left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )^2}\right )}{d^2}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {2 \left (\frac {\left (\frac {\frac {\left (-\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \left (b^{3/2} c^3+13 a \sqrt {b} d^2 c+\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \left (2 \sqrt {c+d x}-\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt {\sqrt {b} c-\sqrt {b c^2+a d^2}}}\right )}{\sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {b c^2+a d^2}}}-\frac {\left (\frac {\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}}{\sqrt {b}}-c \left (b c^2+13 a d^2\right )\right ) \int \frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}-\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{c+d x+\frac {\sqrt {b c^2+a d^2}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt [4]{b}}}d\sqrt {c+d x}}{\sqrt {2}}\right ) d^2}{2 \sqrt {2} \sqrt {b} \sqrt {b c^2+a d^2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}}+\frac {\left (\frac {\left (b c^3+13 a d^2 c-\frac {\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}}{\sqrt {b}}\right ) \int \frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}+\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{c+d x+\frac {\sqrt {b c^2+a d^2}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt [4]{b}}}d\sqrt {c+d x}}{\sqrt {2}}-\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \left (b^{3/2} c^3+13 a \sqrt {b} d^2 c+\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \left (\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}}{\sqrt [4]{b}}+2 \sqrt {c+d x}\right )}{\sqrt {2} \sqrt {\sqrt {b} c-\sqrt {b c^2+a d^2}}}\right )}{\sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {b c^2+a d^2}}}\right ) d^2}{2 \sqrt {2} \sqrt {b} \sqrt {b c^2+a d^2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}}}{4 a \left (b c^2+a d^2\right )}-\frac {\sqrt {c+d x} \left (c \left (b c^2-11 a d^2\right )-\left (b c^2-5 a d^2\right ) (c+d x)\right )}{4 a \left (b c^2+a d^2\right ) \left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )}\right ) d^2}{8 \left (b c^2+a d^2\right )}+\frac {(c-d x) \sqrt {c+d x} d^2}{8 \left (b c^2+a d^2\right ) \left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )^2}\right )}{d^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle -\frac {2 \left (\frac {\left (\frac {\frac {\left (\frac {1}{2} \sqrt [4]{b} \left (\frac {\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}}{\sqrt {b}}-c \left (b c^2+13 a d^2\right )\right ) \log \left (\sqrt {b} (c+d x)-\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}+\sqrt {b c^2+a d^2}\right )-\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \left (b^{3/2} c^3+13 a \sqrt {b} d^2 c+\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \left (2 \sqrt {c+d x}-\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt {\sqrt {b} c-\sqrt {b c^2+a d^2}}}\right )}{\sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {b c^2+a d^2}}}\right ) d^2}{2 \sqrt {2} \sqrt {b} \sqrt {b c^2+a d^2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}}+\frac {\left (\frac {1}{2} \sqrt [4]{b} \left (b c^3+13 a d^2 c-\frac {\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}}{\sqrt {b}}\right ) \log \left (\sqrt {b} (c+d x)+\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}+\sqrt {b c^2+a d^2}\right )-\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \left (b^{3/2} c^3+13 a \sqrt {b} d^2 c+\left (b c^2-5 a d^2\right ) \sqrt {b c^2+a d^2}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \left (\frac {\sqrt {2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}}{\sqrt [4]{b}}+2 \sqrt {c+d x}\right )}{\sqrt {2} \sqrt {\sqrt {b} c-\sqrt {b c^2+a d^2}}}\right )}{\sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {b c^2+a d^2}}}\right ) d^2}{2 \sqrt {2} \sqrt {b} \sqrt {b c^2+a d^2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}}}{4 a \left (b c^2+a d^2\right )}-\frac {\sqrt {c+d x} \left (c \left (b c^2-11 a d^2\right )-\left (b c^2-5 a d^2\right ) (c+d x)\right )}{4 a \left (b c^2+a d^2\right ) \left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )}\right ) d^2}{8 \left (b c^2+a d^2\right )}+\frac {(c-d x) \sqrt {c+d x} d^2}{8 \left (b c^2+a d^2\right ) \left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )^2}\right )}{d^2}\)

Input:

Int[x/(Sqrt[c + d*x]*(a + b*x^2)^3),x]
 

Output:

(-2*((d^2*(c - d*x)*Sqrt[c + d*x])/(8*(b*c^2 + a*d^2)*(a + (b*c^2)/d^2 - ( 
2*b*c*(c + d*x))/d^2 + (b*(c + d*x)^2)/d^2)^2) + (d^2*(-1/4*(Sqrt[c + d*x] 
*(c*(b*c^2 - 11*a*d^2) - (b*c^2 - 5*a*d^2)*(c + d*x)))/(a*(b*c^2 + a*d^2)* 
(a + (b*c^2)/d^2 - (2*b*c*(c + d*x))/d^2 + (b*(c + d*x)^2)/d^2)) + ((d^2*( 
-((Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]]*(b^(3/2)*c^3 + 13*a*Sqrt[b]*c*d^2 
 + (b*c^2 - 5*a*d^2)*Sqrt[b*c^2 + a*d^2])*ArcTanh[(b^(1/4)*(-((Sqrt[2]*Sqr 
t[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]])/b^(1/4)) + 2*Sqrt[c + d*x]))/(Sqrt[2]* 
Sqrt[Sqrt[b]*c - Sqrt[b*c^2 + a*d^2]])])/(b^(1/4)*Sqrt[Sqrt[b]*c - Sqrt[b* 
c^2 + a*d^2]])) + (b^(1/4)*(((b*c^2 - 5*a*d^2)*Sqrt[b*c^2 + a*d^2])/Sqrt[b 
] - c*(b*c^2 + 13*a*d^2))*Log[Sqrt[b*c^2 + a*d^2] - Sqrt[2]*b^(1/4)*Sqrt[S 
qrt[b]*c + Sqrt[b*c^2 + a*d^2]]*Sqrt[c + d*x] + Sqrt[b]*(c + d*x)])/2))/(2 
*Sqrt[2]*Sqrt[b]*Sqrt[b*c^2 + a*d^2]*Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]] 
) + (d^2*(-((Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]]*(b^(3/2)*c^3 + 13*a*Sqr 
t[b]*c*d^2 + (b*c^2 - 5*a*d^2)*Sqrt[b*c^2 + a*d^2])*ArcTanh[(b^(1/4)*((Sqr 
t[2]*Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]])/b^(1/4) + 2*Sqrt[c + d*x]))/(S 
qrt[2]*Sqrt[Sqrt[b]*c - Sqrt[b*c^2 + a*d^2]])])/(b^(1/4)*Sqrt[Sqrt[b]*c - 
Sqrt[b*c^2 + a*d^2]])) + (b^(1/4)*(b*c^3 + 13*a*c*d^2 - ((b*c^2 - 5*a*d^2) 
*Sqrt[b*c^2 + a*d^2])/Sqrt[b])*Log[Sqrt[b*c^2 + a*d^2] + Sqrt[2]*b^(1/4)*S 
qrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]]*Sqrt[c + d*x] + Sqrt[b]*(c + d*x)])/2 
))/(2*Sqrt[2]*Sqrt[b]*Sqrt[b*c^2 + a*d^2]*Sqrt[Sqrt[b]*c + Sqrt[b*c^2 +...
 

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 219

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

rule 561

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{k = Denominator[n]}, Simp[k/d   Subst[Int[x^(k*(n + 1) - 1)*(-c 
/d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), 
 x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac 
tionQ[n] && IntegerQ[p] && IntegerQ[m]
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{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 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 1483

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r)   In 
t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r)   Int[(d*r 
 + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N 
eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
 

rule 1492

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + 
 c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 
 - 4*a*c))   Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 
7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 LtQ[p, -1] && IntegerQ[2*p]
 

Maple [F(-1)]

Timed out.

\[\int \frac {x}{\sqrt {d x +c}\, \left (b \,x^{2}+a \right )^{3}}d x\]

Input:

int(x/(d*x+c)^(1/2)/(b*x^2+a)^3,x)
 

Output:

int(x/(d*x+c)^(1/2)/(b*x^2+a)^3,x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5636 vs. \(2 (556) = 1112\).

Time = 1.01 (sec) , antiderivative size = 5636, normalized size of antiderivative = 8.39 \[ \int \frac {x}{\sqrt {c+d x} \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(x/(d*x+c)^(1/2)/(b*x^2+a)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x}{\sqrt {c+d x} \left (a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x/(d*x+c)**(1/2)/(b*x**2+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x}{\sqrt {c+d x} \left (a+b x^2\right )^3} \, dx=\int { \frac {x}{{\left (b x^{2} + a\right )}^{3} \sqrt {d x + c}} \,d x } \] Input:

integrate(x/(d*x+c)^(1/2)/(b*x^2+a)^3,x, algorithm="maxima")
 

Output:

integrate(x/((b*x^2 + a)^3*sqrt(d*x + c)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1499 vs. \(2 (556) = 1112\).

Time = 0.32 (sec) , antiderivative size = 1499, normalized size of antiderivative = 2.23 \[ \int \frac {x}{\sqrt {c+d x} \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(x/(d*x+c)^(1/2)/(b*x^2+a)^3,x, algorithm="giac")
 

Output:

-1/32*(((a*b^2*c^4*d + 2*a^2*b*c^2*d^3 + a^3*d^5)^2*(b*c^2*d^3 - 5*a*d^5)* 
abs(b) - (sqrt(-a*b)*b^3*c^7*d^3 + 15*sqrt(-a*b)*a*b^2*c^5*d^5 + 27*sqrt(- 
a*b)*a^2*b*c^3*d^7 + 13*sqrt(-a*b)*a^3*c*d^9)*abs(a*b^2*c^4*d + 2*a^2*b*c^ 
2*d^3 + a^3*d^5)*abs(b) + 2*(a*b^6*c^12*d^3 + 8*a^2*b^5*c^10*d^5 + 22*a^3* 
b^4*c^8*d^7 + 28*a^4*b^3*c^6*d^9 + 17*a^5*b^2*c^4*d^11 + 4*a^6*b*c^2*d^13) 
*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(a*b^3*c^5 + 2*a^2*b^2*c^3*d^2 + a^3*b 
*c*d^4 + sqrt((a*b^3*c^5 + 2*a^2*b^2*c^3*d^2 + a^3*b*c*d^4)^2 - (a*b^3*c^6 
 + 3*a^2*b^2*c^4*d^2 + 3*a^3*b*c^2*d^4 + a^4*d^6)*(a*b^3*c^4 + 2*a^2*b^2*c 
^2*d^2 + a^3*b*d^4)))/(a*b^3*c^4 + 2*a^2*b^2*c^2*d^2 + a^3*b*d^4)))/((a^2* 
b^5*c^8*d + 4*a^3*b^4*c^6*d^3 + 6*a^4*b^3*c^4*d^5 + 4*a^5*b^2*c^2*d^7 + a^ 
6*b*d^9 + sqrt(-a*b)*a*b^5*c^9 + 4*sqrt(-a*b)*a^2*b^4*c^7*d^2 + 6*sqrt(-a* 
b)*a^3*b^3*c^5*d^4 + 4*sqrt(-a*b)*a^4*b^2*c^3*d^6 + sqrt(-a*b)*a^5*b*c*d^8 
)*sqrt(-b^2*c - sqrt(-a*b)*b*d)*abs(a*b^2*c^4*d + 2*a^2*b*c^2*d^3 + a^3*d^ 
5)) + ((a*b^2*c^4*d + 2*a^2*b*c^2*d^3 + a^3*d^5)^2*(b*c^2*d^3 - 5*a*d^5)*a 
bs(b) + (sqrt(-a*b)*b^3*c^7*d^3 + 15*sqrt(-a*b)*a*b^2*c^5*d^5 + 27*sqrt(-a 
*b)*a^2*b*c^3*d^7 + 13*sqrt(-a*b)*a^3*c*d^9)*abs(a*b^2*c^4*d + 2*a^2*b*c^2 
*d^3 + a^3*d^5)*abs(b) + 2*(a*b^6*c^12*d^3 + 8*a^2*b^5*c^10*d^5 + 22*a^3*b 
^4*c^8*d^7 + 28*a^4*b^3*c^6*d^9 + 17*a^5*b^2*c^4*d^11 + 4*a^6*b*c^2*d^13)* 
abs(b))*arctan(sqrt(d*x + c)/sqrt(-(a*b^3*c^5 + 2*a^2*b^2*c^3*d^2 + a^3*b* 
c*d^4 - sqrt((a*b^3*c^5 + 2*a^2*b^2*c^3*d^2 + a^3*b*c*d^4)^2 - (a*b^3*c...
 

Mupad [B] (verification not implemented)

Time = 11.59 (sec) , antiderivative size = 8723, normalized size of antiderivative = 12.98 \[ \int \frac {x}{\sqrt {c+d x} \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int(x/((a + b*x^2)^3*(c + d*x)^(1/2)),x)
 

Output:

(((b*c^3*d^2 - 19*a*c*d^4)*(c + d*x)^(1/2))/(16*a*(a*d^2 + b*c^2)) + (3*(c 
 + d*x)^(3/2)*(3*a^2*d^6 - b^2*c^4*d^2 + 10*a*b*c^2*d^4))/(16*a*(a*d^2 + b 
*c^2)^2) + (b*(5*a*d^4 - b*c^2*d^2)*(c + d*x)^(7/2))/(16*a*(a*d^2 + b*c^2) 
^2) - (3*b*c*(7*a*d^4 - b*c^2*d^2)*(c + d*x)^(5/2))/(16*a*(a*d^2 + b*c^2)^ 
2))/(b^2*(c + d*x)^4 + a^2*d^4 + b^2*c^4 + (6*b^2*c^2 + 2*a*b*d^2)*(c + d* 
x)^2 - (4*b^2*c^3 + 4*a*b*c*d^2)*(c + d*x) - 4*b^2*c*(c + d*x)^3 + 2*a*b*c 
^2*d^2) - atan(((((53248*a^6*b^3*c*d^10 + 4096*a^3*b^6*c^7*d^4 + 61440*a^4 
*b^5*c^5*d^6 + 110592*a^5*b^4*c^3*d^8)/(4096*(a^7*d^8 + a^3*b^4*c^8 + 4*a^ 
6*b*c^2*d^6 + 4*a^4*b^3*c^6*d^2 + 6*a^5*b^2*c^4*d^4)) - ((c + d*x)^(1/2)*( 
-(25*a^2*d^9*(-a^9*b^3)^(1/2) - 105*a^6*b^2*c*d^8 + 4*a^3*b^5*c^7*d^2 + 35 
*a^4*b^4*c^5*d^4 + 70*a^5*b^3*c^3*d^6 - 35*b^2*c^4*d^5*(-a^9*b^3)^(1/2) - 
154*a*b*c^2*d^7*(-a^9*b^3)^(1/2))/(4096*(a^6*b^8*c^10 + a^11*b^3*d^10 + 5* 
a^7*b^7*c^8*d^2 + 10*a^8*b^6*c^6*d^4 + 10*a^9*b^5*c^4*d^6 + 5*a^10*b^4*c^2 
*d^8)))^(1/2)*(4096*a^7*b^4*c*d^10 + 4096*a^3*b^8*c^9*d^2 + 16384*a^4*b^7* 
c^7*d^4 + 24576*a^5*b^6*c^5*d^6 + 16384*a^6*b^5*c^3*d^8))/(64*(a^6*d^8 + a 
^2*b^4*c^8 + 4*a^5*b*c^2*d^6 + 4*a^3*b^3*c^6*d^2 + 6*a^4*b^2*c^4*d^4)))*(- 
(25*a^2*d^9*(-a^9*b^3)^(1/2) - 105*a^6*b^2*c*d^8 + 4*a^3*b^5*c^7*d^2 + 35* 
a^4*b^4*c^5*d^4 + 70*a^5*b^3*c^3*d^6 - 35*b^2*c^4*d^5*(-a^9*b^3)^(1/2) - 1 
54*a*b*c^2*d^7*(-a^9*b^3)^(1/2))/(4096*(a^6*b^8*c^10 + a^11*b^3*d^10 + 5*a 
^7*b^7*c^8*d^2 + 10*a^8*b^6*c^6*d^4 + 10*a^9*b^5*c^4*d^6 + 5*a^10*b^4*c...
 

Reduce [B] (verification not implemented)

Time = 5.38 (sec) , antiderivative size = 7573, normalized size of antiderivative = 11.27 \[ \int \frac {x}{\sqrt {c+d x} \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(x/(d*x+c)^(1/2)/(b*x^2+a)^3,x)
 

Output:

( - 10*sqrt(a*d**2 + b*c**2)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqr 
t(2)*atan((sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2) - 2*sqrt(b)*s 
qrt(c + d*x))/(sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)))*a**4*d* 
*4 + 18*sqrt(a*d**2 + b*c**2)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sq 
rt(2)*atan((sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2) - 2*sqrt(b)* 
sqrt(c + d*x))/(sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)))*a**3*b 
*c**2*d**2 - 20*sqrt(a*d**2 + b*c**2)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - 
 b*c)*sqrt(2)*atan((sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2) - 2* 
sqrt(b)*sqrt(c + d*x))/(sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)) 
)*a**3*b*d**4*x**2 + 4*sqrt(a*d**2 + b*c**2)*sqrt(sqrt(b)*sqrt(a*d**2 + b* 
c**2) - b*c)*sqrt(2)*atan((sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt( 
2) - 2*sqrt(b)*sqrt(c + d*x))/(sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*s 
qrt(2)))*a**2*b**2*c**4 + 36*sqrt(a*d**2 + b*c**2)*sqrt(sqrt(b)*sqrt(a*d** 
2 + b*c**2) - b*c)*sqrt(2)*atan((sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) + b*c) 
*sqrt(2) - 2*sqrt(b)*sqrt(c + d*x))/(sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - 
b*c)*sqrt(2)))*a**2*b**2*c**2*d**2*x**2 - 10*sqrt(a*d**2 + b*c**2)*sqrt(sq 
rt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)*atan((sqrt(sqrt(b)*sqrt(a*d**2 
+ b*c**2) + b*c)*sqrt(2) - 2*sqrt(b)*sqrt(c + d*x))/(sqrt(sqrt(b)*sqrt(a*d 
**2 + b*c**2) - b*c)*sqrt(2)))*a**2*b**2*d**4*x**4 + 8*sqrt(a*d**2 + b*c** 
2)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)*atan((sqrt(sqrt(b)...