\(\int \frac {x}{\sqrt {c+d x} (a+b x^2)^2} \, dx\) [686]

Optimal result

Integrand size = 20, antiderivative size = 531 \[ \int \frac {x}{\sqrt {c+d x} \left (a+b x^2\right )^2} \, dx=-\frac {(c-d x) \sqrt {c+d x}}{2 \left (b c^2+a d^2\right ) \left (a+b x^2\right )}+\frac {d^2 \left (2 \sqrt {b} c-\sqrt {b c^2+a d^2}\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}-\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}\right )}{4 \sqrt {2} b^{3/4} \left (b c^2+a d^2\right )^{3/2} \sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}-\frac {d^2 \left (2 \sqrt {b} c-\sqrt {b c^2+a d^2}\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}+\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}\right )}{4 \sqrt {2} b^{3/4} \left (b c^2+a d^2\right )^{3/2} \sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}-\frac {d^2 \left (2 \sqrt {b} c+\sqrt {b c^2+a d^2}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt {b c^2+a d^2}+\sqrt {b} (c+d x)}\right )}{4 \sqrt {2} b^{3/4} \left (b c^2+a d^2\right )^{3/2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}} \] Output:

-1/2*(-d*x+c)*(d*x+c)^(1/2)/(a*d^2+b*c^2)/(b*x^2+a)+1/8*d^2*(2*b^(1/2)*c-( 
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)/b^( 
3/4)/(a*d^2+b*c^2)^(3/2)/(-b^(1/2)*c+(a*d^2+b*c^2)^(1/2))^(1/2)-1/8*d^2*(2 
*b^(1/2)*c-(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)/b^(3/4)/(a*d^2+b*c^2)^(3/2)/(-b^(1/2)*c+(a*d^2+b*c^2)^(1/2))^(1/2 
)-1/8*d^2*(2*b^(1/2)*c+(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)/b^(3/4)/(a*d^2+b*c^2)^(3/2)/(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 = 1.55 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.51 \[ \int \frac {x}{\sqrt {c+d x} \left (a+b x^2\right )^2} \, dx=-\frac {(c-d x) \sqrt {c+d x}}{2 \left (b c^2+a d^2\right ) \left (a+b x^2\right )}-\frac {i d \sqrt {-b c-i \sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c-i \sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+i \sqrt {a} d}\right )}{4 \sqrt {a} b \left (-i \sqrt {b} c+\sqrt {a} d\right )^2}+\frac {i d \sqrt {-b c+i \sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c+i \sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-i \sqrt {a} d}\right )}{4 \sqrt {a} b \left (i \sqrt {b} c+\sqrt {a} d\right )^2} \] Input:

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

Output:

-1/2*((c - d*x)*Sqrt[c + d*x])/((b*c^2 + a*d^2)*(a + b*x^2)) - ((I/4)*d*Sq 
rt[-(b*c) - I*Sqrt[a]*Sqrt[b]*d]*ArcTan[(Sqrt[-(b*c) - I*Sqrt[a]*Sqrt[b]*d 
]*Sqrt[c + d*x])/(Sqrt[b]*c + I*Sqrt[a]*d)])/(Sqrt[a]*b*((-I)*Sqrt[b]*c + 
Sqrt[a]*d)^2) + ((I/4)*d*Sqrt[-(b*c) + I*Sqrt[a]*Sqrt[b]*d]*ArcTan[(Sqrt[- 
(b*c) + I*Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - I*Sqrt[a]*d)])/(S 
qrt[a]*b*(I*Sqrt[b]*c + Sqrt[a]*d)^2)
 

Rubi [A] (verified)

Time = 2.32 (sec) , antiderivative size = 776, normalized size of antiderivative = 1.46, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {561, 25, 27, 1492, 27, 1483, 27, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1492

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1483

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2987 vs. \(2 (424) = 848\).

Time = 0.13 (sec) , antiderivative size = 2987, normalized size of antiderivative = 5.63 \[ \int \frac {x}{\sqrt {c+d x} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*((a*b*c^2 + a^2*d^2 + (b^2*c^2 + a*b*d^2)*x^2)*sqrt(-(b*c^3*d^2 - 3*a* 
c*d^4 + (a*b^4*c^6 + 3*a^2*b^3*c^4*d^2 + 3*a^3*b^2*c^2*d^4 + a^4*b*d^6)*sq 
rt(-(9*b^2*c^4*d^6 - 6*a*b*c^2*d^8 + a^2*d^10)/(a*b^9*c^12 + 6*a^2*b^8*c^1 
0*d^2 + 15*a^3*b^7*c^8*d^4 + 20*a^4*b^6*c^6*d^6 + 15*a^5*b^5*c^4*d^8 + 6*a 
^6*b^4*c^2*d^10 + a^7*b^3*d^12)))/(a*b^4*c^6 + 3*a^2*b^3*c^4*d^2 + 3*a^3*b 
^2*c^2*d^4 + a^4*b*d^6))*log(-(3*b*c^2*d^4 - a*d^6)*sqrt(d*x + c) + (6*a*b 
^2*c^3*d^4 - 2*a^2*b*c*d^6 + (a*b^6*c^8 + 2*a^2*b^5*c^6*d^2 - 2*a^4*b^3*c^ 
2*d^6 - a^5*b^2*d^8)*sqrt(-(9*b^2*c^4*d^6 - 6*a*b*c^2*d^8 + a^2*d^10)/(a*b 
^9*c^12 + 6*a^2*b^8*c^10*d^2 + 15*a^3*b^7*c^8*d^4 + 20*a^4*b^6*c^6*d^6 + 1 
5*a^5*b^5*c^4*d^8 + 6*a^6*b^4*c^2*d^10 + a^7*b^3*d^12)))*sqrt(-(b*c^3*d^2 
- 3*a*c*d^4 + (a*b^4*c^6 + 3*a^2*b^3*c^4*d^2 + 3*a^3*b^2*c^2*d^4 + a^4*b*d 
^6)*sqrt(-(9*b^2*c^4*d^6 - 6*a*b*c^2*d^8 + a^2*d^10)/(a*b^9*c^12 + 6*a^2*b 
^8*c^10*d^2 + 15*a^3*b^7*c^8*d^4 + 20*a^4*b^6*c^6*d^6 + 15*a^5*b^5*c^4*d^8 
 + 6*a^6*b^4*c^2*d^10 + a^7*b^3*d^12)))/(a*b^4*c^6 + 3*a^2*b^3*c^4*d^2 + 3 
*a^3*b^2*c^2*d^4 + a^4*b*d^6))) - (a*b*c^2 + a^2*d^2 + (b^2*c^2 + a*b*d^2) 
*x^2)*sqrt(-(b*c^3*d^2 - 3*a*c*d^4 + (a*b^4*c^6 + 3*a^2*b^3*c^4*d^2 + 3*a^ 
3*b^2*c^2*d^4 + a^4*b*d^6)*sqrt(-(9*b^2*c^4*d^6 - 6*a*b*c^2*d^8 + a^2*d^10 
)/(a*b^9*c^12 + 6*a^2*b^8*c^10*d^2 + 15*a^3*b^7*c^8*d^4 + 20*a^4*b^6*c^6*d 
^6 + 15*a^5*b^5*c^4*d^8 + 6*a^6*b^4*c^2*d^10 + a^7*b^3*d^12)))/(a*b^4*c^6 
+ 3*a^2*b^3*c^4*d^2 + 3*a^3*b^2*c^2*d^4 + a^4*b*d^6))*log(-(3*b*c^2*d^4...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 736, normalized size of antiderivative = 1.39 \[ \int \frac {x}{\sqrt {c+d x} \left (a+b x^2\right )^2} \, dx=-\frac {\frac {{\left ({\left (b c^{2} d + a d^{3}\right )}^{2} a d^{3} {\left | b \right |} + 2 \, {\left (\sqrt {-a b} b c^{3} d^{3} + \sqrt {-a b} a c d^{5}\right )} {\left | -b c^{2} d - a d^{3} \right |} {\left | b \right |} - {\left (b^{3} c^{6} d^{3} + 2 \, a b^{2} c^{4} d^{5} + a^{2} b c^{2} d^{7}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {b^{2} c^{3} + a b c d^{2} + \sqrt {{\left (b^{2} c^{3} + a b c d^{2}\right )}^{2} - {\left (b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}\right )} {\left (b^{2} c^{2} + a b d^{2}\right )}}}{b^{2} c^{2} + a b d^{2}}}}\right )}{{\left (a b^{3} c^{4} d + 2 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b d^{5} - \sqrt {-a b} b^{3} c^{5} - 2 \, \sqrt {-a b} a b^{2} c^{3} d^{2} - \sqrt {-a b} a^{2} b c d^{4}\right )} \sqrt {-b^{2} c + \sqrt {-a b} b d} {\left | -b c^{2} d - a d^{3} \right |}} + \frac {{\left ({\left (b c^{2} d + a d^{3}\right )}^{2} a d^{3} {\left | b \right |} - 2 \, {\left (\sqrt {-a b} b c^{3} d^{3} + \sqrt {-a b} a c d^{5}\right )} {\left | -b c^{2} d - a d^{3} \right |} {\left | b \right |} - {\left (b^{3} c^{6} d^{3} + 2 \, a b^{2} c^{4} d^{5} + a^{2} b c^{2} d^{7}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {b^{2} c^{3} + a b c d^{2} - \sqrt {{\left (b^{2} c^{3} + a b c d^{2}\right )}^{2} - {\left (b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}\right )} {\left (b^{2} c^{2} + a b d^{2}\right )}}}{b^{2} c^{2} + a b d^{2}}}}\right )}{{\left (a b^{3} c^{4} d + 2 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b d^{5} + \sqrt {-a b} b^{3} c^{5} + 2 \, \sqrt {-a b} a b^{2} c^{3} d^{2} + \sqrt {-a b} a^{2} b c d^{4}\right )} \sqrt {-b^{2} c - \sqrt {-a b} b d} {\left | -b c^{2} d - a d^{3} \right |}} - \frac {2 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} d^{3} - 2 \, \sqrt {d x + c} c d^{3}\right )}}{{\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} + a d^{2}\right )} {\left (b c^{2} + a d^{2}\right )}}}{4 \, d} \] Input:

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

Output:

-1/4*(((b*c^2*d + a*d^3)^2*a*d^3*abs(b) + 2*(sqrt(-a*b)*b*c^3*d^3 + sqrt(- 
a*b)*a*c*d^5)*abs(-b*c^2*d - a*d^3)*abs(b) - (b^3*c^6*d^3 + 2*a*b^2*c^4*d^ 
5 + a^2*b*c^2*d^7)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(b^2*c^3 + a*b*c*d^2 
 + sqrt((b^2*c^3 + a*b*c*d^2)^2 - (b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)*(b^2 
*c^2 + a*b*d^2)))/(b^2*c^2 + a*b*d^2)))/((a*b^3*c^4*d + 2*a^2*b^2*c^2*d^3 
+ a^3*b*d^5 - sqrt(-a*b)*b^3*c^5 - 2*sqrt(-a*b)*a*b^2*c^3*d^2 - sqrt(-a*b) 
*a^2*b*c*d^4)*sqrt(-b^2*c + sqrt(-a*b)*b*d)*abs(-b*c^2*d - a*d^3)) + ((b*c 
^2*d + a*d^3)^2*a*d^3*abs(b) - 2*(sqrt(-a*b)*b*c^3*d^3 + sqrt(-a*b)*a*c*d^ 
5)*abs(-b*c^2*d - a*d^3)*abs(b) - (b^3*c^6*d^3 + 2*a*b^2*c^4*d^5 + a^2*b*c 
^2*d^7)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(b^2*c^3 + a*b*c*d^2 - sqrt((b^ 
2*c^3 + a*b*c*d^2)^2 - (b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)*(b^2*c^2 + a*b* 
d^2)))/(b^2*c^2 + a*b*d^2)))/((a*b^3*c^4*d + 2*a^2*b^2*c^2*d^3 + a^3*b*d^5 
 + sqrt(-a*b)*b^3*c^5 + 2*sqrt(-a*b)*a*b^2*c^3*d^2 + sqrt(-a*b)*a^2*b*c*d^ 
4)*sqrt(-b^2*c - sqrt(-a*b)*b*d)*abs(-b*c^2*d - a*d^3)) - 2*((d*x + c)^(3/ 
2)*d^3 - 2*sqrt(d*x + c)*c*d^3)/(((d*x + c)^2*b - 2*(d*x + c)*b*c + b*c^2 
+ a*d^2)*(b*c^2 + a*d^2)))/d
 

Mupad [B] (verification not implemented)

Time = 8.86 (sec) , antiderivative size = 4684, normalized size of antiderivative = 8.82 \[ \int \frac {x}{\sqrt {c+d x} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

((d^2*(c + d*x)^(3/2))/(2*(a*d^2 + b*c^2)) - (c*d^2*(c + d*x)^(1/2))/(a*d^ 
2 + b*c^2))/(b*(c + d*x)^2 + a*d^2 + b*c^2 - 2*b*c*(c + d*x)) + atan(((((1 
28*a*b^4*c^3*d^4 + 128*a^2*b^3*c*d^6)/(8*(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^ 
2)) + ((c + d*x)^(1/2)*(64*a*b^6*c^5*d^2 + 64*a^3*b^4*c*d^6 + 128*a^2*b^5* 
c^3*d^4)*(-(a*d^5*(-a^3*b^3)^(1/2) + a*b^3*c^3*d^2 - 3*a^2*b^2*c*d^4 - 3*b 
*c^2*d^3*(-a^3*b^3)^(1/2))/(64*(a^2*b^6*c^6 + a^5*b^3*d^6 + 3*a^3*b^5*c^4* 
d^2 + 3*a^4*b^4*c^2*d^4)))^(1/2))/(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2))*(-( 
a*d^5*(-a^3*b^3)^(1/2) + a*b^3*c^3*d^2 - 3*a^2*b^2*c*d^4 - 3*b*c^2*d^3*(-a 
^3*b^3)^(1/2))/(64*(a^2*b^6*c^6 + a^5*b^3*d^6 + 3*a^3*b^5*c^4*d^2 + 3*a^4* 
b^4*c^2*d^4)))^(1/2) - ((a*b^2*d^6 - b^3*c^2*d^4)*(c + d*x)^(1/2))/(a^2*d^ 
4 + b^2*c^4 + 2*a*b*c^2*d^2))*(-(a*d^5*(-a^3*b^3)^(1/2) + a*b^3*c^3*d^2 - 
3*a^2*b^2*c*d^4 - 3*b*c^2*d^3*(-a^3*b^3)^(1/2))/(64*(a^2*b^6*c^6 + a^5*b^3 
*d^6 + 3*a^3*b^5*c^4*d^2 + 3*a^4*b^4*c^2*d^4)))^(1/2)*1i - (((128*a*b^4*c^ 
3*d^4 + 128*a^2*b^3*c*d^6)/(8*(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2)) - ((c + 
 d*x)^(1/2)*(64*a*b^6*c^5*d^2 + 64*a^3*b^4*c*d^6 + 128*a^2*b^5*c^3*d^4)*(- 
(a*d^5*(-a^3*b^3)^(1/2) + a*b^3*c^3*d^2 - 3*a^2*b^2*c*d^4 - 3*b*c^2*d^3*(- 
a^3*b^3)^(1/2))/(64*(a^2*b^6*c^6 + a^5*b^3*d^6 + 3*a^3*b^5*c^4*d^2 + 3*a^4 
*b^4*c^2*d^4)))^(1/2))/(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2))*(-(a*d^5*(-a^3 
*b^3)^(1/2) + a*b^3*c^3*d^2 - 3*a^2*b^2*c*d^4 - 3*b*c^2*d^3*(-a^3*b^3)^(1/ 
2))/(64*(a^2*b^6*c^6 + a^5*b^3*d^6 + 3*a^3*b^5*c^4*d^2 + 3*a^4*b^4*c^2*...
 

Reduce [B] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 3219, normalized size of antiderivative = 6.06 \[ \int \frac {x}{\sqrt {c+d x} \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*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)*sq 
rt(c + d*x))/(sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)))*a**2*d** 
2 + 2*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)*sq 
rt(c + d*x))/(sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)))*a*b*c**2 
 - 2*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)*sqr 
t(c + d*x))/(sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)))*a*b*d**2* 
x**2 + 2*sqrt(a*d**2 + b*c**2)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*s 
qrt(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)))*b**2* 
c**2*x**2 + 2*sqrt(b)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)*at 
an((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*c*d**2 + 
2*sqrt(b)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)*atan((sqrt(sqr 
t(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*b*c**3 + 2*sqrt(b)*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)*sqr...