\(\int \frac {x \sqrt {c+d x}}{(a+b x^2)^2} \, dx\) [660]

Optimal result

Integrand size = 20, antiderivative size = 439 \[ \int \frac {x \sqrt {c+d x}}{\left (a+b x^2\right )^2} \, dx=-\frac {\sqrt {c+d x}}{2 b \left (a+b x^2\right )}-\frac {d^2 \arctan \left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}-\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}\right )}{4 \sqrt {2} b^{5/4} \sqrt {b c^2+a d^2} \sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}+\frac {d^2 \arctan \left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}+\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}\right )}{4 \sqrt {2} b^{5/4} \sqrt {b c^2+a d^2} \sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}+\frac {d^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt {b c^2+a d^2}+\sqrt {b} (c+d x)}\right )}{4 \sqrt {2} b^{5/4} \sqrt {b c^2+a d^2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}} \] Output:

-1/2*(d*x+c)^(1/2)/b/(b*x^2+a)-1/8*d^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^(5/4)/(a*d^2+b*c^2)^(1/2)/(-b^(1/2)*c+(a*d^2+b*c^2)^(1/2 
))^(1/2)+1/8*d^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^(5/4 
)/(a*d^2+b*c^2)^(1/2)/(-b^(1/2)*c+(a*d^2+b*c^2)^(1/2))^(1/2)+1/8*d^2*arcta 
nh(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^(5/4)/(a*d^2+b*c^2)^(1/2)/(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 = 0.79 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.56 \[ \int \frac {x \sqrt {c+d x}}{\left (a+b x^2\right )^2} \, dx=-\frac {\sqrt {c+d x}}{2 a b+2 b^2 x^2}-\frac {i d \sqrt {-b c-i \sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c-i \sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+i \sqrt {a} d}\right )}{4 \sqrt {a} b^2 c+4 i a b^{3/2} d}+\frac {i d \sqrt {-b c+i \sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c+i \sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-i \sqrt {a} d}\right )}{4 \sqrt {a} b^2 c-4 i a b^{3/2} d} \] Input:

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

Output:

-(Sqrt[c + d*x]/(2*a*b + 2*b^2*x^2)) - (I*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)])/(4*Sqrt[a]*b^2*c + (4*I)*a*b^(3/2)*d) + (I*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)])/(4*Sqrt[a]*b^2*c - (4*I)*a*b^(3/2)*d)
 

Rubi [A] (verified)

Time = 1.84 (sec) , antiderivative size = 670, normalized size of antiderivative = 1.53, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {561, 25, 27, 1598, 27, 1407, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1598

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1407

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

(-2*((d^2*Sqrt[c + d*x])/(4*b*(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*(-((b^(1/4)*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]]) - (b^(1/4)*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]*Sqrt[b]*Sqrt[b*c^2 + a*d^2]*Sqrt 
[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]]) + (d^2*(-((b^(1/4)*Sqrt[Sqrt[b]*c + Sqr 
t[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]]) + (b^(1/4)*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]*Sqrt[b]*Sqrt[b*c^2 + a*d^2] 
*Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]])))/(4*b)))/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 1407

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

rule 1598

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( 
x_)^4)^(p_.), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(a + b*x^2 + c*x^4)^(p + 1) 
*((b*d - 2*a*e - (b*e - 2*c*d)*x^2)/(2*(p + 1)*(b^2 - 4*a*c))), x] - Simp[f 
^2/(2*(p + 1)*(b^2 - 4*a*c))   Int[(f*x)^(m - 2)*(a + b*x^2 + c*x^4)^(p + 1 
)*Simp[(m - 1)*(b*d - 2*a*e) - (4*p + 4 + m + 1)*(b*e - 2*c*d)*x^2, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && 
 GtQ[m, 1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
 

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 621, normalized size of antiderivative = 1.41

method	result	size
pseudoelliptic	\(-\frac {\frac {\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}\, \left (-b c +\sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}\right ) \left (b \,x^{2}+a \right ) \ln \left (\sqrt {b}\, \left (d x +c \right )-\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}+\sqrt {a \,d^{2}+b \,c^{2}}\right )}{4}-\frac {\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}\, \left (-b c +\sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}\right ) \left (b \,x^{2}+a \right ) \ln \left (\sqrt {b}\, \left (d x +c \right )+\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}+\sqrt {a \,d^{2}+b \,c^{2}}\right )}{4}+\left (2 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {d x +c}\, \sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}+d^{2} \left (b \,x^{2}+a \right ) \left (\arctan \left (\frac {-2 \sqrt {b}\, \sqrt {d x +c}+\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )-\arctan \left (\frac {2 \sqrt {b}\, \sqrt {d x +c}+\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )\right )\right ) b a}{4 \sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}\, \sqrt {a \,d^{2}+b \,c^{2}}\, b^{2} \left (b \,x^{2}+a \right ) a}\)	\(621\)
derivativedivides	\(2 d^{2} \left (-\frac {\sqrt {d x +c}}{4 b \left (b \left (d x +c \right )^{2}-2 b c \left (d x +c \right )+a \,d^{2}+b \,c^{2}\right )}-\frac {-\frac {-\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \ln \left (-\sqrt {b}\, \left (d x +c \right )+\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}-\sqrt {a \,d^{2}+b \,c^{2}}\right )}{2 \sqrt {b}}+\frac {2 \left (-2 a \,d^{2} \sqrt {b}+\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{2 \sqrt {b}}\right ) \arctan \left (\frac {-2 \sqrt {b}\, \sqrt {d x +c}+\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}}{4 \sqrt {b}\, \sqrt {a \,d^{2}+b \,c^{2}}\, a \,d^{2}}-\frac {\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \ln \left (\sqrt {b}\, \left (d x +c \right )+\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}+\sqrt {a \,d^{2}+b \,c^{2}}\right )}{2 \sqrt {b}}+\frac {2 \left (2 a \,d^{2} \sqrt {b}-\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{2 \sqrt {b}}\right ) \arctan \left (\frac {2 \sqrt {b}\, \sqrt {d x +c}+\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}}{4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}\, a \,d^{2}}}{4 b}\right )\)	\(847\)
default	\(2 d^{2} \left (-\frac {\sqrt {d x +c}}{4 b \left (b \left (d x +c \right )^{2}-2 b c \left (d x +c \right )+a \,d^{2}+b \,c^{2}\right )}-\frac {-\frac {-\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \ln \left (-\sqrt {b}\, \left (d x +c \right )+\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}-\sqrt {a \,d^{2}+b \,c^{2}}\right )}{2 \sqrt {b}}+\frac {2 \left (-2 a \,d^{2} \sqrt {b}+\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{2 \sqrt {b}}\right ) \arctan \left (\frac {-2 \sqrt {b}\, \sqrt {d x +c}+\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}}{4 \sqrt {b}\, \sqrt {a \,d^{2}+b \,c^{2}}\, a \,d^{2}}-\frac {\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \ln \left (\sqrt {b}\, \left (d x +c \right )+\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}+\sqrt {a \,d^{2}+b \,c^{2}}\right )}{2 \sqrt {b}}+\frac {2 \left (2 a \,d^{2} \sqrt {b}-\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{2 \sqrt {b}}\right ) \arctan \left (\frac {2 \sqrt {b}\, \sqrt {d x +c}+\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}}{4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}\, a \,d^{2}}}{4 b}\right )\)	\(847\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1163 vs. \(2 (343) = 686\).

Time = 0.10 (sec) , antiderivative size = 1163, normalized size of antiderivative = 2.65 \[ \int \frac {x \sqrt {c+d x}}{\left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/8*((b^2*x^2 + a*b)*sqrt(-(c*d^2 + (a*b^3*c^2 + a^2*b^2*d^2)*sqrt(-d^6/(a 
*b^7*c^4 + 2*a^2*b^6*c^2*d^2 + a^3*b^5*d^4)))/(a*b^3*c^2 + a^2*b^2*d^2))*l 
og(sqrt(d*x + c)*d^4 + (a*b*d^4 + (a*b^5*c^3 + a^2*b^4*c*d^2)*sqrt(-d^6/(a 
*b^7*c^4 + 2*a^2*b^6*c^2*d^2 + a^3*b^5*d^4)))*sqrt(-(c*d^2 + (a*b^3*c^2 + 
a^2*b^2*d^2)*sqrt(-d^6/(a*b^7*c^4 + 2*a^2*b^6*c^2*d^2 + a^3*b^5*d^4)))/(a* 
b^3*c^2 + a^2*b^2*d^2))) - (b^2*x^2 + a*b)*sqrt(-(c*d^2 + (a*b^3*c^2 + a^2 
*b^2*d^2)*sqrt(-d^6/(a*b^7*c^4 + 2*a^2*b^6*c^2*d^2 + a^3*b^5*d^4)))/(a*b^3 
*c^2 + a^2*b^2*d^2))*log(sqrt(d*x + c)*d^4 - (a*b*d^4 + (a*b^5*c^3 + a^2*b 
^4*c*d^2)*sqrt(-d^6/(a*b^7*c^4 + 2*a^2*b^6*c^2*d^2 + a^3*b^5*d^4)))*sqrt(- 
(c*d^2 + (a*b^3*c^2 + a^2*b^2*d^2)*sqrt(-d^6/(a*b^7*c^4 + 2*a^2*b^6*c^2*d^ 
2 + a^3*b^5*d^4)))/(a*b^3*c^2 + a^2*b^2*d^2))) + (b^2*x^2 + a*b)*sqrt(-(c* 
d^2 - (a*b^3*c^2 + a^2*b^2*d^2)*sqrt(-d^6/(a*b^7*c^4 + 2*a^2*b^6*c^2*d^2 + 
 a^3*b^5*d^4)))/(a*b^3*c^2 + a^2*b^2*d^2))*log(sqrt(d*x + c)*d^4 + (a*b*d^ 
4 - (a*b^5*c^3 + a^2*b^4*c*d^2)*sqrt(-d^6/(a*b^7*c^4 + 2*a^2*b^6*c^2*d^2 + 
 a^3*b^5*d^4)))*sqrt(-(c*d^2 - (a*b^3*c^2 + a^2*b^2*d^2)*sqrt(-d^6/(a*b^7* 
c^4 + 2*a^2*b^6*c^2*d^2 + a^3*b^5*d^4)))/(a*b^3*c^2 + a^2*b^2*d^2))) - (b^ 
2*x^2 + a*b)*sqrt(-(c*d^2 - (a*b^3*c^2 + a^2*b^2*d^2)*sqrt(-d^6/(a*b^7*c^4 
 + 2*a^2*b^6*c^2*d^2 + a^3*b^5*d^4)))/(a*b^3*c^2 + a^2*b^2*d^2))*log(sqrt( 
d*x + c)*d^4 - (a*b*d^4 - (a*b^5*c^3 + a^2*b^4*c*d^2)*sqrt(-d^6/(a*b^7*c^4 
 + 2*a^2*b^6*c^2*d^2 + a^3*b^5*d^4)))*sqrt(-(c*d^2 - (a*b^3*c^2 + a^2*b...
 

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]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.66 \[ \int \frac {x \sqrt {c+d x}}{\left (a+b x^2\right )^2} \, dx=-\frac {\sqrt {d x + c} d^{2}}{2 \, {\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} + a d^{2}\right )} b} + \frac {{\left (b^{2} c d^{2} - \sqrt {-a b} d^{2} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {b^{2} c + \sqrt {b^{4} c^{2} - {\left (b^{2} c^{2} + a b d^{2}\right )} b^{2}}}{b^{2}}}}\right )}{4 \, {\left (a b^{2} d + \sqrt {-a b} b^{2} c\right )} \sqrt {-b^{2} c - \sqrt {-a b} b d} {\left | d \right |}} + \frac {{\left (b^{2} c d^{2} + \sqrt {-a b} d^{2} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {b^{2} c - \sqrt {b^{4} c^{2} - {\left (b^{2} c^{2} + a b d^{2}\right )} b^{2}}}{b^{2}}}}\right )}{4 \, {\left (a b^{2} d - \sqrt {-a b} b^{2} c\right )} \sqrt {-b^{2} c + \sqrt {-a b} b d} {\left | d \right |}} \] Input:

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

Output:

-1/2*sqrt(d*x + c)*d^2/(((d*x + c)^2*b - 2*(d*x + c)*b*c + b*c^2 + a*d^2)* 
b) + 1/4*(b^2*c*d^2 - sqrt(-a*b)*d^2*abs(b)*abs(d))*arctan(sqrt(d*x + c)/s 
qrt(-(b^2*c + sqrt(b^4*c^2 - (b^2*c^2 + a*b*d^2)*b^2))/b^2))/((a*b^2*d + s 
qrt(-a*b)*b^2*c)*sqrt(-b^2*c - sqrt(-a*b)*b*d)*abs(d)) + 1/4*(b^2*c*d^2 + 
sqrt(-a*b)*d^2*abs(b)*abs(d))*arctan(sqrt(d*x + c)/sqrt(-(b^2*c - sqrt(b^4 
*c^2 - (b^2*c^2 + a*b*d^2)*b^2))/b^2))/((a*b^2*d - sqrt(-a*b)*b^2*c)*sqrt( 
-b^2*c + sqrt(-a*b)*b*d)*abs(d))
 

Mupad [B] (verification not implemented)

Time = 8.79 (sec) , antiderivative size = 1597, normalized size of antiderivative = 3.64 \[ \int \frac {x \sqrt {c+d x}}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

2*atanh((2*a^2*b^7*c^2*d^4*(- (d^3*(-a^3*b^5)^(1/2))/(64*(a^2*b^6*c^2 + a^ 
3*b^5*d^2)) - (a*b^3*c*d^2)/(64*(a^2*b^6*c^2 + a^3*b^5*d^2)))^(1/2)*(c + d 
*x)^(1/2))/((a^5*b^10*c*d^8)/(4*(a^2*b^6*c^2 + a^3*b^5*d^2)) + (a^4*b^11*c 
^3*d^6)/(4*(a^2*b^6*c^2 + a^3*b^5*d^2)) + (a^4*b^7*d^9*(-a^3*b^5)^(1/2))/( 
4*(a^2*b^6*c^2 + a^3*b^5*d^2)) + (a^3*b^8*c^2*d^7*(-a^3*b^5)^(1/2))/(4*(a^ 
2*b^6*c^2 + a^3*b^5*d^2))) - (2*b*d^4*(- (d^3*(-a^3*b^5)^(1/2))/(64*(a^2*b 
^6*c^2 + a^3*b^5*d^2)) - (a*b^3*c*d^2)/(64*(a^2*b^6*c^2 + a^3*b^5*d^2)))^( 
1/2)*(c + d*x)^(1/2))/((a^2*b^5*c*d^6)/(4*(a^2*b^6*c^2 + a^3*b^5*d^2)) + ( 
a*b^2*d^7*(-a^3*b^5)^(1/2))/(4*(a^2*b^6*c^2 + a^3*b^5*d^2))) + (2*a*b^4*c* 
d^5*(- (d^3*(-a^3*b^5)^(1/2))/(64*(a^2*b^6*c^2 + a^3*b^5*d^2)) - (a*b^3*c* 
d^2)/(64*(a^2*b^6*c^2 + a^3*b^5*d^2)))^(1/2)*(-a^3*b^5)^(1/2)*(c + d*x)^(1 
/2))/((a^5*b^10*c*d^8)/(4*(a^2*b^6*c^2 + a^3*b^5*d^2)) + (a^4*b^11*c^3*d^6 
)/(4*(a^2*b^6*c^2 + a^3*b^5*d^2)) + (a^4*b^7*d^9*(-a^3*b^5)^(1/2))/(4*(a^2 
*b^6*c^2 + a^3*b^5*d^2)) + (a^3*b^8*c^2*d^7*(-a^3*b^5)^(1/2))/(4*(a^2*b^6* 
c^2 + a^3*b^5*d^2))))*(-(d^3*(-a^3*b^5)^(1/2) + a*b^3*c*d^2)/(64*(a^2*b^6* 
c^2 + a^3*b^5*d^2)))^(1/2) - 2*atanh((2*b*d^4*((d^3*(-a^3*b^5)^(1/2))/(64* 
(a^2*b^6*c^2 + a^3*b^5*d^2)) - (a*b^3*c*d^2)/(64*(a^2*b^6*c^2 + a^3*b^5*d^ 
2)))^(1/2)*(c + d*x)^(1/2))/((a^2*b^5*c*d^6)/(4*(a^2*b^6*c^2 + a^3*b^5*d^2 
)) - (a*b^2*d^7*(-a^3*b^5)^(1/2))/(4*(a^2*b^6*c^2 + a^3*b^5*d^2))) - (2*a^ 
2*b^7*c^2*d^4*((d^3*(-a^3*b^5)^(1/2))/(64*(a^2*b^6*c^2 + a^3*b^5*d^2)) ...
 

Reduce [B] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 2322, normalized size of antiderivative = 5.29 \[ \int \frac {x \sqrt {c+d x}}{\left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] 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*b*c - 
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)*sqrt(c 
 + d*x))/(sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)))*b**2*c*x**2 
- 2*sqrt(b)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)*atan((sqrt(s 
qrt(b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2) - 2*sqrt(b)*sqrt(c + d*x))/(sq 
rt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)))*a**2*d**2 - 2*sqrt(b)*sq 
rt(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 
t(a*d**2 + b*c**2) - b*c)*sqrt(2)))*a*b*c**2 - 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)*sqrt(a*d**2 + b*c* 
*2) - b*c)*sqrt(2)))*a*b*d**2*x**2 - 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)*sqr 
t(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(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...