\(\int \frac {x \sqrt {c+d x}}{(a+b x^2)^3} \, dx\) [713]

Optimal result

Integrand size = 20, antiderivative size = 601 \[ \int \frac {x \sqrt {c+d x}}{\left (a+b x^2\right )^3} \, dx=-\frac {\sqrt {c+d x}}{4 b \left (a+b x^2\right )^2}+\frac {d (a d+b c x) \sqrt {c+d x}}{16 a b \left (b c^2+a d^2\right ) \left (a+b x^2\right )}-\frac {d^2 \left (b c^2+3 a d^2+\sqrt {b} c \sqrt {b c^2+a d^2}\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}-\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}\right )}{32 \sqrt {2} a b^{5/4} \left (b c^2+a d^2\right )^{3/2} \sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}+\frac {d^2 \left (b c^2+3 a d^2+\sqrt {b} c \sqrt {b c^2+a d^2}\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}+\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}\right )}{32 \sqrt {2} a b^{5/4} \left (b c^2+a d^2\right )^{3/2} \sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}+\frac {d^2 \left (b c^2+3 a d^2-\sqrt {b} c \sqrt {b c^2+a d^2}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt {b c^2+a d^2}+\sqrt {b} (c+d x)}\right )}{32 \sqrt {2} a b^{5/4} \left (b c^2+a d^2\right )^{3/2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}} \] Output:

-1/4*(d*x+c)^(1/2)/b/(b*x^2+a)^2+1/16*d*(b*c*x+a*d)*(d*x+c)^(1/2)/a/b/(a*d 
^2+b*c^2)/(b*x^2+a)-1/64*d^2*(b*c^2+3*a*d^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)/a/b^(5/4)/(a*d^2+b*c^2 
)^(3/2)/(-b^(1/2)*c+(a*d^2+b*c^2)^(1/2))^(1/2)+1/64*d^2*(b*c^2+3*a*d^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)/a/b^(5/4)/(a*d^2+b*c^2)^(3/2)/(-b^(1/2)*c+(a*d^2+b*c^2)^(1/2))^(1/2)+ 
1/64*d^2*(b*c^2+3*a*d^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)/a/b^(5/4)/(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 = 1.77 (sec) , antiderivative size = 335, normalized size of antiderivative = 0.56 \[ \int \frac {x \sqrt {c+d x}}{\left (a+b x^2\right )^3} \, dx=\frac {d \left (\frac {2 \sqrt {a} \sqrt {c+d x} \left (-3 a^2 d^2+b^2 c d x^3+a b \left (-4 c^2+c d x+d^2 x^2\right )\right )}{\left (b c^2 d+a d^3\right ) \left (a+b x^2\right )^2}+\frac {i \left (2 \sqrt {b} c+3 i \sqrt {a} d\right ) \arctan \left (\frac {\sqrt {-b c-i \sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+i \sqrt {a} d}\right )}{\left (\sqrt {b} c+i \sqrt {a} d\right ) \sqrt {-b c-i \sqrt {a} \sqrt {b} d}}-\frac {i \left (2 \sqrt {b} c-3 i \sqrt {a} d\right ) \arctan \left (\frac {\sqrt {-b c+i \sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-i \sqrt {a} d}\right )}{\left (\sqrt {b} c-i \sqrt {a} d\right ) \sqrt {-b c+i \sqrt {a} \sqrt {b} d}}\right )}{32 a^{3/2} b} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 2.58 (sec) , antiderivative size = 915, normalized size of antiderivative = 1.52, 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, 1598, 27, 1405, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1598

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1405

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1483

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

(-2*((d^2*Sqrt[c + d*x])/(8*b*(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]*(b*c^2 - a*d^2 - b*c*(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*(-((b^(1/4)*Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^ 
2]]*(b*c^2 + 3*a*d^2 + 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)*(b*c^2 + 3*a*d^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]*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 + Sqrt[b*c^2 + a*d^2]]*(b*c^2 + 3*a*d^2 + 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)*(b*c^2 + 3*a*d 
^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]*Sqrt[b]*Sqrt[b*c^2 + a*d^2]*Sqrt[Sqrt[b]*c + Sqrt[b*c^2 
 + a*d^2]]))/(4*a*(b*c^2 + a*d^2))))/(8*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 1405

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 
- 2*a*c + b*c*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[(b^2 - 2*a*c + 2*(p + 1)*( 
b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr 
eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
 

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 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1095\) vs. \(2(487)=974\).

Time = 1.88 (sec) , antiderivative size = 1096, normalized size of antiderivative = 1.82

method	result	size
pseudoelliptic	\(\text {Expression too large to display}\)	\(1096\)
derivativedivides	\(\text {Expression too large to display}\)	\(2678\)
default	\(\text {Expression too large to display}\)	\(2678\)

Input:

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

Output:

-3/32*(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)*((-1/3*b*c*(b*x^2+a)^2*(a*d^2+b*c^2)^(1/2)+(d^2*x^4+2/3*c^2*x^2)*a* 
b^(5/2)+1/3*b^(7/2)*c^2*x^4+a^2*((2*d^2*x^2+1/3*c^2)*b^(3/2)+a*d^2*b^(1/2) 
))*((a*d^2+b*c^2)*b)^(1/2)-(-1/3*b^2*c*(b*x^2+a)^2*(a*d^2+b*c^2)^(1/2)+1/3 
*a^2*(6*d^2*x^2+c^2)*b^(5/2)+(d^2*x^4+2/3*c^2*x^2)*a*b^(7/2)+1/3*b^(9/2)*c 
^2*x^4+a^3*d^2*b^(3/2))*c)*(2*((a*d^2+b*c^2)*b)^(1/2)+2*b*c)^(1/2)*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)*((-1/3*b*c*(b*x^2+a)^2*(a*d^2+b*c^2)^(1/2)+(d^2*x^4+2/3*c^2*x 
^2)*a*b^(5/2)+1/3*b^(7/2)*c^2*x^4+a^2*((2*d^2*x^2+1/3*c^2)*b^(3/2)+a*d^2*b 
^(1/2)))*((a*d^2+b*c^2)*b)^(1/2)-(-1/3*b^2*c*(b*x^2+a)^2*(a*d^2+b*c^2)^(1/ 
2)+1/3*a^2*(6*d^2*x^2+c^2)*b^(5/2)+(d^2*x^4+2/3*c^2*x^2)*a*b^(7/2)+1/3*b^( 
9/2)*c^2*x^4+a^3*d^2*b^(3/2))*c)*(2*((a*d^2+b*c^2)*b)^(1/2)+2*b*c)^(1/2)*l 
n(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*(d*x+c)^(1/2)*(a*d^2+b*c^2)^(1/2)*(4/3*a*(-1/4*d^2*x 
^2-1/4*c*d*x+c^2)*b^(5/2)+d*(-1/3*b^(7/2)*c*x^3+a^2*b^(3/2)*d))*(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))-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^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3602 vs. \(2 (489) = 978\).

Time = 0.24 (sec) , antiderivative size = 3602, normalized size of antiderivative = 5.99 \[ \int \frac {x \sqrt {c+d x}}{\left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] 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]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1025 vs. \(2 (489) = 978\).

Time = 0.28 (sec) , antiderivative size = 1025, normalized size of antiderivative = 1.71 \[ \int \frac {x \sqrt {c+d x}}{\left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/32*((a*b^2*c^2*d + a^2*b*d^3)^2*c*d^2*abs(b) - (sqrt(-a*b)*b^2*c^4*d^2 
+ 4*sqrt(-a*b)*a*b*c^2*d^4 + 3*sqrt(-a*b)*a^2*d^6)*abs(-a*b^2*c^2*d - a^2* 
b*d^3)*abs(b) + (2*a*b^5*c^7*d^2 + 7*a^2*b^4*c^5*d^4 + 8*a^3*b^3*c^3*d^6 + 
 3*a^4*b^2*c*d^8)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(a*b^3*c^3 + a^2*b^2* 
c*d^2 + sqrt((a*b^3*c^3 + a^2*b^2*c*d^2)^2 - (a*b^3*c^4 + 2*a^2*b^2*c^2*d^ 
2 + a^3*b*d^4)*(a*b^3*c^2 + a^2*b^2*d^2)))/(a*b^3*c^2 + a^2*b^2*d^2)))/((a 
^2*b^4*c^4*d + 2*a^3*b^3*c^2*d^3 + a^4*b^2*d^5 - sqrt(-a*b)*a*b^4*c^5 - 2* 
sqrt(-a*b)*a^2*b^3*c^3*d^2 - sqrt(-a*b)*a^3*b^2*c*d^4)*sqrt(-b^2*c + sqrt( 
-a*b)*b*d)*abs(-a*b^2*c^2*d - a^2*b*d^3)) - 1/32*((a*b^2*c^2*d + a^2*b*d^3 
)^2*c*d^2*abs(b) + (sqrt(-a*b)*b^2*c^4*d^2 + 4*sqrt(-a*b)*a*b*c^2*d^4 + 3* 
sqrt(-a*b)*a^2*d^6)*abs(-a*b^2*c^2*d - a^2*b*d^3)*abs(b) + (2*a*b^5*c^7*d^ 
2 + 7*a^2*b^4*c^5*d^4 + 8*a^3*b^3*c^3*d^6 + 3*a^4*b^2*c*d^8)*abs(b))*arcta 
n(sqrt(d*x + c)/sqrt(-(a*b^3*c^3 + a^2*b^2*c*d^2 - sqrt((a*b^3*c^3 + a^2*b 
^2*c*d^2)^2 - (a*b^3*c^4 + 2*a^2*b^2*c^2*d^2 + a^3*b*d^4)*(a*b^3*c^2 + a^2 
*b^2*d^2)))/(a*b^3*c^2 + a^2*b^2*d^2)))/((a^2*b^4*c^4*d + 2*a^3*b^3*c^2*d^ 
3 + a^4*b^2*d^5 + sqrt(-a*b)*a*b^4*c^5 + 2*sqrt(-a*b)*a^2*b^3*c^3*d^2 + sq 
rt(-a*b)*a^3*b^2*c*d^4)*sqrt(-b^2*c - sqrt(-a*b)*b*d)*abs(-a*b^2*c^2*d - a 
^2*b*d^3)) + 1/16*((d*x + c)^(7/2)*b^2*c*d^2 - 3*(d*x + c)^(5/2)*b^2*c^2*d 
^2 + 3*(d*x + c)^(3/2)*b^2*c^3*d^2 - sqrt(d*x + c)*b^2*c^4*d^2 + (d*x + c) 
^(5/2)*a*b*d^4 - (d*x + c)^(3/2)*a*b*c*d^4 - 4*sqrt(d*x + c)*a*b*c^2*d^...
 

Mupad [B] (verification not implemented)

Time = 12.21 (sec) , antiderivative size = 5736, normalized size of antiderivative = 9.54 \[ \int \frac {x \sqrt {c+d x}}{\left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((12288*a^5*b^2*d^8 + 4096*a^3*b^4*c^4*d^4 + 16384*a^4*b^3*c^2*d^6) 
/(4096*(a^5*d^4 + a^3*b^2*c^4 + 2*a^4*b*c^2*d^2)) - ((c + d*x)^(1/2)*(4096 
*a^5*b^4*c*d^6 + 4096*a^3*b^6*c^5*d^2 + 8192*a^4*b^5*c^3*d^4)*(-(15*a^5*b^ 
3*c*d^6 - 9*a*d^7*(-a^9*b^5)^(1/2) + 4*a^3*b^5*c^5*d^2 + 15*a^4*b^4*c^3*d^ 
4 - 5*b*c^2*d^5*(-a^9*b^5)^(1/2))/(4096*(a^6*b^8*c^6 + a^9*b^5*d^6 + 3*a^7 
*b^7*c^4*d^2 + 3*a^8*b^6*c^2*d^4)))^(1/2))/(64*(a^4*d^4 + a^2*b^2*c^4 + 2* 
a^3*b*c^2*d^2)))*(-(15*a^5*b^3*c*d^6 - 9*a*d^7*(-a^9*b^5)^(1/2) + 4*a^3*b^ 
5*c^5*d^2 + 15*a^4*b^4*c^3*d^4 - 5*b*c^2*d^5*(-a^9*b^5)^(1/2))/(4096*(a^6* 
b^8*c^6 + a^9*b^5*d^6 + 3*a^7*b^7*c^4*d^2 + 3*a^8*b^6*c^2*d^4)))^(1/2) - ( 
(c + d*x)^(1/2)*(9*a^2*b*d^8 + 4*b^3*c^4*d^4 + 11*a*b^2*c^2*d^6))/(64*(a^4 
*d^4 + a^2*b^2*c^4 + 2*a^3*b*c^2*d^2)))*(-(15*a^5*b^3*c*d^6 - 9*a*d^7*(-a^ 
9*b^5)^(1/2) + 4*a^3*b^5*c^5*d^2 + 15*a^4*b^4*c^3*d^4 - 5*b*c^2*d^5*(-a^9* 
b^5)^(1/2))/(4096*(a^6*b^8*c^6 + a^9*b^5*d^6 + 3*a^7*b^7*c^4*d^2 + 3*a^8*b 
^6*c^2*d^4)))^(1/2)*1i - (((12288*a^5*b^2*d^8 + 4096*a^3*b^4*c^4*d^4 + 163 
84*a^4*b^3*c^2*d^6)/(4096*(a^5*d^4 + a^3*b^2*c^4 + 2*a^4*b*c^2*d^2)) + ((c 
 + d*x)^(1/2)*(4096*a^5*b^4*c*d^6 + 4096*a^3*b^6*c^5*d^2 + 8192*a^4*b^5*c^ 
3*d^4)*(-(15*a^5*b^3*c*d^6 - 9*a*d^7*(-a^9*b^5)^(1/2) + 4*a^3*b^5*c^5*d^2 
+ 15*a^4*b^4*c^3*d^4 - 5*b*c^2*d^5*(-a^9*b^5)^(1/2))/(4096*(a^6*b^8*c^6 + 
a^9*b^5*d^6 + 3*a^7*b^7*c^4*d^2 + 3*a^8*b^6*c^2*d^4)))^(1/2))/(64*(a^4*d^4 
 + a^2*b^2*c^4 + 2*a^3*b*c^2*d^2)))*(-(15*a^5*b^3*c*d^6 - 9*a*d^7*(-a^9...
 

Reduce [B] (verification not implemented)

Time = 4.82 (sec) , antiderivative size = 6204, normalized size of antiderivative = 10.32 \[ \int \frac {x \sqrt {c+d x}}{\left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 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)*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**3*b*c 
*d**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)*sqrt(2)))*a**2 
*b**2*c**3 - 16*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*d**2*x**2 - 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)*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**3*c**3*x**2 - 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)*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**3*c*d**2*x**4 - 4*sqrt(a*d**2 + b*c**2)*sqrt(sqr 
t(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**4*c**3*x**4 - 6*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**...