\(\int \frac {(e x)^{9/2} (c+d x)}{(a+b x^2)^3} \, dx\) [432]

Optimal result

Integrand size = 22, antiderivative size = 319 \[ \int \frac {(e x)^{9/2} (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {2 d e^4 \sqrt {e x}}{b^3}-\frac {a e^4 \sqrt {e x} (a d-b c x)}{4 b^3 \left (a+b x^2\right )^2}+\frac {e^4 \sqrt {e x} (17 a d-11 b c x)}{16 b^3 \left (a+b x^2\right )}-\frac {3 \left (7 \sqrt {b} c-15 \sqrt {a} d\right ) e^{9/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} \sqrt [4]{a} b^{13/4}}+\frac {3 \left (7 \sqrt {b} c-15 \sqrt {a} d\right ) e^{9/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} \sqrt [4]{a} b^{13/4}}-\frac {3 \left (7 \sqrt {b} c+15 \sqrt {a} d\right ) e^{9/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{32 \sqrt {2} \sqrt [4]{a} b^{13/4}} \] Output:

2*d*e^4*(e*x)^(1/2)/b^3-1/4*a*e^4*(e*x)^(1/2)*(-b*c*x+a*d)/b^3/(b*x^2+a)^2 
+1/16*e^4*(e*x)^(1/2)*(-11*b*c*x+17*a*d)/b^3/(b*x^2+a)-3/64*(7*b^(1/2)*c-1 
5*a^(1/2)*d)*e^(9/2)*arctan(1-2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)) 
*2^(1/2)/a^(1/4)/b^(13/4)+3/64*(7*b^(1/2)*c-15*a^(1/2)*d)*e^(9/2)*arctan(1 
+2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(1/4)/b^(13/4)-3/6 
4*(7*b^(1/2)*c+15*a^(1/2)*d)*e^(9/2)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*(e*x) 
^(1/2)/e^(1/2)/(a^(1/2)+b^(1/2)*x))*2^(1/2)/a^(1/4)/b^(13/4)
 

Mathematica [A] (verified)

Time = 1.02 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.65 \[ \int \frac {(e x)^{9/2} (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {(e x)^{9/2} \left (\frac {4 \sqrt [4]{b} \sqrt {x} \left (45 a^2 d+b^2 x^3 (-11 c+32 d x)+a b x (-7 c+81 d x)\right )}{\left (a+b x^2\right )^2}+\frac {3 \sqrt {2} \left (-7 \sqrt {b} c+15 \sqrt {a} d\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{a}}-\frac {3 \sqrt {2} \left (7 \sqrt {b} c+15 \sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{a}}\right )}{64 b^{13/4} x^{9/2}} \] Input:

Integrate[((e*x)^(9/2)*(c + d*x))/(a + b*x^2)^3,x]
 

Output:

((e*x)^(9/2)*((4*b^(1/4)*Sqrt[x]*(45*a^2*d + b^2*x^3*(-11*c + 32*d*x) + a* 
b*x*(-7*c + 81*d*x)))/(a + b*x^2)^2 + (3*Sqrt[2]*(-7*Sqrt[b]*c + 15*Sqrt[a 
]*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/a^(1 
/4) - (3*Sqrt[2]*(7*Sqrt[b]*c + 15*Sqrt[a]*d)*ArcTanh[(Sqrt[2]*a^(1/4)*b^( 
1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/a^(1/4)))/(64*b^(13/4)*x^(9/2))
 

Rubi [A] (verified)

Time = 1.16 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.26, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {549, 27, 549, 27, 552, 27, 554, 1482, 27, 1476, 1082, 217, 1479, 25, 27, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(e x)^{9/2} (c+d x)}{\left (a+b x^2\right )^3} \, dx\)

\(\Big \downarrow \) 549

\(\displaystyle \frac {e^2 \int \frac {(e x)^{5/2} (7 c+9 d x)}{2 \left (b x^2+a\right )^2}dx}{4 b}-\frac {e (e x)^{7/2} (c+d x)}{4 b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^2 \int \frac {(e x)^{5/2} (7 c+9 d x)}{\left (b x^2+a\right )^2}dx}{8 b}-\frac {e (e x)^{7/2} (c+d x)}{4 b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 549

\(\displaystyle \frac {e^2 \left (\frac {e^2 \int \frac {3 \sqrt {e x} (7 c+15 d x)}{2 \left (b x^2+a\right )}dx}{2 b}-\frac {e (e x)^{3/2} (7 c+9 d x)}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {e (e x)^{7/2} (c+d x)}{4 b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^2 \left (\frac {3 e^2 \int \frac {\sqrt {e x} (7 c+15 d x)}{b x^2+a}dx}{4 b}-\frac {e (e x)^{3/2} (7 c+9 d x)}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {e (e x)^{7/2} (c+d x)}{4 b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 552

\(\displaystyle \frac {e^2 \left (\frac {3 e^2 \left (\frac {30 d \sqrt {e x}}{b}-\frac {2 e \int \frac {15 a d-7 b c x}{2 \sqrt {e x} \left (b x^2+a\right )}dx}{b}\right )}{4 b}-\frac {e (e x)^{3/2} (7 c+9 d x)}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {e (e x)^{7/2} (c+d x)}{4 b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^2 \left (\frac {3 e^2 \left (\frac {30 d \sqrt {e x}}{b}-\frac {e \int \frac {15 a d-7 b c x}{\sqrt {e x} \left (b x^2+a\right )}dx}{b}\right )}{4 b}-\frac {e (e x)^{3/2} (7 c+9 d x)}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {e (e x)^{7/2} (c+d x)}{4 b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 554

\(\displaystyle \frac {e^2 \left (\frac {3 e^2 \left (\frac {30 d \sqrt {e x}}{b}-\frac {2 e \int \frac {15 a d e-7 b c e x}{b x^2 e^2+a e^2}d\sqrt {e x}}{b}\right )}{4 b}-\frac {e (e x)^{3/2} (7 c+9 d x)}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {e (e x)^{7/2} (c+d x)}{4 b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 1482

\(\displaystyle \frac {e^2 \left (\frac {3 e^2 \left (\frac {30 d \sqrt {e x}}{b}-\frac {2 e \left (\frac {1}{2} \left (\frac {15 \sqrt {a} d}{\sqrt {b}}+7 c\right ) \int \frac {\sqrt {b} \left (\sqrt {a} e-\sqrt {b} e x\right )}{b x^2 e^2+a e^2}d\sqrt {e x}-\frac {1}{2} \left (7 c-\frac {15 \sqrt {a} d}{\sqrt {b}}\right ) \int \frac {\sqrt {b} \left (\sqrt {b} x e+\sqrt {a} e\right )}{b x^2 e^2+a e^2}d\sqrt {e x}\right )}{b}\right )}{4 b}-\frac {e (e x)^{3/2} (7 c+9 d x)}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {e (e x)^{7/2} (c+d x)}{4 b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^2 \left (\frac {3 e^2 \left (\frac {30 d \sqrt {e x}}{b}-\frac {2 e \left (\frac {1}{2} \sqrt {b} \left (\frac {15 \sqrt {a} d}{\sqrt {b}}+7 c\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}-\frac {1}{2} \sqrt {b} \left (7 c-\frac {15 \sqrt {a} d}{\sqrt {b}}\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{b x^2 e^2+a e^2}d\sqrt {e x}\right )}{b}\right )}{4 b}-\frac {e (e x)^{3/2} (7 c+9 d x)}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {e (e x)^{7/2} (c+d x)}{4 b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {e^2 \left (\frac {3 e^2 \left (\frac {30 d \sqrt {e x}}{b}-\frac {2 e \left (\frac {1}{2} \sqrt {b} \left (\frac {15 \sqrt {a} d}{\sqrt {b}}+7 c\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}-\frac {1}{2} \sqrt {b} \left (7 c-\frac {15 \sqrt {a} d}{\sqrt {b}}\right ) \left (\frac {\int \frac {1}{x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {b}}\right )\right )}{b}\right )}{4 b}-\frac {e (e x)^{3/2} (7 c+9 d x)}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {e (e x)^{7/2} (c+d x)}{4 b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {e^2 \left (\frac {3 e^2 \left (\frac {30 d \sqrt {e x}}{b}-\frac {2 e \left (\frac {1}{2} \sqrt {b} \left (\frac {15 \sqrt {a} d}{\sqrt {b}}+7 c\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}-\frac {1}{2} \sqrt {b} \left (7 c-\frac {15 \sqrt {a} d}{\sqrt {b}}\right ) \left (\frac {\int \frac {1}{-e x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\int \frac {1}{-e x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )\right )}{b}\right )}{4 b}-\frac {e (e x)^{3/2} (7 c+9 d x)}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {e (e x)^{7/2} (c+d x)}{4 b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {e^2 \left (\frac {3 e^2 \left (\frac {30 d \sqrt {e x}}{b}-\frac {2 e \left (\frac {1}{2} \sqrt {b} \left (\frac {15 \sqrt {a} d}{\sqrt {b}}+7 c\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}-\frac {1}{2} \sqrt {b} \left (7 c-\frac {15 \sqrt {a} d}{\sqrt {b}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )\right )}{b}\right )}{4 b}-\frac {e (e x)^{3/2} (7 c+9 d x)}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {e (e x)^{7/2} (c+d x)}{4 b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {e^2 \left (\frac {3 e^2 \left (\frac {30 d \sqrt {e x}}{b}-\frac {2 e \left (\frac {1}{2} \sqrt {b} \left (\frac {15 \sqrt {a} d}{\sqrt {b}}+7 c\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}\right )}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )-\frac {1}{2} \sqrt {b} \left (7 c-\frac {15 \sqrt {a} d}{\sqrt {b}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )\right )}{b}\right )}{4 b}-\frac {e (e x)^{3/2} (7 c+9 d x)}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {e (e x)^{7/2} (c+d x)}{4 b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {e^2 \left (\frac {3 e^2 \left (\frac {30 d \sqrt {e x}}{b}-\frac {2 e \left (\frac {1}{2} \sqrt {b} \left (\frac {15 \sqrt {a} d}{\sqrt {b}}+7 c\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}\right )}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )-\frac {1}{2} \sqrt {b} \left (7 c-\frac {15 \sqrt {a} d}{\sqrt {b}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )\right )}{b}\right )}{4 b}-\frac {e (e x)^{3/2} (7 c+9 d x)}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {e (e x)^{7/2} (c+d x)}{4 b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^2 \left (\frac {3 e^2 \left (\frac {30 d \sqrt {e x}}{b}-\frac {2 e \left (\frac {1}{2} \sqrt {b} \left (\frac {15 \sqrt {a} d}{\sqrt {b}}+7 c\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {e}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {e}}\right )-\frac {1}{2} \sqrt {b} \left (7 c-\frac {15 \sqrt {a} d}{\sqrt {b}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )\right )}{b}\right )}{4 b}-\frac {e (e x)^{3/2} (7 c+9 d x)}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {e (e x)^{7/2} (c+d x)}{4 b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {e^2 \left (\frac {3 e^2 \left (\frac {30 d \sqrt {e x}}{b}-\frac {2 e \left (\frac {1}{2} \sqrt {b} \left (\frac {15 \sqrt {a} d}{\sqrt {b}}+7 c\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e} \sqrt {e x}+\sqrt {a} e+\sqrt {b} e x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e} \sqrt {e x}+\sqrt {a} e+\sqrt {b} e x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )-\frac {1}{2} \sqrt {b} \left (7 c-\frac {15 \sqrt {a} d}{\sqrt {b}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )\right )}{b}\right )}{4 b}-\frac {e (e x)^{3/2} (7 c+9 d x)}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {e (e x)^{7/2} (c+d x)}{4 b \left (a+b x^2\right )^2}\)

Input:

Int[((e*x)^(9/2)*(c + d*x))/(a + b*x^2)^3,x]
 

Output:

-1/4*(e*(e*x)^(7/2)*(c + d*x))/(b*(a + b*x^2)^2) + (e^2*(-1/2*(e*(e*x)^(3/ 
2)*(7*c + 9*d*x))/(b*(a + b*x^2)) + (3*e^2*((30*d*Sqrt[e*x])/b - (2*e*(-1/ 
2*(Sqrt[b]*(7*c - (15*Sqrt[a]*d)/Sqrt[b])*(-(ArcTan[1 - (Sqrt[2]*b^(1/4)*S 
qrt[e*x])/(a^(1/4)*Sqrt[e])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e])) + ArcTan[1 
 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])]/(Sqrt[2]*a^(1/4)*b^(1/4) 
*Sqrt[e]))) + (Sqrt[b]*(7*c + (15*Sqrt[a]*d)/Sqrt[b])*(-1/2*Log[Sqrt[a]*e 
+ Sqrt[b]*e*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e]*Sqrt[e*x]]/(Sqrt[2]*a^(1/4 
)*b^(1/4)*Sqrt[e]) + Log[Sqrt[a]*e + Sqrt[b]*e*x + Sqrt[2]*a^(1/4)*b^(1/4) 
*Sqrt[e]*Sqrt[e*x]]/(2*Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e])))/2))/b))/(4*b)))/ 
(8*b)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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

rule 549

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

rule 552

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

rule 554

Int[((c_) + (d_.)*(x_))/(Sqrt[(e_.)*(x_)]*((a_) + (b_.)*(x_)^2)), x_Symbol] 
 :> Simp[2   Subst[Int[(e*c + d*x^2)/(a*e^2 + b*x^4), x], x, Sqrt[e*x]], x] 
 /; FreeQ[{a, b, c, d, e}, x]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 1482

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

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.15

method	result	size
derivativedivides	\(2 e^{4} \left (\frac {d \sqrt {e x}}{b^{3}}-\frac {e \left (\frac {\frac {11 b^{2} c \left (e x \right )^{\frac {7}{2}}}{32}-\frac {17 a b d e \left (e x \right )^{\frac {5}{2}}}{32}+\frac {7 a b c \,e^{2} \left (e x \right )^{\frac {3}{2}}}{32}-\frac {13 a^{2} d \,e^{3} \sqrt {e x}}{32}}{\left (b \,e^{2} x^{2}+a \,e^{2}\right )^{2}}+\frac {45 d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 e}-\frac {21 c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{3}}\right )\)	\(366\)
default	\(2 e^{4} \left (\frac {d \sqrt {e x}}{b^{3}}-\frac {e \left (\frac {\frac {11 b^{2} c \left (e x \right )^{\frac {7}{2}}}{32}-\frac {17 a b d e \left (e x \right )^{\frac {5}{2}}}{32}+\frac {7 a b c \,e^{2} \left (e x \right )^{\frac {3}{2}}}{32}-\frac {13 a^{2} d \,e^{3} \sqrt {e x}}{32}}{\left (b \,e^{2} x^{2}+a \,e^{2}\right )^{2}}+\frac {45 d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 e}-\frac {21 c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{3}}\right )\)	\(366\)
risch	\(\frac {2 d x \,e^{5}}{b^{3} \sqrt {e x}}+\frac {\left (\frac {-\frac {11 b^{2} c \left (e x \right )^{\frac {7}{2}}}{16}+\frac {17 a b d e \left (e x \right )^{\frac {5}{2}}}{16}-\frac {7 a b c \,e^{2} \left (e x \right )^{\frac {3}{2}}}{16}+\frac {13 a^{2} d \,e^{3} \sqrt {e x}}{16}}{\left (b \,e^{2} x^{2}+a \,e^{2}\right )^{2}}-\frac {45 d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 e}+\frac {21 c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right ) e^{5}}{b^{3}}\)	\(368\)
pseudoelliptic	\(-\frac {45 \left (-\frac {7 \sqrt {2}\, c e \left (b \,x^{2}+a \right )^{2} \ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )}{30}+\frac {\sqrt {2}\, d \sqrt {\frac {a \,e^{2}}{b}}\, \left (b \,x^{2}+a \right )^{2} \ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )}{2}+\left (\sqrt {\frac {a \,e^{2}}{b}}\, d -\frac {7 c e}{15}\right ) \sqrt {2}\, \left (b \,x^{2}+a \right )^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )+\left (\sqrt {\frac {a \,e^{2}}{b}}\, d -\frac {7 c e}{15}\right ) \sqrt {2}\, \left (b \,x^{2}+a \right )^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )-4 \sqrt {e x}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \left (-\frac {11 x^{3} \left (-\frac {32 d x}{11}+c \right ) b^{2}}{45}-\frac {7 x \left (-\frac {81 d x}{7}+c \right ) a b}{45}+a^{2} d \right )\right ) e^{4}}{64 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} b^{3} \left (b \,x^{2}+a \right )^{2}}\)	\(386\)

Input:

int((e*x)^(9/2)*(d*x+c)/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

2*e^4*(d/b^3*(e*x)^(1/2)-e/b^3*((11/32*b^2*c*(e*x)^(7/2)-17/32*a*b*d*e*(e* 
x)^(5/2)+7/32*a*b*c*e^2*(e*x)^(3/2)-13/32*a^2*d*e^3*(e*x)^(1/2))/(b*e^2*x^ 
2+a*e^2)^2+45/256*d/e*(a*e^2/b)^(1/4)*2^(1/2)*(ln((e*x+(a*e^2/b)^(1/4)*(e* 
x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2))/(e*x-(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2) 
+(a*e^2/b)^(1/2)))+2*arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2)+1)+2*arcta 
n(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2)-1))-21/256*c/(a*e^2/b)^(1/4)*2^(1/2) 
*(ln((e*x-(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2))/(e*x+(a*e^2 
/b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2)))+2*arctan(2^(1/2)/(a*e^2/b) 
^(1/4)*(e*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2)-1))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1086 vs. \(2 (233) = 466\).

Time = 0.34 (sec) , antiderivative size = 1086, normalized size of antiderivative = 3.40 \[ \int \frac {(e x)^{9/2} (c+d x)}{\left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((e*x)^(9/2)*(d*x+c)/(b*x^2+a)^3,x, algorithm="fricas")
 

Output:

-1/64*(3*(b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)*sqrt((210*c*d*e^9 + sqrt(-(2401 
*b^2*c^4 - 22050*a*b*c^2*d^2 + 50625*a^2*d^4)*e^18/(a*b^13))*b^6)/b^6)*log 
(-27*(2401*b^2*c^4 - 50625*a^2*d^4)*sqrt(e*x)*e^13 + 27*(7*sqrt(-(2401*b^2 
*c^4 - 22050*a*b*c^2*d^2 + 50625*a^2*d^4)*e^18/(a*b^13))*a*b^10*c - 15*(49 
*a*b^4*c^2*d - 225*a^2*b^3*d^3)*e^9)*sqrt((210*c*d*e^9 + sqrt(-(2401*b^2*c 
^4 - 22050*a*b*c^2*d^2 + 50625*a^2*d^4)*e^18/(a*b^13))*b^6)/b^6)) - 3*(b^5 
*x^4 + 2*a*b^4*x^2 + a^2*b^3)*sqrt((210*c*d*e^9 + sqrt(-(2401*b^2*c^4 - 22 
050*a*b*c^2*d^2 + 50625*a^2*d^4)*e^18/(a*b^13))*b^6)/b^6)*log(-27*(2401*b^ 
2*c^4 - 50625*a^2*d^4)*sqrt(e*x)*e^13 - 27*(7*sqrt(-(2401*b^2*c^4 - 22050* 
a*b*c^2*d^2 + 50625*a^2*d^4)*e^18/(a*b^13))*a*b^10*c - 15*(49*a*b^4*c^2*d 
- 225*a^2*b^3*d^3)*e^9)*sqrt((210*c*d*e^9 + sqrt(-(2401*b^2*c^4 - 22050*a* 
b*c^2*d^2 + 50625*a^2*d^4)*e^18/(a*b^13))*b^6)/b^6)) - 3*(b^5*x^4 + 2*a*b^ 
4*x^2 + a^2*b^3)*sqrt((210*c*d*e^9 - sqrt(-(2401*b^2*c^4 - 22050*a*b*c^2*d 
^2 + 50625*a^2*d^4)*e^18/(a*b^13))*b^6)/b^6)*log(-27*(2401*b^2*c^4 - 50625 
*a^2*d^4)*sqrt(e*x)*e^13 + 27*(7*sqrt(-(2401*b^2*c^4 - 22050*a*b*c^2*d^2 + 
 50625*a^2*d^4)*e^18/(a*b^13))*a*b^10*c + 15*(49*a*b^4*c^2*d - 225*a^2*b^3 
*d^3)*e^9)*sqrt((210*c*d*e^9 - sqrt(-(2401*b^2*c^4 - 22050*a*b*c^2*d^2 + 5 
0625*a^2*d^4)*e^18/(a*b^13))*b^6)/b^6)) + 3*(b^5*x^4 + 2*a*b^4*x^2 + a^2*b 
^3)*sqrt((210*c*d*e^9 - sqrt(-(2401*b^2*c^4 - 22050*a*b*c^2*d^2 + 50625*a^ 
2*d^4)*e^18/(a*b^13))*b^6)/b^6)*log(-27*(2401*b^2*c^4 - 50625*a^2*d^4)*...
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 155.15 (sec) , antiderivative size = 6293, normalized size of antiderivative = 19.73 \[ \int \frac {(e x)^{9/2} (c+d x)}{\left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((e*x)**(9/2)*(d*x+c)/(b*x**2+a)**3,x)
 

Output:

c*(-231*a**(27/4)*b**3*e**(9/2)*x**(23/2)*log(1 - b**(1/4)*sqrt(x)*exp_pol 
ar(I*pi/4)/a**(1/4))*gamma(11/4)/(256*a**7*b**(23/4)*x**(23/2)*exp(3*I*pi/ 
4)*gamma(15/4) + 768*a**6*b**(27/4)*x**(27/2)*exp(3*I*pi/4)*gamma(15/4) + 
768*a**5*b**(31/4)*x**(31/2)*exp(3*I*pi/4)*gamma(15/4) + 256*a**4*b**(35/4 
)*x**(35/2)*exp(3*I*pi/4)*gamma(15/4)) - 231*I*a**(27/4)*b**3*e**(9/2)*x** 
(23/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(3*I*pi/4)/a**(1/4))*gamma(11/4)/ 
(256*a**7*b**(23/4)*x**(23/2)*exp(3*I*pi/4)*gamma(15/4) + 768*a**6*b**(27/ 
4)*x**(27/2)*exp(3*I*pi/4)*gamma(15/4) + 768*a**5*b**(31/4)*x**(31/2)*exp( 
3*I*pi/4)*gamma(15/4) + 256*a**4*b**(35/4)*x**(35/2)*exp(3*I*pi/4)*gamma(1 
5/4)) + 231*a**(27/4)*b**3*e**(9/2)*x**(23/2)*log(1 - b**(1/4)*sqrt(x)*exp 
_polar(5*I*pi/4)/a**(1/4))*gamma(11/4)/(256*a**7*b**(23/4)*x**(23/2)*exp(3 
*I*pi/4)*gamma(15/4) + 768*a**6*b**(27/4)*x**(27/2)*exp(3*I*pi/4)*gamma(15 
/4) + 768*a**5*b**(31/4)*x**(31/2)*exp(3*I*pi/4)*gamma(15/4) + 256*a**4*b* 
*(35/4)*x**(35/2)*exp(3*I*pi/4)*gamma(15/4)) + 231*I*a**(27/4)*b**3*e**(9/ 
2)*x**(23/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(7*I*pi/4)/a**(1/4))*gamma( 
11/4)/(256*a**7*b**(23/4)*x**(23/2)*exp(3*I*pi/4)*gamma(15/4) + 768*a**6*b 
**(27/4)*x**(27/2)*exp(3*I*pi/4)*gamma(15/4) + 768*a**5*b**(31/4)*x**(31/2 
)*exp(3*I*pi/4)*gamma(15/4) + 256*a**4*b**(35/4)*x**(35/2)*exp(3*I*pi/4)*g 
amma(15/4)) - 693*a**(23/4)*b**4*e**(9/2)*x**(27/2)*log(1 - b**(1/4)*sqrt( 
x)*exp_polar(I*pi/4)/a**(1/4))*gamma(11/4)/(256*a**7*b**(23/4)*x**(23/2...
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e x)^{9/2} (c+d x)}{\left (a+b x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x)^(9/2)*(d*x+c)/(b*x^2+a)^3,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.34 \[ \int \frac {(e x)^{9/2} (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {2 \, \sqrt {e x} d e^{4}}{b^{3}} - \frac {3 \, \sqrt {2} {\left (15 \, \left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e^{4} - 7 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a b^{5}} - \frac {3 \, \sqrt {2} {\left (15 \, \left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e^{4} - 7 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c e^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a b^{5}} - \frac {3 \, \sqrt {2} {\left (15 \, \left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e^{4} + 7 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c e^{3}\right )} \log \left (e x + \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{128 \, a b^{5}} + \frac {3 \, \sqrt {2} {\left (15 \, \left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e^{4} + 7 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c e^{3}\right )} \log \left (e x - \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{128 \, a b^{5}} - \frac {11 \, \sqrt {e x} b^{2} c e^{8} x^{3} - 17 \, \sqrt {e x} a b d e^{8} x^{2} + 7 \, \sqrt {e x} a b c e^{8} x - 13 \, \sqrt {e x} a^{2} d e^{8}}{16 \, {\left (b e^{2} x^{2} + a e^{2}\right )}^{2} b^{3}} \] Input:

integrate((e*x)^(9/2)*(d*x+c)/(b*x^2+a)^3,x, algorithm="giac")
 

Output:

2*sqrt(e*x)*d*e^4/b^3 - 3/64*sqrt(2)*(15*(a*b^3*e^2)^(1/4)*a*b*d*e^4 - 7*( 
a*b^3*e^2)^(3/4)*c*e^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*e^2/b)^(1/4) + 2*sq 
rt(e*x))/(a*e^2/b)^(1/4))/(a*b^5) - 3/64*sqrt(2)*(15*(a*b^3*e^2)^(1/4)*a*b 
*d*e^4 - 7*(a*b^3*e^2)^(3/4)*c*e^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*e^2/b) 
^(1/4) - 2*sqrt(e*x))/(a*e^2/b)^(1/4))/(a*b^5) - 3/128*sqrt(2)*(15*(a*b^3* 
e^2)^(1/4)*a*b*d*e^4 + 7*(a*b^3*e^2)^(3/4)*c*e^3)*log(e*x + sqrt(2)*(a*e^2 
/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/(a*b^5) + 3/128*sqrt(2)*(15*(a*b^3*e^ 
2)^(1/4)*a*b*d*e^4 + 7*(a*b^3*e^2)^(3/4)*c*e^3)*log(e*x - sqrt(2)*(a*e^2/b 
)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/(a*b^5) - 1/16*(11*sqrt(e*x)*b^2*c*e^8* 
x^3 - 17*sqrt(e*x)*a*b*d*e^8*x^2 + 7*sqrt(e*x)*a*b*c*e^8*x - 13*sqrt(e*x)* 
a^2*d*e^8)/((b*e^2*x^2 + a*e^2)^2*b^3)
 

Mupad [B] (verification not implemented)

Time = 7.23 (sec) , antiderivative size = 810, normalized size of antiderivative = 2.54 \[ \int \frac {(e x)^{9/2} (c+d x)}{\left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(((e*x)^(9/2)*(c + d*x))/(a + b*x^2)^3,x)
 

Output:

atan((a*c^2*e^12*(e*x)^(1/2)*((945*c*d*e^9)/(2048*b^6) - (2025*d^2*e^9*(-a 
*b^13)^(1/2))/(4096*b^13) + (441*c^2*e^9*(-a*b^13)^(1/2))/(4096*a*b^12))^( 
1/2)*441i)/(32*((9261*a*c^3*e^17)/(2048*b^3) + (91125*a^2*d^3*e^17*(-a*b^1 
3)^(1/2))/(2048*b^11) - (42525*a^2*c*d^2*e^17)/(2048*b^4) - (19845*a*c^2*d 
*e^17*(-a*b^13)^(1/2))/(2048*b^10))) - (a^2*d^2*e^12*(e*x)^(1/2)*((945*c*d 
*e^9)/(2048*b^6) - (2025*d^2*e^9*(-a*b^13)^(1/2))/(4096*b^13) + (441*c^2*e 
^9*(-a*b^13)^(1/2))/(4096*a*b^12))^(1/2)*2025i)/(32*((9261*a*c^3*e^17)/(20 
48*b^2) + (91125*a^2*d^3*e^17*(-a*b^13)^(1/2))/(2048*b^10) - (42525*a^2*c* 
d^2*e^17)/(2048*b^3) - (19845*a*c^2*d*e^17*(-a*b^13)^(1/2))/(2048*b^9))))* 
((9*(49*b*c^2*e^9*(-a*b^13)^(1/2) - 225*a*d^2*e^9*(-a*b^13)^(1/2) + 210*a* 
b^7*c*d*e^9))/(4096*a*b^13))^(1/2)*2i + atan((a*c^2*e^12*(e*x)^(1/2)*((202 
5*d^2*e^9*(-a*b^13)^(1/2))/(4096*b^13) + (945*c*d*e^9)/(2048*b^6) - (441*c 
^2*e^9*(-a*b^13)^(1/2))/(4096*a*b^12))^(1/2)*441i)/(32*((9261*a*c^3*e^17)/ 
(2048*b^3) - (91125*a^2*d^3*e^17*(-a*b^13)^(1/2))/(2048*b^11) - (42525*a^2 
*c*d^2*e^17)/(2048*b^4) + (19845*a*c^2*d*e^17*(-a*b^13)^(1/2))/(2048*b^10) 
)) - (a^2*d^2*e^12*(e*x)^(1/2)*((2025*d^2*e^9*(-a*b^13)^(1/2))/(4096*b^13) 
 + (945*c*d*e^9)/(2048*b^6) - (441*c^2*e^9*(-a*b^13)^(1/2))/(4096*a*b^12)) 
^(1/2)*2025i)/(32*((9261*a*c^3*e^17)/(2048*b^2) - (91125*a^2*d^3*e^17*(-a* 
b^13)^(1/2))/(2048*b^10) - (42525*a^2*c*d^2*e^17)/(2048*b^3) + (19845*a*c^ 
2*d*e^17*(-a*b^13)^(1/2))/(2048*b^9))))*((9*(225*a*d^2*e^9*(-a*b^13)^(1...
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 975, normalized size of antiderivative = 3.06 \[ \int \frac {(e x)^{9/2} (c+d x)}{\left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((e*x)^(9/2)*(d*x+c)/(b*x^2+a)^3,x)
 

Output:

(sqrt(e)*e**4*( - 42*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqr 
t(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b*c - 84*b**(1 
/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/ 
(b**(1/4)*a**(1/4)*sqrt(2)))*a*b**2*c*x**2 - 42*b**(1/4)*a**(3/4)*sqrt(2)* 
atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sq 
rt(2)))*b**3*c*x**4 + 90*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4) 
*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**3*d + 180*b* 
*(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b 
))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b*d*x**2 + 90*b**(3/4)*a**(1/4)*sqrt( 
2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4) 
*sqrt(2)))*a*b**2*d*x**4 + 42*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a** 
(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b*c + 
 84*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)* 
sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b**2*c*x**2 + 42*b**(1/4)*a**(3/4) 
*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a* 
*(1/4)*sqrt(2)))*b**3*c*x**4 - 90*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4) 
*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**3*d 
 - 180*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt( 
x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b*d*x**2 - 90*b**(3/4)*a**(1 
/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1...