\(\int \frac {A+B x^3}{x^{3/2} (a+b x^3)^3} \, dx\) [156]

Optimal result

Integrand size = 22, antiderivative size = 270 \[ \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )^3} \, dx=-\frac {2 A}{a^3 \sqrt {x}}-\frac {(A b-a B) x^{5/2}}{6 a^2 \left (a+b x^3\right )^2}-\frac {(19 A b-7 a B) x^{5/2}}{36 a^3 \left (a+b x^3\right )}+\frac {7 (13 A b-a B) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{216 a^{19/6} b^{5/6}}-\frac {7 (13 A b-a B) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{216 a^{19/6} b^{5/6}}-\frac {7 (13 A b-a B) \arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{108 a^{19/6} b^{5/6}}+\frac {7 (13 A b-a B) \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{a}+\sqrt [3]{b} x}\right )}{72 \sqrt {3} a^{19/6} b^{5/6}} \] Output:

-2*A/a^3/x^(1/2)-1/6*(A*b-B*a)*x^(5/2)/a^2/(b*x^3+a)^2-1/36*(19*A*b-7*B*a) 
*x^(5/2)/a^3/(b*x^3+a)-7/216*(13*A*b-B*a)*arctan(-3^(1/2)+2*b^(1/6)*x^(1/2 
)/a^(1/6))/a^(19/6)/b^(5/6)-7/216*(13*A*b-B*a)*arctan(3^(1/2)+2*b^(1/6)*x^ 
(1/2)/a^(1/6))/a^(19/6)/b^(5/6)-7/108*(13*A*b-B*a)*arctan(b^(1/6)*x^(1/2)/ 
a^(1/6))/a^(19/6)/b^(5/6)+7/216*(13*A*b-B*a)*arctanh(3^(1/2)*a^(1/6)*b^(1/ 
6)*x^(1/2)/(a^(1/3)+b^(1/3)*x))*3^(1/2)/a^(19/6)/b^(5/6)
 

Mathematica [A] (verified)

Time = 1.23 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.77 \[ \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )^3} \, dx=\frac {-\frac {6 \sqrt [6]{a} \left (91 A b^2 x^6+a^2 \left (72 A-13 B x^3\right )+a b x^3 \left (169 A-7 B x^3\right )\right )}{\sqrt {x} \left (a+b x^3\right )^2}+\frac {14 (-13 A b+a B) \arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{b^{5/6}}+\frac {7 (13 A b-a B) \arctan \left (\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}\right )}{b^{5/6}}+\frac {7 \sqrt {3} (13 A b-a B) \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{a}+\sqrt [3]{b} x}\right )}{b^{5/6}}}{216 a^{19/6}} \] Input:

Integrate[(A + B*x^3)/(x^(3/2)*(a + b*x^3)^3),x]
 

Output:

((-6*a^(1/6)*(91*A*b^2*x^6 + a^2*(72*A - 13*B*x^3) + a*b*x^3*(169*A - 7*B* 
x^3)))/(Sqrt[x]*(a + b*x^3)^2) + (14*(-13*A*b + a*B)*ArcTan[(b^(1/6)*Sqrt[ 
x])/a^(1/6)])/b^(5/6) + (7*(13*A*b - a*B)*ArcTan[(a^(1/3) - b^(1/3)*x)/(a^ 
(1/6)*b^(1/6)*Sqrt[x])])/b^(5/6) + (7*Sqrt[3]*(13*A*b - a*B)*ArcTanh[(Sqrt 
[3]*a^(1/6)*b^(1/6)*Sqrt[x])/(a^(1/3) + b^(1/3)*x)])/b^(5/6))/(216*a^(19/6 
))
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.24, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {957, 819, 847, 851, 824, 27, 218, 1142, 25, 27, 1082, 217, 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.

\(\displaystyle \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )^3} \, dx\)

\(\Big \downarrow \) 957

\(\displaystyle \frac {(13 A b-a B) \int \frac {1}{x^{3/2} \left (b x^3+a\right )^2}dx}{12 a b}+\frac {A b-a B}{6 a b \sqrt {x} \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 819

\(\displaystyle \frac {(13 A b-a B) \left (\frac {7 \int \frac {1}{x^{3/2} \left (b x^3+a\right )}dx}{6 a}+\frac {1}{3 a \sqrt {x} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b \sqrt {x} \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 847

\(\displaystyle \frac {(13 A b-a B) \left (\frac {7 \left (-\frac {b \int \frac {x^{3/2}}{b x^3+a}dx}{a}-\frac {2}{a \sqrt {x}}\right )}{6 a}+\frac {1}{3 a \sqrt {x} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b \sqrt {x} \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {(13 A b-a B) \left (\frac {7 \left (-\frac {2 b \int \frac {x^2}{b x^3+a}d\sqrt {x}}{a}-\frac {2}{a \sqrt {x}}\right )}{6 a}+\frac {1}{3 a \sqrt {x} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b \sqrt {x} \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 824

\(\displaystyle \frac {(13 A b-a B) \left (\frac {7 \left (-\frac {2 b \left (\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}d\sqrt {x}}{3 b^{2/3}}+\frac {\int -\frac {\sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} \sqrt {x}}{2 \left (\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}\right )}d\sqrt {x}}{3 \sqrt [6]{a} b^{2/3}}+\frac {\int -\frac {\sqrt {3} \sqrt [6]{b} \sqrt {x}+\sqrt [6]{a}}{2 \left (\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}\right )}d\sqrt {x}}{3 \sqrt [6]{a} b^{2/3}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{6 a}+\frac {1}{3 a \sqrt {x} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b \sqrt {x} \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(13 A b-a B) \left (\frac {7 \left (-\frac {2 b \left (\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}d\sqrt {x}}{3 b^{2/3}}-\frac {\int \frac {\sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\int \frac {\sqrt {3} \sqrt [6]{b} \sqrt {x}+\sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{6 a}+\frac {1}{3 a \sqrt {x} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b \sqrt {x} \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {(13 A b-a B) \left (\frac {7 \left (-\frac {2 b \left (-\frac {\int \frac {\sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\int \frac {\sqrt {3} \sqrt [6]{b} \sqrt {x}+\sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{6 a}+\frac {1}{3 a \sqrt {x} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b \sqrt {x} \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {(13 A b-a B) \left (\frac {7 \left (-\frac {2 b \left (-\frac {-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {\sqrt {3} \int -\frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}\right )}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{6 a}+\frac {1}{3 a \sqrt {x} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b \sqrt {x} \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(13 A b-a B) \left (\frac {7 \left (-\frac {2 b \left (-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}\right )}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{6 a}+\frac {1}{3 a \sqrt {x} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b \sqrt {x} \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(13 A b-a B) \left (\frac {7 \left (-\frac {2 b \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{6 a}+\frac {1}{3 a \sqrt {x} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b \sqrt {x} \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {(13 A b-a B) \left (\frac {7 \left (-\frac {2 b \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {\int \frac {1}{-x-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )}{\sqrt {3} \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {\int \frac {1}{-x-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}+1\right )}{\sqrt {3} \sqrt [6]{b}}+\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{6 a}+\frac {1}{3 a \sqrt {x} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b \sqrt {x} \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {(13 A b-a B) \left (\frac {7 \left (-\frac {2 b \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}+\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}+1\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{6 a}+\frac {1}{3 a \sqrt {x} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b \sqrt {x} \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {(13 A b-a B) \left (\frac {7 \left (-\frac {2 b \left (\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}-\frac {\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )\right )}{\sqrt [6]{b}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \sqrt [6]{b}}-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}+1\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{6 a}+\frac {1}{3 a \sqrt {x} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b \sqrt {x} \left (a+b x^3\right )^2}\)

Input:

Int[(A + B*x^3)/(x^(3/2)*(a + b*x^3)^3),x]
 

Output:

(A*b - a*B)/(6*a*b*Sqrt[x]*(a + b*x^3)^2) + ((13*A*b - a*B)*(1/(3*a*Sqrt[x 
]*(a + b*x^3)) + (7*(-2/(a*Sqrt[x]) - (2*b*(ArcTan[(b^(1/6)*Sqrt[x])/a^(1/ 
6)]/(3*a^(1/6)*b^(5/6)) - (ArcTan[Sqrt[3]*(1 - (2*b^(1/6)*Sqrt[x])/(Sqrt[3 
]*a^(1/6)))]/b^(1/6) - (Sqrt[3]*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt 
[x] + b^(1/3)*x])/(2*b^(1/6)))/(6*a^(1/6)*b^(2/3)) - (-(ArcTan[Sqrt[3]*(1 
+ (2*b^(1/6)*Sqrt[x])/(Sqrt[3]*a^(1/6)))]/b^(1/6)) + (Sqrt[3]*Log[a^(1/3) 
+ Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(2*b^(1/6)))/(6*a^(1/6)*b^ 
(2/3))))/a))/(6*a)))/(12*a*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 218

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

rule 819

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( 
c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 
1) + 1)/(a*n*(p + 1))   Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a 
, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p 
, x]
 

rule 824

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator 
[Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k 
- 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 
 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k 
- 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] 
; 2*(-1)^(m/2)*(r^(m + 2)/(a*n*s^m))   Int[1/(r^2 + s^2*x^2), x] + 2*(r^(m 
+ 1)/(a*n*s^m))   Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGt 
Q[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]
 

rule 847

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x 
)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) 
+ 1)/(a*c^n*(m + 1)))   Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a 
, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p 
, x]
 

rule 851

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = 
 Denominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ 
n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && 
FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
 

rule 957

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a 
*b*e*n*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b*n* 
(p + 1))   Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, 
 m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && N 
eQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1 
, m, (-n)*(p + 1)]))
 

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 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]
 

Maple [A] (verified)

Time = 0.85 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(-\frac {2 A}{a^{3} \sqrt {x}}-\frac {2 \left (\frac {\left (\frac {19}{72} b^{2} A -\frac {7}{72} a b B \right ) x^{\frac {11}{2}}+\frac {a \left (25 A b -13 B a \right ) x^{\frac {5}{2}}}{72}}{\left (b \,x^{3}+a \right )^{2}}+\left (\frac {91 A b}{72}-\frac {7 B a}{72}\right ) \left (\frac {\arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )\right )}{a^{3}}\)	\(236\)
default	\(-\frac {2 A}{a^{3} \sqrt {x}}-\frac {2 \left (\frac {\left (\frac {19}{72} b^{2} A -\frac {7}{72} a b B \right ) x^{\frac {11}{2}}+\frac {a \left (25 A b -13 B a \right ) x^{\frac {5}{2}}}{72}}{\left (b \,x^{3}+a \right )^{2}}+\left (\frac {91 A b}{72}-\frac {7 B a}{72}\right ) \left (\frac {\arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )\right )}{a^{3}}\)	\(236\)
risch	\(-\frac {2 A}{a^{3} \sqrt {x}}-\frac {\frac {2 \left (\frac {19}{72} b^{2} A -\frac {7}{72} a b B \right ) x^{\frac {11}{2}}+\frac {a \left (25 A b -13 B a \right ) x^{\frac {5}{2}}}{36}}{\left (b \,x^{3}+a \right )^{2}}+2 \left (\frac {91 A b}{72}-\frac {7 B a}{72}\right ) \left (\frac {\arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{a^{3}}\)	\(238\)

Input:

int((B*x^3+A)/x^(3/2)/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
 

Output:

-2*A/a^3/x^(1/2)-2/a^3*(((19/72*b^2*A-7/72*a*b*B)*x^(11/2)+1/72*a*(25*A*b- 
13*B*a)*x^(5/2))/(b*x^3+a)^2+(91/72*A*b-7/72*B*a)*(1/3/b/(a/b)^(1/6)*arcta 
n(x^(1/2)/(a/b)^(1/6))+1/12/a*3^(1/2)*(a/b)^(5/6)*ln(3^(1/2)*(a/b)^(1/6)*x 
^(1/2)-x-(a/b)^(1/3))+1/6/b/(a/b)^(1/6)*arctan(2*x^(1/2)/(a/b)^(1/6)-3^(1/ 
2))-1/12/a*3^(1/2)*(a/b)^(5/6)*ln(x+3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3 
))+1/6/b/(a/b)^(1/6)*arctan(2*x^(1/2)/(a/b)^(1/6)+3^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1934 vs. \(2 (198) = 396\).

Time = 0.13 (sec) , antiderivative size = 1934, normalized size of antiderivative = 7.16 \[ \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((B*x^3+A)/x^(3/2)/(b*x^3+a)^3,x, algorithm="fricas")
 

Output:

-1/432*(14*(a^3*b^2*x^7 + 2*a^4*b*x^4 + a^5*x)*(-(B^6*a^6 - 78*A*B^5*a^5*b 
 + 2535*A^2*B^4*a^4*b^2 - 43940*A^3*B^3*a^3*b^3 + 428415*A^4*B^2*a^2*b^4 - 
 2227758*A^5*B*a*b^5 + 4826809*A^6*b^6)/(a^19*b^5))^(1/6)*log(16807*a^16*b 
^4*(-(B^6*a^6 - 78*A*B^5*a^5*b + 2535*A^2*B^4*a^4*b^2 - 43940*A^3*B^3*a^3* 
b^3 + 428415*A^4*B^2*a^2*b^4 - 2227758*A^5*B*a*b^5 + 4826809*A^6*b^6)/(a^1 
9*b^5))^(5/6) - 16807*(B^5*a^5 - 65*A*B^4*a^4*b + 1690*A^2*B^3*a^3*b^2 - 2 
1970*A^3*B^2*a^2*b^3 + 142805*A^4*B*a*b^4 - 371293*A^5*b^5)*sqrt(x)) - 14* 
(a^3*b^2*x^7 + 2*a^4*b*x^4 + a^5*x)*(-(B^6*a^6 - 78*A*B^5*a^5*b + 2535*A^2 
*B^4*a^4*b^2 - 43940*A^3*B^3*a^3*b^3 + 428415*A^4*B^2*a^2*b^4 - 2227758*A^ 
5*B*a*b^5 + 4826809*A^6*b^6)/(a^19*b^5))^(1/6)*log(-16807*a^16*b^4*(-(B^6* 
a^6 - 78*A*B^5*a^5*b + 2535*A^2*B^4*a^4*b^2 - 43940*A^3*B^3*a^3*b^3 + 4284 
15*A^4*B^2*a^2*b^4 - 2227758*A^5*B*a*b^5 + 4826809*A^6*b^6)/(a^19*b^5))^(5 
/6) - 16807*(B^5*a^5 - 65*A*B^4*a^4*b + 1690*A^2*B^3*a^3*b^2 - 21970*A^3*B 
^2*a^2*b^3 + 142805*A^4*B*a*b^4 - 371293*A^5*b^5)*sqrt(x)) + 7*(a^3*b^2*x^ 
7 + 2*a^4*b*x^4 + a^5*x - sqrt(-3)*(a^3*b^2*x^7 + 2*a^4*b*x^4 + a^5*x))*(- 
(B^6*a^6 - 78*A*B^5*a^5*b + 2535*A^2*B^4*a^4*b^2 - 43940*A^3*B^3*a^3*b^3 + 
 428415*A^4*B^2*a^2*b^4 - 2227758*A^5*B*a*b^5 + 4826809*A^6*b^6)/(a^19*b^5 
))^(1/6)*log(16807/2*(sqrt(-3)*a^16*b^4 + a^16*b^4)*(-(B^6*a^6 - 78*A*B^5* 
a^5*b + 2535*A^2*B^4*a^4*b^2 - 43940*A^3*B^3*a^3*b^3 + 428415*A^4*B^2*a^2* 
b^4 - 2227758*A^5*B*a*b^5 + 4826809*A^6*b^6)/(a^19*b^5))^(5/6) - 16807*...
 

Sympy [F(-1)]

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

integrate((B*x**3+A)/x**(3/2)/(b*x**3+a)**3,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.01 \[ \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )^3} \, dx=\frac {7 \, {\left (B a b - 13 \, A b^{2}\right )} x^{6} + 13 \, {\left (B a^{2} - 13 \, A a b\right )} x^{3} - 72 \, A a^{2}}{36 \, {\left (a^{3} b^{2} x^{\frac {13}{2}} + 2 \, a^{4} b x^{\frac {7}{2}} + a^{5} \sqrt {x}\right )}} - \frac {7 \, {\left (B a - 13 \, A b\right )} {\left (\frac {\sqrt {3} \log \left (\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {\sqrt {3} \log \left (-\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {2 \, \arctan \left (\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} + 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {2 \, \arctan \left (-\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} - 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {4 \, \arctan \left (\frac {b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}}{432 \, a^{3}} \] Input:

integrate((B*x^3+A)/x^(3/2)/(b*x^3+a)^3,x, algorithm="maxima")
 

Output:

1/36*(7*(B*a*b - 13*A*b^2)*x^6 + 13*(B*a^2 - 13*A*a*b)*x^3 - 72*A*a^2)/(a^ 
3*b^2*x^(13/2) + 2*a^4*b*x^(7/2) + a^5*sqrt(x)) - 7/432*(B*a - 13*A*b)*(sq 
rt(3)*log(sqrt(3)*a^(1/6)*b^(1/6)*sqrt(x) + b^(1/3)*x + a^(1/3))/(a^(1/6)* 
b^(5/6)) - sqrt(3)*log(-sqrt(3)*a^(1/6)*b^(1/6)*sqrt(x) + b^(1/3)*x + a^(1 
/3))/(a^(1/6)*b^(5/6)) - 2*arctan((sqrt(3)*a^(1/6)*b^(1/6) + 2*b^(1/3)*sqr 
t(x))/sqrt(a^(1/3)*b^(1/3)))/(b^(2/3)*sqrt(a^(1/3)*b^(1/3))) - 2*arctan(-( 
sqrt(3)*a^(1/6)*b^(1/6) - 2*b^(1/3)*sqrt(x))/sqrt(a^(1/3)*b^(1/3)))/(b^(2/ 
3)*sqrt(a^(1/3)*b^(1/3))) - 4*arctan(b^(1/3)*sqrt(x)/sqrt(a^(1/3)*b^(1/3)) 
)/(b^(2/3)*sqrt(a^(1/3)*b^(1/3))))/a^3
 

Giac [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.11 \[ \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )^3} \, dx=\frac {7 \, {\left (B a - 13 \, A b\right )} \arctan \left (\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} + 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{216 \, \left (a b^{5}\right )^{\frac {1}{6}} a^{3}} + \frac {7 \, {\left (B a - 13 \, A b\right )} \arctan \left (-\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} - 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{216 \, \left (a b^{5}\right )^{\frac {1}{6}} a^{3}} + \frac {7 \, {\left (B a - 13 \, A b\right )} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{108 \, \left (a b^{5}\right )^{\frac {1}{6}} a^{3}} - \frac {2 \, A}{a^{3} \sqrt {x}} + \frac {7 \, B a b x^{\frac {11}{2}} - 19 \, A b^{2} x^{\frac {11}{2}} + 13 \, B a^{2} x^{\frac {5}{2}} - 25 \, A a b x^{\frac {5}{2}}}{36 \, {\left (b x^{3} + a\right )}^{2} a^{3}} - \frac {7 \, \sqrt {3} {\left (\left (a b^{5}\right )^{\frac {5}{6}} B a - 13 \, \left (a b^{5}\right )^{\frac {5}{6}} A b\right )} \log \left (\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{432 \, a^{4} b^{5}} + \frac {7 \, \sqrt {3} {\left (\left (a b^{5}\right )^{\frac {5}{6}} B a - 13 \, \left (a b^{5}\right )^{\frac {5}{6}} A b\right )} \log \left (-\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{432 \, a^{4} b^{5}} \] Input:

integrate((B*x^3+A)/x^(3/2)/(b*x^3+a)^3,x, algorithm="giac")
 

Output:

7/216*(B*a - 13*A*b)*arctan((sqrt(3)*(a/b)^(1/6) + 2*sqrt(x))/(a/b)^(1/6)) 
/((a*b^5)^(1/6)*a^3) + 7/216*(B*a - 13*A*b)*arctan(-(sqrt(3)*(a/b)^(1/6) - 
 2*sqrt(x))/(a/b)^(1/6))/((a*b^5)^(1/6)*a^3) + 7/108*(B*a - 13*A*b)*arctan 
(sqrt(x)/(a/b)^(1/6))/((a*b^5)^(1/6)*a^3) - 2*A/(a^3*sqrt(x)) + 1/36*(7*B* 
a*b*x^(11/2) - 19*A*b^2*x^(11/2) + 13*B*a^2*x^(5/2) - 25*A*a*b*x^(5/2))/(( 
b*x^3 + a)^2*a^3) - 7/432*sqrt(3)*((a*b^5)^(5/6)*B*a - 13*(a*b^5)^(5/6)*A* 
b)*log(sqrt(3)*sqrt(x)*(a/b)^(1/6) + x + (a/b)^(1/3))/(a^4*b^5) + 7/432*sq 
rt(3)*((a*b^5)^(5/6)*B*a - 13*(a*b^5)^(5/6)*A*b)*log(-sqrt(3)*sqrt(x)*(a/b 
)^(1/6) + x + (a/b)^(1/3))/(a^4*b^5)
 

Mupad [B] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 1786, normalized size of antiderivative = 6.61 \[ \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )^3} \, dx=\text {Too large to display} \] Input:

int((A + B*x^3)/(x^(3/2)*(a + b*x^3)^3),x)
 

Output:

(atan((((13*A*b - B*a)^2*(28229306112*B^3*a^24*b^3 - 62019785528064*A^3*a^ 
21*b^6 - 1100942938368*A*B^2*a^23*b^4 + 14312258198784*A^2*B*a^22*b^5 + (3 
43*x^(1/2)*(13*A*b - B*a)*(140169666861858816*A^2*a^24*b^6 + 8294063127920 
64*B^2*a^26*b^4 - 21564564132593664*A*B*a^25*b^5))/(10077696*(-a)^(19/6)*b 
^(5/6)))*1i)/((-a)^(19/3)*b^(5/3)) + ((13*A*b - B*a)^2*(62019785528064*A^3 
*a^21*b^6 - 28229306112*B^3*a^24*b^3 + 1100942938368*A*B^2*a^23*b^4 - 1431 
2258198784*A^2*B*a^22*b^5 + (343*x^(1/2)*(13*A*b - B*a)*(14016966686185881 
6*A^2*a^24*b^6 + 829406312792064*B^2*a^26*b^4 - 21564564132593664*A*B*a^25 
*b^5))/(10077696*(-a)^(19/6)*b^(5/6)))*1i)/((-a)^(19/3)*b^(5/3)))/(((13*A* 
b - B*a)^2*(28229306112*B^3*a^24*b^3 - 62019785528064*A^3*a^21*b^6 - 11009 
42938368*A*B^2*a^23*b^4 + 14312258198784*A^2*B*a^22*b^5 + (343*x^(1/2)*(13 
*A*b - B*a)*(140169666861858816*A^2*a^24*b^6 + 829406312792064*B^2*a^26*b^ 
4 - 21564564132593664*A*B*a^25*b^5))/(10077696*(-a)^(19/6)*b^(5/6))))/((-a 
)^(19/3)*b^(5/3)) - ((13*A*b - B*a)^2*(62019785528064*A^3*a^21*b^6 - 28229 
306112*B^3*a^24*b^3 + 1100942938368*A*B^2*a^23*b^4 - 14312258198784*A^2*B* 
a^22*b^5 + (343*x^(1/2)*(13*A*b - B*a)*(140169666861858816*A^2*a^24*b^6 + 
829406312792064*B^2*a^26*b^4 - 21564564132593664*A*B*a^25*b^5))/(10077696* 
(-a)^(19/6)*b^(5/6))))/((-a)^(19/3)*b^(5/3))))*(13*A*b - B*a)*7i)/(108*(-a 
)^(19/6)*b^(5/6)) - ((2*A)/a + (13*x^3*(13*A*b - B*a))/(36*a^2) + (7*b*x^6 
*(13*A*b - B*a))/(36*a^3))/(a^2*x^(1/2) + b^2*x^(13/2) + 2*a*b*x^(7/2))...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.34 \[ \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )^3} \, dx=\frac {14 \sqrt {x}\, b^{\frac {1}{6}} a^{\frac {7}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 \sqrt {x}\, b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )+14 \sqrt {x}\, b^{\frac {7}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 \sqrt {x}\, b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) x^{3}-14 \sqrt {x}\, b^{\frac {1}{6}} a^{\frac {7}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 \sqrt {x}\, b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )-14 \sqrt {x}\, b^{\frac {7}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 \sqrt {x}\, b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) x^{3}-28 \sqrt {x}\, b^{\frac {1}{6}} a^{\frac {7}{6}} \mathit {atan} \left (\frac {\sqrt {x}\, b^{\frac {1}{6}}}{a^{\frac {1}{6}}}\right )-28 \sqrt {x}\, b^{\frac {7}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {\sqrt {x}\, b^{\frac {1}{6}}}{a^{\frac {1}{6}}}\right ) x^{3}-7 \sqrt {x}\, b^{\frac {1}{6}} a^{\frac {7}{6}} \sqrt {3}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )-7 \sqrt {x}\, b^{\frac {7}{6}} a^{\frac {1}{6}} \sqrt {3}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) x^{3}+7 \sqrt {x}\, b^{\frac {1}{6}} a^{\frac {7}{6}} \sqrt {3}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )+7 \sqrt {x}\, b^{\frac {7}{6}} a^{\frac {1}{6}} \sqrt {3}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) x^{3}-72 a^{\frac {4}{3}}-84 a^{\frac {1}{3}} b \,x^{3}}{36 \sqrt {x}\, a^{\frac {7}{3}} \left (b \,x^{3}+a \right )} \] Input:

int((B*x^3+A)/x^(3/2)/(b*x^3+a)^3,x)
 

Output:

(14*sqrt(x)*b**(1/6)*a**(1/6)*atan((b**(1/6)*a**(1/6)*sqrt(3) - 2*sqrt(x)* 
b**(1/3))/(b**(1/6)*a**(1/6)))*a + 14*sqrt(x)*b**(1/6)*a**(1/6)*atan((b**( 
1/6)*a**(1/6)*sqrt(3) - 2*sqrt(x)*b**(1/3))/(b**(1/6)*a**(1/6)))*b*x**3 - 
14*sqrt(x)*b**(1/6)*a**(1/6)*atan((b**(1/6)*a**(1/6)*sqrt(3) + 2*sqrt(x)*b 
**(1/3))/(b**(1/6)*a**(1/6)))*a - 14*sqrt(x)*b**(1/6)*a**(1/6)*atan((b**(1 
/6)*a**(1/6)*sqrt(3) + 2*sqrt(x)*b**(1/3))/(b**(1/6)*a**(1/6)))*b*x**3 - 2 
8*sqrt(x)*b**(1/6)*a**(1/6)*atan((sqrt(x)*b**(1/3))/(b**(1/6)*a**(1/6)))*a 
 - 28*sqrt(x)*b**(1/6)*a**(1/6)*atan((sqrt(x)*b**(1/3))/(b**(1/6)*a**(1/6) 
))*b*x**3 - 7*sqrt(x)*b**(1/6)*a**(1/6)*sqrt(3)*log( - sqrt(x)*b**(1/6)*a* 
*(1/6)*sqrt(3) + a**(1/3) + b**(1/3)*x)*a - 7*sqrt(x)*b**(1/6)*a**(1/6)*sq 
rt(3)*log( - sqrt(x)*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + b**(1/3)*x)*b* 
x**3 + 7*sqrt(x)*b**(1/6)*a**(1/6)*sqrt(3)*log(sqrt(x)*b**(1/6)*a**(1/6)*s 
qrt(3) + a**(1/3) + b**(1/3)*x)*a + 7*sqrt(x)*b**(1/6)*a**(1/6)*sqrt(3)*lo 
g(sqrt(x)*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + b**(1/3)*x)*b*x**3 - 72*a 
**(1/3)*a - 84*a**(1/3)*b*x**3)/(36*sqrt(x)*a**(1/3)*a**2*(a + b*x**3))