\(\int \frac {(e x)^{3/2} (A+B x^3)}{(a+b x^3)^{5/2}} \, dx\) [283]

Optimal result

Integrand size = 26, antiderivative size = 596 \[ \int \frac {(e x)^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx=\frac {2 (A b-a B) (e x)^{5/2}}{9 a b e \left (a+b x^3\right )^{3/2}}+\frac {2 (4 A b+5 a B) (e x)^{5/2}}{27 a^2 b e \sqrt {a+b x^3}}-\frac {2 \left (1+\sqrt {3}\right ) (4 A b+5 a B) e \sqrt {e x} \sqrt {a+b x^3}}{27 a^2 b^{5/3} \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )}+\frac {2 (4 A b+5 a B) e \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{9\ 3^{3/4} a^{5/3} b^{5/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {\left (1-\sqrt {3}\right ) (4 A b+5 a B) e \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{27 \sqrt [4]{3} a^{5/3} b^{5/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:

2/9*(A*b-B*a)*(e*x)^(5/2)/a/b/e/(b*x^3+a)^(3/2)+2/27*(4*A*b+5*B*a)*(e*x)^( 
5/2)/a^2/b/e/(b*x^3+a)^(1/2)-2/27*(1+3^(1/2))*(4*A*b+5*B*a)*e*(e*x)^(1/2)* 
(b*x^3+a)^(1/2)/a^2/b^(5/3)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)+2/27*(4*A*b+5* 
B*a)*e*(e*x)^(1/2)*(a^(1/3)+b^(1/3)*x)*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3) 
*x^2)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)^2)^(1/2)*EllipticE((1-(a^(1/3)+(1-3^ 
(1/2))*b^(1/3)*x)^2/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)^2)^(1/2),1/4*6^(1/2)+1 
/4*2^(1/2))*3^(1/4)/a^(5/3)/b^(5/3)/(b^(1/3)*x*(a^(1/3)+b^(1/3)*x)/(a^(1/3 
)+(1+3^(1/2))*b^(1/3)*x)^2)^(1/2)/(b*x^3+a)^(1/2)+1/81*(1-3^(1/2))*(4*A*b+ 
5*B*a)*e*(e*x)^(1/2)*(a^(1/3)+b^(1/3)*x)*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/ 
3)*x^2)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)^2)^(1/2)*InverseJacobiAM(arccos((a 
^(1/3)+(1-3^(1/2))*b^(1/3)*x)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)),1/4*6^(1/2) 
+1/4*2^(1/2))*3^(3/4)/a^(5/3)/b^(5/3)/(b^(1/3)*x*(a^(1/3)+b^(1/3)*x)/(a^(1 
/3)+(1+3^(1/2))*b^(1/3)*x)^2)^(1/2)/(b*x^3+a)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.14 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.14 \[ \int \frac {(e x)^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx=\frac {x (e x)^{3/2} \left (-5 a^2 B+(4 A b+5 a B) \left (a+b x^3\right ) \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {5}{6},\frac {5}{2},\frac {11}{6},-\frac {b x^3}{a}\right )\right )}{10 a^2 b \left (a+b x^3\right )^{3/2}} \] Input:

Integrate[((e*x)^(3/2)*(A + B*x^3))/(a + b*x^3)^(5/2),x]
 

Output:

(x*(e*x)^(3/2)*(-5*a^2*B + (4*A*b + 5*a*B)*(a + b*x^3)*Sqrt[1 + (b*x^3)/a] 
*Hypergeometric2F1[5/6, 5/2, 11/6, -((b*x^3)/a)]))/(10*a^2*b*(a + b*x^3)^( 
3/2))
 

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 643, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {957, 819, 851, 837, 25, 766, 2420}

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.

\(\displaystyle \int \frac {(e x)^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 957

\(\displaystyle \frac {(5 a B+4 A b) \int \frac {(e x)^{3/2}}{\left (b x^3+a\right )^{3/2}}dx}{9 a b}+\frac {2 (e x)^{5/2} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}}\)

\(\Big \downarrow \) 819

\(\displaystyle \frac {(5 a B+4 A b) \left (\frac {2 (e x)^{5/2}}{3 a e \sqrt {a+b x^3}}-\frac {2 \int \frac {(e x)^{3/2}}{\sqrt {b x^3+a}}dx}{3 a}\right )}{9 a b}+\frac {2 (e x)^{5/2} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}}\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {(5 a B+4 A b) \left (\frac {2 (e x)^{5/2}}{3 a e \sqrt {a+b x^3}}-\frac {4 \int \frac {e^2 x^2}{\sqrt {b x^3+a}}d\sqrt {e x}}{3 a e}\right )}{9 a b}+\frac {2 (e x)^{5/2} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}}\)

\(\Big \downarrow \) 837

\(\displaystyle \frac {(5 a B+4 A b) \left (\frac {2 (e x)^{5/2}}{3 a e \sqrt {a+b x^3}}-\frac {4 \left (-\frac {\left (1-\sqrt {3}\right ) a^{2/3} e^2 \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {e x}}{2 b^{2/3}}-\frac {\int -\frac {2 b^{2/3} x^2 e^2+\left (1-\sqrt {3}\right ) a^{2/3} e^2}{\sqrt {b x^3+a}}d\sqrt {e x}}{2 b^{2/3}}\right )}{3 a e}\right )}{9 a b}+\frac {2 (e x)^{5/2} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(5 a B+4 A b) \left (\frac {2 (e x)^{5/2}}{3 a e \sqrt {a+b x^3}}-\frac {4 \left (\frac {\int \frac {2 b^{2/3} x^2 e^2+\left (1-\sqrt {3}\right ) a^{2/3} e^2}{\sqrt {b x^3+a}}d\sqrt {e x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) a^{2/3} e^2 \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {e x}}{2 b^{2/3}}\right )}{3 a e}\right )}{9 a b}+\frac {2 (e x)^{5/2} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}}\)

\(\Big \downarrow \) 766

\(\displaystyle \frac {(5 a B+4 A b) \left (\frac {2 (e x)^{5/2}}{3 a e \sqrt {a+b x^3}}-\frac {4 \left (\frac {\int \frac {2 b^{2/3} x^2 e^2+\left (1-\sqrt {3}\right ) a^{2/3} e^2}{\sqrt {b x^3+a}}d\sqrt {e x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} e \sqrt {e x} \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right ) \sqrt {\frac {a^{2/3} e^2-\sqrt [3]{a} \sqrt [3]{b} e^2 x+b^{2/3} e^2 x^2}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} e x \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right )}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}}}\right )}{3 a e}\right )}{9 a b}+\frac {2 (e x)^{5/2} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}}\)

\(\Big \downarrow \) 2420

\(\displaystyle \frac {(5 a B+4 A b) \left (\frac {2 (e x)^{5/2}}{3 a e \sqrt {a+b x^3}}-\frac {4 \left (\frac {\frac {\left (1+\sqrt {3}\right ) e^3 \sqrt {e x} \sqrt {a+b x^3}}{\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x}-\frac {\sqrt [4]{3} \sqrt [3]{a} e \sqrt {e x} \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right ) \sqrt {\frac {a^{2/3} e^2-\sqrt [3]{a} \sqrt [3]{b} e^2 x+b^{2/3} e^2 x^2}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}} E\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} e x \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right )}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}}}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} e \sqrt {e x} \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right ) \sqrt {\frac {a^{2/3} e^2-\sqrt [3]{a} \sqrt [3]{b} e^2 x+b^{2/3} e^2 x^2}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} e x \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right )}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}}}\right )}{3 a e}\right )}{9 a b}+\frac {2 (e x)^{5/2} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}}\)

Input:

Int[((e*x)^(3/2)*(A + B*x^3))/(a + b*x^3)^(5/2),x]
 

Output:

(2*(A*b - a*B)*(e*x)^(5/2))/(9*a*b*e*(a + b*x^3)^(3/2)) + ((4*A*b + 5*a*B) 
*((2*(e*x)^(5/2))/(3*a*e*Sqrt[a + b*x^3]) - (4*((((1 + Sqrt[3])*e^3*Sqrt[e 
*x]*Sqrt[a + b*x^3])/(a^(1/3)*e + (1 + Sqrt[3])*b^(1/3)*e*x) - (3^(1/4)*a^ 
(1/3)*e*Sqrt[e*x]*(a^(1/3)*e + b^(1/3)*e*x)*Sqrt[(a^(2/3)*e^2 - a^(1/3)*b^ 
(1/3)*e^2*x + b^(2/3)*e^2*x^2)/(a^(1/3)*e + (1 + Sqrt[3])*b^(1/3)*e*x)^2]* 
EllipticE[ArcCos[(a^(1/3)*e + (1 - Sqrt[3])*b^(1/3)*e*x)/(a^(1/3)*e + (1 + 
 Sqrt[3])*b^(1/3)*e*x)], (2 + Sqrt[3])/4])/(Sqrt[(b^(1/3)*e*x*(a^(1/3)*e + 
 b^(1/3)*e*x))/(a^(1/3)*e + (1 + Sqrt[3])*b^(1/3)*e*x)^2]*Sqrt[a + b*x^3]) 
)/(2*b^(2/3)) - ((1 - Sqrt[3])*a^(1/3)*e*Sqrt[e*x]*(a^(1/3)*e + b^(1/3)*e* 
x)*Sqrt[(a^(2/3)*e^2 - a^(1/3)*b^(1/3)*e^2*x + b^(2/3)*e^2*x^2)/(a^(1/3)*e 
 + (1 + Sqrt[3])*b^(1/3)*e*x)^2]*EllipticF[ArcCos[(a^(1/3)*e + (1 - Sqrt[3 
])*b^(1/3)*e*x)/(a^(1/3)*e + (1 + Sqrt[3])*b^(1/3)*e*x)], (2 + Sqrt[3])/4] 
)/(4*3^(1/4)*b^(2/3)*Sqrt[(b^(1/3)*e*x*(a^(1/3)*e + b^(1/3)*e*x))/(a^(1/3) 
*e + (1 + Sqrt[3])*b^(1/3)*e*x)^2]*Sqrt[a + b*x^3])))/(3*a*e)))/(9*a*b)
 

Defintions of rubi rules used

rule 25

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

rule 766

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[x*(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/ 
(s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*((s + 
r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])* 
r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x 
]
 

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 837

Int[(x_)^4/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 
3]], s = Denom[Rt[b/a, 3]]}, Simp[(Sqrt[3] - 1)*(s^2/(2*r^2))   Int[1/Sqrt[ 
a + b*x^6], x], x] - Simp[1/(2*r^2)   Int[((Sqrt[3] - 1)*s^2 - 2*r^2*x^4)/S 
qrt[a + b*x^6], x], x]] /; FreeQ[{a, b}, 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 2420

Int[((c_) + (d_.)*(x_)^4)/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = 
 Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(1 + Sqrt[3])*d*s^3*x*(Sqr 
t[a + b*x^6]/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2))), x] - Simp[3^(1/4)*d*s*x* 
(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]/(2 
*r^2*Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*x^2)^2]*Sqrt[a + b*x^6]) 
)*EllipticE[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 
 + Sqrt[3])/4], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 
- Sqrt[3])*d, 0]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 5.36 (sec) , antiderivative size = 1190, normalized size of antiderivative = 2.00

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1190\)
default	\(\text {Expression too large to display}\)	\(10786\)

Input:

int((e*x)^(3/2)*(B*x^3+A)/(b*x^3+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/e/x*(e*x)^(1/2)/(b*x^3+a)^(1/2)*((b*x^3+a)*e*x)^(1/2)*(2/9*e/a/b^3*x^2*( 
A*b-B*a)*(b*e*x^4+a*e*x)^(1/2)/(x^3+a/b)^2+2/27/b*e^2*x^3/a^2*(4*A*b+5*B*a 
)/((x^3+a/b)*b*e*x)^(1/2)-2/27/b/a^2*e^2*(4*A*b+5*B*a)*(x*(x+1/2/b*(-a*b^2 
)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1 
/2)/b*(-a*b^2)^(1/3))+(1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3) 
)*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*x/(-1/2/b*(-a*b^ 
2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2)*(x- 
1/b*(-a*b^2)^(1/3))^2*(1/b*(-a*b^2)^(1/3)*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^ 
(1/2)/b*(-a*b^2)^(1/3))/(-1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1 
/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2)*(1/b*(-a*b^2)^(1/3)*(x+1/2/b*(-a*b^2)^( 
1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(-1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/ 
b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2)*(((-1/2/b*(-a*b^2)^(1/3)+1 
/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/b*(-a*b^2)^(1/3)+1/b^2*(-a*b^2)^(2/3))/(-3/ 
2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*b/(-a*b^2)^(1/3)*Ellipt 
icF(((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*x/(-1/2/b*(-a* 
b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2),( 
(3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*(1/2/b*(-a*b^2)^(1/3 
)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(- 
a*b^2)^(1/3))/(3/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2) 
)+(1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*EllipticE(((-3/...
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.28 \[ \int \frac {(e x)^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx=-\frac {2 \, {\left ({\left ({\left (5 \, B a b^{2} + 4 \, A b^{3}\right )} e x^{7} + 2 \, {\left (5 \, B a^{2} b + 4 \, A a b^{2}\right )} e x^{4} + {\left (5 \, B a^{3} + 4 \, A a^{2} b\right )} e x\right )} \sqrt {a e} {\rm weierstrassZeta}\left (0, -\frac {4 \, b}{a}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, b}{a}, \frac {1}{x}\right )\right ) + {\left ({\left (8 \, B a^{2} b + A a b^{2}\right )} e x^{3} + {\left (5 \, B a^{3} + 4 \, A a^{2} b\right )} e\right )} \sqrt {b x^{3} + a} \sqrt {e x}\right )}}{27 \, {\left (a^{2} b^{4} x^{7} + 2 \, a^{3} b^{3} x^{4} + a^{4} b^{2} x\right )}} \] Input:

integrate((e*x)^(3/2)*(B*x^3+A)/(b*x^3+a)^(5/2),x, algorithm="fricas")
 

Output:

-2/27*(((5*B*a*b^2 + 4*A*b^3)*e*x^7 + 2*(5*B*a^2*b + 4*A*a*b^2)*e*x^4 + (5 
*B*a^3 + 4*A*a^2*b)*e*x)*sqrt(a*e)*weierstrassZeta(0, -4*b/a, weierstrassP 
Inverse(0, -4*b/a, 1/x)) + ((8*B*a^2*b + A*a*b^2)*e*x^3 + (5*B*a^3 + 4*A*a 
^2*b)*e)*sqrt(b*x^3 + a)*sqrt(e*x))/(a^2*b^4*x^7 + 2*a^3*b^3*x^4 + a^4*b^2 
*x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((e*x)**(3/2)*(B*x**3+A)/(b*x**3+a)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(e x)^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx=\int { \frac {{\left (B x^{3} + A\right )} \left (e x\right )^{\frac {3}{2}}}{{\left (b x^{3} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((e*x)^(3/2)*(B*x^3+A)/(b*x^3+a)^(5/2),x, algorithm="maxima")
 

Output:

integrate((B*x^3 + A)*(e*x)^(3/2)/(b*x^3 + a)^(5/2), x)
 

Giac [F]

\[ \int \frac {(e x)^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx=\int { \frac {{\left (B x^{3} + A\right )} \left (e x\right )^{\frac {3}{2}}}{{\left (b x^{3} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((e*x)^(3/2)*(B*x^3+A)/(b*x^3+a)^(5/2),x, algorithm="giac")
 

Output:

integrate((B*x^3 + A)*(e*x)^(3/2)/(b*x^3 + a)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx=\int \frac {\left (B\,x^3+A\right )\,{\left (e\,x\right )}^{3/2}}{{\left (b\,x^3+a\right )}^{5/2}} \,d x \] Input:

int(((A + B*x^3)*(e*x)^(3/2))/(a + b*x^3)^(5/2),x)
 

Output:

int(((A + B*x^3)*(e*x)^(3/2))/(a + b*x^3)^(5/2), x)
 

Reduce [F]

\[ \int \frac {(e x)^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx=\sqrt {e}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{3}+a}\, x}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) e \] Input:

int((e*x)^(3/2)*(B*x^3+A)/(b*x^3+a)^(5/2),x)
 

Output:

sqrt(e)*int((sqrt(x)*sqrt(a + b*x**3)*x)/(a**2 + 2*a*b*x**3 + b**2*x**6),x 
)*e