∫ A+Bx3 ____________________________(ex)3∕2 (a+bx3)5∕2 dx [564]

Optimal. Leaf size=624 \[ -\frac {2 A}{a e \sqrt {e x} \left (a+b x^3\right )^{3/2}}-\frac {2 (10 A b-a B) (e x)^{5/2}}{9 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac {8 (10 A b-a B) (e x)^{5/2}}{27 a^3 e^4 \sqrt {a+b x^3}}+\frac {8 \left (1+\sqrt {3}\right ) (10 A b-a B) \sqrt {e x} \sqrt {a+b x^3}}{27 a^3 b^{2/3} e^2 \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )}-\frac {8 (10 A b-a B) \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\cos ^{-1}\left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{9\ 3^{3/4} a^{8/3} b^{2/3} e^2 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {4 \left (1-\sqrt {3}\right ) (10 A b-a B) \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{27 \sqrt [4]{3} a^{8/3} b^{2/3} e^2 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]

-2/9*(10*A*b-B*a)*(e*x)^(5/2)/a^2/e^4/(b*x^3+a)^(3/2)-2*A/a/e/(b*x^3+a)^(3/2)/(e*x)^(1/2)-8/27*(10*A*b-B*a)*(e

*x)^(5/2)/a^3/e^4/(b*x^3+a)^(1/2)+8/27*(10*A*b-B*a)*(1+3^(1/2))*(e*x)^(1/2)*(b*x^3+a)^(1/2)/a^3/b^(2/3)/e^2/(a

^(1/3)+b^(1/3)*x*(1+3^(1/2)))-8/27*(10*A*b-B*a)*(a^(1/3)+b^(1/3)*x)*((a^(1/3)+b^(1/3)*x*(1-3^(1/2)))^2/(a^(1/3

)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2)/(a^(1/3)+b^(1/3)*x*(1-3^(1/2)))*(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))*EllipticE((1

-(a^(1/3)+b^(1/3)*x*(1-3^(1/2)))^2/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))*(e*x)^(1/

2)*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2)*3^(1/4)/a^(8/3)/b^(2/3)/e

^2/(b*x^3+a)^(1/2)/(b^(1/3)*x*(a^(1/3)+b^(1/3)*x)/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2)-4/81*(10*A*b-B*a)*(

a^(1/3)+b^(1/3)*x)*((a^(1/3)+b^(1/3)*x*(1-3^(1/2)))^2/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2)/(a^(1/3)+b^(1/3

)*x*(1-3^(1/2)))*(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))*EllipticF((1-(a^(1/3)+b^(1/3)*x*(1-3^(1/2)))^2/(a^(1/3)+b^(1/

3)*x*(1+3^(1/2)))^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))*(1-3^(1/2))*(e*x)^(1/2)*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3

)*x^2)/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2)*3^(3/4)/a^(8/3)/b^(2/3)/e^2/(b*x^3+a)^(1/2)/(b^(1/3)*x*(a^(1/3

)+b^(1/3)*x)/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.49, antiderivative size = 624, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {464, 296, 335, 314, 231, 1895} \begin {gather*} -\frac {4 \left (1-\sqrt {3}\right ) \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}} (10 A b-a B) F\left (\text {ArcCos}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{27 \sqrt [4]{3} a^{8/3} b^{2/3} e^2 \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 {8 \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}} (10 A b-a B) E\left (\text {ArcCos}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{9\ 3^{3/4} a^{8/3} b^{2/3} e^2 \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 {8 \left (1+\sqrt {3}\right ) \sqrt {e x} \sqrt {a+b x^3} (10 A b-a B)}{27 a^3 b^{2/3} e^2 \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )}-\frac {8 (e x)^{5/2} (10 A b-a B)}{27 a^3 e^4 \sqrt {a+b x^3}}-\frac {2 (e x)^{5/2} (10 A b-a B)}{9 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac {2 A}{a e \sqrt {e x} \left (a+b x^3\right )^{3/2}} \end {gather*}

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

(-2*A)/(a*e*Sqrt[e*x]*(a + b*x^3)^(3/2)) - (2*(10*A*b - a*B)*(e*x)^(5/2))/(9*a^2*e^4*(a + b*x^3)^(3/2)) - (8*(

10*A*b - a*B)*(e*x)^(5/2))/(27*a^3*e^4*Sqrt[a + b*x^3]) + (8*(1 + Sqrt[3])*(10*A*b - a*B)*Sqrt[e*x]*Sqrt[a + b

*x^3])/(27*a^3*b^(2/3)*e^2*(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)) - (8*(10*A*b - a*B)*Sqrt[e*x]*(a^(1/3) + b^(1/

3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*EllipticE[ArcCos

[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)], (2 + Sqrt[3])/4])/(9*3^(3/4)*a^(8/3

)*b^(2/3)*e^2*Sqrt[(b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*Sqrt[a + b*x^3]) -

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

^2)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*EllipticF[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x)/(a^(1/3) + (1

+ Sqrt[3])*b^(1/3)*x)], (2 + Sqrt[3])/4])/(27*3^(1/4)*a^(8/3)*b^(2/3)*e^2*Sqrt[(b^(1/3)*x*(a^(1/3) + b^(1/3)*x

))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*Sqrt[a + b*x^3])

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]

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] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free

Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Int[(x_)^4/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(

Sqrt[3] - 1)*(s^2/(2*r^2)), Int[1/Sqrt[a + b*x^6], x], x] - Dist[1/(2*r^2), Int[((Sqrt[3] - 1)*s^2 - 2*r^2*x^4

)/Sqrt[a + b*x^6], x], x]] /; FreeQ[{a, b}, x]

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I

nt[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]

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m +

1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In

t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||

GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]

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*(Sqrt[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]

\begin {align*} \int \frac {A+B x^3}{(e x)^{3/2} \left (a+b x^3\right )^{5/2}} \, dx &=-\frac {2 A}{a e \sqrt {e x} \left (a+b x^3\right )^{3/2}}-\frac {(10 A b-a B) \int \frac {(e x)^{3/2}}{\left (a+b x^3\right )^{5/2}} \, dx}{a e^3}\\ &=-\frac {2 A}{a e \sqrt {e x} \left (a+b x^3\right )^{3/2}}-\frac {2 (10 A b-a B) (e x)^{5/2}}{9 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac {(4 (10 A b-a B)) \int \frac {(e x)^{3/2}}{\left (a+b x^3\right )^{3/2}} \, dx}{9 a^2 e^3}\\ &=-\frac {2 A}{a e \sqrt {e x} \left (a+b x^3\right )^{3/2}}-\frac {2 (10 A b-a B) (e x)^{5/2}}{9 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac {8 (10 A b-a B) (e x)^{5/2}}{27 a^3 e^4 \sqrt {a+b x^3}}+\frac {(8 (10 A b-a B)) \int \frac {(e x)^{3/2}}{\sqrt {a+b x^3}} \, dx}{27 a^3 e^3}\\ &=-\frac {2 A}{a e \sqrt {e x} \left (a+b x^3\right )^{3/2}}-\frac {2 (10 A b-a B) (e x)^{5/2}}{9 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac {8 (10 A b-a B) (e x)^{5/2}}{27 a^3 e^4 \sqrt {a+b x^3}}+\frac {(16 (10 A b-a B)) \text {Subst}\left (\int \frac {x^4}{\sqrt {a+\frac {b x^6}{e^3}}} \, dx,x,\sqrt {e x}\right )}{27 a^3 e^4}\\ &=-\frac {2 A}{a e \sqrt {e x} \left (a+b x^3\right )^{3/2}}-\frac {2 (10 A b-a B) (e x)^{5/2}}{9 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac {8 (10 A b-a B) (e x)^{5/2}}{27 a^3 e^4 \sqrt {a+b x^3}}-\frac {(8 (10 A b-a B)) \text {Subst}\left (\int \frac {\left (-1+\sqrt {3}\right ) a^{2/3} e^2-2 b^{2/3} x^4}{\sqrt {a+\frac {b x^6}{e^3}}} \, dx,x,\sqrt {e x}\right )}{27 a^3 b^{2/3} e^4}-\frac {\left (8 \left (1-\sqrt {3}\right ) (10 A b-a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^6}{e^3}}} \, dx,x,\sqrt {e x}\right )}{27 a^{7/3} b^{2/3} e^2}\\ &=-\frac {2 A}{a e \sqrt {e x} \left (a+b x^3\right )^{3/2}}-\frac {2 (10 A b-a B) (e x)^{5/2}}{9 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac {8 (10 A b-a B) (e x)^{5/2}}{27 a^3 e^4 \sqrt {a+b x^3}}+\frac {8 \left (1+\sqrt {3}\right ) (10 A b-a B) \sqrt {e x} \sqrt {a+b x^3}}{27 a^3 b^{2/3} e^2 \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )}-\frac {8 (10 A b-a B) \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 (\cos ^{-1}\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^{8/3} b^{2/3} e^2 \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 {4 \left (1-\sqrt {3}\right ) (10 A b-a B) \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}} F\left (\cos ^{-1}\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^{8/3} b^{2/3} e^2 \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}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.05, size = 85, normalized size = 0.14 \begin {gather*} \frac {2 x \left (-5 a^2 A+(-10 A b+a B) x^3 \left (a+b x^3\right ) \sqrt {1+\frac {b x^3}{a}} \, _2F_1\left (\frac {5}{6},\frac {5}{2};\frac {11}{6};-\frac {b x^3}{a}\right )\right )}{5 a^3 (e x)^{3/2} \left (a+b x^3\right )^{3/2}} \end {gather*}

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

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

3)/a)]))/(5*a^3*(e*x)^(3/2)*(a + b*x^3)^(3/2))

________________________________________________________________________________________

Maple [C] Result contains complex when optimal does not.
time = 0.43, size = 10961, normalized size = 17.57


method	result	size

elliptic	\(\text {Expression too large to display}\)	\(1225\)
risch	\(\text {Expression too large to display}\)	\(3336\)
default	\(\text {Expression too large to display}\)	\(10961\)

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.50, size = 156, normalized size = 0.25 \begin {gather*} -\frac {2 \, {\left (4 \, {\left ({\left (B a b^{2} - 10 \, A b^{3}\right )} x^{7} + 2 \, {\left (B a^{2} b - 10 \, A a b^{2}\right )} x^{4} + {\left (B a^{3} - 10 \, A a^{2} b\right )} x\right )} \sqrt {a} {\rm weierstrassZeta}\left (0, -\frac {4 \, b}{a}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, b}{a}, \frac {1}{x}\right )\right ) + {\left (4 \, B a^{3} - 13 \, A a^{2} b + {\left (B a^{2} b - 10 \, A a b^{2}\right )} x^{3}\right )} \sqrt {b x^{3} + a} \sqrt {x}\right )} e^{\left (-\frac {3}{2}\right )}}{27 \, {\left (a^{3} b^{3} x^{7} + 2 \, a^{4} b^{2} x^{4} + a^{5} b x\right )}} \end {gather*}

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

-2/27*(4*((B*a*b^2 - 10*A*b^3)*x^7 + 2*(B*a^2*b - 10*A*a*b^2)*x^4 + (B*a^3 - 10*A*a^2*b)*x)*sqrt(a)*weierstras

sZeta(0, -4*b/a, weierstrassPInverse(0, -4*b/a, 1/x)) + (4*B*a^3 - 13*A*a^2*b + (B*a^2*b - 10*A*a*b^2)*x^3)*sq

rt(b*x^3 + a)*sqrt(x))*e^(-3/2)/(a^3*b^3*x^7 + 2*a^4*b^2*x^4 + a^5*b*x)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {B\,x^3+A}{{\left (e\,x\right )}^{3/2}\,{\left (b\,x^3+a\right )}^{5/2}} \,d x \end {gather*}

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

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

________________________________________________________________________________________

3.6.64 \(\int \frac {A+B x^3}{(e x)^{3/2} (a+b x^3)^{5/2}} \, dx\) [564]