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

Optimal. Leaf size=188 \[ \frac{5 a^2 (a B+6 A b) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{24 \sqrt{b} e^{5/2}}+\frac{(e x)^{3/2} \left (a+b x^3\right )^{5/2} (a B+6 A b)}{9 a e^4}+\frac{5 (e x)^{3/2} \left (a+b x^3\right )^{3/2} (a B+6 A b)}{36 e^4}+\frac{5 a (e x)^{3/2} \sqrt{a+b x^3} (a B+6 A b)}{24 e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{3 a e (e x)^{3/2}} \]

[Out]

(5*a*(6*A*b + a*B)*(e*x)^(3/2)*Sqrt[a + b*x^3])/(24*e^4) + (5*(6*A*b + a*B)*(e*x
)^(3/2)*(a + b*x^3)^(3/2))/(36*e^4) + ((6*A*b + a*B)*(e*x)^(3/2)*(a + b*x^3)^(5/
2))/(9*a*e^4) - (2*A*(a + b*x^3)^(7/2))/(3*a*e*(e*x)^(3/2)) + (5*a^2*(6*A*b + a*
B)*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/(e^(3/2)*Sqrt[a + b*x^3])])/(24*Sqrt[b]*e^(5/2)
)

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Rubi [A]  time = 0.366636, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{5 a^2 (a B+6 A b) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{24 \sqrt{b} e^{5/2}}+\frac{(e x)^{3/2} \left (a+b x^3\right )^{5/2} (a B+6 A b)}{9 a e^4}+\frac{5 (e x)^{3/2} \left (a+b x^3\right )^{3/2} (a B+6 A b)}{36 e^4}+\frac{5 a (e x)^{3/2} \sqrt{a+b x^3} (a B+6 A b)}{24 e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{3 a e (e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(5*a*(6*A*b + a*B)*(e*x)^(3/2)*Sqrt[a + b*x^3])/(24*e^4) + (5*(6*A*b + a*B)*(e*x
)^(3/2)*(a + b*x^3)^(3/2))/(36*e^4) + ((6*A*b + a*B)*(e*x)^(3/2)*(a + b*x^3)^(5/
2))/(9*a*e^4) - (2*A*(a + b*x^3)^(7/2))/(3*a*e*(e*x)^(3/2)) + (5*a^2*(6*A*b + a*
B)*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/(e^(3/2)*Sqrt[a + b*x^3])])/(24*Sqrt[b]*e^(5/2)
)

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Rubi in Sympy [A]  time = 33.9601, size = 177, normalized size = 0.94 \[ - \frac{2 A \left (a + b x^{3}\right )^{\frac{7}{2}}}{3 a e \left (e x\right )^{\frac{3}{2}}} + \frac{5 a^{2} \left (6 A b + B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{e^{\frac{3}{2}} \sqrt{a + b x^{3}}} \right )}}{24 \sqrt{b} e^{\frac{5}{2}}} + \frac{5 a \left (e x\right )^{\frac{3}{2}} \sqrt{a + b x^{3}} \left (6 A b + B a\right )}{24 e^{4}} + \frac{5 \left (e x\right )^{\frac{3}{2}} \left (a + b x^{3}\right )^{\frac{3}{2}} \left (6 A b + B a\right )}{36 e^{4}} + \frac{\left (e x\right )^{\frac{3}{2}} \left (a + b x^{3}\right )^{\frac{5}{2}} \left (6 A b + B a\right )}{9 a e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*(a + b*x**3)**(7/2)/(3*a*e*(e*x)**(3/2)) + 5*a**2*(6*A*b + B*a)*atanh(sqrt(
b)*(e*x)**(3/2)/(e**(3/2)*sqrt(a + b*x**3)))/(24*sqrt(b)*e**(5/2)) + 5*a*(e*x)**
(3/2)*sqrt(a + b*x**3)*(6*A*b + B*a)/(24*e**4) + 5*(e*x)**(3/2)*(a + b*x**3)**(3
/2)*(6*A*b + B*a)/(36*e**4) + (e*x)**(3/2)*(a + b*x**3)**(5/2)*(6*A*b + B*a)/(9*
a*e**4)

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Mathematica [A]  time = 0.277901, size = 137, normalized size = 0.73 \[ \frac{x \left (\sqrt{b} \left (a+b x^3\right ) \left (a^2 \left (33 B x^3-48 A\right )+a \left (54 A b x^3+26 b B x^6\right )+4 b^2 x^6 \left (3 A+2 B x^3\right )\right )+15 a^2 x^3 \sqrt{\frac{a}{x^3}+b} (a B+6 A b) \tanh ^{-1}\left (\frac{\sqrt{\frac{a}{x^3}+b}}{\sqrt{b}}\right )\right )}{72 \sqrt{b} (e x)^{5/2} \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(Sqrt[b]*(a + b*x^3)*(4*b^2*x^6*(3*A + 2*B*x^3) + a^2*(-48*A + 33*B*x^3) + a*
(54*A*b*x^3 + 26*b*B*x^6)) + 15*a^2*(6*A*b + a*B)*Sqrt[b + a/x^3]*x^3*ArcTanh[Sq
rt[b + a/x^3]/Sqrt[b]]))/(72*Sqrt[b]*(e*x)^(5/2)*Sqrt[a + b*x^3])

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Maple [C]  time = 0.052, size = 7544, normalized size = 40.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.673287, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (B a^{3} + 6 \, A a^{2} b\right )} e x^{2} \log \left (-4 \,{\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt{b x^{3} + a} \sqrt{e x} -{\left (8 \, b^{2} x^{6} + 8 \, a b x^{3} + a^{2}\right )} \sqrt{b e}\right ) + 4 \,{\left (8 \, B b^{2} x^{9} + 2 \,{\left (13 \, B a b + 6 \, A b^{2}\right )} x^{6} + 3 \,{\left (11 \, B a^{2} + 18 \, A a b\right )} x^{3} - 48 \, A a^{2}\right )} \sqrt{b x^{3} + a} \sqrt{b e} \sqrt{e x}}{288 \, \sqrt{b e} e^{3} x^{2}}, \frac{15 \,{\left (B a^{3} + 6 \, A a^{2} b\right )} e x^{2} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} \sqrt{-b e} \sqrt{e x} x}{2 \, b e x^{3} + a e}\right ) + 2 \,{\left (8 \, B b^{2} x^{9} + 2 \,{\left (13 \, B a b + 6 \, A b^{2}\right )} x^{6} + 3 \,{\left (11 \, B a^{2} + 18 \, A a b\right )} x^{3} - 48 \, A a^{2}\right )} \sqrt{b x^{3} + a} \sqrt{-b e} \sqrt{e x}}{144 \, \sqrt{-b e} e^{3} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/288*(15*(B*a^3 + 6*A*a^2*b)*e*x^2*log(-4*(2*b^2*x^4 + a*b*x)*sqrt(b*x^3 + a)*
sqrt(e*x) - (8*b^2*x^6 + 8*a*b*x^3 + a^2)*sqrt(b*e)) + 4*(8*B*b^2*x^9 + 2*(13*B*
a*b + 6*A*b^2)*x^6 + 3*(11*B*a^2 + 18*A*a*b)*x^3 - 48*A*a^2)*sqrt(b*x^3 + a)*sqr
t(b*e)*sqrt(e*x))/(sqrt(b*e)*e^3*x^2), 1/144*(15*(B*a^3 + 6*A*a^2*b)*e*x^2*arcta
n(2*sqrt(b*x^3 + a)*sqrt(-b*e)*sqrt(e*x)*x/(2*b*e*x^3 + a*e)) + 2*(8*B*b^2*x^9 +
2*(13*B*a*b + 6*A*b^2)*x^6 + 3*(11*B*a^2 + 18*A*a*b)*x^3 - 48*A*a^2)*sqrt(b*x^3
+ a)*sqrt(-b*e)*sqrt(e*x))/(sqrt(-b*e)*e^3*x^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{5}{2}}}{\left (e x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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