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

Optimal. Leaf size=152 \[ \frac{a (a B+4 A b) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{4 \sqrt{b} e^{5/2}}+\frac{(e x)^{3/2} \left (a+b x^3\right )^{3/2} (a B+4 A b)}{6 a e^4}+\frac{(e x)^{3/2} \sqrt{a+b x^3} (a B+4 A b)}{4 e^4}-\frac{2 A \left (a+b x^3\right )^{5/2}}{3 a e (e x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.306196, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{a (a B+4 A b) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{4 \sqrt{b} e^{5/2}}+\frac{(e x)^{3/2} \left (a+b x^3\right )^{3/2} (a B+4 A b)}{6 a e^4}+\frac{(e x)^{3/2} \sqrt{a+b x^3} (a B+4 A b)}{4 e^4}-\frac{2 A \left (a+b x^3\right )^{5/2}}{3 a e (e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.4392, size = 138, normalized size = 0.91 \[ - \frac{2 A \left (a + b x^{3}\right )^{\frac{5}{2}}}{3 a e \left (e x\right )^{\frac{3}{2}}} + \frac{a \left (4 A b + B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{e^{\frac{3}{2}} \sqrt{a + b x^{3}}} \right )}}{4 \sqrt{b} e^{\frac{5}{2}}} + \frac{\left (e x\right )^{\frac{3}{2}} \sqrt{a + b x^{3}} \left (4 A b + B a\right )}{4 e^{4}} + \frac{\left (e x\right )^{\frac{3}{2}} \left (a + b x^{3}\right )^{\frac{3}{2}} \left (4 A b + B a\right )}{6 a e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.239739, size = 111, normalized size = 0.73 \[ \frac{x \left (3 a x^3 \sqrt{\frac{a}{x^3}+b} (a B+4 A b) \tanh ^{-1}\left (\frac{\sqrt{\frac{a}{x^3}+b}}{\sqrt{b}}\right )+\sqrt{b} \left (a+b x^3\right ) \left (-8 a A+5 a B x^3+4 A b x^3+2 b B x^6\right )\right )}{12 \sqrt{b} (e x)^{5/2} \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(Sqrt[b]*(a + b*x^3)*(-8*a*A + 4*A*b*x^3 + 5*a*B*x^3 + 2*b*B*x^6) + 3*a*(4*A*
b + a*B)*Sqrt[b + a/x^3]*x^3*ArcTanh[Sqrt[b + a/x^3]/Sqrt[b]]))/(12*Sqrt[b]*(e*x
)^(5/2)*Sqrt[a + b*x^3])

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Maple [C]  time = 0.048, size = 7108, normalized size = 46.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.658036, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (B a^{2} + 4 \, A a b\right )} e x^{2} \log \left (-4 \,{\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt{b x^{3} + a} \sqrt{e x} -{\left (8 \, b^{2} x^{6} + 8 \, a b x^{3} + a^{2}\right )} \sqrt{b e}\right ) + 4 \,{\left (2 \, B b x^{6} +{\left (5 \, B a + 4 \, A b\right )} x^{3} - 8 \, A a\right )} \sqrt{b x^{3} + a} \sqrt{b e} \sqrt{e x}}{48 \, \sqrt{b e} e^{3} x^{2}}, \frac{3 \,{\left (B a^{2} + 4 \, A a b\right )} e x^{2} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} \sqrt{-b e} \sqrt{e x} x}{2 \, b e x^{3} + a e}\right ) + 2 \,{\left (2 \, B b x^{6} +{\left (5 \, B a + 4 \, A b\right )} x^{3} - 8 \, A a\right )} \sqrt{b x^{3} + a} \sqrt{-b e} \sqrt{e x}}{24 \, \sqrt{-b e} e^{3} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(3*(B*a^2 + 4*A*a*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*(2*B*b*x^6 + (5*B*a + 4*A*b
)*x^3 - 8*A*a)*sqrt(b*x^3 + a)*sqrt(b*e)*sqrt(e*x))/(sqrt(b*e)*e^3*x^2), 1/24*(3
*(B*a^2 + 4*A*a*b)*e*x^2*arctan(2*sqrt(b*x^3 + a)*sqrt(-b*e)*sqrt(e*x)*x/(2*b*e*
x^3 + a*e)) + 2*(2*B*b*x^6 + (5*B*a + 4*A*b)*x^3 - 8*A*a)*sqrt(b*x^3 + a)*sqrt(-
b*e)*sqrt(e*x))/(sqrt(-b*e)*e^3*x^2)]

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Sympy [A]  time = 130.744, size = 289, normalized size = 1.9 \[ - \frac{2 A a^{\frac{3}{2}}}{3 e^{\frac{5}{2}} x^{\frac{3}{2}} \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{A \sqrt{a} b x^{\frac{3}{2}} \sqrt{1 + \frac{b x^{3}}{a}}}{3 e^{\frac{5}{2}}} - \frac{2 A \sqrt{a} b x^{\frac{3}{2}}}{3 e^{\frac{5}{2}} \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{A a \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a}} \right )}}{e^{\frac{5}{2}}} + \frac{B a^{\frac{3}{2}} x^{\frac{3}{2}} \sqrt{1 + \frac{b x^{3}}{a}}}{3 e^{\frac{5}{2}}} + \frac{B a^{\frac{3}{2}} x^{\frac{3}{2}}}{12 e^{\frac{5}{2}} \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{B \sqrt{a} b x^{\frac{9}{2}}}{4 e^{\frac{5}{2}} \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{B a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a}} \right )}}{4 \sqrt{b} e^{\frac{5}{2}}} + \frac{B b^{2} x^{\frac{15}{2}}}{6 \sqrt{a} e^{\frac{5}{2}} \sqrt{1 + \frac{b x^{3}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a**(3/2)/(3*e**(5/2)*x**(3/2)*sqrt(1 + b*x**3/a)) + A*sqrt(a)*b*x**(3/2)*sq
rt(1 + b*x**3/a)/(3*e**(5/2)) - 2*A*sqrt(a)*b*x**(3/2)/(3*e**(5/2)*sqrt(1 + b*x*
*3/a)) + A*a*sqrt(b)*asinh(sqrt(b)*x**(3/2)/sqrt(a))/e**(5/2) + B*a**(3/2)*x**(3
/2)*sqrt(1 + b*x**3/a)/(3*e**(5/2)) + B*a**(3/2)*x**(3/2)/(12*e**(5/2)*sqrt(1 +
b*x**3/a)) + B*sqrt(a)*b*x**(9/2)/(4*e**(5/2)*sqrt(1 + b*x**3/a)) + B*a**2*asinh
(sqrt(b)*x**(3/2)/sqrt(a))/(4*sqrt(b)*e**(5/2)) + B*b**2*x**(15/2)/(6*sqrt(a)*e*
*(5/2)*sqrt(1 + b*x**3/a))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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