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

Optimal. Leaf size=118 \[ \frac{(a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{3 \sqrt{b} e^{5/2}}+\frac{(e x)^{3/2} \sqrt{a+b x^3} (a B+2 A b)}{3 a e^4}-\frac{2 A \left (a+b x^3\right )^{3/2}}{3 a e (e x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.247429, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{(a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{3 \sqrt{b} e^{5/2}}+\frac{(e x)^{3/2} \sqrt{a+b x^3} (a B+2 A b)}{3 a e^4}-\frac{2 A \left (a+b x^3\right )^{3/2}}{3 a e (e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.0637, size = 109, normalized size = 0.92 \[ - \frac{2 A \left (a + b x^{3}\right )^{\frac{3}{2}}}{3 a e \left (e x\right )^{\frac{3}{2}}} + \frac{2 \left (A b + \frac{B a}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{e^{\frac{3}{2}} \sqrt{a + b x^{3}}} \right )}}{3 \sqrt{b} e^{\frac{5}{2}}} + \frac{2 \left (e x\right )^{\frac{3}{2}} \sqrt{a + b x^{3}} \left (A b + \frac{B a}{2}\right )}{3 a e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**3+A)*(b*x**3+a)**(1/2)/(e*x)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.246398, size = 92, normalized size = 0.78 \[ \frac{x \left (\sqrt{b} \left (a+b x^3\right ) \left (B x^3-2 A\right )+x^3 \sqrt{\frac{a}{x^3}+b} (a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{\frac{a}{x^3}+b}}{\sqrt{b}}\right )\right )}{3 \sqrt{b} (e x)^{5/2} \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(Sqrt[b]*(a + b*x^3)*(-2*A + B*x^3) + (2*A*b + a*B)*Sqrt[b + a/x^3]*x^3*ArcTa
nh[Sqrt[b + a/x^3]/Sqrt[b]]))/(3*Sqrt[b]*(e*x)^(5/2)*Sqrt[a + b*x^3])

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Maple [C]  time = 0.07, size = 6668, normalized size = 56.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)*(b*x^3+a)^(1/2)/(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)*sqrt(b*x^3 + a)/(e*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.677675, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (B a + 2 \, A b\right )} e x^{2} \log \left (-4 \,{\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt{b x^{3} + a} \sqrt{e x} -{\left (8 \, b^{2} x^{6} + 8 \, a b x^{3} + a^{2}\right )} \sqrt{b e}\right ) + 4 \,{\left (B x^{3} - 2 \, A\right )} \sqrt{b x^{3} + a} \sqrt{b e} \sqrt{e x}}{12 \, \sqrt{b e} e^{3} x^{2}}, \frac{{\left (B a + 2 \, A b\right )} e x^{2} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} \sqrt{-b e} \sqrt{e x} x}{2 \, b e x^{3} + a e}\right ) + 2 \,{\left (B x^{3} - 2 \, A\right )} \sqrt{b x^{3} + a} \sqrt{-b e} \sqrt{e x}}{6 \, \sqrt{-b e} e^{3} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*((B*a + 2*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*(B*x^3 - 2*A)*sqrt(b*x^3 + a)*sqr
t(b*e)*sqrt(e*x))/(sqrt(b*e)*e^3*x^2), 1/6*((B*a + 2*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*(B*x^3 - 2*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 = 31.6847, size = 160, normalized size = 1.36 \[ - \frac{2 A \sqrt{a}}{3 e^{\frac{5}{2}} x^{\frac{3}{2}} \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{2 A \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a}} \right )}}{3 e^{\frac{5}{2}}} - \frac{2 A b x^{\frac{3}{2}}}{3 \sqrt{a} e^{\frac{5}{2}} \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{B \sqrt{a} x^{\frac{3}{2}} \sqrt{1 + \frac{b x^{3}}{a}}}{3 e^{\frac{5}{2}}} + \frac{B a \operatorname{asinh}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a}} \right )}}{3 \sqrt{b} e^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**3+A)*(b*x**3+a)**(1/2)/(e*x)**(5/2),x)

[Out]

-2*A*sqrt(a)/(3*e**(5/2)*x**(3/2)*sqrt(1 + b*x**3/a)) + 2*A*sqrt(b)*asinh(sqrt(b
)*x**(3/2)/sqrt(a))/(3*e**(5/2)) - 2*A*b*x**(3/2)/(3*sqrt(a)*e**(5/2)*sqrt(1 + b
*x**3/a)) + B*sqrt(a)*x**(3/2)*sqrt(1 + b*x**3/a)/(3*e**(5/2)) + B*a*asinh(sqrt(
b)*x**(3/2)/sqrt(a))/(3*sqrt(b)*e**(5/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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