3.200 \(\int \frac{\left (a+b x^3\right )^{3/2} \left (A+B x^3\right )}{x^4} \, dx\)

Optimal. Leaf size=110 \[ \frac{\left (a+b x^3\right )^{3/2} (2 a B+3 A b)}{9 a}+\frac{1}{3} \sqrt{a+b x^3} (2 a B+3 A b)-\frac{1}{3} \sqrt{a} (2 a B+3 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )-\frac{A \left (a+b x^3\right )^{5/2}}{3 a x^3} \]

[Out]

((3*A*b + 2*a*B)*Sqrt[a + b*x^3])/3 + ((3*A*b + 2*a*B)*(a + b*x^3)^(3/2))/(9*a)
- (A*(a + b*x^3)^(5/2))/(3*a*x^3) - (Sqrt[a]*(3*A*b + 2*a*B)*ArcTanh[Sqrt[a + b*
x^3]/Sqrt[a]])/3

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Rubi [A]  time = 0.240579, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\left (a+b x^3\right )^{3/2} (2 a B+3 A b)}{9 a}+\frac{1}{3} \sqrt{a+b x^3} (2 a B+3 A b)-\frac{1}{3} \sqrt{a} (2 a B+3 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )-\frac{A \left (a+b x^3\right )^{5/2}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^3)^(3/2)*(A + B*x^3))/x^4,x]

[Out]

((3*A*b + 2*a*B)*Sqrt[a + b*x^3])/3 + ((3*A*b + 2*a*B)*(a + b*x^3)^(3/2))/(9*a)
- (A*(a + b*x^3)^(5/2))/(3*a*x^3) - (Sqrt[a]*(3*A*b + 2*a*B)*ArcTanh[Sqrt[a + b*
x^3]/Sqrt[a]])/3

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Rubi in Sympy [A]  time = 16.7957, size = 99, normalized size = 0.9 \[ - \frac{A \left (a + b x^{3}\right )^{\frac{5}{2}}}{3 a x^{3}} - \frac{2 \sqrt{a} \left (\frac{3 A b}{2} + B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{3} + \sqrt{a + b x^{3}} \left (A b + \frac{2 B a}{3}\right ) + \frac{2 \left (a + b x^{3}\right )^{\frac{3}{2}} \left (\frac{3 A b}{2} + B a\right )}{9 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**(3/2)*(B*x**3+A)/x**4,x)

[Out]

-A*(a + b*x**3)**(5/2)/(3*a*x**3) - 2*sqrt(a)*(3*A*b/2 + B*a)*atanh(sqrt(a + b*x
**3)/sqrt(a))/3 + sqrt(a + b*x**3)*(A*b + 2*B*a/3) + 2*(a + b*x**3)**(3/2)*(3*A*
b/2 + B*a)/(9*a)

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Mathematica [A]  time = 0.37275, size = 82, normalized size = 0.75 \[ \frac{1}{3} \sqrt{a+b x^3} \left (-\frac{(2 a B+3 A b) \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{\sqrt{\frac{b x^3}{a}+1}}-\frac{a A}{x^3}+\frac{8 a B}{3}+2 A b+\frac{2}{3} b B x^3\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^3)^(3/2)*(A + B*x^3))/x^4,x]

[Out]

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

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Maple [A]  time = 0.013, size = 101, normalized size = 0.9 \[ A \left ( -{\frac{a}{3\,{x}^{3}}\sqrt{b{x}^{3}+a}}+{\frac{2\,b}{3}\sqrt{b{x}^{3}+a}}-\sqrt{a}b{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ) \right ) +B \left ({\frac{2\,b{x}^{3}}{9}\sqrt{b{x}^{3}+a}}+{\frac{8\,a}{9}\sqrt{b{x}^{3}+a}}-{\frac{2}{3}{a}^{{\frac{3}{2}}}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(3/2)*(B*x^3+A)/x^4,x)

[Out]

A*(-1/3*a*(b*x^3+a)^(1/2)/x^3+2/3*b*(b*x^3+a)^(1/2)-a^(1/2)*b*arctanh((b*x^3+a)^
(1/2)/a^(1/2)))+B*(2/9*b*x^3*(b*x^3+a)^(1/2)+8/9*a*(b*x^3+a)^(1/2)-2/3*a^(3/2)*a
rctanh((b*x^3+a)^(1/2)/a^(1/2)))

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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)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.254448, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (2 \, B a + 3 \, A b\right )} \sqrt{a} x^{3} \log \left (\frac{b x^{3} - 2 \, \sqrt{b x^{3} + a} \sqrt{a} + 2 \, a}{x^{3}}\right ) + 2 \,{\left (2 \, B b x^{6} + 2 \,{\left (4 \, B a + 3 \, A b\right )} x^{3} - 3 \, A a\right )} \sqrt{b x^{3} + a}}{18 \, x^{3}}, -\frac{3 \,{\left (2 \, B a + 3 \, A b\right )} \sqrt{-a} x^{3} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right ) -{\left (2 \, B b x^{6} + 2 \,{\left (4 \, B a + 3 \, A b\right )} x^{3} - 3 \, A a\right )} \sqrt{b x^{3} + a}}{9 \, x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^(3/2)/x^4,x, algorithm="fricas")

[Out]

[1/18*(3*(2*B*a + 3*A*b)*sqrt(a)*x^3*log((b*x^3 - 2*sqrt(b*x^3 + a)*sqrt(a) + 2*
a)/x^3) + 2*(2*B*b*x^6 + 2*(4*B*a + 3*A*b)*x^3 - 3*A*a)*sqrt(b*x^3 + a))/x^3, -1
/9*(3*(2*B*a + 3*A*b)*sqrt(-a)*x^3*arctan(sqrt(b*x^3 + a)/sqrt(-a)) - (2*B*b*x^6
+ 2*(4*B*a + 3*A*b)*x^3 - 3*A*a)*sqrt(b*x^3 + a))/x^3]

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Sympy [A]  time = 42.0701, size = 223, normalized size = 2.03 \[ - A \sqrt{a} b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )} - \frac{A a \sqrt{b} \sqrt{\frac{a}{b x^{3}} + 1}}{3 x^{\frac{3}{2}}} + \frac{2 A a \sqrt{b}}{3 x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{2 A b^{\frac{3}{2}} x^{\frac{3}{2}}}{3 \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{2 B a^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3} + \frac{2 B a^{2}}{3 \sqrt{b} x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{2 B a \sqrt{b} x^{\frac{3}{2}}}{3 \sqrt{\frac{a}{b x^{3}} + 1}} + B b \left (\begin{cases} \frac{\sqrt{a} x^{3}}{3} & \text{for}\: b = 0 \\\frac{2 \left (a + b x^{3}\right )^{\frac{3}{2}}}{9 b} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**(3/2)*(B*x**3+A)/x**4,x)

[Out]

-A*sqrt(a)*b*asinh(sqrt(a)/(sqrt(b)*x**(3/2))) - A*a*sqrt(b)*sqrt(a/(b*x**3) + 1
)/(3*x**(3/2)) + 2*A*a*sqrt(b)/(3*x**(3/2)*sqrt(a/(b*x**3) + 1)) + 2*A*b**(3/2)*
x**(3/2)/(3*sqrt(a/(b*x**3) + 1)) - 2*B*a**(3/2)*asinh(sqrt(a)/(sqrt(b)*x**(3/2)
))/3 + 2*B*a**2/(3*sqrt(b)*x**(3/2)*sqrt(a/(b*x**3) + 1)) + 2*B*a*sqrt(b)*x**(3/
2)/(3*sqrt(a/(b*x**3) + 1)) + B*b*Piecewise((sqrt(a)*x**3/3, Eq(b, 0)), (2*(a +
b*x**3)**(3/2)/(9*b), True))

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GIAC/XCAS [A]  time = 0.219951, size = 139, normalized size = 1.26 \[ \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} B b + 6 \, \sqrt{b x^{3} + a} B a b + 6 \, \sqrt{b x^{3} + a} A b^{2} + \frac{3 \,{\left (2 \, B a^{2} b + 3 \, A a b^{2}\right )} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{3 \, \sqrt{b x^{3} + a} A a b}{x^{3}}}{9 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^(3/2)/x^4,x, algorithm="giac")

[Out]

1/9*(2*(b*x^3 + a)^(3/2)*B*b + 6*sqrt(b*x^3 + a)*B*a*b + 6*sqrt(b*x^3 + a)*A*b^2
+ 3*(2*B*a^2*b + 3*A*a*b^2)*arctan(sqrt(b*x^3 + a)/sqrt(-a))/sqrt(-a) - 3*sqrt(
b*x^3 + a)*A*a*b/x^3)/b