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

Optimal. Leaf size=81 \[ -\frac{2}{3} a^{3/2} A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )+\frac{2}{9} A \left (a+b x^3\right )^{3/2}+\frac{2}{3} a A \sqrt{a+b x^3}+\frac{2 B \left (a+b x^3\right )^{5/2}}{15 b} \]

[Out]

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

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Rubi [A]  time = 0.202637, antiderivative size = 81, 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 \[ -\frac{2}{3} a^{3/2} A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )+\frac{2}{9} A \left (a+b x^3\right )^{3/2}+\frac{2}{3} a A \sqrt{a+b x^3}+\frac{2 B \left (a+b x^3\right )^{5/2}}{15 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.0243, size = 75, normalized size = 0.93 \[ - \frac{2 A a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{3} + \frac{2 A a \sqrt{a + b x^{3}}}{3} + \frac{2 A \left (a + b x^{3}\right )^{\frac{3}{2}}}{9} + \frac{2 B \left (a + b x^{3}\right )^{\frac{5}{2}}}{15 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a**(3/2)*atanh(sqrt(a + b*x**3)/sqrt(a))/3 + 2*A*a*sqrt(a + b*x**3)/3 + 2*A
*(a + b*x**3)**(3/2)/9 + 2*B*(a + b*x**3)**(5/2)/(15*b)

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Mathematica [A]  time = 0.379946, size = 83, normalized size = 1.02 \[ \frac{2}{45} \sqrt{a+b x^3} \left (x^3 (6 a B+5 A b)+\frac{a (3 a B+20 A b)}{b}-\frac{15 a A \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{\sqrt{\frac{b x^3}{a}+1}}+3 b B x^6\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a + b*x^3]*((a*(20*A*b + 3*a*B))/b + (5*A*b + 6*a*B)*x^3 + 3*b*B*x^6 - (
15*a*A*ArcTanh[Sqrt[1 + (b*x^3)/a]])/Sqrt[1 + (b*x^3)/a]))/45

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Maple [A]  time = 0.01, size = 66, normalized size = 0.8 \[ A \left ({\frac{2\,b{x}^{3}}{9}\sqrt{b{x}^{3}+a}}+{\frac{8\,a}{9}\sqrt{b{x}^{3}+a}}-{\frac{2}{3}{a}^{{\frac{3}{2}}}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ) } \right ) +{\frac{2\,B}{15\,b} \left ( b{x}^{3}+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(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)*arctanh((b*x^3+a)
^(1/2)/a^(1/2)))+2/15*B*(b*x^3+a)^(5/2)/b

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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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.263777, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, A a^{\frac{3}{2}} b \log \left (\frac{b x^{3} - 2 \, \sqrt{b x^{3} + a} \sqrt{a} + 2 \, a}{x^{3}}\right ) + 2 \,{\left (3 \, B b^{2} x^{6} +{\left (6 \, B a b + 5 \, A b^{2}\right )} x^{3} + 3 \, B a^{2} + 20 \, A a b\right )} \sqrt{b x^{3} + a}}{45 \, b}, -\frac{2 \,{\left (15 \, A \sqrt{-a} a b \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right ) -{\left (3 \, B b^{2} x^{6} +{\left (6 \, B a b + 5 \, A b^{2}\right )} x^{3} + 3 \, B a^{2} + 20 \, A a b\right )} \sqrt{b x^{3} + a}\right )}}{45 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/45*(15*A*a^(3/2)*b*log((b*x^3 - 2*sqrt(b*x^3 + a)*sqrt(a) + 2*a)/x^3) + 2*(3*
B*b^2*x^6 + (6*B*a*b + 5*A*b^2)*x^3 + 3*B*a^2 + 20*A*a*b)*sqrt(b*x^3 + a))/b, -2
/45*(15*A*sqrt(-a)*a*b*arctan(sqrt(b*x^3 + a)/sqrt(-a)) - (3*B*b^2*x^6 + (6*B*a*
b + 5*A*b^2)*x^3 + 3*B*a^2 + 20*A*a*b)*sqrt(b*x^3 + a))/b]

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Sympy [A]  time = 19.8162, size = 144, normalized size = 1.78 \[ - \frac{2 A a^{2} \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b x^{3} \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b x^{3} \wedge - a < 0 \end{cases}\right )}{3} + \frac{2 A a \sqrt{a + b x^{3}}}{3} + \frac{2 A \left (a + b x^{3}\right )^{\frac{3}{2}}}{9} + \frac{2 B \left (a + b x^{3}\right )^{\frac{5}{2}}}{15 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a**2*Piecewise((-atan(sqrt(a + b*x**3)/sqrt(-a))/sqrt(-a), -a > 0), (acoth(
sqrt(a + b*x**3)/sqrt(a))/sqrt(a), (-a < 0) & (a < a + b*x**3)), (atanh(sqrt(a +
b*x**3)/sqrt(a))/sqrt(a), (-a < 0) & (a > a + b*x**3)))/3 + 2*A*a*sqrt(a + b*x*
*3)/3 + 2*A*(a + b*x**3)**(3/2)/9 + 2*B*(a + b*x**3)**(5/2)/(15*b)

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GIAC/XCAS [A]  time = 0.218349, size = 108, normalized size = 1.33 \[ \frac{2 \, A a^{2} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a}} + \frac{2 \,{\left (3 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} B b^{4} + 5 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} A b^{5} + 15 \, \sqrt{b x^{3} + a} A a b^{5}\right )}}{45 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*A*a^2*arctan(sqrt(b*x^3 + a)/sqrt(-a))/sqrt(-a) + 2/45*(3*(b*x^3 + a)^(5/2)*
B*b^4 + 5*(b*x^3 + a)^(3/2)*A*b^5 + 15*sqrt(b*x^3 + a)*A*a*b^5)/b^5