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

Optimal. Leaf size=84 \[ \frac{\sqrt{a+b x^3} (2 a B+A b)}{3 a}-\frac{(2 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a}}-\frac{A \left (a+b x^3\right )^{3/2}}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.199463, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\sqrt{a+b x^3} (2 a B+A b)}{3 a}-\frac{(2 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a}}-\frac{A \left (a+b x^3\right )^{3/2}}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.1199, size = 75, normalized size = 0.89 \[ - \frac{A \left (a + b x^{3}\right )^{\frac{3}{2}}}{3 a x^{3}} + \frac{2 \sqrt{a + b x^{3}} \left (\frac{A b}{2} + B a\right )}{3 a} - \frac{2 \left (\frac{A b}{2} + B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{3 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**3+A)*(b*x**3+a)**(1/2)/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.262504, size = 67, normalized size = 0.8 \[ \frac{1}{3} \sqrt{a+b x^3} \left (-\frac{(2 a B+A b) \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{a \sqrt{\frac{b x^3}{a}+1}}-\frac{A}{x^3}+2 B\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 72, normalized size = 0.9 \[ A \left ( -{\frac{1}{3\,{x}^{3}}\sqrt{b{x}^{3}+a}}-{\frac{b}{3}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}} \right ) +B \left ({\frac{2}{3}\sqrt{b{x}^{3}+a}}-{\frac{2}{3}\sqrt{a}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)*(b*x^3+a)^(1/2)/x^4,x)

[Out]

A*(-1/3*(b*x^3+a)^(1/2)/x^3-1/3*b*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a^(1/2))+B*(2
/3*(b*x^3+a)^(1/2)-2/3*a^(1/2)*arctanh((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)*sqrt(b*x^3 + a)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.250217, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (2 \, B a + A b\right )} x^{3} \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) + 2 \,{\left (2 \, B x^{3} - A\right )} \sqrt{b x^{3} + a} \sqrt{a}}{6 \, \sqrt{a} x^{3}}, \frac{{\left (2 \, B a + A b\right )} x^{3} \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) +{\left (2 \, B x^{3} - A\right )} \sqrt{b x^{3} + a} \sqrt{-a}}{3 \, \sqrt{-a} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 30.966, size = 134, normalized size = 1.6 \[ - \frac{A \sqrt{b} \sqrt{\frac{a}{b x^{3}} + 1}}{3 x^{\frac{3}{2}}} - \frac{A b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3 \sqrt{a}} - \frac{2 B \sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3} + \frac{2 B a}{3 \sqrt{b} x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{2 B \sqrt{b} x^{\frac{3}{2}}}{3 \sqrt{\frac{a}{b x^{3}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**3+A)*(b*x**3+a)**(1/2)/x**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.219514, size = 92, normalized size = 1.1 \[ \frac{2 \, \sqrt{b x^{3} + a} B b + \frac{{\left (2 \, B a b + A b^{2}\right )} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{\sqrt{b x^{3} + a} A b}{x^{3}}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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