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

Optimal. Leaf size=64 \[ \frac{2}{3} A \sqrt{a+b x^3}-\frac{2}{3} \sqrt{a} A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b} \]

[Out]

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

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Rubi [A]  time = 0.144048, antiderivative size = 64, 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 \[ \frac{2}{3} A \sqrt{a+b x^3}-\frac{2}{3} \sqrt{a} A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.9359, size = 58, normalized size = 0.91 \[ - \frac{2 A \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{3} + \frac{2 A \sqrt{a + b x^{3}}}{3} + \frac{2 B \left (a + b x^{3}\right )^{\frac{3}{2}}}{9 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.260072, size = 62, normalized size = 0.97 \[ \frac{2}{9} \sqrt{a+b x^3} \left (-\frac{3 A \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{\sqrt{\frac{b x^3}{a}+1}}+\frac{a B}{b}+3 A+B x^3\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a + b*x^3]*(3*A + (a*B)/b + B*x^3 - (3*A*ArcTanh[Sqrt[1 + (b*x^3)/a]])/S
qrt[1 + (b*x^3)/a]))/9

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Maple [A]  time = 0.01, size = 50, normalized size = 0.8 \[ A \left ({\frac{2}{3}\sqrt{b{x}^{3}+a}}-{\frac{2}{3}\sqrt{a}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ) } \right ) +{\frac{2\,B}{9\,b} \left ( b{x}^{3}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(2/3*(b*x^3+a)^(1/2)-2/3*a^(1/2)*arctanh((b*x^3+a)^(1/2)/a^(1/2)))+2/9*B*(b*x^
3+a)^(3/2)/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)*sqrt(b*x^3 + a)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.275826, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, A \sqrt{a} b \log \left (\frac{b x^{3} - 2 \, \sqrt{b x^{3} + a} \sqrt{a} + 2 \, a}{x^{3}}\right ) + 2 \,{\left (B b x^{3} + B a + 3 \, A b\right )} \sqrt{b x^{3} + a}}{9 \, b}, -\frac{2 \,{\left (3 \, A \sqrt{-a} b \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right ) -{\left (B b x^{3} + B a + 3 \, A b\right )} \sqrt{b x^{3} + a}\right )}}{9 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 12.1838, size = 126, normalized size = 1.97 \[ - \frac{2 A a \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b x^{3} \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b x^{3} \wedge - a < 0 \end{cases}\right )}{3} + \frac{2 A \sqrt{a + b x^{3}}}{3} + \frac{2 B \left (a + b x^{3}\right )^{\frac{3}{2}}}{9 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a*Piecewise((-atan(sqrt(a + b*x**3)/sqrt(-a))/sqrt(-a), -a > 0), (acoth(sqr
t(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*sqrt(a + b*x**3)/3
+ 2*B*(a + b*x**3)**(3/2)/(9*b)

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GIAC/XCAS [A]  time = 0.221373, size = 82, normalized size = 1.28 \[ \frac{2 \, A a \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a}} + \frac{2 \,{\left ({\left (b x^{3} + a\right )}^{\frac{3}{2}} B b^{2} + 3 \, \sqrt{b x^{3} + a} A b^{3}\right )}}{9 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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