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

Optimal. Leaf size=53 \[ \frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{3 \sqrt{a} b^{3/2}}+\frac{2 B x^{3/2}}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.108199, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{3 \sqrt{a} b^{3/2}}+\frac{2 B x^{3/2}}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.5483, size = 48, normalized size = 0.91 \[ \frac{2 B x^{\frac{3}{2}}}{3 b} + \frac{2 \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a}} \right )}}{3 \sqrt{a} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.114264, size = 139, normalized size = 2.62 \[ \frac{2 \left ((a B-A b) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )+(A b-a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )-A b \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )+\sqrt{a} \sqrt{b} B x^{3/2}+a B \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )\right )}{3 \sqrt{a} b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(Sqrt[a]*Sqrt[b]*B*x^(3/2) + (-(A*b) + a*B)*ArcTan[Sqrt[3] - (2*b^(1/6)*Sqrt[
x])/a^(1/6)] + (A*b - a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)] - A*b*A
rcTan[(b^(1/6)*Sqrt[x])/a^(1/6)] + a*B*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)]))/(3*Sq
rt[a]*b^(3/2))

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Maple [A]  time = 0.01, size = 53, normalized size = 1. \[{\frac{2\,B}{3\,b}{x}^{{\frac{3}{2}}}}+{\frac{2\,A}{3}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{2\,Ba}{3\,b}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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(x)/(b*x^3 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245341, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{-a b} B x^{\frac{3}{2}} -{\left (B a - A b\right )} \log \left (\frac{2 \, a b x^{\frac{3}{2}} +{\left (b x^{3} - a\right )} \sqrt{-a b}}{b x^{3} + a}\right )}{3 \, \sqrt{-a b} b}, \frac{2 \,{\left (\sqrt{a b} B x^{\frac{3}{2}} -{\left (B a - A b\right )} \arctan \left (\frac{\sqrt{a b} x^{\frac{3}{2}}}{a}\right )\right )}}{3 \, \sqrt{a b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.21394, size = 53, normalized size = 1. \[ \frac{2 \, B x^{\frac{3}{2}}}{3 \, b} - \frac{2 \,{\left (B a - A b\right )} \arctan \left (\frac{b x^{\frac{3}{2}}}{\sqrt{a b}}\right )}{3 \, \sqrt{a b} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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