3.507 \(\int \frac{x^2}{\sqrt{a+b x^3} \sqrt{c+d x^3}} \, dx\)

Optimal. Leaf size=48 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^3}}{\sqrt{b} \sqrt{c+d x^3}}\right )}{3 \sqrt{b} \sqrt{d}} \]

[Out]

(2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^3])/(Sqrt[b]*Sqrt[c + d*x^3])])/(3*Sqrt[b]*Sqrt
[d])

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Rubi [A]  time = 0.157188, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^3}}{\sqrt{b} \sqrt{c+d x^3}}\right )}{3 \sqrt{b} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]

[Out]

(2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^3])/(Sqrt[b]*Sqrt[c + d*x^3])])/(3*Sqrt[b]*Sqrt
[d])

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Rubi in Sympy [A]  time = 13.5054, size = 44, normalized size = 0.92 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{d} \sqrt{a + b x^{3}}} \right )}}{3 \sqrt{b} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*atanh(sqrt(b)*sqrt(c + d*x**3)/(sqrt(d)*sqrt(a + b*x**3)))/(3*sqrt(b)*sqrt(d))

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Mathematica [A]  time = 0.0431235, size = 63, normalized size = 1.31 \[ \frac{\log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x^3} \sqrt{c+d x^3}+a d+b c+2 b d x^3\right )}{3 \sqrt{b} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]

[Out]

Log[b*c + a*d + 2*b*d*x^3 + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x^3]*Sqrt[c + d*x^3]]/(
3*Sqrt[b]*Sqrt[d])

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Maple [F]  time = 0.075, size = 0, normalized size = 0. \[ \int{{x}^{2}{\frac{1}{\sqrt{b{x}^{3}+a}}}{\frac{1}{\sqrt{d{x}^{3}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(x^2/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x)

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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(x^2/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236023, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (4 \,{\left (2 \, b^{2} d^{2} x^{3} + b^{2} c d + a b d^{2}\right )} \sqrt{b x^{3} + a} \sqrt{d x^{3} + c} +{\left (8 \, b^{2} d^{2} x^{6} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x^{3}\right )} \sqrt{b d}\right )}{6 \, \sqrt{b d}}, \frac{\arctan \left (\frac{{\left (2 \, b d x^{3} + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x^{3} + a} \sqrt{d x^{3} + c} b d}\right )}{3 \, \sqrt{-b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*log(4*(2*b^2*d^2*x^3 + b^2*c*d + a*b*d^2)*sqrt(b*x^3 + a)*sqrt(d*x^3 + c) +
(8*b^2*d^2*x^6 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b^2*c*d + a*b*d^2)*x^3)*sqr
t(b*d))/sqrt(b*d), 1/3*arctan(1/2*(2*b*d*x^3 + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x^3
+ a)*sqrt(d*x^3 + c)*b*d))/sqrt(-b*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{a + b x^{3}} \sqrt{c + d x^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2/(sqrt(a + b*x**3)*sqrt(c + d*x**3)), x)

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GIAC/XCAS [A]  time = 0.230133, size = 73, normalized size = 1.52 \[ -\frac{2 \, b{\rm ln}\left ({\left | -\sqrt{b x^{3} + a} \sqrt{b d} + \sqrt{b^{2} c +{\left (b x^{3} + a\right )} b d - a b d} \right |}\right )}{3 \, \sqrt{b d}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*b*ln(abs(-sqrt(b*x^3 + a)*sqrt(b*d) + sqrt(b^2*c + (b*x^3 + a)*b*d - a*b*d)
))/(sqrt(b*d)*abs(b))