3.506 \(\int \frac{x^5}{\sqrt{a+b x^3} \sqrt{c+d x^3}} \, dx\)

Optimal. Leaf size=88 \[ \frac{\sqrt{a+b x^3} \sqrt{c+d x^3}}{3 b d}-\frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^3}}{\sqrt{b} \sqrt{c+d x^3}}\right )}{3 b^{3/2} d^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.264225, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\sqrt{a+b x^3} \sqrt{c+d x^3}}{3 b d}-\frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^3}}{\sqrt{b} \sqrt{c+d x^3}}\right )}{3 b^{3/2} d^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]

[Out]

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

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Rubi in Sympy [A]  time = 18.3217, size = 75, normalized size = 0.85 \[ \frac{\sqrt{a + b x^{3}} \sqrt{c + d x^{3}}}{3 b d} - \frac{\left (a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x^{3}}}{\sqrt{b} \sqrt{c + d x^{3}}} \right )}}{3 b^{\frac{3}{2}} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x**3+a)**(1/2)/(d*x**3+c)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.121773, size = 103, normalized size = 1.17 \[ \frac{\sqrt{a+b x^3} \sqrt{c+d x^3}}{3 b d}-\frac{(a d+b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x^3} \sqrt{c+d x^3}+a d+b c+2 b d x^3\right )}{6 b^{3/2} d^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]

[Out]

(Sqrt[a + b*x^3]*Sqrt[c + d*x^3])/(3*b*d) - ((b*c + a*d)*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]])/(6*b^(3/2)*d^(3/2))

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{{x}^{5}{\frac{1}{\sqrt{b{x}^{3}+a}}}{\frac{1}{\sqrt{d{x}^{3}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x)

[Out]

int(x^5/(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^5/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.257652, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (b c + a d\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x^{3} + b^{2} c d + a b d^{2}\right )} \sqrt{b x^{3} + a} \sqrt{d x^{3} + c} +{\left (8 \, b^{2} d^{2} x^{6} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x^{3}\right )} \sqrt{b d}\right ) + 4 \, \sqrt{b x^{3} + a} \sqrt{d x^{3} + c} \sqrt{b d}}{12 \, \sqrt{b d} b d}, -\frac{{\left (b c + a d\right )} \arctan \left (\frac{{\left (2 \, b d x^{3} + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x^{3} + a} \sqrt{d x^{3} + c} b d}\right ) - 2 \, \sqrt{b x^{3} + a} \sqrt{d x^{3} + c} \sqrt{-b d}}{6 \, \sqrt{-b d} b d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)),x, algorithm="fricas")

[Out]

[1/12*((b*c + a*d)*log(-4*(2*b^2*d^2*x^3 + b^2*c*d + a*b*d^2)*sqrt(b*x^3 + a)*sq
rt(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)*sqrt(b*d)) + 4*sqrt(b*x^3 + a)*sqrt(d*x^3 + c)*sqrt(b*d))/(sqrt(b*d)
*b*d), -1/6*((b*c + a*d)*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)) - 2*sqrt(b*x^3 + a)*sqrt(d*x^3 + c)*sqrt(-b*d))/(s
qrt(-b*d)*b*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\sqrt{a + b x^{3}} \sqrt{c + d x^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x**3+a)**(1/2)/(d*x**3+c)**(1/2),x)

[Out]

Integral(x**5/(sqrt(a + b*x**3)*sqrt(c + d*x**3)), x)

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GIAC/XCAS [A]  time = 0.235647, size = 140, normalized size = 1.59 \[ \frac{\frac{{\left (b c + a d\right )}{\rm ln}\left ({\left | -\sqrt{b x^{3} + a} \sqrt{b d} + \sqrt{b^{2} c +{\left (b x^{3} + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d} + \frac{\sqrt{b x^{3} + a} \sqrt{b^{2} c +{\left (b x^{3} + a\right )} b d - a b d}}{b d}}{3 \,{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)),x, algorithm="giac")

[Out]

1/3*((b*c + a*d)*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)*d) + sqrt(b*x^3 + a)*sqrt(b^2*c + (b*x^3 + a)*b*d - a*b*
d)/(b*d))/abs(b)