3.124 \(\int \frac{1}{\sqrt [5]{a+b x^5} \left (c+d x^5\right )} \, dx\)

Optimal. Leaf size=545 \[ -\frac{\log \left (\sqrt [5]{c}-\frac{x \sqrt [5]{b c-a d}}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}-\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}-\frac{2 \sqrt{\frac{2}{5+\sqrt{5}}} x \sqrt [5]{b c-a d}}{\sqrt [5]{c} \sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} x \sqrt [5]{b c-a d}}{\sqrt [5]{c} \sqrt [5]{a+b x^5}}+\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac{\left (1-\sqrt{5}\right ) \log \left (\frac{2 c^{2/5} \left (a+b x^5\right )^{2/5}-\sqrt{5} \sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}+\sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}+2 x^2 (b c-a d)^{2/5}}{\left (a+b x^5\right )^{2/5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}+\frac{\left (1+\sqrt{5}\right ) \log \left (\frac{2 c^{2/5} \left (a+b x^5\right )^{2/5}+\sqrt{5} \sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}+\sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}+2 x^2 (b c-a d)^{2/5}}{\left (a+b x^5\right )^{2/5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}} \]

[Out]

-(Sqrt[(5 + Sqrt[5])/2]*ArcTan[Sqrt[(5 - 2*Sqrt[5])/5] - (2*Sqrt[2/(5 + Sqrt[5])
]*(b*c - a*d)^(1/5)*x)/(c^(1/5)*(a + b*x^5)^(1/5))])/(5*c^(4/5)*(b*c - a*d)^(1/5
)) + (Sqrt[(5 - Sqrt[5])/2]*ArcTan[Sqrt[(5 + 2*Sqrt[5])/5] + (Sqrt[(2*(5 + Sqrt[
5]))/5]*(b*c - a*d)^(1/5)*x)/(c^(1/5)*(a + b*x^5)^(1/5))])/(5*c^(4/5)*(b*c - a*d
)^(1/5)) - Log[c^(1/5) - ((b*c - a*d)^(1/5)*x)/(a + b*x^5)^(1/5)]/(5*c^(4/5)*(b*
c - a*d)^(1/5)) + ((1 - Sqrt[5])*Log[(2*(b*c - a*d)^(2/5)*x^2 + c^(1/5)*(b*c - a
*d)^(1/5)*x*(a + b*x^5)^(1/5) - Sqrt[5]*c^(1/5)*(b*c - a*d)^(1/5)*x*(a + b*x^5)^
(1/5) + 2*c^(2/5)*(a + b*x^5)^(2/5))/(a + b*x^5)^(2/5)])/(20*c^(4/5)*(b*c - a*d)
^(1/5)) + ((1 + Sqrt[5])*Log[(2*(b*c - a*d)^(2/5)*x^2 + c^(1/5)*(b*c - a*d)^(1/5
)*x*(a + b*x^5)^(1/5) + Sqrt[5]*c^(1/5)*(b*c - a*d)^(1/5)*x*(a + b*x^5)^(1/5) +
2*c^(2/5)*(a + b*x^5)^(2/5))/(a + b*x^5)^(2/5)])/(20*c^(4/5)*(b*c - a*d)^(1/5))

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Rubi [A]  time = 2.3109, antiderivative size = 545, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{\log \left (\sqrt [5]{c}-\frac{x \sqrt [5]{b c-a d}}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}-\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}-\frac{2 \sqrt{\frac{2}{5+\sqrt{5}}} x \sqrt [5]{b c-a d}}{\sqrt [5]{c} \sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} x \sqrt [5]{b c-a d}}{\sqrt [5]{c} \sqrt [5]{a+b x^5}}+\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac{\left (1-\sqrt{5}\right ) \log \left (\frac{2 c^{2/5} \left (a+b x^5\right )^{2/5}-\sqrt{5} \sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}+\sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}+2 x^2 (b c-a d)^{2/5}}{\left (a+b x^5\right )^{2/5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}+\frac{\left (1+\sqrt{5}\right ) \log \left (\frac{2 c^{2/5} \left (a+b x^5\right )^{2/5}+\sqrt{5} \sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}+\sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}+2 x^2 (b c-a d)^{2/5}}{\left (a+b x^5\right )^{2/5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[(5 + Sqrt[5])/2]*ArcTan[Sqrt[(5 - 2*Sqrt[5])/5] - (2*Sqrt[2/(5 + Sqrt[5])
]*(b*c - a*d)^(1/5)*x)/(c^(1/5)*(a + b*x^5)^(1/5))])/(5*c^(4/5)*(b*c - a*d)^(1/5
)) + (Sqrt[(5 - Sqrt[5])/2]*ArcTan[Sqrt[(5 + 2*Sqrt[5])/5] + (Sqrt[(2*(5 + Sqrt[
5]))/5]*(b*c - a*d)^(1/5)*x)/(c^(1/5)*(a + b*x^5)^(1/5))])/(5*c^(4/5)*(b*c - a*d
)^(1/5)) - Log[c^(1/5) - ((b*c - a*d)^(1/5)*x)/(a + b*x^5)^(1/5)]/(5*c^(4/5)*(b*
c - a*d)^(1/5)) + ((1 - Sqrt[5])*Log[(2*(b*c - a*d)^(2/5)*x^2 + c^(1/5)*(b*c - a
*d)^(1/5)*x*(a + b*x^5)^(1/5) - Sqrt[5]*c^(1/5)*(b*c - a*d)^(1/5)*x*(a + b*x^5)^
(1/5) + 2*c^(2/5)*(a + b*x^5)^(2/5))/(a + b*x^5)^(2/5)])/(20*c^(4/5)*(b*c - a*d)
^(1/5)) + ((1 + Sqrt[5])*Log[(2*(b*c - a*d)^(2/5)*x^2 + c^(1/5)*(b*c - a*d)^(1/5
)*x*(a + b*x^5)^(1/5) + Sqrt[5]*c^(1/5)*(b*c - a*d)^(1/5)*x*(a + b*x^5)^(1/5) +
2*c^(2/5)*(a + b*x^5)^(2/5))/(a + b*x^5)^(2/5)])/(20*c^(4/5)*(b*c - a*d)^(1/5))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 1.6263, size = 379, normalized size = 0.7 \[ \frac{-\left (\sqrt{5}-1\right ) \log \left (-\frac{\left (\sqrt{5}-1\right ) \sqrt [5]{c} x \sqrt [5]{b c-a d}}{2 \sqrt [5]{a x^5+b}}+\frac{x^2 (b c-a d)^{2/5}}{\left (a x^5+b\right )^{2/5}}+c^{2/5}\right )+\left (1+\sqrt{5}\right ) \log \left (\frac{\left (1+\sqrt{5}\right ) \sqrt [5]{c} x \sqrt [5]{b c-a d}}{2 \sqrt [5]{a x^5+b}}+\frac{x^2 (b c-a d)^{2/5}}{\left (a x^5+b\right )^{2/5}}+c^{2/5}\right )-4 \log \left (\sqrt [5]{c}-\frac{x \sqrt [5]{b c-a d}}{\sqrt [5]{a x^5+b}}\right )+2 \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{2 \sqrt{\frac{2}{5+\sqrt{5}}} \left (\frac{x \sqrt [5]{b c-a d}}{\sqrt [5]{a x^5+b}}-\frac{1}{4} \left (\sqrt{5}-1\right ) \sqrt [5]{c}\right )}{\sqrt [5]{c}}\right )+2 \sqrt{10-2 \sqrt{5}} \tan ^{-1}\left (\frac{\sqrt{2+\frac{2}{\sqrt{5}}} \left (\frac{x \sqrt [5]{b c-a d}}{\sqrt [5]{a x^5+b}}+\frac{1}{4} \left (1+\sqrt{5}\right ) \sqrt [5]{c}\right )}{\sqrt [5]{c}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*Sqrt[2*(5 + Sqrt[5])]*ArcTan[(2*Sqrt[2/(5 + Sqrt[5])]*(-((-1 + Sqrt[5])*c^(1/
5))/4 + ((b*c - a*d)^(1/5)*x)/(b + a*x^5)^(1/5)))/c^(1/5)] + 2*Sqrt[10 - 2*Sqrt[
5]]*ArcTan[(Sqrt[2 + 2/Sqrt[5]]*(((1 + Sqrt[5])*c^(1/5))/4 + ((b*c - a*d)^(1/5)*
x)/(b + a*x^5)^(1/5)))/c^(1/5)] - 4*Log[c^(1/5) - ((b*c - a*d)^(1/5)*x)/(b + a*x
^5)^(1/5)] - (-1 + Sqrt[5])*Log[c^(2/5) + ((b*c - a*d)^(2/5)*x^2)/(b + a*x^5)^(2
/5) - ((-1 + Sqrt[5])*c^(1/5)*(b*c - a*d)^(1/5)*x)/(2*(b + a*x^5)^(1/5))] + (1 +
Sqrt[5])*Log[c^(2/5) + ((b*c - a*d)^(2/5)*x^2)/(b + a*x^5)^(2/5) + ((1 + Sqrt[5
])*c^(1/5)*(b*c - a*d)^(1/5)*x)/(2*(b + a*x^5)^(1/5))])/(20*c^(4/5)*(b*c - a*d)^
(1/5))

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \[ \int{\frac{1}{d{x}^{5}+c}{\frac{1}{\sqrt [5]{b{x}^{5}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(1/(b*x^5+a)^(1/5)/(d*x^5+c),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{5} + a\right )}^{\frac{1}{5}}{\left (d x^{5} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + a)^(1/5)*(d*x^5 + c)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^5 + a)^(1/5)*(d*x^5 + c)), x)

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Fricas [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(1/((b*x^5 + a)^(1/5)*(d*x^5 + c)),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{5} + a\right )}^{\frac{1}{5}}{\left (d x^{5} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + a)^(1/5)*(d*x^5 + c)),x, algorithm="giac")

[Out]

integrate(1/((b*x^5 + a)^(1/5)*(d*x^5 + c)), x)