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

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

[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 = 1.08507, 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, Rules used = {377, 202, 634, 618, 204, 628, 31} \[ -\frac{\log \left (\sqrt [5]{c}-\frac{x \sqrt [5]{b c-a d}}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac{\left (1-\sqrt{5}\right ) \log \left (\frac{2 c^{2/5} \left (a+b x^5\right )^{2/5}+2 x^2 (b c-a d)^{2/5}-\sqrt{5} \sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}+\sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}}{\left (a+b x^5\right )^{2/5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}+\frac{\left (1+\sqrt{5}\right ) \log \left (\frac{2 c^{2/5} \left (a+b x^5\right )^{2/5}+2 x^2 (b c-a d)^{2/5}+\sqrt{5} \sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}+\sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}}{\left (a+b x^5\right )^{2/5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}-\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}-\frac{2 \sqrt{\frac{2}{5+\sqrt{5}}} x \sqrt [5]{b c-a d}}{\sqrt [5]{c} \sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} x \sqrt [5]{b c-a d}}{\sqrt [5]{c} \sqrt [5]{a+b x^5}}+\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}} \]

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))

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 202

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-(a/b), n]], s = Denominator[Rt[-(a/b
), n]], k, u}, Simp[u = Int[(r + s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 + 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x
]; (r*Int[1/(r - s*x), x])/(a*n) + Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] &&
IGtQ[(n - 3)/2, 0] && NegQ[a/b]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt [5]{a+b x^5} \left (c+d x^5\right )} \, dx &=\operatorname{Subst}\left (\int \frac{1}{c-(b c-a d) x^5} \, dx,x,\frac{x}{\sqrt [5]{a+b x^5}}\right )\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{\sqrt [5]{c}+\frac{1}{4} \left (1-\sqrt{5}\right ) \sqrt [5]{b c-a d} x}{c^{2/5}+\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x+(b c-a d)^{2/5} x^2} \, dx,x,\frac{x}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5}}+\frac{2 \operatorname{Subst}\left (\int \frac{\sqrt [5]{c}+\frac{1}{4} \left (1+\sqrt{5}\right ) \sqrt [5]{b c-a d} x}{c^{2/5}+\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x+(b c-a d)^{2/5} x^2} \, dx,x,\frac{x}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5}}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [5]{c}-\sqrt [5]{b c-a d} x} \, dx,x,\frac{x}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5}}\\ &=-\frac{\log \left (\sqrt [5]{c}-\frac{\sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac{\left (5-\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{c^{2/5}+\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x+(b c-a d)^{2/5} x^2} \, dx,x,\frac{x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{3/5}}+\frac{\left (5+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{c^{2/5}+\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x+(b c-a d)^{2/5} x^2} \, dx,x,\frac{x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{3/5}}+\frac{\left (1-\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d}+2 (b c-a d)^{2/5} x}{c^{2/5}+\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x+(b c-a d)^{2/5} x^2} \, dx,x,\frac{x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}+\frac{\left (1+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d}+2 (b c-a d)^{2/5} x}{c^{2/5}+\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x+(b c-a d)^{2/5} x^2} \, dx,x,\frac{x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}\\ &=-\frac{\log \left (\sqrt [5]{c}-\frac{\sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac{\left (1-\sqrt{5}\right ) \log \left (2 c^{2/5}+\frac{2 (b c-a d)^{2/5} x^2}{\left (a+b x^5\right )^{2/5}}+\frac{\sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}-\frac{\sqrt{5} \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}+\frac{\left (1+\sqrt{5}\right ) \log \left (2 c^{2/5}+\frac{2 (b c-a d)^{2/5} x^2}{\left (a+b x^5\right )^{2/5}}+\frac{\sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}+\frac{\sqrt{5} \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}-\frac{\left (5-\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2} \left (5-\sqrt{5}\right ) c^{2/5} (b c-a d)^{2/5}-x^2} \, dx,x,\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d}+\frac{2 (b c-a d)^{2/5} x}{\sqrt [5]{a+b x^5}}\right )}{10 c^{3/5}}-\frac{\left (5+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2} \left (5+\sqrt{5}\right ) c^{2/5} (b c-a d)^{2/5}-x^2} \, dx,x,\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d}+\frac{2 (b c-a d)^{2/5} x}{\sqrt [5]{a+b x^5}}\right )}{10 c^{3/5}}\\ &=\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\left (1-\sqrt{5}\right ) \sqrt [5]{c}+\frac{4 \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}}{\sqrt{2 \left (5+\sqrt{5}\right )} \sqrt [5]{c}}\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{5+\sqrt{5}} \left (\left (1+\sqrt{5}\right ) \sqrt [5]{c}+\frac{4 \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{2 \sqrt{10} \sqrt [5]{c}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}-\frac{\log \left (\sqrt [5]{c}-\frac{\sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac{\left (1-\sqrt{5}\right ) \log \left (2 c^{2/5}+\frac{2 (b c-a d)^{2/5} x^2}{\left (a+b x^5\right )^{2/5}}+\frac{\sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}-\frac{\sqrt{5} \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}+\frac{\left (1+\sqrt{5}\right ) \log \left (2 c^{2/5}+\frac{2 (b c-a d)^{2/5} x^2}{\left (a+b x^5\right )^{2/5}}+\frac{\sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}+\frac{\sqrt{5} \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}\\ \end{align*}

Mathematica [C]  time = 0.0196107, size = 49, normalized size = 0.09 \[ \frac{x \, _2F_1\left (\frac{1}{5},1;\frac{6}{5};-\frac{(a d-b c) x^5}{c \left (b x^5+a\right )}\right )}{c \sqrt [5]{a+b x^5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Hypergeometric2F1[1/5, 1, 6/5, -(((-(b*c) + a*d)*x^5)/(c*(a + b*x^5)))])/(c*(a + b*x^5)^(1/5))

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Maple [F]  time = 0.427, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{5}+c}{\frac{1}{\sqrt [5]{b{x}^{5}+a}}}}\, dx \end{align*}

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. \begin{align*} \int \frac{1}{{\left (b x^{5} + a\right )}^{\frac{1}{5}}{\left (d x^{5} + c\right )}}\,{d x} \end{align*}

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. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

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: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [5]{a + b x^{5}} \left (c + d x^{5}\right )}\, dx \end{align*}

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]

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

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

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)