3.730 \(\int \frac{1}{x^2 \sqrt [3]{a+b x^3} (c+d x^3)} \, dx\)

Optimal. Leaf size=62 \[ -\frac{\sqrt [3]{\frac{b x^3}{a}+1} F_1\left (-\frac{1}{3};\frac{1}{3},1;\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c x \sqrt [3]{a+b x^3}} \]

[Out]

-(((1 + (b*x^3)/a)^(1/3)*AppellF1[-1/3, 1/3, 1, 2/3, -((b*x^3)/a), -((d*x^3)/c)])/(c*x*(a + b*x^3)^(1/3)))

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Rubi [A]  time = 0.0547463, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {511, 510} \[ -\frac{\sqrt [3]{\frac{b x^3}{a}+1} F_1\left (-\frac{1}{3};\frac{1}{3},1;\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c x \sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((1 + (b*x^3)/a)^(1/3)*AppellF1[-1/3, 1/3, 1, 2/3, -((b*x^3)/a), -((d*x^3)/c)])/(c*x*(a + b*x^3)^(1/3)))

Rule 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^2 \sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx &=\frac{\sqrt [3]{1+\frac{b x^3}{a}} \int \frac{1}{x^2 \sqrt [3]{1+\frac{b x^3}{a}} \left (c+d x^3\right )} \, dx}{\sqrt [3]{a+b x^3}}\\ &=-\frac{\sqrt [3]{1+\frac{b x^3}{a}} F_1\left (-\frac{1}{3};\frac{1}{3},1;\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c x \sqrt [3]{a+b x^3}}\\ \end{align*}

Mathematica [B]  time = 0.0965735, size = 141, normalized size = 2.27 \[ \frac{2 b d x^6 \sqrt [3]{\frac{b x^3}{a}+1} F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+5 x^3 \sqrt [3]{\frac{b x^3}{a}+1} (b c-a d) F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )-10 c \left (a+b x^3\right )}{10 a c^2 x \sqrt [3]{a+b x^3}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-10*c*(a + b*x^3) + 5*(b*c - a*d)*x^3*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), -((d*x^3
)/c)] + 2*b*d*x^6*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)])/(10*a*c^2*x*(a
+ b*x^3)^(1/3))

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( d{x}^{3}+c \right ) }{\frac{1}{\sqrt [3]{b{x}^{3}+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (d x^{3} + c\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x^3+a)^(1/3)/(d*x^3+c),x, algorithm="maxima")

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x^3+a)^(1/3)/(d*x^3+c),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \sqrt [3]{a + b x^{3}} \left (c + d x^{3}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (d x^{3} + c\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x^3+a)^(1/3)/(d*x^3+c),x, algorithm="giac")

[Out]

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