3.695 \(\int \frac{(a+b x^3)^{2/3}}{x^5 (c+d x^3)} \, dx\)

Optimal. Leaf size=64 \[ -\frac{\left (a+b x^3\right )^{2/3} F_1\left (-\frac{4}{3};-\frac{2}{3},1;-\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{4 c x^4 \left (\frac{b x^3}{a}+1\right )^{2/3}} \]

[Out]

-((a + b*x^3)^(2/3)*AppellF1[-4/3, -2/3, 1, -1/3, -((b*x^3)/a), -((d*x^3)/c)])/(4*c*x^4*(1 + (b*x^3)/a)^(2/3))

________________________________________________________________________________________

Rubi [A]  time = 0.0605682, antiderivative size = 64, 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} \[ -\frac{\left (a+b x^3\right )^{2/3} F_1\left (-\frac{4}{3};-\frac{2}{3},1;-\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{4 c x^4 \left (\frac{b x^3}{a}+1\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + b*x^3)^(2/3)*AppellF1[-4/3, -2/3, 1, -1/3, -((b*x^3)/a), -((d*x^3)/c)])/(4*c*x^4*(1 + (b*x^3)/a)^(2/3))

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{\left (a+b x^3\right )^{2/3}}{x^5 \left (c+d x^3\right )} \, dx &=\frac{\left (a+b x^3\right )^{2/3} \int \frac{\left (1+\frac{b x^3}{a}\right )^{2/3}}{x^5 \left (c+d x^3\right )} \, dx}{\left (1+\frac{b x^3}{a}\right )^{2/3}}\\ &=-\frac{\left (a+b x^3\right )^{2/3} F_1\left (-\frac{4}{3};-\frac{2}{3},1;-\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{4 c x^4 \left (1+\frac{b x^3}{a}\right )^{2/3}}\\ \end{align*}

Mathematica [B]  time = 0.156636, size = 181, normalized size = 2.83 \[ \frac{5 x^6 \sqrt [3]{\frac{b x^3}{a}+1} \left (2 a^2 d^2-4 a b c d+b^2 c^2\right ) F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+2 b d x^9 \sqrt [3]{\frac{b x^3}{a}+1} (b c-2 a d) F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )-5 c \left (a+b x^3\right ) \left (a \left (c-4 d x^3\right )+2 b c x^3\right )}{20 a c^3 x^4 \sqrt [3]{a+b x^3}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-5*c*(a + b*x^3)*(2*b*c*x^3 + a*(c - 4*d*x^3)) + 5*(b^2*c^2 - 4*a*b*c*d + 2*a^2*d^2)*x^6*(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*(b*c - 2*a*d)*x^9*(1 + (b*x^3)/a)^(1/3)*Appel
lF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)])/(20*a*c^3*x^4*(a + b*x^3)^(1/3))

________________________________________________________________________________________

Maple [F]  time = 0.06, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5} \left ( d{x}^{3}+c \right ) } \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{{\left (d x^{3} + c\right )} x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^5/(d*x^3+c),x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(2/3)/((d*x^3 + c)*x^5), x)

________________________________________________________________________________________

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((b*x^3+a)^(2/3)/x^5/(d*x^3+c),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{3}\right )^{\frac{2}{3}}}{x^{5} \left (c + d x^{3}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**(2/3)/x**5/(d*x**3+c),x)

[Out]

Integral((a + b*x**3)**(2/3)/(x**5*(c + d*x**3)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{{\left (d x^{3} + c\right )} x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^5/(d*x^3+c),x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(2/3)/((d*x^3 + c)*x^5), x)