3.766 \(\int \frac{1}{x^5 (a+b x^3)^{4/3} (c+d x^3)} \, dx\)

Optimal. Leaf size=67 \[ -\frac{\sqrt [3]{\frac{b x^3}{a}+1} F_1\left (-\frac{4}{3};\frac{4}{3},1;-\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{4 a c x^4 \sqrt [3]{a+b x^3}} \]

[Out]

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

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Rubi [A]  time = 0.0583748, antiderivative size = 67, 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} \[ -\frac{\sqrt [3]{\frac{b x^3}{a}+1} F_1\left (-\frac{4}{3};\frac{4}{3},1;-\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{4 a c x^4 \sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a + b*x^3)^(4/3)*(c + d*x^3)),x]

[Out]

-((1 + (b*x^3)/a)^(1/3)*AppellF1[-4/3, 4/3, 1, -1/3, -((b*x^3)/a), -((d*x^3)/c)])/(4*a*c*x^4*(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^5 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx &=\frac{\sqrt [3]{1+\frac{b x^3}{a}} \int \frac{1}{x^5 \left (1+\frac{b x^3}{a}\right )^{4/3} \left (c+d x^3\right )} \, dx}{a \sqrt [3]{a+b x^3}}\\ &=-\frac{\sqrt [3]{1+\frac{b x^3}{a}} F_1\left (-\frac{4}{3};\frac{4}{3},1;-\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{4 a c x^4 \sqrt [3]{a+b x^3}}\\ \end{align*}

Mathematica [B]  time = 0.284689, size = 264, normalized size = 3.94 \[ \frac{-2 b d x^9 \sqrt [3]{\frac{b x^3}{a}+1} \left (2 a^2 d^2+a b c d-5 b^2 c^2\right ) F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+5 x^6 \sqrt [3]{\frac{b x^3}{a}+1} \left (-2 a^2 b c d^2+2 a^3 d^3-a b^2 c^2 d+5 b^3 c^3\right ) F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+5 c \left (a^2 b \left (c^2+c d x^3+4 d^2 x^6\right )+a^3 d \left (4 d x^3-c\right )+a b^2 c x^3 \left (2 d x^3-5 c\right )-10 b^3 c^2 x^6\right )}{20 a^3 c^3 x^4 \sqrt [3]{a+b x^3} (a d-b c)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x^5*(a + b*x^3)^(4/3)*(c + d*x^3)),x]

[Out]

(5*c*(-10*b^3*c^2*x^6 + a*b^2*c*x^3*(-5*c + 2*d*x^3) + a^3*d*(-c + 4*d*x^3) + a^2*b*(c^2 + c*d*x^3 + 4*d^2*x^6
)) + 5*(5*b^3*c^3 - a*b^2*c^2*d - 2*a^2*b*c*d^2 + 2*a^3*d^3)*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*(-5*b^2*c^2 + a*b*c*d + 2*a^2*d^2)*x^9*(1 + (b*x^3)/a)^(1/3)*AppellF1[
5/3, 1/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)])/(20*a^3*c^3*(-(b*c) + a*d)*x^4*(a + b*x^3)^(1/3))

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Maple [F]  time = 0.066, 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{4}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(b*x^3+a)^(4/3)/(d*x^3+c),x)

[Out]

int(1/x^5/(b*x^3+a)^(4/3)/(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{4}{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(1/x^5/(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="maxima")

[Out]

integrate(1/((b*x^3 + a)^(4/3)*(d*x^3 + c)*x^5), 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^5/(b*x^3+a)^(4/3)/(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^{5} \left (a + b x^{3}\right )^{\frac{4}{3}} \left (c + d x^{3}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(b*x**3+a)**(4/3)/(d*x**3+c),x)

[Out]

Integral(1/(x**5*(a + b*x**3)**(4/3)*(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{4}{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(1/x^5/(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(4/3)*(d*x^3 + c)*x^5), x)