3.731 \(\int \frac {1}{x^5 \sqrt [3]{a+b x^3} (c+d x^3)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 64, normalized size of antiderivative = 1.00, 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} \[ -\frac {\sqrt [3]{\frac {b x^3}{a}+1} F_1\left (-\frac {4}{3};\frac {1}{3},1;-\frac {1}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{4 c x^4 \sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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])

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])

Rubi steps

\begin {align*} \int \frac {1}{x^5 \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^5 \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 {4}{3};\frac {1}{3},1;-\frac {1}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{4 c x^4 \sqrt [3]{a+b x^3}}\\ \end {align*}

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Mathematica [B]  time = 0.22, size = 183, normalized size = 2.86 \[ \frac {5 x^6 \sqrt [3]{\frac {b x^3}{a}+1} \left (2 a^2 d^2-2 a b c d-b^2 c^2\right ) F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-2 b d x^9 \sqrt [3]{\frac {b x^3}{a}+1} (2 a d+b c) F_1\left (\frac {5}{3};\frac {1}{3},1;\frac {8}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+5 c \left (a+b x^3\right ) \left (-a c+4 a d x^3+2 b c x^3\right )}{20 a^2 c^3 x^4 \sqrt [3]{a+b x^3}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x^3+a)^(1/3)/(d*x^3+c),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x^{3} + c\right )} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(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^5), x)

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maple [F]  time = 0.59, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (d \,x^{3}+c \right ) x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x^{3} + c\right )} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(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^5), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^5\,{\left (b\,x^3+a\right )}^{1/3}\,\left (d\,x^3+c\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{5} \sqrt [3]{a + b x^{3}} \left (c + d x^{3}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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