3.691 \(\int \frac{x^7 (a+b x^3)^{2/3}}{c+d x^3} \, dx\)

Optimal. Leaf size=64 \[ \frac{x^8 \left (a+b x^3\right )^{2/3} F_1\left (\frac{8}{3};-\frac{2}{3},1;\frac{11}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{8 c \left (\frac{b x^3}{a}+1\right )^{2/3}} \]

[Out]

(x^8*(a + b*x^3)^(2/3)*AppellF1[8/3, -2/3, 1, 11/3, -((b*x^3)/a), -((d*x^3)/c)])/(8*c*(1 + (b*x^3)/a)^(2/3))

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Rubi [A]  time = 0.0579732, 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} \[ \frac{x^8 \left (a+b x^3\right )^{2/3} F_1\left (\frac{8}{3};-\frac{2}{3},1;\frac{11}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{8 c \left (\frac{b x^3}{a}+1\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[(x^7*(a + b*x^3)^(2/3))/(c + d*x^3),x]

[Out]

(x^8*(a + b*x^3)^(2/3)*AppellF1[8/3, -2/3, 1, 11/3, -((b*x^3)/a), -((d*x^3)/c)])/(8*c*(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{x^7 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx &=\frac{\left (a+b x^3\right )^{2/3} \int \frac{x^7 \left (1+\frac{b x^3}{a}\right )^{2/3}}{c+d x^3} \, dx}{\left (1+\frac{b x^3}{a}\right )^{2/3}}\\ &=\frac{x^8 \left (a+b x^3\right )^{2/3} F_1\left (\frac{8}{3};-\frac{2}{3},1;\frac{11}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{8 c \left (1+\frac{b x^3}{a}\right )^{2/3}}\\ \end{align*}

Mathematica [B]  time = 0.200559, size = 181, normalized size = 2.83 \[ \frac{x^2 \left (-2 x^3 \sqrt [3]{\frac{b x^3}{a}+1} \left (2 a^2 d^2+7 a b c d-14 b^2 c^2\right ) F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+5 a c \sqrt [3]{\frac{b x^3}{a}+1} (7 b c-2 a d) F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+5 c \left (a+b x^3\right ) \left (2 a d-7 b c+4 b d x^3\right )\right )}{140 b c d^2 \sqrt [3]{a+b x^3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^7*(a + b*x^3)^(2/3))/(c + d*x^3),x]

[Out]

(x^2*(5*c*(a + b*x^3)*(-7*b*c + 2*a*d + 4*b*d*x^3) + 5*a*c*(7*b*c - 2*a*d)*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3,
1/3, 1, 5/3, -((b*x^3)/a), -((d*x^3)/c)] - 2*(-14*b^2*c^2 + 7*a*b*c*d + 2*a^2*d^2)*x^3*(1 + (b*x^3)/a)^(1/3)*
AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)]))/(140*b*c*d^2*(a + b*x^3)^(1/3))

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{7}}{d{x}^{3}+c} \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(b*x^3+a)^(2/3)/(d*x^3+c),x)

[Out]

int(x^7*(b*x^3+a)^(2/3)/(d*x^3+c),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}} x^{7}}{d x^{3} + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^3+a)^(2/3)/(d*x^3+c),x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(2/3)*x^7/(d*x^3 + c), 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(x^7*(b*x^3+a)^(2/3)/(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{x^{7} \left (a + b x^{3}\right )^{\frac{2}{3}}}{c + d x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(b*x**3+a)**(2/3)/(d*x**3+c),x)

[Out]

Integral(x**7*(a + b*x**3)**(2/3)/(c + d*x**3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}} x^{7}}{d x^{3} + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^3+a)^(2/3)/(d*x^3+c),x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(2/3)*x^7/(d*x^3 + c), x)