3.709 \(\int \frac{x^6 (a+b x^3)^{4/3}}{c+d x^3} \, dx\)

Optimal. Leaf size=65 \[ \frac{a x^7 \sqrt [3]{a+b x^3} F_1\left (\frac{7}{3};-\frac{4}{3},1;\frac{10}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{7 c \sqrt [3]{\frac{b x^3}{a}+1}} \]

[Out]

(a*x^7*(a + b*x^3)^(1/3)*AppellF1[7/3, -4/3, 1, 10/3, -((b*x^3)/a), -((d*x^3)/c)])/(7*c*(1 + (b*x^3)/a)^(1/3))

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Rubi [A]  time = 0.0559373, antiderivative size = 65, 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} \[ \frac{a x^7 \sqrt [3]{a+b x^3} F_1\left (\frac{7}{3};-\frac{4}{3},1;\frac{10}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{7 c \sqrt [3]{\frac{b x^3}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Int[(x^6*(a + b*x^3)^(4/3))/(c + d*x^3),x]

[Out]

(a*x^7*(a + b*x^3)^(1/3)*AppellF1[7/3, -4/3, 1, 10/3, -((b*x^3)/a), -((d*x^3)/c)])/(7*c*(1 + (b*x^3)/a)^(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{x^6 \left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx &=\frac{\left (a \sqrt [3]{a+b x^3}\right ) \int \frac{x^6 \left (1+\frac{b x^3}{a}\right )^{4/3}}{c+d x^3} \, dx}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ &=\frac{a x^7 \sqrt [3]{a+b x^3} F_1\left (\frac{7}{3};-\frac{4}{3},1;\frac{10}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{7 c \sqrt [3]{1+\frac{b x^3}{a}}}\\ \end{align*}

Mathematica [B]  time = 0.603063, size = 343, normalized size = 5.28 \[ \frac{x \left (-\frac{x^3 \left (\frac{b x^3}{a}+1\right )^{2/3} \left (8 a^2 b c d^2+a^3 d^3-30 a b^2 c^2 d+20 b^3 c^3\right ) F_1\left (\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c}+\frac{16 a^2 c^2 \left (a^2 d^2-12 a b c d+10 b^2 c^2\right ) F_1\left (\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{\left (c+d x^3\right ) \left (x^3 \left (3 a d F_1\left (\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+2 b c F_1\left (\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )-4 a c F_1\left (\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )}+2 \left (a+b x^3\right ) \left (2 a^2 d^2+3 a b d \left (3 d x^3-8 c\right )+b^2 \left (20 c^2-8 c d x^3+5 d^2 x^6\right )\right )\right )}{80 b d^3 \left (a+b x^3\right )^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^6*(a + b*x^3)^(4/3))/(c + d*x^3),x]

[Out]

(x*(2*(a + b*x^3)*(2*a^2*d^2 + 3*a*b*d*(-8*c + 3*d*x^3) + b^2*(20*c^2 - 8*c*d*x^3 + 5*d^2*x^6)) - ((20*b^3*c^3
- 30*a*b^2*c^2*d + 8*a^2*b*c*d^2 + a^3*d^3)*x^3*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a)
, -((d*x^3)/c)])/c + (16*a^2*c^2*(10*b^2*c^2 - 12*a*b*c*d + a^2*d^2)*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a),
-((d*x^3)/c)])/((c + d*x^3)*(-4*a*c*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + x^3*(3*a*d*Appell
F1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*c*AppellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)
])))))/(80*b*d^3*(a + b*x^3)^(2/3))

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{6}}{d{x}^{3}+c} \left ( b{x}^{3}+a \right ) ^{{\frac{4}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(b*x^3+a)^(4/3)/(d*x^3+c),x)

[Out]

int(x^6*(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{{\left (b x^{3} + a\right )}^{\frac{4}{3}} x^{6}}{d x^{3} + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(4/3)*x^6/(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^6*(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="fricas")

[Out]

Timed out

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Sympy [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**6*(b*x**3+a)**(4/3)/(d*x**3+c),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{4}{3}} x^{6}}{d x^{3} + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(4/3)*x^6/(d*x^3 + c), x)