3.672 \(\int \frac{x^6 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx\)

Optimal. Leaf size=64 \[ \frac{x^7 \sqrt [3]{a+b x^3} F_1\left (\frac{7}{3};-\frac{1}{3},1;\frac{10}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{7 c \sqrt [3]{\frac{b x^3}{a}+1}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0509833, 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} \[ \frac{x^7 \sqrt [3]{a+b x^3} F_1\left (\frac{7}{3};-\frac{1}{3},1;\frac{10}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{7 c \sqrt [3]{\frac{b x^3}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Int[(x^6*(a + b*x^3)^(1/3))/(c + d*x^3),x]

[Out]

(x^7*(a + b*x^3)^(1/3)*AppellF1[7/3, -1/3, 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 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx &=\frac{\sqrt [3]{a+b x^3} \int \frac{x^6 \sqrt [3]{1+\frac{b x^3}{a}}}{c+d x^3} \, dx}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ &=\frac{x^7 \sqrt [3]{a+b x^3} F_1\left (\frac{7}{3};-\frac{1}{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 = 0.542318, size = 281, normalized size = 4.39 \[ \frac{x \left (-\frac{x^3 \left (\frac{b x^3}{a}+1\right )^{2/3} \left (a^2 d^2+5 a b c d-10 b^2 c^2\right ) F_1\left (\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c}+\frac{16 a^2 c^2 (a d-5 b c) F_1\left (\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{\left (c+d x^3\right ) \left (x^3 \left (3 a d F_1\left (\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+2 b c F_1\left (\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )-4 a c F_1\left (\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )}+4 \left (a+b x^3\right ) \left (a d-5 b c+2 b d x^3\right )\right )}{40 b d^2 \left (a+b x^3\right )^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^6*(a + b*x^3)^(1/3))/(c + d*x^3),x]

[Out]

(x*(4*(a + b*x^3)*(-5*b*c + a*d + 2*b*d*x^3) - ((-10*b^2*c^2 + 5*a*b*c*d + a^2*d^2)*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*(-5*b*c + a*d)*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*AppellF1[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)])))))/(40*b*d^2*(a + b*x^3)^(2/3))

________________________________________________________________________________________

Maple [F]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{6}}{d{x}^{3}+c}\sqrt [3]{b{x}^{3}+a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(b*x^3+a)^(1/3)/(d*x^3+c),x)

[Out]

int(x^6*(b*x^3+a)^(1/3)/(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{1}{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)^(1/3)/(d*x^3+c),x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(1/3)*x^6/(d*x^3 + c), 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(x^6*(b*x^3+a)^(1/3)/(d*x^3+c),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6} \sqrt [3]{a + b x^{3}}}{c + d x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(b*x**3+a)**(1/3)/(d*x**3+c),x)

[Out]

Integral(x**6*(a + b*x**3)**(1/3)/(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{1}{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)^(1/3)/(d*x^3+c),x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(1/3)*x^6/(d*x^3 + c), x)