3.8.9 \(\int \frac {x^6 (a+b x^3)^{4/3}}{c+d x^3} \, dx\) [709]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 65, 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 = {525, 524} \begin {gather*} \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}} \end {gather*}

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 524

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)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], 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 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[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 {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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(343\) vs. \(2(65)=130\).
time = 8.25, size = 343, normalized size = 5.28 \begin {gather*} \frac {x \left (2 \left (a+b x^3\right ) \left (2 a^2 d^2+3 a b d \left (-8 c+3 d x^3\right )+b^2 \left (20 c^2-8 c d x^3+5 d^2 x^6\right )\right )-\frac {\left (20 b^3 c^3-30 a b^2 c^2 d+8 a^2 b c d^2+a^3 d^3\right ) x^3 \left (1+\frac {b x^3}{a}\right )^{2/3} 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 (10 b^2 c^2-12 a b c d+a^2 d^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 (-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 )+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 )\right )}\right )}{80 b d^3 \left (a+b x^3\right )^{2/3}} \end {gather*}

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 = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{6} \left (b \,x^{3}+a \right )^{\frac {4}{3}}}{d \,x^{3}+c}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} \left (a + b x^{3}\right )^{\frac {4}{3}}}{c + d x^{3}}\, dx \end {gather*}

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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^6\,{\left (b\,x^3+a\right )}^{4/3}}{d\,x^3+c} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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