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

Optimal. Leaf size=512 \[ \frac {a^{7/3} \log \left (\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 \sqrt [3]{2} b^{8/3} d}-\frac {2^{2/3} a^{7/3} \log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 b^{8/3} d}-\frac {a^{7/3} \log \left (\frac {\sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}-\frac {2^{2/3} \sqrt [3]{b} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{2 \sqrt [3]{2} b^{8/3} d}+\frac {2^{2/3} a^{7/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^{8/3} d}+\frac {a^{7/3} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} b^{8/3} d}+\frac {a^{7/3} \log \left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a}\right )}{6 \sqrt [3]{2} b^{8/3} d}-\frac {19 a^2 x^2 \sqrt [3]{\frac {b x^3}{a}+1} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )}{28 b^2 d \sqrt [3]{a+b x^3}}-\frac {9 a x^2 \left (a+b x^3\right )^{2/3}}{28 b^2 d}-\frac {x^5 \left (a+b x^3\right )^{2/3}}{7 b d} \]

[Out]

-9/28*a*x^2*(b*x^3+a)^(2/3)/b^2/d-1/7*x^5*(b*x^3+a)^(2/3)/b/d-19/28*a^2*x^2*(1+b*x^3/a)^(1/3)*hypergeom([1/3,
2/3],[5/3],-b*x^3/a)/b^2/d/(b*x^3+a)^(1/3)+1/12*a^(7/3)*ln((a^(1/3)-b^(1/3)*x)^2*(a^(1/3)+b^(1/3)*x)/a)*2^(2/3
)/b^(8/3)/d+1/6*a^(7/3)*ln(1+2^(2/3)*(a^(1/3)+b^(1/3)*x)^2/(b*x^3+a)^(2/3)-2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+
a)^(1/3))*2^(2/3)/b^(8/3)/d-1/3*2^(2/3)*a^(7/3)*ln(1+2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))/b^(8/3)/d-1/
4*a^(7/3)*ln(b^(1/3)*(a^(1/3)+b^(1/3)*x)/a^(1/3)-2^(2/3)*b^(1/3)*(b*x^3+a)^(1/3)/a^(1/3))*2^(2/3)/b^(8/3)/d+1/
3*2^(2/3)*a^(7/3)*arctan(1/3*(1-2*2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*3^(1/2))/b^(8/3)/d*3^(1/2)+1/6*
a^(7/3)*arctan(1/3*(1+2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*3^(1/2))*2^(2/3)/b^(8/3)/d*3^(1/2)

________________________________________________________________________________________

Rubi [C]  time = 0.07, antiderivative size = 66, normalized size of antiderivative = 0.13, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {511, 510} \[ \frac {x^8 \left (a+b x^3\right )^{2/3} F_1\left (\frac {8}{3};-\frac {2}{3},1;\frac {11}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{8 a d \left (\frac {b x^3}{a}+1\right )^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

Int[(x^7*(a + b*x^3)^(2/3))/(a*d - b*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), (b*x^3)/a])/(8*a*d*(1 + (b*x^3)/a)^(2/3))

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 {x^7 \left (a+b x^3\right )^{2/3}}{a d-b 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}}{a d-b 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 {b x^3}{a}\right )}{8 a d \left (1+\frac {b x^3}{a}\right )^{2/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.11, size = 147, normalized size = 0.29 \[ \frac {45 a^2 x^2 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )-5 \left (9 a^2 x^2+13 a b x^5+4 b^2 x^8\right )+38 a b x^5 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {5}{3};\frac {1}{3},1;\frac {8}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{140 b^2 d \sqrt [3]{a+b x^3}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-5*(9*a^2*x^2 + 13*a*b*x^5 + 4*b^2*x^8) + 45*a^2*x^2*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^
3)/a), (b*x^3)/a] + 38*a*b*x^5*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), (b*x^3)/a])/(140
*b^2*d*(a + b*x^3)^(1/3))

________________________________________________________________________________________

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(x^7*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} x^{7}}{b d x^{3} - a d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.60, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x^{7}}{-b d \,x^{3}+a d}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} x^{7}}{b d x^{3} - a d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^7\,{\left (b\,x^3+a\right )}^{2/3}}{a\,d-b\,d\,x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {x^{7} \left (a + b x^{3}\right )^{\frac {2}{3}}}{- a + b x^{3}}\, dx}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________