3.593 \(\int \frac{x^6 (a+b x^3)^{2/3}}{a d-b d x^3} \, dx\)

Optimal. Leaf size=264 \[ \frac{a^2 \log \left (a d-b d x^3\right )}{3 \sqrt [3]{2} b^{7/3} d}-\frac{a^2 \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^{7/3} d}+\frac{7 a^2 \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{9 b^{7/3} d}-\frac{14 a^2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} b^{7/3} d}+\frac{2^{2/3} a^2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} b^{7/3} d}-\frac{4 a x \left (a+b x^3\right )^{2/3}}{9 b^2 d}-\frac{x^4 \left (a+b x^3\right )^{2/3}}{6 b d} \]

[Out]

(-4*a*x*(a + b*x^3)^(2/3))/(9*b^2*d) - (x^4*(a + b*x^3)^(2/3))/(6*b*d) - (14*a^2*ArcTan[(1 + (2*b^(1/3)*x)/(a
+ b*x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]*b^(7/3)*d) + (2^(2/3)*a^2*ArcTan[(1 + (2*2^(1/3)*b^(1/3)*x)/(a + b*x^3)^(
1/3))/Sqrt[3]])/(Sqrt[3]*b^(7/3)*d) + (a^2*Log[a*d - b*d*x^3])/(3*2^(1/3)*b^(7/3)*d) - (a^2*Log[2^(1/3)*b^(1/3
)*x - (a + b*x^3)^(1/3)])/(2^(1/3)*b^(7/3)*d) + (7*a^2*Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)])/(9*b^(7/3)*d)

________________________________________________________________________________________

Rubi [C]  time = 0.0632764, antiderivative size = 66, normalized size of antiderivative = 0.25, 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} \[ \frac{x^7 \left (a+b x^3\right )^{2/3} F_1\left (\frac{7}{3};-\frac{2}{3},1;\frac{10}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{7 a d \left (\frac{b x^3}{a}+1\right )^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

Int[(x^6*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x]

[Out]

(x^7*(a + b*x^3)^(2/3)*AppellF1[7/3, -2/3, 1, 10/3, -((b*x^3)/a), (b*x^3)/a])/(7*a*d*(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^6 \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^6 \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^7 \left (a+b x^3\right )^{2/3} F_1\left (\frac{7}{3};-\frac{2}{3},1;\frac{10}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{7 a d \left (1+\frac{b x^3}{a}\right )^{2/3}}\\ \end{align*}

Mathematica [C]  time = 0.263719, size = 244, normalized size = 0.92 \[ \frac{2\ 2^{2/3} a^2 \sqrt [3]{a+b x^3} \left (\log \left (\frac{2^{2/3} b^{2/3} x^2}{\left (a x^3+b\right )^{2/3}}+\frac{\sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a x^3+b}}+1\right )-2 \log \left (1-\frac{\sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a x^3+b}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a x^3+b}}+1}{\sqrt{3}}\right )\right )+21 a b^{4/3} x^4 \sqrt [3]{\frac{b x^3}{a}+1} F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )-3 \sqrt [3]{b} \left (a+b x^3\right ) \left (8 a x+3 b x^4\right )}{54 b^{7/3} d \sqrt [3]{a+b x^3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^6*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x]

[Out]

(-3*b^(1/3)*(a + b*x^3)*(8*a*x + 3*b*x^4) + 21*a*b^(4/3)*x^4*(1 + (b*x^3)/a)^(1/3)*AppellF1[4/3, 1/3, 1, 7/3,
-((b*x^3)/a), (b*x^3)/a] + 2*2^(2/3)*a^2*(a + b*x^3)^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*2^(1/3)*b^(1/3)*x)/(b + a
*x^3)^(1/3))/Sqrt[3]] - 2*Log[1 - (2^(1/3)*b^(1/3)*x)/(b + a*x^3)^(1/3)] + Log[1 + (2^(2/3)*b^(2/3)*x^2)/(b +
a*x^3)^(2/3) + (2^(1/3)*b^(1/3)*x)/(b + a*x^3)^(1/3)]))/(54*b^(7/3)*d*(a + b*x^3)^(1/3))

________________________________________________________________________________________

Maple [F]  time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{6}}{-bd{x}^{3}+ad} \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^6*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x)

[Out]

int(x^6*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}} x^{6}}{b d x^{3} - a d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x, algorithm="maxima")

[Out]

-integrate((b*x^3 + a)^(2/3)*x^6/(b*d*x^3 - a*d), x)

________________________________________________________________________________________

Fricas [A]  time = 1.69421, size = 1875, normalized size = 7.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x, algorithm="fricas")

[Out]

[-1/54*(18*4^(1/3)*sqrt(3)*a^2*b*(-1/b)^(1/3)*arctan(-1/3*(sqrt(3)*x - 4^(1/3)*sqrt(3)*(b*x^3 + a)^(1/3)*(-1/b
)^(1/3))/x) - 42*sqrt(1/3)*a^2*b*sqrt(-1/b^(2/3))*log(3*b*x^3 - 3*(b*x^3 + a)^(1/3)*b^(2/3)*x^2 - 3*sqrt(1/3)*
(b^(4/3)*x^3 + (b*x^3 + a)^(1/3)*b*x^2 - 2*(b*x^3 + a)^(2/3)*b^(2/3)*x)*sqrt(-1/b^(2/3)) + 2*a) - 18*4^(1/3)*a
^2*b*(-1/b)^(1/3)*log(-(4^(2/3)*b*x*(-1/b)^(2/3) - 2*(b*x^3 + a)^(1/3))/x) + 9*4^(1/3)*a^2*b*(-1/b)^(1/3)*log(
-(2*4^(1/3)*b*x^2*(-1/b)^(1/3) - 4^(2/3)*(b*x^3 + a)^(1/3)*b*x*(-1/b)^(2/3) - 2*(b*x^3 + a)^(2/3))/x^2) - 28*a
^2*b^(2/3)*log(-(b^(1/3)*x - (b*x^3 + a)^(1/3))/x) + 14*a^2*b^(2/3)*log((b^(2/3)*x^2 + (b*x^3 + a)^(1/3)*b^(1/
3)*x + (b*x^3 + a)^(2/3))/x^2) + 3*(3*b^2*x^4 + 8*a*b*x)*(b*x^3 + a)^(2/3))/(b^3*d), -1/54*(18*4^(1/3)*sqrt(3)
*a^2*b*(-1/b)^(1/3)*arctan(-1/3*(sqrt(3)*x - 4^(1/3)*sqrt(3)*(b*x^3 + a)^(1/3)*(-1/b)^(1/3))/x) - 18*4^(1/3)*a
^2*b*(-1/b)^(1/3)*log(-(4^(2/3)*b*x*(-1/b)^(2/3) - 2*(b*x^3 + a)^(1/3))/x) + 9*4^(1/3)*a^2*b*(-1/b)^(1/3)*log(
-(2*4^(1/3)*b*x^2*(-1/b)^(1/3) - 4^(2/3)*(b*x^3 + a)^(1/3)*b*x*(-1/b)^(2/3) - 2*(b*x^3 + a)^(2/3))/x^2) - 84*s
qrt(1/3)*a^2*b^(2/3)*arctan(sqrt(1/3)*(b^(1/3)*x + 2*(b*x^3 + a)^(1/3))/(b^(1/3)*x)) - 28*a^2*b^(2/3)*log(-(b^
(1/3)*x - (b*x^3 + a)^(1/3))/x) + 14*a^2*b^(2/3)*log((b^(2/3)*x^2 + (b*x^3 + a)^(1/3)*b^(1/3)*x + (b*x^3 + a)^
(2/3))/x^2) + 3*(3*b^2*x^4 + 8*a*b*x)*(b*x^3 + a)^(2/3))/(b^3*d)]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{x^{6} \left (a + b x^{3}\right )^{\frac{2}{3}}}{- a + b x^{3}}\, dx}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(b*x**3+a)**(2/3)/(-b*d*x**3+a*d),x)

[Out]

-Integral(x**6*(a + b*x**3)**(2/3)/(-a + b*x**3), x)/d

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}} x^{6}}{b d x^{3} - a d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x, algorithm="giac")

[Out]

integrate(-(b*x^3 + a)^(2/3)*x^6/(b*d*x^3 - a*d), x)