3.586 \(\int \frac{x^{11} (a+b x^3)^{2/3}}{a d-b d x^3} \, dx\)

Optimal. Leaf size=223 \[ -\frac{a^3 \left (a+b x^3\right )^{2/3}}{2 b^4 d}-\frac{a^2 \left (a+b x^3\right )^{5/3}}{5 b^4 d}+\frac{a^{11/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b^4 d}-\frac{a^{11/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^4 d}-\frac{2^{2/3} a^{11/3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^4 d}+\frac{a \left (a+b x^3\right )^{8/3}}{8 b^4 d}-\frac{\left (a+b x^3\right )^{11/3}}{11 b^4 d} \]

[Out]

-(a^3*(a + b*x^3)^(2/3))/(2*b^4*d) - (a^2*(a + b*x^3)^(5/3))/(5*b^4*d) + (a*(a + b*x^3)^(8/3))/(8*b^4*d) - (a
+ b*x^3)^(11/3)/(11*b^4*d) - (2^(2/3)*a^(11/3)*ArcTan[(a^(1/3) + 2^(2/3)*(a + b*x^3)^(1/3))/(Sqrt[3]*a^(1/3))]
)/(Sqrt[3]*b^4*d) + (a^(11/3)*Log[a - b*x^3])/(3*2^(1/3)*b^4*d) - (a^(11/3)*Log[2^(1/3)*a^(1/3) - (a + b*x^3)^
(1/3)])/(2^(1/3)*b^4*d)

________________________________________________________________________________________

Rubi [A]  time = 0.250387, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {446, 88, 50, 55, 617, 204, 31} \[ -\frac{a^3 \left (a+b x^3\right )^{2/3}}{2 b^4 d}-\frac{a^2 \left (a+b x^3\right )^{5/3}}{5 b^4 d}+\frac{a^{11/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b^4 d}-\frac{a^{11/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^4 d}-\frac{2^{2/3} a^{11/3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^4 d}+\frac{a \left (a+b x^3\right )^{8/3}}{8 b^4 d}-\frac{\left (a+b x^3\right )^{11/3}}{11 b^4 d} \]

Antiderivative was successfully verified.

[In]

Int[(x^11*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x]

[Out]

-(a^3*(a + b*x^3)^(2/3))/(2*b^4*d) - (a^2*(a + b*x^3)^(5/3))/(5*b^4*d) + (a*(a + b*x^3)^(8/3))/(8*b^4*d) - (a
+ b*x^3)^(11/3)/(11*b^4*d) - (2^(2/3)*a^(11/3)*ArcTan[(a^(1/3) + 2^(2/3)*(a + b*x^3)^(1/3))/(Sqrt[3]*a^(1/3))]
)/(Sqrt[3]*b^4*d) + (a^(11/3)*Log[a - b*x^3])/(3*2^(1/3)*b^4*d) - (a^(11/3)*Log[2^(1/3)*a^(1/3) - (a + b*x^3)^
(1/3)])/(2^(1/3)*b^4*d)

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 55

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ
[(b*c - a*d)/b]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{x^{11} \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^3 (a+b x)^{2/3}}{a d-b d x} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a^2 (a+b x)^{2/3}}{b^3 d}+\frac{a (a+b x)^{5/3}}{b^3 d}-\frac{(a+b x)^{8/3}}{b^3 d}+\frac{a^3 (a+b x)^{2/3}}{b^3 (a d-b d x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{a^2 \left (a+b x^3\right )^{5/3}}{5 b^4 d}+\frac{a \left (a+b x^3\right )^{8/3}}{8 b^4 d}-\frac{\left (a+b x^3\right )^{11/3}}{11 b^4 d}+\frac{a^3 \operatorname{Subst}\left (\int \frac{(a+b x)^{2/3}}{a d-b d x} \, dx,x,x^3\right )}{3 b^3}\\ &=-\frac{a^3 \left (a+b x^3\right )^{2/3}}{2 b^4 d}-\frac{a^2 \left (a+b x^3\right )^{5/3}}{5 b^4 d}+\frac{a \left (a+b x^3\right )^{8/3}}{8 b^4 d}-\frac{\left (a+b x^3\right )^{11/3}}{11 b^4 d}+\frac{\left (2 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a+b x} (a d-b d x)} \, dx,x,x^3\right )}{3 b^3}\\ &=-\frac{a^3 \left (a+b x^3\right )^{2/3}}{2 b^4 d}-\frac{a^2 \left (a+b x^3\right )^{5/3}}{5 b^4 d}+\frac{a \left (a+b x^3\right )^{8/3}}{8 b^4 d}-\frac{\left (a+b x^3\right )^{11/3}}{11 b^4 d}+\frac{a^{11/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b^4 d}+\frac{a^{11/3} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{2} \sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^4 d}-\frac{a^4 \operatorname{Subst}\left (\int \frac{1}{2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{b^4 d}\\ &=-\frac{a^3 \left (a+b x^3\right )^{2/3}}{2 b^4 d}-\frac{a^2 \left (a+b x^3\right )^{5/3}}{5 b^4 d}+\frac{a \left (a+b x^3\right )^{8/3}}{8 b^4 d}-\frac{\left (a+b x^3\right )^{11/3}}{11 b^4 d}+\frac{a^{11/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b^4 d}-\frac{a^{11/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^4 d}+\frac{\left (2^{2/3} a^{11/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{b^4 d}\\ &=-\frac{a^3 \left (a+b x^3\right )^{2/3}}{2 b^4 d}-\frac{a^2 \left (a+b x^3\right )^{5/3}}{5 b^4 d}+\frac{a \left (a+b x^3\right )^{8/3}}{8 b^4 d}-\frac{\left (a+b x^3\right )^{11/3}}{11 b^4 d}-\frac{2^{2/3} a^{11/3} \tan ^{-1}\left (\frac{1+\frac{2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} b^4 d}+\frac{a^{11/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b^4 d}-\frac{a^{11/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^4 d}\\ \end{align*}

Mathematica [A]  time = 0.252983, size = 163, normalized size = 0.73 \[ -\frac{3 \left (a+b x^3\right )^{2/3} \left (98 a^2 b x^3+293 a^3+65 a b^2 x^6+40 b^3 x^9\right )-220\ 2^{2/3} a^{11/3} \log \left (a-b x^3\right )+660\ 2^{2/3} a^{11/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )+440\ 2^{2/3} \sqrt{3} a^{11/3} \tan ^{-1}\left (\frac{\frac{2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )}{1320 b^4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^11*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x]

[Out]

-(3*(a + b*x^3)^(2/3)*(293*a^3 + 98*a^2*b*x^3 + 65*a*b^2*x^6 + 40*b^3*x^9) + 440*2^(2/3)*Sqrt[3]*a^(11/3)*ArcT
an[(1 + (2^(2/3)*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 220*2^(2/3)*a^(11/3)*Log[a - b*x^3] + 660*2^(2/3)*a^(1
1/3)*Log[2^(1/3)*a^(1/3) - (a + b*x^3)^(1/3)])/(1320*b^4*d)

________________________________________________________________________________________

Maple [F]  time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{11}}{-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^11*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x)

[Out]

int(x^11*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.60753, size = 559, normalized size = 2.51 \begin{align*} -\frac{440 \cdot 4^{\frac{1}{3}} \sqrt{3} \left (-a^{2}\right )^{\frac{1}{3}} a^{3} \arctan \left (\frac{4^{\frac{1}{3}} \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{1}{3}} - \sqrt{3} a}{3 \, a}\right ) + 220 \cdot 4^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{1}{3}} a^{3} \log \left (4^{\frac{2}{3}}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{2}{3}} + 2 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} a - 2 \cdot 4^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{1}{3}} a\right ) - 440 \cdot 4^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{1}{3}} a^{3} \log \left (-4^{\frac{2}{3}} \left (-a^{2}\right )^{\frac{2}{3}} + 2 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} a\right ) + 3 \,{\left (40 \, b^{3} x^{9} + 65 \, a b^{2} x^{6} + 98 \, a^{2} b x^{3} + 293 \, a^{3}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{1320 \, b^{4} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x, algorithm="fricas")

[Out]

-1/1320*(440*4^(1/3)*sqrt(3)*(-a^2)^(1/3)*a^3*arctan(1/3*(4^(1/3)*sqrt(3)*(b*x^3 + a)^(1/3)*(-a^2)^(1/3) - sqr
t(3)*a)/a) + 220*4^(1/3)*(-a^2)^(1/3)*a^3*log(4^(2/3)*(b*x^3 + a)^(1/3)*(-a^2)^(2/3) + 2*(b*x^3 + a)^(2/3)*a -
2*4^(1/3)*(-a^2)^(1/3)*a) - 440*4^(1/3)*(-a^2)^(1/3)*a^3*log(-4^(2/3)*(-a^2)^(2/3) + 2*(b*x^3 + a)^(1/3)*a) +
3*(40*b^3*x^9 + 65*a*b^2*x^6 + 98*a^2*b*x^3 + 293*a^3)*(b*x^3 + a)^(2/3))/(b^4*d)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x, algorithm="giac")

[Out]

sage0*x