3.658 \(\int \frac{x^{11} \sqrt [3]{a+b x^3}}{c+d x^3} \, dx\)

Optimal. Leaf size=264 \[ \frac{\left (a+b x^3\right )^{4/3} \left (a^2 d^2+a b c d+b^2 c^2\right )}{4 b^3 d^3}-\frac{\left (a+b x^3\right )^{7/3} (2 a d+b c)}{7 b^3 d^2}+\frac{\left (a+b x^3\right )^{10/3}}{10 b^3 d}-\frac{c^3 \sqrt [3]{a+b x^3}}{d^4}-\frac{c^3 \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 d^{13/3}}+\frac{c^3 \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{13/3}}-\frac{c^3 \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} d^{13/3}} \]

[Out]

-((c^3*(a + b*x^3)^(1/3))/d^4) + ((b^2*c^2 + a*b*c*d + a^2*d^2)*(a + b*x^3)^(4/3))/(4*b^3*d^3) - ((b*c + 2*a*d
)*(a + b*x^3)^(7/3))/(7*b^3*d^2) + (a + b*x^3)^(10/3)/(10*b^3*d) - (c^3*(b*c - a*d)^(1/3)*ArcTan[(1 - (2*d^(1/
3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]])/(Sqrt[3]*d^(13/3)) - (c^3*(b*c - a*d)^(1/3)*Log[c + d*x^3])
/(6*d^(13/3)) + (c^3*(b*c - a*d)^(1/3)*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)])/(2*d^(13/3))

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Rubi [A]  time = 0.386141, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {446, 88, 50, 58, 617, 204, 31} \[ \frac{\left (a+b x^3\right )^{4/3} \left (a^2 d^2+a b c d+b^2 c^2\right )}{4 b^3 d^3}-\frac{\left (a+b x^3\right )^{7/3} (2 a d+b c)}{7 b^3 d^2}+\frac{\left (a+b x^3\right )^{10/3}}{10 b^3 d}-\frac{c^3 \sqrt [3]{a+b x^3}}{d^4}-\frac{c^3 \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 d^{13/3}}+\frac{c^3 \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{13/3}}-\frac{c^3 \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} d^{13/3}} \]

Antiderivative was successfully verified.

[In]

Int[(x^11*(a + b*x^3)^(1/3))/(c + d*x^3),x]

[Out]

-((c^3*(a + b*x^3)^(1/3))/d^4) + ((b^2*c^2 + a*b*c*d + a^2*d^2)*(a + b*x^3)^(4/3))/(4*b^3*d^3) - ((b*c + 2*a*d
)*(a + b*x^3)^(7/3))/(7*b^3*d^2) + (a + b*x^3)^(10/3)/(10*b^3*d) - (c^3*(b*c - a*d)^(1/3)*ArcTan[(1 - (2*d^(1/
3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]])/(Sqrt[3]*d^(13/3)) - (c^3*(b*c - a*d)^(1/3)*Log[c + d*x^3])
/(6*d^(13/3)) + (c^3*(b*c - a*d)^(1/3)*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)])/(2*d^(13/3))

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 58

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, -Sim
p[Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (Dist[3/(2*b*q), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d
*x)^(1/3)], x] + Dist[3/(2*b*q^2), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x
] && NegQ[(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} \sqrt [3]{a+b x^3}}{c+d x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^3 \sqrt [3]{a+b x}}{c+d x} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) \sqrt [3]{a+b x}}{b^2 d^3}+\frac{(-b c-2 a d) (a+b x)^{4/3}}{b^2 d^2}+\frac{(a+b x)^{7/3}}{b^2 d}-\frac{c^3 \sqrt [3]{a+b x}}{d^3 (c+d x)}\right ) \, dx,x,x^3\right )\\ &=\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) \left (a+b x^3\right )^{4/3}}{4 b^3 d^3}-\frac{(b c+2 a d) \left (a+b x^3\right )^{7/3}}{7 b^3 d^2}+\frac{\left (a+b x^3\right )^{10/3}}{10 b^3 d}-\frac{c^3 \operatorname{Subst}\left (\int \frac{\sqrt [3]{a+b x}}{c+d x} \, dx,x,x^3\right )}{3 d^3}\\ &=-\frac{c^3 \sqrt [3]{a+b x^3}}{d^4}+\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) \left (a+b x^3\right )^{4/3}}{4 b^3 d^3}-\frac{(b c+2 a d) \left (a+b x^3\right )^{7/3}}{7 b^3 d^2}+\frac{\left (a+b x^3\right )^{10/3}}{10 b^3 d}+\frac{\left (c^3 (b c-a d)\right ) \operatorname{Subst}\left (\int \frac{1}{(a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )}{3 d^4}\\ &=-\frac{c^3 \sqrt [3]{a+b x^3}}{d^4}+\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) \left (a+b x^3\right )^{4/3}}{4 b^3 d^3}-\frac{(b c+2 a d) \left (a+b x^3\right )^{7/3}}{7 b^3 d^2}+\frac{\left (a+b x^3\right )^{10/3}}{10 b^3 d}-\frac{c^3 \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 d^{13/3}}+\frac{\left (c^3 \sqrt [3]{b c-a d}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt [3]{b c-a d}}{\sqrt [3]{d}}+x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 d^{13/3}}+\frac{\left (c^3 (b c-a d)^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{(b c-a d)^{2/3}}{d^{2/3}}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{d}}+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 d^{14/3}}\\ &=-\frac{c^3 \sqrt [3]{a+b x^3}}{d^4}+\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) \left (a+b x^3\right )^{4/3}}{4 b^3 d^3}-\frac{(b c+2 a d) \left (a+b x^3\right )^{7/3}}{7 b^3 d^2}+\frac{\left (a+b x^3\right )^{10/3}}{10 b^3 d}-\frac{c^3 \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 d^{13/3}}+\frac{c^3 \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{13/3}}+\frac{\left (c^3 \sqrt [3]{b c-a d}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}\right )}{d^{13/3}}\\ &=-\frac{c^3 \sqrt [3]{a+b x^3}}{d^4}+\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) \left (a+b x^3\right )^{4/3}}{4 b^3 d^3}-\frac{(b c+2 a d) \left (a+b x^3\right )^{7/3}}{7 b^3 d^2}+\frac{\left (a+b x^3\right )^{10/3}}{10 b^3 d}-\frac{c^3 \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} d^{13/3}}-\frac{c^3 \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 d^{13/3}}+\frac{c^3 \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{13/3}}\\ \end{align*}

Mathematica [A]  time = 0.534551, size = 270, normalized size = 1.02 \[ \frac{\frac{105 d \left (a+b x^3\right )^{4/3} \left (a^2 d^2+a b c d+b^2 c^2\right )}{b^3}-\frac{60 d^2 \left (a+b x^3\right )^{7/3} (2 a d+b c)}{b^3}+\frac{42 d^3 \left (a+b x^3\right )^{10/3}}{b^3}-\frac{70 c^3 \sqrt [3]{b c-a d} \left (\log \left (-\sqrt [3]{d} \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )-2 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )\right )}{\sqrt [3]{d}}-420 c^3 \sqrt [3]{a+b x^3}}{420 d^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^11*(a + b*x^3)^(1/3))/(c + d*x^3),x]

[Out]

(-420*c^3*(a + b*x^3)^(1/3) + (105*d*(b^2*c^2 + a*b*c*d + a^2*d^2)*(a + b*x^3)^(4/3))/b^3 - (60*d^2*(b*c + 2*a
*d)*(a + b*x^3)^(7/3))/b^3 + (42*d^3*(a + b*x^3)^(10/3))/b^3 - (70*c^3*(b*c - a*d)^(1/3)*(2*Sqrt[3]*ArcTan[(1
- (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]] - 2*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1
/3)] + Log[(b*c - a*d)^(2/3) - d^(1/3)*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3) + d^(2/3)*(a + b*x^3)^(2/3)]))/d^(1
/3))/(420*d^4)

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Maple [F]  time = 0.058, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{11}}{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^11*(b*x^3+a)^(1/3)/(d*x^3+c),x)

[Out]

int(x^11*(b*x^3+a)^(1/3)/(d*x^3+c),x)

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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)^(1/3)/(d*x^3+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.77748, size = 749, normalized size = 2.84 \begin{align*} -\frac{140 \, \sqrt{3} b^{3} c^{3} \left (\frac{b c - a d}{d}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} d \left (\frac{b c - a d}{d}\right )^{\frac{2}{3}} - \sqrt{3}{\left (b c - a d\right )}}{3 \,{\left (b c - a d\right )}}\right ) + 70 \, b^{3} c^{3} \left (\frac{b c - a d}{d}\right )^{\frac{1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (\frac{b c - a d}{d}\right )^{\frac{1}{3}} + \left (\frac{b c - a d}{d}\right )^{\frac{2}{3}}\right ) - 140 \, b^{3} c^{3} \left (\frac{b c - a d}{d}\right )^{\frac{1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} + \left (\frac{b c - a d}{d}\right )^{\frac{1}{3}}\right ) - 3 \,{\left (14 \, b^{3} d^{3} x^{9} - 2 \,{\left (10 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{6} - 140 \, b^{3} c^{3} + 35 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} + 9 \, a^{3} d^{3} +{\left (35 \, b^{3} c^{2} d - 5 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3}\right )} x^{3}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{420 \, b^{3} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^3+a)^(1/3)/(d*x^3+c),x, algorithm="fricas")

[Out]

-1/420*(140*sqrt(3)*b^3*c^3*((b*c - a*d)/d)^(1/3)*arctan(-1/3*(2*sqrt(3)*(b*x^3 + a)^(1/3)*d*((b*c - a*d)/d)^(
2/3) - sqrt(3)*(b*c - a*d))/(b*c - a*d)) + 70*b^3*c^3*((b*c - a*d)/d)^(1/3)*log((b*x^3 + a)^(2/3) - (b*x^3 + a
)^(1/3)*((b*c - a*d)/d)^(1/3) + ((b*c - a*d)/d)^(2/3)) - 140*b^3*c^3*((b*c - a*d)/d)^(1/3)*log((b*x^3 + a)^(1/
3) + ((b*c - a*d)/d)^(1/3)) - 3*(14*b^3*d^3*x^9 - 2*(10*b^3*c*d^2 - a*b^2*d^3)*x^6 - 140*b^3*c^3 + 35*a*b^2*c^
2*d + 15*a^2*b*c*d^2 + 9*a^3*d^3 + (35*b^3*c^2*d - 5*a*b^2*c*d^2 - 3*a^2*b*d^3)*x^3)*(b*x^3 + a)^(1/3))/(b^3*d
^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{11} \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**11*(b*x**3+a)**(1/3)/(d*x**3+c),x)

[Out]

Integral(x**11*(a + b*x**3)**(1/3)/(c + d*x**3), x)

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Giac [A]  time = 1.26662, size = 512, normalized size = 1.94 \begin{align*} -\frac{{\left (b^{34} c^{4} d^{6} - a b^{33} c^{3} d^{7}\right )} \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}} \log \left ({\left |{\left (b x^{3} + a\right )}^{\frac{1}{3}} - \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b^{34} c d^{10} - a b^{33} d^{11}\right )}} + \frac{\sqrt{3}{\left (-b c d^{2} + a d^{3}\right )}^{\frac{1}{3}} c^{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} + \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}}}\right )}{3 \, d^{5}} + \frac{{\left (-b c d^{2} + a d^{3}\right )}^{\frac{1}{3}} c^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}} + \left (-\frac{b c - a d}{d}\right )^{\frac{2}{3}}\right )}{6 \, d^{5}} - \frac{140 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} b^{30} c^{3} d^{6} - 35 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} b^{29} c^{2} d^{7} + 20 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} b^{28} c d^{8} - 35 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} a b^{28} c d^{8} - 14 \,{\left (b x^{3} + a\right )}^{\frac{10}{3}} b^{27} d^{9} + 40 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} a b^{27} d^{9} - 35 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} a^{2} b^{27} d^{9}}{140 \, b^{30} d^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^3+a)^(1/3)/(d*x^3+c),x, algorithm="giac")

[Out]

-1/3*(b^34*c^4*d^6 - a*b^33*c^3*d^7)*(-(b*c - a*d)/d)^(1/3)*log(abs((b*x^3 + a)^(1/3) - (-(b*c - a*d)/d)^(1/3)
))/(b^34*c*d^10 - a*b^33*d^11) + 1/3*sqrt(3)*(-b*c*d^2 + a*d^3)^(1/3)*c^3*arctan(1/3*sqrt(3)*(2*(b*x^3 + a)^(1
/3) + (-(b*c - a*d)/d)^(1/3))/(-(b*c - a*d)/d)^(1/3))/d^5 + 1/6*(-b*c*d^2 + a*d^3)^(1/3)*c^3*log((b*x^3 + a)^(
2/3) + (b*x^3 + a)^(1/3)*(-(b*c - a*d)/d)^(1/3) + (-(b*c - a*d)/d)^(2/3))/d^5 - 1/140*(140*(b*x^3 + a)^(1/3)*b
^30*c^3*d^6 - 35*(b*x^3 + a)^(4/3)*b^29*c^2*d^7 + 20*(b*x^3 + a)^(7/3)*b^28*c*d^8 - 35*(b*x^3 + a)^(4/3)*a*b^2
8*c*d^8 - 14*(b*x^3 + a)^(10/3)*b^27*d^9 + 40*(b*x^3 + a)^(7/3)*a*b^27*d^9 - 35*(b*x^3 + a)^(4/3)*a^2*b^27*d^9
)/(b^30*d^10)