3.659 \(\int \frac{x^8 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx\)

Optimal. Leaf size=220 \[ -\frac{\left (a+b x^3\right )^{4/3} (a d+b c)}{4 b^2 d^2}+\frac{\left (a+b x^3\right )^{7/3}}{7 b^2 d}+\frac{c^2 \sqrt [3]{a+b x^3}}{d^3}+\frac{c^2 \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 d^{10/3}}-\frac{c^2 \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{10/3}}+\frac{c^2 \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} d^{10/3}} \]

[Out]

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

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Rubi [A]  time = 0.258599, antiderivative size = 220, 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} \[ -\frac{\left (a+b x^3\right )^{4/3} (a d+b c)}{4 b^2 d^2}+\frac{\left (a+b x^3\right )^{7/3}}{7 b^2 d}+\frac{c^2 \sqrt [3]{a+b x^3}}{d^3}+\frac{c^2 \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 d^{10/3}}-\frac{c^2 \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{10/3}}+\frac{c^2 \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} d^{10/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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^8 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2 \sqrt [3]{a+b x}}{c+d x} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{(-b c-a d) \sqrt [3]{a+b x}}{b d^2}+\frac{(a+b x)^{4/3}}{b d}+\frac{c^2 \sqrt [3]{a+b x}}{d^2 (c+d x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{(b c+a d) \left (a+b x^3\right )^{4/3}}{4 b^2 d^2}+\frac{\left (a+b x^3\right )^{7/3}}{7 b^2 d}+\frac{c^2 \operatorname{Subst}\left (\int \frac{\sqrt [3]{a+b x}}{c+d x} \, dx,x,x^3\right )}{3 d^2}\\ &=\frac{c^2 \sqrt [3]{a+b x^3}}{d^3}-\frac{(b c+a d) \left (a+b x^3\right )^{4/3}}{4 b^2 d^2}+\frac{\left (a+b x^3\right )^{7/3}}{7 b^2 d}-\frac{\left (c^2 (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^3}\\ &=\frac{c^2 \sqrt [3]{a+b x^3}}{d^3}-\frac{(b c+a d) \left (a+b x^3\right )^{4/3}}{4 b^2 d^2}+\frac{\left (a+b x^3\right )^{7/3}}{7 b^2 d}+\frac{c^2 \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 d^{10/3}}-\frac{\left (c^2 \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^{10/3}}-\frac{\left (c^2 (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^{11/3}}\\ &=\frac{c^2 \sqrt [3]{a+b x^3}}{d^3}-\frac{(b c+a d) \left (a+b x^3\right )^{4/3}}{4 b^2 d^2}+\frac{\left (a+b x^3\right )^{7/3}}{7 b^2 d}+\frac{c^2 \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 d^{10/3}}-\frac{c^2 \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^{10/3}}-\frac{\left (c^2 \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^{10/3}}\\ &=\frac{c^2 \sqrt [3]{a+b x^3}}{d^3}-\frac{(b c+a d) \left (a+b x^3\right )^{4/3}}{4 b^2 d^2}+\frac{\left (a+b x^3\right )^{7/3}}{7 b^2 d}+\frac{c^2 \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^{10/3}}+\frac{c^2 \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 d^{10/3}}-\frac{c^2 \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^{10/3}}\\ \end{align*}

Mathematica [A]  time = 0.33104, size = 230, normalized size = 1.05 \[ \frac{-\frac{21 d \left (a+b x^3\right )^{4/3} (a d+b c)}{b^2}+\frac{12 d^2 \left (a+b x^3\right )^{7/3}}{b^2}+\frac{14 c^2 \sqrt [3]{b c-a d} \left (\log \left (-\sqrt [3]{d} \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )-2 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )\right )}{\sqrt [3]{d}}+84 c^2 \sqrt [3]{a+b x^3}}{84 d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^8*(a + b*x^3)^(1/3))/(c + d*x^3),x]

[Out]

(84*c^2*(a + b*x^3)^(1/3) - (21*d*(b*c + a*d)*(a + b*x^3)^(4/3))/b^2 + (12*d^2*(a + b*x^3)^(7/3))/b^2 + (14*c^
2*(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))/(84*d^3)

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Maple [F]  time = 0.045, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{8}}{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^8*(b*x^3+a)^(1/3)/(d*x^3+c),x)

[Out]

int(x^8*(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^8*(b*x^3+a)^(1/3)/(d*x^3+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.7464, size = 644, normalized size = 2.93 \begin{align*} -\frac{28 \, \sqrt{3} b^{2} c^{2} \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 ) + 14 \, b^{2} c^{2} \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 ) - 28 \, b^{2} c^{2} \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 (4 \, b^{2} d^{2} x^{6} + 28 \, b^{2} c^{2} - 7 \, a b c d - 3 \, a^{2} d^{2} -{\left (7 \, b^{2} c d - a b d^{2}\right )} x^{3}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{84 \, b^{2} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x^3+a)^(1/3)/(d*x^3+c),x, algorithm="fricas")

[Out]

-1/84*(28*sqrt(3)*b^2*c^2*(-(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)) + 14*b^2*c^2*(-(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)) - 28*b^2*c^2*(-(b*c - a*d)/d)^(1/3)*log((b*x^3 + a)^
(1/3) - (-(b*c - a*d)/d)^(1/3)) - 3*(4*b^2*d^2*x^6 + 28*b^2*c^2 - 7*a*b*c*d - 3*a^2*d^2 - (7*b^2*c*d - a*b*d^2
)*x^3)*(b*x^3 + a)^(1/3))/(b^2*d^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{8} \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**8*(b*x**3+a)**(1/3)/(d*x**3+c),x)

[Out]

Integral(x**8*(a + b*x**3)**(1/3)/(c + d*x**3), x)

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Giac [A]  time = 1.22617, size = 432, normalized size = 1.96 \begin{align*} \frac{{\left (b^{17} c^{3} d^{4} - a b^{16} c^{2} d^{5}\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^{17} c d^{7} - a b^{16} d^{8}\right )}} - \frac{\sqrt{3}{\left (-b c d^{2} + a d^{3}\right )}^{\frac{1}{3}} c^{2} \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^{4}} - \frac{{\left (-b c d^{2} + a d^{3}\right )}^{\frac{1}{3}} c^{2} \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^{4}} + \frac{28 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} b^{14} c^{2} d^{4} - 7 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} b^{13} c d^{5} + 4 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} b^{12} d^{6} - 7 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} a b^{12} d^{6}}{28 \, b^{14} d^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x^3+a)^(1/3)/(d*x^3+c),x, algorithm="giac")

[Out]

1/3*(b^17*c^3*d^4 - a*b^16*c^2*d^5)*(-(b*c - a*d)/d)^(1/3)*log(abs((b*x^3 + a)^(1/3) - (-(b*c - a*d)/d)^(1/3))
)/(b^17*c*d^7 - a*b^16*d^8) - 1/3*sqrt(3)*(-b*c*d^2 + a*d^3)^(1/3)*c^2*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^4 - 1/6*(-b*c*d^2 + a*d^3)^(1/3)*c^2*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^4 + 1/28*(28*(b*x^3 + a)^(1/3)*b^14*c
^2*d^4 - 7*(b*x^3 + a)^(4/3)*b^13*c*d^5 + 4*(b*x^3 + a)^(7/3)*b^12*d^6 - 7*(b*x^3 + a)^(4/3)*a*b^12*d^6)/(b^14
*d^7)