3.714 \(\int \frac{x^{14}}{\sqrt [3]{a+b x^3} (c+d x^3)} \, dx\)

Optimal. Leaf size=290 \[ \frac{\left (a+b x^3\right )^{5/3} \left (3 a^2 d^2+2 a b c d+b^2 c^2\right )}{5 b^4 d^3}-\frac{\left (a+b x^3\right )^{2/3} (a d+b c) \left (a^2 d^2+b^2 c^2\right )}{2 b^4 d^4}-\frac{\left (a+b x^3\right )^{8/3} (3 a d+b c)}{8 b^4 d^2}+\frac{\left (a+b x^3\right )^{11/3}}{11 b^4 d}+\frac{c^4 \log \left (c+d x^3\right )}{6 d^{14/3} \sqrt [3]{b c-a d}}-\frac{c^4 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{14/3} \sqrt [3]{b c-a d}}-\frac{c^4 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} d^{14/3} \sqrt [3]{b c-a d}} \]

[Out]

-((b*c + a*d)*(b^2*c^2 + a^2*d^2)*(a + b*x^3)^(2/3))/(2*b^4*d^4) + ((b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*(a + b*x
^3)^(5/3))/(5*b^4*d^3) - ((b*c + 3*a*d)*(a + b*x^3)^(8/3))/(8*b^4*d^2) + (a + b*x^3)^(11/3)/(11*b^4*d) - (c^4*
ArcTan[(1 - (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]])/(Sqrt[3]*d^(14/3)*(b*c - a*d)^(1/3)) +
(c^4*Log[c + d*x^3])/(6*d^(14/3)*(b*c - a*d)^(1/3)) - (c^4*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)])
/(2*d^(14/3)*(b*c - a*d)^(1/3))

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Rubi [A]  time = 0.316778, antiderivative size = 290, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {446, 88, 56, 617, 204, 31} \[ \frac{\left (a+b x^3\right )^{5/3} \left (3 a^2 d^2+2 a b c d+b^2 c^2\right )}{5 b^4 d^3}-\frac{\left (a+b x^3\right )^{2/3} (a d+b c) \left (a^2 d^2+b^2 c^2\right )}{2 b^4 d^4}-\frac{\left (a+b x^3\right )^{8/3} (3 a d+b c)}{8 b^4 d^2}+\frac{\left (a+b x^3\right )^{11/3}}{11 b^4 d}+\frac{c^4 \log \left (c+d x^3\right )}{6 d^{14/3} \sqrt [3]{b c-a d}}-\frac{c^4 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{14/3} \sqrt [3]{b c-a d}}-\frac{c^4 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} d^{14/3} \sqrt [3]{b c-a d}} \]

Antiderivative was successfully verified.

[In]

Int[x^14/((a + b*x^3)^(1/3)*(c + d*x^3)),x]

[Out]

-((b*c + a*d)*(b^2*c^2 + a^2*d^2)*(a + b*x^3)^(2/3))/(2*b^4*d^4) + ((b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*(a + b*x
^3)^(5/3))/(5*b^4*d^3) - ((b*c + 3*a*d)*(a + b*x^3)^(8/3))/(8*b^4*d^2) + (a + b*x^3)^(11/3)/(11*b^4*d) - (c^4*
ArcTan[(1 - (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]])/(Sqrt[3]*d^(14/3)*(b*c - a*d)^(1/3)) +
(c^4*Log[c + d*x^3])/(6*d^(14/3)*(b*c - a*d)^(1/3)) - (c^4*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)])
/(2*d^(14/3)*(b*c - a*d)^(1/3))

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 56

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, Simp
[Log[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] && Ne
gQ[(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^{14}}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^4}{\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) \left (-b^2 c^2-a^2 d^2\right )}{b^3 d^4 \sqrt [3]{a+b x}}+\frac{\left (b^2 c^2+2 a b c d+3 a^2 d^2\right ) (a+b x)^{2/3}}{b^3 d^3}+\frac{(-b c-3 a d) (a+b x)^{5/3}}{b^3 d^2}+\frac{(a+b x)^{8/3}}{b^3 d}+\frac{c^4}{d^4 \sqrt [3]{a+b x} (c+d x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{(b c+a d) \left (b^2 c^2+a^2 d^2\right ) \left (a+b x^3\right )^{2/3}}{2 b^4 d^4}+\frac{\left (b^2 c^2+2 a b c d+3 a^2 d^2\right ) \left (a+b x^3\right )^{5/3}}{5 b^4 d^3}-\frac{(b c+3 a d) \left (a+b x^3\right )^{8/3}}{8 b^4 d^2}+\frac{\left (a+b x^3\right )^{11/3}}{11 b^4 d}+\frac{c^4 \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a+b x} (c+d x)} \, dx,x,x^3\right )}{3 d^4}\\ &=-\frac{(b c+a d) \left (b^2 c^2+a^2 d^2\right ) \left (a+b x^3\right )^{2/3}}{2 b^4 d^4}+\frac{\left (b^2 c^2+2 a b c d+3 a^2 d^2\right ) \left (a+b x^3\right )^{5/3}}{5 b^4 d^3}-\frac{(b c+3 a d) \left (a+b x^3\right )^{8/3}}{8 b^4 d^2}+\frac{\left (a+b x^3\right )^{11/3}}{11 b^4 d}+\frac{c^4 \log \left (c+d x^3\right )}{6 d^{14/3} \sqrt [3]{b c-a d}}+\frac{c^4 \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^5}-\frac{c^4 \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^{14/3} \sqrt [3]{b c-a d}}\\ &=-\frac{(b c+a d) \left (b^2 c^2+a^2 d^2\right ) \left (a+b x^3\right )^{2/3}}{2 b^4 d^4}+\frac{\left (b^2 c^2+2 a b c d+3 a^2 d^2\right ) \left (a+b x^3\right )^{5/3}}{5 b^4 d^3}-\frac{(b c+3 a d) \left (a+b x^3\right )^{8/3}}{8 b^4 d^2}+\frac{\left (a+b x^3\right )^{11/3}}{11 b^4 d}+\frac{c^4 \log \left (c+d x^3\right )}{6 d^{14/3} \sqrt [3]{b c-a d}}-\frac{c^4 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{14/3} \sqrt [3]{b c-a d}}+\frac{c^4 \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^{14/3} \sqrt [3]{b c-a d}}\\ &=-\frac{(b c+a d) \left (b^2 c^2+a^2 d^2\right ) \left (a+b x^3\right )^{2/3}}{2 b^4 d^4}+\frac{\left (b^2 c^2+2 a b c d+3 a^2 d^2\right ) \left (a+b x^3\right )^{5/3}}{5 b^4 d^3}-\frac{(b c+3 a d) \left (a+b x^3\right )^{8/3}}{8 b^4 d^2}+\frac{\left (a+b x^3\right )^{11/3}}{11 b^4 d}-\frac{c^4 \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^{14/3} \sqrt [3]{b c-a d}}+\frac{c^4 \log \left (c+d x^3\right )}{6 d^{14/3} \sqrt [3]{b c-a d}}-\frac{c^4 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{14/3} \sqrt [3]{b c-a d}}\\ \end{align*}

Mathematica [C]  time = 0.203575, size = 157, normalized size = 0.54 \[ \frac{\left (a+b x^3\right )^{2/3} \left (\frac{9 a^2 b d^2 \left (6 d x^3-11 c\right )-81 a^3 d^3-3 a b^2 d \left (44 c^2-22 c d x^3+15 d^2 x^6\right )+b^3 \left (88 c^2 d x^3-220 c^3-55 c d^2 x^6+40 d^3 x^9\right )}{b^4}+\frac{220 c^4 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{d \left (b x^3+a\right )}{a d-b c}\right )}{b c-a d}\right )}{440 d^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^14/((a + b*x^3)^(1/3)*(c + d*x^3)),x]

[Out]

((a + b*x^3)^(2/3)*((-81*a^3*d^3 + 9*a^2*b*d^2*(-11*c + 6*d*x^3) - 3*a*b^2*d*(44*c^2 - 22*c*d*x^3 + 15*d^2*x^6
) + b^3*(-220*c^3 + 88*c^2*d*x^3 - 55*c*d^2*x^6 + 40*d^3*x^9))/b^4 + (220*c^4*Hypergeometric2F1[2/3, 1, 5/3, (
d*(a + b*x^3))/(-(b*c) + a*d)])/(b*c - a*d)))/(440*d^4)

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{14}}{d{x}^{3}+c}{\frac{1}{\sqrt [3]{b{x}^{3}+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^14/(b*x^3+a)^(1/3)/(d*x^3+c),x)

[Out]

int(x^14/(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^14/(b*x^3+a)^(1/3)/(d*x^3+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.96175, size = 2228, normalized size = 7.68 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14/(b*x^3+a)^(1/3)/(d*x^3+c),x, algorithm="fricas")

[Out]

[1/1320*(220*(-b*c*d^2 + a*d^3)^(2/3)*b^4*c^4*log((b*x^3 + a)^(2/3)*d^2 + (-b*c*d^2 + a*d^3)^(1/3)*(b*x^3 + a)
^(1/3)*d + (-b*c*d^2 + a*d^3)^(2/3)) - 440*(-b*c*d^2 + a*d^3)^(2/3)*b^4*c^4*log((b*x^3 + a)^(1/3)*d - (-b*c*d^
2 + a*d^3)^(1/3)) + 660*sqrt(1/3)*(b^5*c^5*d - a*b^4*c^4*d^2)*sqrt((-b*c*d^2 + a*d^3)^(1/3)/(b*c - a*d))*log((
2*b*d^2*x^3 - b*c*d + 3*a*d^2 + 3*sqrt(1/3)*(2*(-b*c*d^2 + a*d^3)^(2/3)*(b*x^3 + a)^(2/3) + (b*x^3 + a)^(1/3)*
(b*c*d - a*d^2) + (-b*c*d^2 + a*d^3)^(1/3)*(b*c - a*d))*sqrt((-b*c*d^2 + a*d^3)^(1/3)/(b*c - a*d)) - 3*(-b*c*d
^2 + a*d^3)^(2/3)*(b*x^3 + a)^(1/3))/(d*x^3 + c)) - 3*(220*b^4*c^4*d^2 - 88*a*b^3*c^3*d^3 - 33*a^2*b^2*c^2*d^4
- 18*a^3*b*c*d^5 - 81*a^4*d^6 - 40*(b^4*c*d^5 - a*b^3*d^6)*x^9 + 5*(11*b^4*c^2*d^4 - 2*a*b^3*c*d^5 - 9*a^2*b^
2*d^6)*x^6 - 2*(44*b^4*c^3*d^3 - 11*a*b^3*c^2*d^4 - 6*a^2*b^2*c*d^5 - 27*a^3*b*d^6)*x^3)*(b*x^3 + a)^(2/3))/(b
^5*c*d^6 - a*b^4*d^7), 1/1320*(220*(-b*c*d^2 + a*d^3)^(2/3)*b^4*c^4*log((b*x^3 + a)^(2/3)*d^2 + (-b*c*d^2 + a*
d^3)^(1/3)*(b*x^3 + a)^(1/3)*d + (-b*c*d^2 + a*d^3)^(2/3)) - 440*(-b*c*d^2 + a*d^3)^(2/3)*b^4*c^4*log((b*x^3 +
a)^(1/3)*d - (-b*c*d^2 + a*d^3)^(1/3)) + 1320*sqrt(1/3)*(b^5*c^5*d - a*b^4*c^4*d^2)*sqrt(-(-b*c*d^2 + a*d^3)^
(1/3)/(b*c - a*d))*arctan(sqrt(1/3)*(2*(b*x^3 + a)^(1/3)*d + (-b*c*d^2 + a*d^3)^(1/3))*sqrt(-(-b*c*d^2 + a*d^3
)^(1/3)/(b*c - a*d))/d) - 3*(220*b^4*c^4*d^2 - 88*a*b^3*c^3*d^3 - 33*a^2*b^2*c^2*d^4 - 18*a^3*b*c*d^5 - 81*a^4
*d^6 - 40*(b^4*c*d^5 - a*b^3*d^6)*x^9 + 5*(11*b^4*c^2*d^4 - 2*a*b^3*c*d^5 - 9*a^2*b^2*d^6)*x^6 - 2*(44*b^4*c^3
*d^3 - 11*a*b^3*c^2*d^4 - 6*a^2*b^2*c*d^5 - 27*a^3*b*d^6)*x^3)*(b*x^3 + a)^(2/3))/(b^5*c*d^6 - a*b^4*d^7)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{14}}{\sqrt [3]{a + b x^{3}} \left (c + d x^{3}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**14/(b*x**3+a)**(1/3)/(d*x**3+c),x)

[Out]

Integral(x**14/((a + b*x**3)**(1/3)*(c + d*x**3)), x)

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Giac [A]  time = 1.26883, size = 613, normalized size = 2.11 \begin{align*} -\frac{b^{48} c^{4} d^{7} \left (-\frac{b c - a d}{d}\right )^{\frac{2}{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^{49} c d^{11} - a b^{48} d^{12}\right )}} - \frac{{\left (-b c d^{2} + a d^{3}\right )}^{\frac{2}{3}} c^{4} \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 )}{\sqrt{3} b c d^{6} - \sqrt{3} a d^{7}} + \frac{{\left (-b c d^{2} + a d^{3}\right )}^{\frac{2}{3}} c^{4} \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 \,{\left (b c d^{6} - a d^{7}\right )}} - \frac{220 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} b^{43} c^{3} d^{7} - 88 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} b^{42} c^{2} d^{8} + 220 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} a b^{42} c^{2} d^{8} + 55 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}} b^{41} c d^{9} - 176 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} a b^{41} c d^{9} + 220 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} a^{2} b^{41} c d^{9} - 40 \,{\left (b x^{3} + a\right )}^{\frac{11}{3}} b^{40} d^{10} + 165 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}} a b^{40} d^{10} - 264 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} a^{2} b^{40} d^{10} + 220 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} a^{3} b^{40} d^{10}}{440 \, b^{44} d^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14/(b*x^3+a)^(1/3)/(d*x^3+c),x, algorithm="giac")

[Out]

-1/3*b^48*c^4*d^7*(-(b*c - a*d)/d)^(2/3)*log(abs((b*x^3 + a)^(1/3) - (-(b*c - a*d)/d)^(1/3)))/(b^49*c*d^11 - a
*b^48*d^12) - (-b*c*d^2 + a*d^3)^(2/3)*c^4*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))/(sqrt(3)*b*c*d^6 - sqrt(3)*a*d^7) + 1/6*(-b*c*d^2 + a*d^3)^(2/3)*c^4*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))/(b*c*d^6 - a*d^7) - 1/440*(220*(b*x^3
+ a)^(2/3)*b^43*c^3*d^7 - 88*(b*x^3 + a)^(5/3)*b^42*c^2*d^8 + 220*(b*x^3 + a)^(2/3)*a*b^42*c^2*d^8 + 55*(b*x^
3 + a)^(8/3)*b^41*c*d^9 - 176*(b*x^3 + a)^(5/3)*a*b^41*c*d^9 + 220*(b*x^3 + a)^(2/3)*a^2*b^41*c*d^9 - 40*(b*x^
3 + a)^(11/3)*b^40*d^10 + 165*(b*x^3 + a)^(8/3)*a*b^40*d^10 - 264*(b*x^3 + a)^(5/3)*a^2*b^40*d^10 + 220*(b*x^3
+ a)^(2/3)*a^3*b^40*d^10)/(b^44*d^11)