3.747 \(\int \frac{x^{14}}{(a+b x^3)^{4/3} (c+d x^3)} \, dx\)

Optimal. Leaf size=347 \[ \frac{\left (a+b x^3\right )^{2/3} \left (a^2 d^2+a b c d+b^2 c^2\right )}{2 b^4 d^3}-\frac{a^4}{b^4 \sqrt [3]{a+b x^3} (b c-a d)}+\frac{a^2 \left (a+b x^3\right )^{2/3}}{2 b^4 d}+\frac{a \left (a+b x^3\right )^{2/3} (a d+b c)}{2 b^4 d^2}-\frac{\left (a+b x^3\right )^{5/3} (a d+b c)}{5 b^4 d^2}-\frac{2 a \left (a+b x^3\right )^{5/3}}{5 b^4 d}+\frac{\left (a+b x^3\right )^{8/3}}{8 b^4 d}-\frac{c^4 \log \left (c+d x^3\right )}{6 d^{11/3} (b c-a d)^{4/3}}+\frac{c^4 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{11/3} (b c-a d)^{4/3}}+\frac{c^4 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} d^{11/3} (b c-a d)^{4/3}} \]

[Out]

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

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Rubi [A]  time = 0.440589, antiderivative size = 347, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {446, 87, 43, 56, 617, 204, 31} \[ \frac{\left (a+b x^3\right )^{2/3} \left (a^2 d^2+a b c d+b^2 c^2\right )}{2 b^4 d^3}-\frac{a^4}{b^4 \sqrt [3]{a+b x^3} (b c-a d)}+\frac{a^2 \left (a+b x^3\right )^{2/3}}{2 b^4 d}+\frac{a \left (a+b x^3\right )^{2/3} (a d+b c)}{2 b^4 d^2}-\frac{\left (a+b x^3\right )^{5/3} (a d+b c)}{5 b^4 d^2}-\frac{2 a \left (a+b x^3\right )^{5/3}}{5 b^4 d}+\frac{\left (a+b x^3\right )^{8/3}}{8 b^4 d}-\frac{c^4 \log \left (c+d x^3\right )}{6 d^{11/3} (b c-a d)^{4/3}}+\frac{c^4 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{11/3} (b c-a d)^{4/3}}+\frac{c^4 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} d^{11/3} (b c-a d)^{4/3}} \]

Antiderivative was successfully verified.

[In]

Int[x^14/((a + b*x^3)^(4/3)*(c + d*x^3)),x]

[Out]

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

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 87

Int[(((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_))/((a_.) + (b_.)*(x_)), x_Symbol] :> Int[ExpandIntegr
and[(e + f*x)^FractionalPart[p], ((c + d*x)^n*(e + f*x)^IntegerPart[p])/(a + b*x), x], x] /; FreeQ[{a, b, c, d
, e, f}, x] && IGtQ[n, 0] && LtQ[p, -1] && FractionQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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}}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^4}{(a+b x)^{4/3} (c+d x)} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^4}{b^3 (b c-a d) (a+b x)^{4/3}}+\frac{b^2 c^2+a b c d+a^2 d^2}{b^3 d^3 \sqrt [3]{a+b x}}-\frac{(b c+a d) x}{b^2 d^2 \sqrt [3]{a+b x}}+\frac{x^2}{b d \sqrt [3]{a+b x}}+\frac{c^4}{d^3 (-b c+a d) \sqrt [3]{a+b x} (c+d x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{a^4}{b^4 (b c-a d) \sqrt [3]{a+b x^3}}+\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) \left (a+b x^3\right )^{2/3}}{2 b^4 d^3}+\frac{\operatorname{Subst}\left (\int \frac{x^2}{\sqrt [3]{a+b x}} \, dx,x,x^3\right )}{3 b 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^3 (b c-a d)}-\frac{(b c+a d) \operatorname{Subst}\left (\int \frac{x}{\sqrt [3]{a+b x}} \, dx,x,x^3\right )}{3 b^2 d^2}\\ &=-\frac{a^4}{b^4 (b c-a d) \sqrt [3]{a+b x^3}}+\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) \left (a+b x^3\right )^{2/3}}{2 b^4 d^3}-\frac{c^4 \log \left (c+d x^3\right )}{6 d^{11/3} (b c-a d)^{4/3}}+\frac{\operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 \sqrt [3]{a+b x}}-\frac{2 a (a+b x)^{2/3}}{b^2}+\frac{(a+b x)^{5/3}}{b^2}\right ) \, dx,x,x^3\right )}{3 b d}+\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^{11/3} (b c-a d)^{4/3}}-\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^4 (b c-a d)}-\frac{(b c+a d) \operatorname{Subst}\left (\int \left (-\frac{a}{b \sqrt [3]{a+b x}}+\frac{(a+b x)^{2/3}}{b}\right ) \, dx,x,x^3\right )}{3 b^2 d^2}\\ &=-\frac{a^4}{b^4 (b c-a d) \sqrt [3]{a+b x^3}}+\frac{a^2 \left (a+b x^3\right )^{2/3}}{2 b^4 d}+\frac{a (b c+a d) \left (a+b x^3\right )^{2/3}}{2 b^4 d^2}+\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) \left (a+b x^3\right )^{2/3}}{2 b^4 d^3}-\frac{2 a \left (a+b x^3\right )^{5/3}}{5 b^4 d}-\frac{(b c+a d) \left (a+b x^3\right )^{5/3}}{5 b^4 d^2}+\frac{\left (a+b x^3\right )^{8/3}}{8 b^4 d}-\frac{c^4 \log \left (c+d x^3\right )}{6 d^{11/3} (b c-a d)^{4/3}}+\frac{c^4 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{11/3} (b c-a d)^{4/3}}-\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^{11/3} (b c-a d)^{4/3}}\\ &=-\frac{a^4}{b^4 (b c-a d) \sqrt [3]{a+b x^3}}+\frac{a^2 \left (a+b x^3\right )^{2/3}}{2 b^4 d}+\frac{a (b c+a d) \left (a+b x^3\right )^{2/3}}{2 b^4 d^2}+\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) \left (a+b x^3\right )^{2/3}}{2 b^4 d^3}-\frac{2 a \left (a+b x^3\right )^{5/3}}{5 b^4 d}-\frac{(b c+a d) \left (a+b x^3\right )^{5/3}}{5 b^4 d^2}+\frac{\left (a+b x^3\right )^{8/3}}{8 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^{11/3} (b c-a d)^{4/3}}-\frac{c^4 \log \left (c+d x^3\right )}{6 d^{11/3} (b c-a d)^{4/3}}+\frac{c^4 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{11/3} (b c-a d)^{4/3}}\\ \end{align*}

Mathematica [C]  time = 0.213262, size = 157, normalized size = 0.45 \[ \frac{\frac{9 a^2 b d^2 \left (8 c+3 d x^3\right )+81 a^3 d^3+3 a b^2 d \left (20 c^2+8 c d x^3-3 d^2 x^6\right )+b^3 \left (20 c^2 d x^3+40 c^3-8 c d^2 x^6+5 d^3 x^9\right )}{b^4}-\frac{40 c^4 \, _2F_1\left (-\frac{1}{3},1;\frac{2}{3};\frac{d \left (b x^3+a\right )}{a d-b c}\right )}{b c-a d}}{40 d^4 \sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^14/((a + b*x^3)^(4/3)*(c + d*x^3)),x]

[Out]

((81*a^3*d^3 + 9*a^2*b*d^2*(8*c + 3*d*x^3) + 3*a*b^2*d*(20*c^2 + 8*c*d*x^3 - 3*d^2*x^6) + b^3*(40*c^3 + 20*c^2
*d*x^3 - 8*c*d^2*x^6 + 5*d^3*x^9))/b^4 - (40*c^4*Hypergeometric2F1[-1/3, 1, 2/3, (d*(a + b*x^3))/(-(b*c) + a*d
)])/(b*c - a*d))/(40*d^4*(a + b*x^3)^(1/3))

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{14}}{d{x}^{3}+c} \left ( b{x}^{3}+a \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^14/(b*x^3+a)^(4/3)/(d*x^3+c),x)

[Out]

int(x^14/(b*x^3+a)^(4/3)/(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)^(4/3)/(d*x^3+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.24325, size = 2768, normalized size = 7.98 \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)^(4/3)/(d*x^3+c),x, algorithm="fricas")

[Out]

[-1/120*(60*sqrt(1/3)*(a*b^5*c^5*d - a^2*b^4*c^4*d^2 + (b^6*c^5*d - a*b^5*c^4*d^2)*x^3)*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)) + 20*(b^5*c^4*x^3 + a*b^4*c^4)*(-b
*c*d^2 + a*d^3)^(2/3)*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)) - 40*(b^5*c^4*x^3 + a*b^4*c^4)*(-b*c*d^2 + a*d^3)^(2/3)*log((b*x^3 + a)^(1/3)*d - (-b*c*d^2 + a*d
^3)^(1/3)) - 3*(20*a*b^4*c^4*d^2 - 8*a^2*b^3*c^3*d^3 - 3*a^3*b^2*c^2*d^4 - 90*a^4*b*c*d^5 + 81*a^5*d^6 + 5*(b^
5*c^2*d^4 - 2*a*b^4*c*d^5 + a^2*b^3*d^6)*x^9 - (8*b^5*c^3*d^3 - 7*a*b^4*c^2*d^4 - 10*a^2*b^3*c*d^5 + 9*a^3*b^2
*d^6)*x^6 + (20*b^5*c^4*d^2 - 16*a*b^4*c^3*d^3 - a^2*b^3*c^2*d^4 - 30*a^3*b^2*c*d^5 + 27*a^4*b*d^6)*x^3)*(b*x^
3 + a)^(2/3))/(a*b^6*c^2*d^5 - 2*a^2*b^5*c*d^6 + a^3*b^4*d^7 + (b^7*c^2*d^5 - 2*a*b^6*c*d^6 + a^2*b^5*d^7)*x^3
), -1/120*(120*sqrt(1/3)*(a*b^5*c^5*d - a^2*b^4*c^4*d^2 + (b^6*c^5*d - a*b^5*c^4*d^2)*x^3)*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) + 20*(b^5*c^4*x^3 + a*b^4*c^4)*(-b*c*d^2 + a*d^3)^(2/3)*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)) - 40*(b^5*c^4*x^3 + a*b^4*c^4)*(-
b*c*d^2 + a*d^3)^(2/3)*log((b*x^3 + a)^(1/3)*d - (-b*c*d^2 + a*d^3)^(1/3)) - 3*(20*a*b^4*c^4*d^2 - 8*a^2*b^3*c
^3*d^3 - 3*a^3*b^2*c^2*d^4 - 90*a^4*b*c*d^5 + 81*a^5*d^6 + 5*(b^5*c^2*d^4 - 2*a*b^4*c*d^5 + a^2*b^3*d^6)*x^9 -
(8*b^5*c^3*d^3 - 7*a*b^4*c^2*d^4 - 10*a^2*b^3*c*d^5 + 9*a^3*b^2*d^6)*x^6 + (20*b^5*c^4*d^2 - 16*a*b^4*c^3*d^3
- a^2*b^3*c^2*d^4 - 30*a^3*b^2*c*d^5 + 27*a^4*b*d^6)*x^3)*(b*x^3 + a)^(2/3))/(a*b^6*c^2*d^5 - 2*a^2*b^5*c*d^6
+ a^3*b^4*d^7 + (b^7*c^2*d^5 - 2*a*b^6*c*d^6 + a^2*b^5*d^7)*x^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{14}}{\left (a + b x^{3}\right )^{\frac{4}{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)**(4/3)/(d*x**3+c),x)

[Out]

Integral(x**14/((a + b*x**3)**(4/3)*(c + d*x**3)), x)

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Giac [A]  time = 1.22618, size = 582, normalized size = 1.68 \begin{align*} \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^{2} c^{2} d^{5} - 2 \, \sqrt{3} a b c d^{6} + \sqrt{3} a^{2} 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^{2} c^{2} d^{5} - 2 \, a b c d^{6} + a^{2} d^{7}\right )}} + \frac{c^{4} \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^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )}} - \frac{a^{4}}{{\left (b^{5} c - a b^{4} d\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}} + \frac{20 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} b^{30} c^{2} d^{5} - 8 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} b^{29} c d^{6} + 40 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} a b^{29} c d^{6} + 5 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}} b^{28} d^{7} - 24 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} a b^{28} d^{7} + 60 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} a^{2} b^{28} d^{7}}{40 \, b^{32} d^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14/(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="giac")

[Out]

(-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^2*c^2*d^5 - 2*sqrt(3)*a*b*c*d^6 + sqrt(3)*a^2*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^2*c^2*d^5 - 2*a*b*c*
d^6 + a^2*d^7) + 1/3*c^4*(-(b*c - a*d)/d)^(2/3)*log(abs((b*x^3 + a)^(1/3) - (-(b*c - a*d)/d)^(1/3)))/(b^2*c^2*
d^3 - 2*a*b*c*d^4 + a^2*d^5) - a^4/((b^5*c - a*b^4*d)*(b*x^3 + a)^(1/3)) + 1/40*(20*(b*x^3 + a)^(2/3)*b^30*c^2
*d^5 - 8*(b*x^3 + a)^(5/3)*b^29*c*d^6 + 40*(b*x^3 + a)^(2/3)*a*b^29*c*d^6 + 5*(b*x^3 + a)^(8/3)*b^28*d^7 - 24*
(b*x^3 + a)^(5/3)*a*b^28*d^7 + 60*(b*x^3 + a)^(2/3)*a^2*b^28*d^7)/(b^32*d^8)