3.664 \(\int \frac{\sqrt [3]{a+b x^3}}{x^7 (c+d x^3)} \, dx\)

Optimal. Leaf size=370 \[ -\frac{\left (-9 a^2 d^2+3 a b c d+b^2 c^2\right ) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{18 a^{5/3} c^3}+\frac{\left (-9 a^2 d^2+3 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{5/3} c^3}+\frac{\log (x) \left (-9 a^2 d^2+3 a b c d+b^2 c^2\right )}{18 a^{5/3} c^3}-\frac{d^{5/3} \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 c^3}+\frac{d^{5/3} \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^3}-\frac{d^{5/3} \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} c^3}+\frac{\sqrt [3]{a+b x^3} (3 a d+b c)}{9 a c^2 x^3}-\frac{\left (a+b x^3\right )^{4/3}}{6 a c x^6} \]

[Out]

((b*c + 3*a*d)*(a + b*x^3)^(1/3))/(9*a*c^2*x^3) - (a + b*x^3)^(4/3)/(6*a*c*x^6) + ((b^2*c^2 + 3*a*b*c*d - 9*a^
2*d^2)*ArcTan[(a^(1/3) + 2*(a + b*x^3)^(1/3))/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(5/3)*c^3) - (d^(5/3)*(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]*c^3) + ((b^2*c^2 + 3*
a*b*c*d - 9*a^2*d^2)*Log[x])/(18*a^(5/3)*c^3) - (d^(5/3)*(b*c - a*d)^(1/3)*Log[c + d*x^3])/(6*c^3) - ((b^2*c^2
+ 3*a*b*c*d - 9*a^2*d^2)*Log[a^(1/3) - (a + b*x^3)^(1/3)])/(18*a^(5/3)*c^3) + (d^(5/3)*(b*c - a*d)^(1/3)*Log[
(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)])/(2*c^3)

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Rubi [A]  time = 0.488675, antiderivative size = 370, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {446, 103, 149, 156, 57, 617, 204, 31, 58} \[ -\frac{\left (-9 a^2 d^2+3 a b c d+b^2 c^2\right ) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{18 a^{5/3} c^3}+\frac{\left (-9 a^2 d^2+3 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{5/3} c^3}+\frac{\log (x) \left (-9 a^2 d^2+3 a b c d+b^2 c^2\right )}{18 a^{5/3} c^3}-\frac{d^{5/3} \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 c^3}+\frac{d^{5/3} \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^3}-\frac{d^{5/3} \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} c^3}+\frac{\sqrt [3]{a+b x^3} (3 a d+b c)}{9 a c^2 x^3}-\frac{\left (a+b x^3\right )^{4/3}}{6 a c x^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^(1/3)/(x^7*(c + d*x^3)),x]

[Out]

((b*c + 3*a*d)*(a + b*x^3)^(1/3))/(9*a*c^2*x^3) - (a + b*x^3)^(4/3)/(6*a*c*x^6) + ((b^2*c^2 + 3*a*b*c*d - 9*a^
2*d^2)*ArcTan[(a^(1/3) + 2*(a + b*x^3)^(1/3))/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(5/3)*c^3) - (d^(5/3)*(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]*c^3) + ((b^2*c^2 + 3*
a*b*c*d - 9*a^2*d^2)*Log[x])/(18*a^(5/3)*c^3) - (d^(5/3)*(b*c - a*d)^(1/3)*Log[c + d*x^3])/(6*c^3) - ((b^2*c^2
+ 3*a*b*c*d - 9*a^2*d^2)*Log[a^(1/3) - (a + b*x^3)^(1/3)])/(18*a^(5/3)*c^3) + (d^(5/3)*(b*c - a*d)^(1/3)*Log[
(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)])/(2*c^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 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 149

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 57

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

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]

Rubi steps

\begin{align*} \int \frac{\sqrt [3]{a+b x^3}}{x^7 \left (c+d x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{\sqrt [3]{a+b x}}{x^3 (c+d x)} \, dx,x,x^3\right )\\ &=-\frac{\left (a+b x^3\right )^{4/3}}{6 a c x^6}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt [3]{a+b x} \left (\frac{2}{3} (b c+3 a d)+\frac{2 b d x}{3}\right )}{x^2 (c+d x)} \, dx,x,x^3\right )}{6 a c}\\ &=\frac{(b c+3 a d) \sqrt [3]{a+b x^3}}{9 a c^2 x^3}-\frac{\left (a+b x^3\right )^{4/3}}{6 a c x^6}-\frac{\operatorname{Subst}\left (\int \frac{\frac{2}{9} \left (b^2 c^2+3 a b c d-9 a^2 d^2\right )+\frac{2}{9} b d (b c-6 a d) x}{x (a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )}{6 a c^2}\\ &=\frac{(b c+3 a d) \sqrt [3]{a+b x^3}}{9 a c^2 x^3}-\frac{\left (a+b x^3\right )^{4/3}}{6 a c x^6}+\frac{\left (d^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 c^3}-\frac{\left (b^2 c^2+3 a b c d-9 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{2/3}} \, dx,x,x^3\right )}{27 a c^3}\\ &=\frac{(b c+3 a d) \sqrt [3]{a+b x^3}}{9 a c^2 x^3}-\frac{\left (a+b x^3\right )^{4/3}}{6 a c x^6}+\frac{\left (b^2 c^2+3 a b c d-9 a^2 d^2\right ) \log (x)}{18 a^{5/3} c^3}-\frac{d^{5/3} \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 c^3}+\frac{\left (d^{5/3} \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 c^3}+\frac{\left (d^{4/3} (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 c^3}+\frac{\left (b^2 c^2+3 a b c d-9 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{18 a^{5/3} c^3}+\frac{\left (b^2 c^2+3 a b c d-9 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{18 a^{4/3} c^3}\\ &=\frac{(b c+3 a d) \sqrt [3]{a+b x^3}}{9 a c^2 x^3}-\frac{\left (a+b x^3\right )^{4/3}}{6 a c x^6}+\frac{\left (b^2 c^2+3 a b c d-9 a^2 d^2\right ) \log (x)}{18 a^{5/3} c^3}-\frac{d^{5/3} \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 c^3}-\frac{\left (b^2 c^2+3 a b c d-9 a^2 d^2\right ) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{18 a^{5/3} c^3}+\frac{d^{5/3} \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 c^3}+\frac{\left (d^{5/3} \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 )}{c^3}-\frac{\left (b^2 c^2+3 a b c d-9 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{9 a^{5/3} c^3}\\ &=\frac{(b c+3 a d) \sqrt [3]{a+b x^3}}{9 a c^2 x^3}-\frac{\left (a+b x^3\right )^{4/3}}{6 a c x^6}+\frac{\left (b^2 c^2+3 a b c d-9 a^2 d^2\right ) \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{9 \sqrt{3} a^{5/3} c^3}-\frac{d^{5/3} \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} c^3}+\frac{\left (b^2 c^2+3 a b c d-9 a^2 d^2\right ) \log (x)}{18 a^{5/3} c^3}-\frac{d^{5/3} \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 c^3}-\frac{\left (b^2 c^2+3 a b c d-9 a^2 d^2\right ) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{18 a^{5/3} c^3}+\frac{d^{5/3} \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 c^3}\\ \end{align*}

Mathematica [A]  time = 1.35555, size = 411, normalized size = 1.11 \[ -\frac{\frac{2 \left (-9 a^2 d^2+3 a b c d+b^2 c^2\right ) \left (3 \sqrt [3]{a+b x^3}-\frac{1}{2} \sqrt [3]{a} \left (\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )-2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )\right )\right )}{9 a c^2}+\frac{a d^{5/3} \left (\sqrt [3]{b c-a d} \log \left (-\sqrt [3]{d} \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )-2 \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )-2 \sqrt{3} \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}-1}{\sqrt{3}}\right )+6 \sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{c^2}-\frac{2 \left (a+b x^3\right )^{4/3} (3 a d+b c)}{3 a c x^3}+\frac{\left (a+b x^3\right )^{4/3}}{x^6}}{6 a c} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^(1/3)/(x^7*(c + d*x^3)),x]

[Out]

-((a + b*x^3)^(4/3)/x^6 - (2*(b*c + 3*a*d)*(a + b*x^3)^(4/3))/(3*a*c*x^3) + (2*(b^2*c^2 + 3*a*b*c*d - 9*a^2*d^
2)*(3*(a + b*x^3)^(1/3) - (a^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 2*Log[a^(1
/3) - (a + b*x^3)^(1/3)] + Log[a^(2/3) + a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)]))/2))/(9*a*c^2) + (a*d
^(5/3)*(6*d^(1/3)*(a + b*x^3)^(1/3) - 2*Sqrt[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]] - 2*(b*c - a*d)^(1/3)*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)] + (b*c - a
*d)^(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)]))/c
^2)/(6*a*c)

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{7} \left ( d{x}^{3}+c \right ) }\sqrt [3]{b{x}^{3}+a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(1/3)/x^7/(d*x^3+c),x)

[Out]

int((b*x^3+a)^(1/3)/x^7/(d*x^3+c),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{{\left (d x^{3} + c\right )} x^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(1/3)/x^7/(d*x^3+c),x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(1/3)/((d*x^3 + c)*x^7), x)

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Fricas [A]  time = 11.634, size = 1142, normalized size = 3.09 \begin{align*} -\frac{18 \, \sqrt{3}{\left (b c d^{2} - a d^{3}\right )}^{\frac{1}{3}} a^{3} d x^{6} \arctan \left (-\frac{2 \, \sqrt{3}{\left (b c d^{2} - a d^{3}\right )}^{\frac{2}{3}}{\left (b x^{3} + a\right )}^{\frac{1}{3}} - \sqrt{3}{\left (b c d - a d^{2}\right )}}{3 \,{\left (b c d - a d^{2}\right )}}\right ) + 9 \,{\left (b c d^{2} - a d^{3}\right )}^{\frac{1}{3}} a^{3} d x^{6} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} d^{2} -{\left (b c d^{2} - a d^{3}\right )}^{\frac{1}{3}}{\left (b x^{3} + a\right )}^{\frac{1}{3}} d +{\left (b c d^{2} - a d^{3}\right )}^{\frac{2}{3}}\right ) - 18 \,{\left (b c d^{2} - a d^{3}\right )}^{\frac{1}{3}} a^{3} d x^{6} \log \left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} d +{\left (b c d^{2} - a d^{3}\right )}^{\frac{1}{3}}\right ) - 2 \, \sqrt{3}{\left (a b^{2} c^{2} + 3 \, a^{2} b c d - 9 \, a^{3} d^{2}\right )}{\left (a^{2}\right )}^{\frac{1}{6}} x^{6} \arctan \left (\frac{{\left (a^{2}\right )}^{\frac{1}{6}}{\left (\sqrt{3}{\left (a^{2}\right )}^{\frac{1}{3}} a + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{2}{3}}\right )}}{3 \, a^{2}}\right ) -{\left (b^{2} c^{2} + 3 \, a b c d - 9 \, a^{2} d^{2}\right )}{\left (a^{2}\right )}^{\frac{2}{3}} x^{6} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} a +{\left (a^{2}\right )}^{\frac{1}{3}} a +{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{2}{3}}\right ) + 2 \,{\left (b^{2} c^{2} + 3 \, a b c d - 9 \, a^{2} d^{2}\right )}{\left (a^{2}\right )}^{\frac{2}{3}} x^{6} \log \left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} a -{\left (a^{2}\right )}^{\frac{2}{3}}\right ) + 3 \,{\left (3 \, a^{3} c^{2} +{\left (a^{2} b c^{2} - 6 \, a^{3} c d\right )} x^{3}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{54 \, a^{3} c^{3} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(1/3)/x^7/(d*x^3+c),x, algorithm="fricas")

[Out]

-1/54*(18*sqrt(3)*(b*c*d^2 - a*d^3)^(1/3)*a^3*d*x^6*arctan(-1/3*(2*sqrt(3)*(b*c*d^2 - a*d^3)^(2/3)*(b*x^3 + a)
^(1/3) - sqrt(3)*(b*c*d - a*d^2))/(b*c*d - a*d^2)) + 9*(b*c*d^2 - a*d^3)^(1/3)*a^3*d*x^6*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)) - 18*(b*c*d^2 - a*d^3)^(1/3)*a^3
*d*x^6*log((b*x^3 + a)^(1/3)*d + (b*c*d^2 - a*d^3)^(1/3)) - 2*sqrt(3)*(a*b^2*c^2 + 3*a^2*b*c*d - 9*a^3*d^2)*(a
^2)^(1/6)*x^6*arctan(1/3*(a^2)^(1/6)*(sqrt(3)*(a^2)^(1/3)*a + 2*sqrt(3)*(b*x^3 + a)^(1/3)*(a^2)^(2/3))/a^2) -
(b^2*c^2 + 3*a*b*c*d - 9*a^2*d^2)*(a^2)^(2/3)*x^6*log((b*x^3 + a)^(2/3)*a + (a^2)^(1/3)*a + (b*x^3 + a)^(1/3)*
(a^2)^(2/3)) + 2*(b^2*c^2 + 3*a*b*c*d - 9*a^2*d^2)*(a^2)^(2/3)*x^6*log((b*x^3 + a)^(1/3)*a - (a^2)^(2/3)) + 3*
(3*a^3*c^2 + (a^2*b*c^2 - 6*a^3*c*d)*x^3)*(b*x^3 + a)^(1/3))/(a^3*c^3*x^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [3]{a + b x^{3}}}{x^{7} \left (c + d x^{3}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**(1/3)/x**7/(d*x**3+c),x)

[Out]

Integral((a + b*x**3)**(1/3)/(x**7*(c + d*x**3)), x)

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Giac [A]  time = 2.73655, size = 653, normalized size = 1.76 \begin{align*} -\frac{1}{54} \,{\left (\frac{18 \,{\left (b c d^{2} - a d^{3}\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 )}{b^{4} c^{4} - a b^{3} c^{3} d} - \frac{18 \, \sqrt{3}{\left (-b c d^{2} + a d^{3}\right )}^{\frac{1}{3}} d \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 )}{b^{3} c^{3}} - \frac{9 \,{\left (-b c d^{2} + a d^{3}\right )}^{\frac{1}{3}} d \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 )}{b^{3} c^{3}} - \frac{2 \, \sqrt{3}{\left (a^{\frac{1}{3}} b^{2} c^{2} + 3 \, a^{\frac{4}{3}} b c d - 9 \, a^{\frac{7}{3}} d^{2}\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{2} b^{3} c^{3}} + \frac{2 \,{\left (b^{2} c^{2} + 3 \, a b c d - 9 \, a^{2} d^{2}\right )} \log \left ({\left |{\left (b x^{3} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{5}{3}} b^{3} c^{3}} - \frac{{\left (a^{\frac{1}{3}} b^{2} c^{2} + 3 \, a^{\frac{4}{3}} b c d - 9 \, a^{\frac{7}{3}} d^{2}\right )} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{a^{2} b^{3} c^{3}} + \frac{3 \,{\left ({\left (b x^{3} + a\right )}^{\frac{4}{3}} b c + 2 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} a b c - 6 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} a d + 6 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{2} d\right )}}{a b^{4} c^{2} x^{6}}\right )} b^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(1/3)/x^7/(d*x^3+c),x, algorithm="giac")

[Out]

-1/54*(18*(b*c*d^2 - a*d^3)*(-(b*c - a*d)/d)^(1/3)*log(abs((b*x^3 + a)^(1/3) - (-(b*c - a*d)/d)^(1/3)))/(b^4*c
^4 - a*b^3*c^3*d) - 18*sqrt(3)*(-b*c*d^2 + a*d^3)^(1/3)*d*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))/(b^3*c^3) - 9*(-b*c*d^2 + a*d^3)^(1/3)*d*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^3*c^3) - 2*sqrt(3)*(a^(1/3)*b^2*c^2 + 3*a^(4/3)
*b*c*d - 9*a^(7/3)*d^2)*arctan(1/3*sqrt(3)*(2*(b*x^3 + a)^(1/3) + a^(1/3))/a^(1/3))/(a^2*b^3*c^3) + 2*(b^2*c^2
+ 3*a*b*c*d - 9*a^2*d^2)*log(abs((b*x^3 + a)^(1/3) - a^(1/3)))/(a^(5/3)*b^3*c^3) - (a^(1/3)*b^2*c^2 + 3*a^(4/
3)*b*c*d - 9*a^(7/3)*d^2)*log((b*x^3 + a)^(2/3) + (b*x^3 + a)^(1/3)*a^(1/3) + a^(2/3))/(a^2*b^3*c^3) + 3*((b*x
^3 + a)^(4/3)*b*c + 2*(b*x^3 + a)^(1/3)*a*b*c - 6*(b*x^3 + a)^(4/3)*a*d + 6*(b*x^3 + a)^(1/3)*a^2*d)/(a*b^4*c^
2*x^6))*b^3