3.6.73 \(\int \frac {\sqrt [3]{a+b x^3}}{x^7 (a d-b d x^3)} \, dx\) [573]

Optimal. Leaf size=283 \[ -\frac {2 b \sqrt [3]{a+b x^3}}{9 a^2 d x^3}-\frac {\left (a+b x^3\right )^{4/3}}{6 a^2 d x^6}-\frac {11 b^2 \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3} d}+\frac {\sqrt [3]{2} b^2 \tan ^{-1}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{8/3} d}-\frac {11 b^2 \log (x)}{18 a^{8/3} d}+\frac {b^2 \log \left (a-b x^3\right )}{3\ 2^{2/3} a^{8/3} d}+\frac {11 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{18 a^{8/3} d}-\frac {b^2 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} a^{8/3} d} \]

[Out]

-2/9*b*(b*x^3+a)^(1/3)/a^2/d/x^3-1/6*(b*x^3+a)^(4/3)/a^2/d/x^6-11/18*b^2*ln(x)/a^(8/3)/d+1/6*b^2*ln(-b*x^3+a)*
2^(1/3)/a^(8/3)/d+11/18*b^2*ln(a^(1/3)-(b*x^3+a)^(1/3))/a^(8/3)/d-1/2*b^2*ln(2^(1/3)*a^(1/3)-(b*x^3+a)^(1/3))*
2^(1/3)/a^(8/3)/d-11/27*b^2*arctan(1/3*(a^(1/3)+2*(b*x^3+a)^(1/3))/a^(1/3)*3^(1/2))/a^(8/3)/d*3^(1/2)+1/3*2^(1
/3)*b^2*arctan(1/3*(a^(1/3)+2^(2/3)*(b*x^3+a)^(1/3))/a^(1/3)*3^(1/2))/a^(8/3)/d*3^(1/2)

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Rubi [A]
time = 0.19, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {457, 105, 154, 162, 59, 631, 210, 31} \begin {gather*} -\frac {11 b^2 \text {ArcTan}\left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3} d}+\frac {\sqrt [3]{2} b^2 \text {ArcTan}\left (\frac {2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{8/3} d}+\frac {b^2 \log \left (a-b x^3\right )}{3\ 2^{2/3} a^{8/3} d}+\frac {11 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{18 a^{8/3} d}-\frac {b^2 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} a^{8/3} d}-\frac {11 b^2 \log (x)}{18 a^{8/3} d}-\frac {2 b \sqrt [3]{a+b x^3}}{9 a^2 d x^3}-\frac {\left (a+b x^3\right )^{4/3}}{6 a^2 d x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*(a + b*x^3)^(1/3))/(9*a^2*d*x^3) - (a + b*x^3)^(4/3)/(6*a^2*d*x^6) - (11*b^2*ArcTan[(a^(1/3) + 2*(a + b*
x^3)^(1/3))/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(8/3)*d) + (2^(1/3)*b^2*ArcTan[(a^(1/3) + 2^(2/3)*(a + b*x^3)^(1/
3))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(8/3)*d) - (11*b^2*Log[x])/(18*a^(8/3)*d) + (b^2*Log[a - b*x^3])/(3*2^(2/3)
*a^(8/3)*d) + (11*b^2*Log[a^(1/3) - (a + b*x^3)^(1/3)])/(18*a^(8/3)*d) - (b^2*Log[2^(1/3)*a^(1/3) - (a + b*x^3
)^(1/3)])/(2^(2/3)*a^(8/3)*d)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 59

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 105

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] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 154

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] && ILtQ[m, -1] && GtQ[n, 0]

Rule 162

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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 457

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 631

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]

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{a+b x^3}}{x^7 \left (a d-b d x^3\right )} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {\sqrt [3]{a+b x}}{x^3 (a d-b d x)} \, dx,x,x^3\right )\\ &=-\frac {\left (a+b x^3\right )^{4/3}}{6 a^2 d x^6}-\frac {\text {Subst}\left (\int \frac {\sqrt [3]{a+b x} \left (-\frac {4}{3} a b d-\frac {2}{3} b^2 d x\right )}{x^2 (a d-b d x)} \, dx,x,x^3\right )}{6 a^2 d}\\ &=-\frac {2 b \sqrt [3]{a+b x^3}}{9 a^2 d x^3}-\frac {\left (a+b x^3\right )^{4/3}}{6 a^2 d x^6}-\frac {\text {Subst}\left (\int \frac {-\frac {22}{9} a^2 b^2 d^2-\frac {14}{9} a b^3 d^2 x}{x (a+b x)^{2/3} (a d-b d x)} \, dx,x,x^3\right )}{6 a^3 d^2}\\ &=-\frac {2 b \sqrt [3]{a+b x^3}}{9 a^2 d x^3}-\frac {\left (a+b x^3\right )^{4/3}}{6 a^2 d x^6}+\frac {\left (2 b^3\right ) \text {Subst}\left (\int \frac {1}{(a+b x)^{2/3} (a d-b d x)} \, dx,x,x^3\right )}{3 a^2}+\frac {\left (11 b^2\right ) \text {Subst}\left (\int \frac {1}{x (a+b x)^{2/3}} \, dx,x,x^3\right )}{27 a^2 d}\\ &=-\frac {2 b \sqrt [3]{a+b x^3}}{9 a^2 d x^3}-\frac {\left (a+b x^3\right )^{4/3}}{6 a^2 d x^6}-\frac {11 b^2 \log (x)}{18 a^{8/3} d}+\frac {b^2 \log \left (a-b x^3\right )}{3\ 2^{2/3} a^{8/3} d}-\frac {\left (11 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{18 a^{8/3} d}+\frac {b^2 \text {Subst}\left (\int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2^{2/3} a^{8/3} d}-\frac {\left (11 b^2\right ) \text {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^{7/3} d}+\frac {b^2 \text {Subst}\left (\int \frac {1}{2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^{7/3} d}\\ &=-\frac {2 b \sqrt [3]{a+b x^3}}{9 a^2 d x^3}-\frac {\left (a+b x^3\right )^{4/3}}{6 a^2 d x^6}-\frac {11 b^2 \log (x)}{18 a^{8/3} d}+\frac {b^2 \log \left (a-b x^3\right )}{3\ 2^{2/3} a^{8/3} d}+\frac {11 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{18 a^{8/3} d}-\frac {b^2 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} a^{8/3} d}+\frac {\left (11 b^2\right ) \text {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^{8/3} d}-\frac {\left (\sqrt [3]{2} b^2\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{a^{8/3} d}\\ &=-\frac {2 b \sqrt [3]{a+b x^3}}{9 a^2 d x^3}-\frac {\left (a+b x^3\right )^{4/3}}{6 a^2 d x^6}-\frac {11 b^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{9 \sqrt {3} a^{8/3} d}+\frac {\sqrt [3]{2} b^2 \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{8/3} d}-\frac {11 b^2 \log (x)}{18 a^{8/3} d}+\frac {b^2 \log \left (a-b x^3\right )}{3\ 2^{2/3} a^{8/3} d}+\frac {11 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{18 a^{8/3} d}-\frac {b^2 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} a^{8/3} d}\\ \end {align*}

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Mathematica [A]
time = 0.55, size = 316, normalized size = 1.12 \begin {gather*} -\frac {9 a^{5/3} \sqrt [3]{a+b x^3}+21 a^{2/3} b x^3 \sqrt [3]{a+b x^3}+22 \sqrt {3} b^2 x^6 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-18 \sqrt [3]{2} \sqrt {3} b^2 x^6 \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-22 b^2 x^6 \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )+18 \sqrt [3]{2} b^2 x^6 \log \left (-2 \sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}\right )+11 b^2 x^6 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )-9 \sqrt [3]{2} b^2 x^6 \log \left (2 a^{2/3}+2^{2/3} \sqrt [3]{a} \sqrt [3]{a+b x^3}+\sqrt [3]{2} \left (a+b x^3\right )^{2/3}\right )}{54 a^{8/3} d x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/54*(9*a^(5/3)*(a + b*x^3)^(1/3) + 21*a^(2/3)*b*x^3*(a + b*x^3)^(1/3) + 22*Sqrt[3]*b^2*x^6*ArcTan[(1 + (2*(a
+ b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 18*2^(1/3)*Sqrt[3]*b^2*x^6*ArcTan[(1 + (2^(2/3)*(a + b*x^3)^(1/3))/a^(1/3
))/Sqrt[3]] - 22*b^2*x^6*Log[-a^(1/3) + (a + b*x^3)^(1/3)] + 18*2^(1/3)*b^2*x^6*Log[-2*a^(1/3) + 2^(2/3)*(a +
b*x^3)^(1/3)] + 11*b^2*x^6*Log[a^(2/3) + a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)] - 9*2^(1/3)*b^2*x^6*Lo
g[2*a^(2/3) + 2^(2/3)*a^(1/3)*(a + b*x^3)^(1/3) + 2^(1/3)*(a + b*x^3)^(2/3)])/(a^(8/3)*d*x^6)

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x^{7} \left (-b d \,x^{3}+a d \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 3.19, size = 345, normalized size = 1.22 \begin {gather*} -\frac {18 \, \sqrt {3} 2^{\frac {1}{3}} a^{2} b^{2} x^{6} \left (-\frac {1}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} 2^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} a \left (-\frac {1}{a^{2}}\right )^{\frac {2}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + 9 \cdot 2^{\frac {1}{3}} a^{2} b^{2} x^{6} \left (-\frac {1}{a^{2}}\right )^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} a^{2} \left (-\frac {1}{a^{2}}\right )^{\frac {2}{3}} - 2^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} a \left (-\frac {1}{a^{2}}\right )^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}}\right ) - 18 \cdot 2^{\frac {1}{3}} a^{2} b^{2} x^{6} \left (-\frac {1}{a^{2}}\right )^{\frac {1}{3}} \log \left (2^{\frac {1}{3}} a \left (-\frac {1}{a^{2}}\right )^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right ) + 22 \, \sqrt {3} {\left (a^{2}\right )}^{\frac {1}{6}} a b^{2} 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 ) + 11 \, {\left (a^{2}\right )}^{\frac {2}{3}} b^{2} 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 ) - 22 \, {\left (a^{2}\right )}^{\frac {2}{3}} b^{2} x^{6} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} a - {\left (a^{2}\right )}^{\frac {2}{3}}\right ) + 3 \, {\left (7 \, a^{2} b x^{3} + 3 \, a^{3}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{54 \, a^{4} d x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/54*(18*sqrt(3)*2^(1/3)*a^2*b^2*x^6*(-1/a^2)^(1/3)*arctan(1/3*sqrt(3)*2^(2/3)*(b*x^3 + a)^(1/3)*a*(-1/a^2)^(
2/3) + 1/3*sqrt(3)) + 9*2^(1/3)*a^2*b^2*x^6*(-1/a^2)^(1/3)*log(2^(2/3)*a^2*(-1/a^2)^(2/3) - 2^(1/3)*(b*x^3 + a
)^(1/3)*a*(-1/a^2)^(1/3) + (b*x^3 + a)^(2/3)) - 18*2^(1/3)*a^2*b^2*x^6*(-1/a^2)^(1/3)*log(2^(1/3)*a*(-1/a^2)^(
1/3) + (b*x^3 + a)^(1/3)) + 22*sqrt(3)*(a^2)^(1/6)*a*b^2*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) + 11*(a^2)^(2/3)*b^2*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)) - 22*(a^2)^(2/3)*b^2*x^6*log((b*x^3 + a)^(1/3)*a - (a^2)^(2/3)) + 3*(7*a^2*b*
x^3 + 3*a^3)*(b*x^3 + a)^(1/3))/(a^4*d*x^6)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {\sqrt [3]{a + b x^{3}}}{- a x^{7} + b x^{10}}\, dx}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral((a + b*x**3)**(1/3)/(-a*x**7 + b*x**10), x)/d

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Algebraic extensions not allowed in a ro
otofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a rootofAlgebraic extensio
ns not allowed in a r

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Mupad [B]
time = 5.44, size = 490, normalized size = 1.73 \begin {gather*} \frac {\frac {2\,b^2\,{\left (b\,x^3+a\right )}^{1/3}}{9\,a}-\frac {7\,b^2\,{\left (b\,x^3+a\right )}^{4/3}}{18\,a^2}}{d\,{\left (b\,x^3+a\right )}^2+a^2\,d-2\,a\,d\,\left (b\,x^3+a\right )}+\frac {11\,\ln \left (b^2\,{\left (b\,x^3+a\right )}^{1/3}-a^3\,d\,{\left (\frac {b^6}{a^8\,d^3}\right )}^{1/3}\right )\,{\left (\frac {b^6}{a^8\,d^3}\right )}^{1/3}}{27}+\ln \left (b^2\,{\left (b\,x^3+a\right )}^{1/3}+2^{1/3}\,a^3\,d\,{\left (-\frac {b^6}{a^8\,d^3}\right )}^{1/3}\right )\,{\left (-\frac {2\,b^6}{27\,a^8\,d^3}\right )}^{1/3}-\ln \left (2^{1/3}\,a^3\,d\,{\left (-\frac {b^6}{a^8\,d^3}\right )}^{1/3}-2\,b^2\,{\left (b\,x^3+a\right )}^{1/3}+2^{1/3}\,\sqrt {3}\,a^3\,d\,{\left (-\frac {b^6}{a^8\,d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {2\,b^6}{27\,a^8\,d^3}\right )}^{1/3}+\ln \left (2\,b^2\,{\left (b\,x^3+a\right )}^{1/3}-2^{1/3}\,a^3\,d\,{\left (-\frac {b^6}{a^8\,d^3}\right )}^{1/3}+2^{1/3}\,\sqrt {3}\,a^3\,d\,{\left (-\frac {b^6}{a^8\,d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {2\,b^6}{27\,a^8\,d^3}\right )}^{1/3}+\frac {11\,\ln \left (2\,b^2\,{\left (b\,x^3+a\right )}^{1/3}+a^3\,d\,{\left (\frac {b^6}{a^8\,d^3}\right )}^{1/3}-\sqrt {3}\,a^3\,d\,{\left (\frac {b^6}{a^8\,d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {b^6}{a^8\,d^3}\right )}^{1/3}}{54}-\frac {11\,\ln \left (2\,b^2\,{\left (b\,x^3+a\right )}^{1/3}+a^3\,d\,{\left (\frac {b^6}{a^8\,d^3}\right )}^{1/3}+\sqrt {3}\,a^3\,d\,{\left (\frac {b^6}{a^8\,d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {b^6}{a^8\,d^3}\right )}^{1/3}}{54} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*b^2*(a + b*x^3)^(1/3))/(9*a) - (7*b^2*(a + b*x^3)^(4/3))/(18*a^2))/(d*(a + b*x^3)^2 + a^2*d - 2*a*d*(a + b
*x^3)) + (11*log(b^2*(a + b*x^3)^(1/3) - a^3*d*(b^6/(a^8*d^3))^(1/3))*(b^6/(a^8*d^3))^(1/3))/27 + log(b^2*(a +
b*x^3)^(1/3) + 2^(1/3)*a^3*d*(-b^6/(a^8*d^3))^(1/3))*(-(2*b^6)/(27*a^8*d^3))^(1/3) - log(2^(1/3)*a^3*d*(-b^6/
(a^8*d^3))^(1/3) - 2*b^2*(a + b*x^3)^(1/3) + 2^(1/3)*3^(1/2)*a^3*d*(-b^6/(a^8*d^3))^(1/3)*1i)*((3^(1/2)*1i)/2
+ 1/2)*(-(2*b^6)/(27*a^8*d^3))^(1/3) + log(2*b^2*(a + b*x^3)^(1/3) - 2^(1/3)*a^3*d*(-b^6/(a^8*d^3))^(1/3) + 2^
(1/3)*3^(1/2)*a^3*d*(-b^6/(a^8*d^3))^(1/3)*1i)*((3^(1/2)*1i)/2 - 1/2)*(-(2*b^6)/(27*a^8*d^3))^(1/3) + (11*log(
2*b^2*(a + b*x^3)^(1/3) + a^3*d*(b^6/(a^8*d^3))^(1/3) - 3^(1/2)*a^3*d*(b^6/(a^8*d^3))^(1/3)*1i)*(3^(1/2)*1i -
1)*(b^6/(a^8*d^3))^(1/3))/54 - (11*log(2*b^2*(a + b*x^3)^(1/3) + a^3*d*(b^6/(a^8*d^3))^(1/3) + 3^(1/2)*a^3*d*(
b^6/(a^8*d^3))^(1/3)*1i)*(3^(1/2)*1i + 1)*(b^6/(a^8*d^3))^(1/3))/54

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