3.405 \(\int \frac{1}{x^3 \sqrt [3]{-a+b x}} \, dx\)
Optimal. Leaf size=136 \[ \frac{b^2 \log (x)}{9 a^{7/3}}-\frac{b^2 \log \left (\sqrt [3]{b x-a}+\sqrt [3]{a}\right )}{3 a^{7/3}}-\frac{2 b^2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b x-a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{7/3}}+\frac{2 b (b x-a)^{2/3}}{3 a^2 x}+\frac{(b x-a)^{2/3}}{2 a x^2} \]
[Out]
(-a + b*x)^(2/3)/(2*a*x^2) + (2*b*(-a + b*x)^(2/3))/(3*a^2*x) - (2*b^2*ArcTan[(a^(1/3) - 2*(-a + b*x)^(1/3))/(
Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(7/3)) + (b^2*Log[x])/(9*a^(7/3)) - (b^2*Log[a^(1/3) + (-a + b*x)^(1/3)])/(3*a
^(7/3))
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Rubi [A] time = 0.0440761, antiderivative size = 136, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used =
{51, 56, 617, 204, 31} \[ \frac{b^2 \log (x)}{9 a^{7/3}}-\frac{b^2 \log \left (\sqrt [3]{b x-a}+\sqrt [3]{a}\right )}{3 a^{7/3}}-\frac{2 b^2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b x-a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{7/3}}+\frac{2 b (b x-a)^{2/3}}{3 a^2 x}+\frac{(b x-a)^{2/3}}{2 a x^2} \]
Antiderivative was successfully verified.
[In]
Int[1/(x^3*(-a + b*x)^(1/3)),x]
[Out]
(-a + b*x)^(2/3)/(2*a*x^2) + (2*b*(-a + b*x)^(2/3))/(3*a^2*x) - (2*b^2*ArcTan[(a^(1/3) - 2*(-a + b*x)^(1/3))/(
Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(7/3)) + (b^2*Log[x])/(9*a^(7/3)) - (b^2*Log[a^(1/3) + (-a + b*x)^(1/3)])/(3*a
^(7/3))
Rule 51
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]
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{1}{x^3 \sqrt [3]{-a+b x}} \, dx &=\frac{(-a+b x)^{2/3}}{2 a x^2}+\frac{(2 b) \int \frac{1}{x^2 \sqrt [3]{-a+b x}} \, dx}{3 a}\\ &=\frac{(-a+b x)^{2/3}}{2 a x^2}+\frac{2 b (-a+b x)^{2/3}}{3 a^2 x}+\frac{\left (2 b^2\right ) \int \frac{1}{x \sqrt [3]{-a+b x}} \, dx}{9 a^2}\\ &=\frac{(-a+b x)^{2/3}}{2 a x^2}+\frac{2 b (-a+b x)^{2/3}}{3 a^2 x}+\frac{b^2 \log (x)}{9 a^{7/3}}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+x} \, dx,x,\sqrt [3]{-a+b x}\right )}{3 a^{7/3}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{a^{2/3}-\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{-a+b x}\right )}{3 a^2}\\ &=\frac{(-a+b x)^{2/3}}{2 a x^2}+\frac{2 b (-a+b x)^{2/3}}{3 a^2 x}+\frac{b^2 \log (x)}{9 a^{7/3}}-\frac{b^2 \log \left (\sqrt [3]{a}+\sqrt [3]{-a+b x}\right )}{3 a^{7/3}}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{-a+b x}}{\sqrt [3]{a}}\right )}{3 a^{7/3}}\\ &=\frac{(-a+b x)^{2/3}}{2 a x^2}+\frac{2 b (-a+b x)^{2/3}}{3 a^2 x}-\frac{2 b^2 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{-a+b x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{3 \sqrt{3} a^{7/3}}+\frac{b^2 \log (x)}{9 a^{7/3}}-\frac{b^2 \log \left (\sqrt [3]{a}+\sqrt [3]{-a+b x}\right )}{3 a^{7/3}}\\ \end{align*}
Mathematica [C] time = 0.0125004, size = 38, normalized size = 0.28 \[ \frac{3 b^2 (b x-a)^{2/3} \, _2F_1\left (\frac{2}{3},3;\frac{5}{3};1-\frac{b x}{a}\right )}{2 a^3} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(x^3*(-a + b*x)^(1/3)),x]
[Out]
(3*b^2*(-a + b*x)^(2/3)*Hypergeometric2F1[2/3, 3, 5/3, 1 - (b*x)/a])/(2*a^3)
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Maple [A] time = 0.006, size = 128, normalized size = 0.9 \begin{align*}{\frac{1}{2\,a{x}^{2}} \left ( bx-a \right ) ^{{\frac{2}{3}}}}+{\frac{2\,b}{3\,{a}^{2}x} \left ( bx-a \right ) ^{{\frac{2}{3}}}}-{\frac{2\,{b}^{2}}{9}\ln \left ( \sqrt [3]{a}+\sqrt [3]{bx-a} \right ){a}^{-{\frac{7}{3}}}}+{\frac{{b}^{2}}{9}\ln \left ( \left ( bx-a \right ) ^{{\frac{2}{3}}}-\sqrt [3]{a}\sqrt [3]{bx-a}+{a}^{{\frac{2}{3}}} \right ){a}^{-{\frac{7}{3}}}}+{\frac{2\,{b}^{2}\sqrt{3}}{9}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{bx-a}}{\sqrt [3]{a}}}-1 \right ) } \right ){a}^{-{\frac{7}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^3/(b*x-a)^(1/3),x)
[Out]
1/2*(b*x-a)^(2/3)/a/x^2+2/3*b*(b*x-a)^(2/3)/a^2/x-2/9*b^2*ln(a^(1/3)+(b*x-a)^(1/3))/a^(7/3)+1/9*b^2/a^(7/3)*ln
((b*x-a)^(2/3)-a^(1/3)*(b*x-a)^(1/3)+a^(2/3))+2/9*b^2/a^(7/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/a^(1/3)*(b*x-a)^(1
/3)-1))
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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(1/x^3/(b*x-a)^(1/3),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 1.64782, size = 956, normalized size = 7.03 \begin{align*} \left [\frac{6 \, \sqrt{\frac{1}{3}} a b^{2} x^{2} \sqrt{\frac{\left (-a\right )^{\frac{1}{3}}}{a}} \log \left (\frac{2 \, b x + 3 \, \sqrt{\frac{1}{3}}{\left (2 \,{\left (b x - a\right )}^{\frac{2}{3}} \left (-a\right )^{\frac{2}{3}} +{\left (b x - a\right )}^{\frac{1}{3}} a + \left (-a\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (-a\right )^{\frac{1}{3}}}{a}} - 3 \,{\left (b x - a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{2}{3}} - 3 \, a}{x}\right ) + 2 \, \left (-a\right )^{\frac{2}{3}} b^{2} x^{2} \log \left ({\left (b x - a\right )}^{\frac{2}{3}} +{\left (b x - a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{1}{3}} + \left (-a\right )^{\frac{2}{3}}\right ) - 4 \, \left (-a\right )^{\frac{2}{3}} b^{2} x^{2} \log \left ({\left (b x - a\right )}^{\frac{1}{3}} - \left (-a\right )^{\frac{1}{3}}\right ) + 3 \,{\left (4 \, a b x + 3 \, a^{2}\right )}{\left (b x - a\right )}^{\frac{2}{3}}}{18 \, a^{3} x^{2}}, \frac{12 \, \sqrt{\frac{1}{3}} a b^{2} x^{2} \sqrt{-\frac{\left (-a\right )^{\frac{1}{3}}}{a}} \arctan \left (\sqrt{\frac{1}{3}}{\left (2 \,{\left (b x - a\right )}^{\frac{1}{3}} + \left (-a\right )^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (-a\right )^{\frac{1}{3}}}{a}}\right ) + 2 \, \left (-a\right )^{\frac{2}{3}} b^{2} x^{2} \log \left ({\left (b x - a\right )}^{\frac{2}{3}} +{\left (b x - a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{1}{3}} + \left (-a\right )^{\frac{2}{3}}\right ) - 4 \, \left (-a\right )^{\frac{2}{3}} b^{2} x^{2} \log \left ({\left (b x - a\right )}^{\frac{1}{3}} - \left (-a\right )^{\frac{1}{3}}\right ) + 3 \,{\left (4 \, a b x + 3 \, a^{2}\right )}{\left (b x - a\right )}^{\frac{2}{3}}}{18 \, a^{3} x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^3/(b*x-a)^(1/3),x, algorithm="fricas")
[Out]
[1/18*(6*sqrt(1/3)*a*b^2*x^2*sqrt((-a)^(1/3)/a)*log((2*b*x + 3*sqrt(1/3)*(2*(b*x - a)^(2/3)*(-a)^(2/3) + (b*x
- a)^(1/3)*a + (-a)^(1/3)*a)*sqrt((-a)^(1/3)/a) - 3*(b*x - a)^(1/3)*(-a)^(2/3) - 3*a)/x) + 2*(-a)^(2/3)*b^2*x^
2*log((b*x - a)^(2/3) + (b*x - a)^(1/3)*(-a)^(1/3) + (-a)^(2/3)) - 4*(-a)^(2/3)*b^2*x^2*log((b*x - a)^(1/3) -
(-a)^(1/3)) + 3*(4*a*b*x + 3*a^2)*(b*x - a)^(2/3))/(a^3*x^2), 1/18*(12*sqrt(1/3)*a*b^2*x^2*sqrt(-(-a)^(1/3)/a)
*arctan(sqrt(1/3)*(2*(b*x - a)^(1/3) + (-a)^(1/3))*sqrt(-(-a)^(1/3)/a)) + 2*(-a)^(2/3)*b^2*x^2*log((b*x - a)^(
2/3) + (b*x - a)^(1/3)*(-a)^(1/3) + (-a)^(2/3)) - 4*(-a)^(2/3)*b^2*x^2*log((b*x - a)^(1/3) - (-a)^(1/3)) + 3*(
4*a*b*x + 3*a^2)*(b*x - a)^(2/3))/(a^3*x^2)]
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Sympy [C] time = 4.01392, size = 2744, normalized size = 20.18 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**3/(b*x-a)**(1/3),x)
[Out]
-4*a**(14/3)*b**(10/3)*(-a/b + x)**(4/3)*log(1 - b**(1/3)*(-a/b + x)**(1/3)*exp_polar(I*pi/3)/a**(1/3))*gamma(
2/3)/(27*a**7*b**(4/3)*(-a/b + x)**(4/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**6*b**(7/3)*(-a/b + x)**(7/3)*exp(2*I
*pi/3)*gamma(5/3) + 81*a**5*b**(10/3)*(-a/b + x)**(10/3)*exp(2*I*pi/3)*gamma(5/3) + 27*a**4*b**(13/3)*(-a/b +
x)**(13/3)*exp(2*I*pi/3)*gamma(5/3)) - 4*a**(14/3)*b**(10/3)*(-a/b + x)**(4/3)*exp(2*I*pi/3)*log(1 - b**(1/3)*
(-a/b + x)**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(2/3)/(27*a**7*b**(4/3)*(-a/b + x)**(4/3)*exp(2*I*pi/3)*gamma
(5/3) + 81*a**6*b**(7/3)*(-a/b + x)**(7/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**5*b**(10/3)*(-a/b + x)**(10/3)*exp
(2*I*pi/3)*gamma(5/3) + 27*a**4*b**(13/3)*(-a/b + x)**(13/3)*exp(2*I*pi/3)*gamma(5/3)) - 4*a**(14/3)*b**(10/3)
*(-a/b + x)**(4/3)*exp(-2*I*pi/3)*log(1 - b**(1/3)*(-a/b + x)**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(2/3)/
(27*a**7*b**(4/3)*(-a/b + x)**(4/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**6*b**(7/3)*(-a/b + x)**(7/3)*exp(2*I*pi/3
)*gamma(5/3) + 81*a**5*b**(10/3)*(-a/b + x)**(10/3)*exp(2*I*pi/3)*gamma(5/3) + 27*a**4*b**(13/3)*(-a/b + x)**(
13/3)*exp(2*I*pi/3)*gamma(5/3)) - 12*a**(11/3)*b**(13/3)*(-a/b + x)**(7/3)*log(1 - b**(1/3)*(-a/b + x)**(1/3)*
exp_polar(I*pi/3)/a**(1/3))*gamma(2/3)/(27*a**7*b**(4/3)*(-a/b + x)**(4/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**6*
b**(7/3)*(-a/b + x)**(7/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**5*b**(10/3)*(-a/b + x)**(10/3)*exp(2*I*pi/3)*gamma
(5/3) + 27*a**4*b**(13/3)*(-a/b + x)**(13/3)*exp(2*I*pi/3)*gamma(5/3)) - 12*a**(11/3)*b**(13/3)*(-a/b + x)**(7
/3)*exp(2*I*pi/3)*log(1 - b**(1/3)*(-a/b + x)**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(2/3)/(27*a**7*b**(4/3)*(-
a/b + x)**(4/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**6*b**(7/3)*(-a/b + x)**(7/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a*
*5*b**(10/3)*(-a/b + x)**(10/3)*exp(2*I*pi/3)*gamma(5/3) + 27*a**4*b**(13/3)*(-a/b + x)**(13/3)*exp(2*I*pi/3)*
gamma(5/3)) - 12*a**(11/3)*b**(13/3)*(-a/b + x)**(7/3)*exp(-2*I*pi/3)*log(1 - b**(1/3)*(-a/b + x)**(1/3)*exp_p
olar(5*I*pi/3)/a**(1/3))*gamma(2/3)/(27*a**7*b**(4/3)*(-a/b + x)**(4/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**6*b**
(7/3)*(-a/b + x)**(7/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**5*b**(10/3)*(-a/b + x)**(10/3)*exp(2*I*pi/3)*gamma(5/
3) + 27*a**4*b**(13/3)*(-a/b + x)**(13/3)*exp(2*I*pi/3)*gamma(5/3)) - 12*a**(8/3)*b**(16/3)*(-a/b + x)**(10/3)
*log(1 - b**(1/3)*(-a/b + x)**(1/3)*exp_polar(I*pi/3)/a**(1/3))*gamma(2/3)/(27*a**7*b**(4/3)*(-a/b + x)**(4/3)
*exp(2*I*pi/3)*gamma(5/3) + 81*a**6*b**(7/3)*(-a/b + x)**(7/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**5*b**(10/3)*(-
a/b + x)**(10/3)*exp(2*I*pi/3)*gamma(5/3) + 27*a**4*b**(13/3)*(-a/b + x)**(13/3)*exp(2*I*pi/3)*gamma(5/3)) - 1
2*a**(8/3)*b**(16/3)*(-a/b + x)**(10/3)*exp(2*I*pi/3)*log(1 - b**(1/3)*(-a/b + x)**(1/3)*exp_polar(I*pi)/a**(1
/3))*gamma(2/3)/(27*a**7*b**(4/3)*(-a/b + x)**(4/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**6*b**(7/3)*(-a/b + x)**(7
/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**5*b**(10/3)*(-a/b + x)**(10/3)*exp(2*I*pi/3)*gamma(5/3) + 27*a**4*b**(13/
3)*(-a/b + x)**(13/3)*exp(2*I*pi/3)*gamma(5/3)) - 12*a**(8/3)*b**(16/3)*(-a/b + x)**(10/3)*exp(-2*I*pi/3)*log(
1 - b**(1/3)*(-a/b + x)**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(2/3)/(27*a**7*b**(4/3)*(-a/b + x)**(4/3)*ex
p(2*I*pi/3)*gamma(5/3) + 81*a**6*b**(7/3)*(-a/b + x)**(7/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**5*b**(10/3)*(-a/b
+ x)**(10/3)*exp(2*I*pi/3)*gamma(5/3) + 27*a**4*b**(13/3)*(-a/b + x)**(13/3)*exp(2*I*pi/3)*gamma(5/3)) - 4*a*
*(5/3)*b**(19/3)*(-a/b + x)**(13/3)*log(1 - b**(1/3)*(-a/b + x)**(1/3)*exp_polar(I*pi/3)/a**(1/3))*gamma(2/3)/
(27*a**7*b**(4/3)*(-a/b + x)**(4/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**6*b**(7/3)*(-a/b + x)**(7/3)*exp(2*I*pi/3
)*gamma(5/3) + 81*a**5*b**(10/3)*(-a/b + x)**(10/3)*exp(2*I*pi/3)*gamma(5/3) + 27*a**4*b**(13/3)*(-a/b + x)**(
13/3)*exp(2*I*pi/3)*gamma(5/3)) - 4*a**(5/3)*b**(19/3)*(-a/b + x)**(13/3)*exp(2*I*pi/3)*log(1 - b**(1/3)*(-a/b
+ x)**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(2/3)/(27*a**7*b**(4/3)*(-a/b + x)**(4/3)*exp(2*I*pi/3)*gamma(5/3)
+ 81*a**6*b**(7/3)*(-a/b + x)**(7/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**5*b**(10/3)*(-a/b + x)**(10/3)*exp(2*I*
pi/3)*gamma(5/3) + 27*a**4*b**(13/3)*(-a/b + x)**(13/3)*exp(2*I*pi/3)*gamma(5/3)) - 4*a**(5/3)*b**(19/3)*(-a/b
+ x)**(13/3)*exp(-2*I*pi/3)*log(1 - b**(1/3)*(-a/b + x)**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(2/3)/(27*a
**7*b**(4/3)*(-a/b + x)**(4/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**6*b**(7/3)*(-a/b + x)**(7/3)*exp(2*I*pi/3)*gam
ma(5/3) + 81*a**5*b**(10/3)*(-a/b + x)**(10/3)*exp(2*I*pi/3)*gamma(5/3) + 27*a**4*b**(13/3)*(-a/b + x)**(13/3)
*exp(2*I*pi/3)*gamma(5/3)) + 21*a**4*b**4*(-a/b + x)**2*exp(2*I*pi/3)*gamma(2/3)/(27*a**7*b**(4/3)*(-a/b + x)*
*(4/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**6*b**(7/3)*(-a/b + x)**(7/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**5*b**(10
/3)*(-a/b + x)**(10/3)*exp(2*I*pi/3)*gamma(5/3) + 27*a**4*b**(13/3)*(-a/b + x)**(13/3)*exp(2*I*pi/3)*gamma(5/3
)) + 33*a**3*b**5*(-a/b + x)**3*exp(2*I*pi/3)*gamma(2/3)/(27*a**7*b**(4/3)*(-a/b + x)**(4/3)*exp(2*I*pi/3)*gam
ma(5/3) + 81*a**6*b**(7/3)*(-a/b + x)**(7/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**5*b**(10/3)*(-a/b + x)**(10/3)*e
xp(2*I*pi/3)*gamma(5/3) + 27*a**4*b**(13/3)*(-a/b + x)**(13/3)*exp(2*I*pi/3)*gamma(5/3)) + 12*a**2*b**6*(-a/b
+ x)**4*exp(2*I*pi/3)*gamma(2/3)/(27*a**7*b**(4/3)*(-a/b + x)**(4/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**6*b**(7/
3)*(-a/b + x)**(7/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**5*b**(10/3)*(-a/b + x)**(10/3)*exp(2*I*pi/3)*gamma(5/3)
+ 27*a**4*b**(13/3)*(-a/b + x)**(13/3)*exp(2*I*pi/3)*gamma(5/3))
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Giac [A] time = 1.75651, size = 225, normalized size = 1.65 \begin{align*} -\frac{\frac{4 \, \sqrt{3} \left (-a\right )^{\frac{2}{3}} b^{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x - a\right )}^{\frac{1}{3}} + \left (-a\right )^{\frac{1}{3}}\right )}}{3 \, \left (-a\right )^{\frac{1}{3}}}\right )}{a^{3}} - \frac{2 \, \left (-a\right )^{\frac{2}{3}} b^{3} \log \left ({\left (b x - a\right )}^{\frac{2}{3}} +{\left (b x - a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{1}{3}} + \left (-a\right )^{\frac{2}{3}}\right )}{a^{3}} + \frac{4 \, \left (-a\right )^{\frac{2}{3}} b^{3} \log \left ({\left |{\left (b x - a\right )}^{\frac{1}{3}} - \left (-a\right )^{\frac{1}{3}} \right |}\right )}{a^{3}} - \frac{3 \,{\left (4 \,{\left (b x - a\right )}^{\frac{5}{3}} b^{3} + 7 \,{\left (b x - a\right )}^{\frac{2}{3}} a b^{3}\right )}}{a^{2} b^{2} x^{2}}}{18 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^3/(b*x-a)^(1/3),x, algorithm="giac")
[Out]
-1/18*(4*sqrt(3)*(-a)^(2/3)*b^3*arctan(1/3*sqrt(3)*(2*(b*x - a)^(1/3) + (-a)^(1/3))/(-a)^(1/3))/a^3 - 2*(-a)^(
2/3)*b^3*log((b*x - a)^(2/3) + (b*x - a)^(1/3)*(-a)^(1/3) + (-a)^(2/3))/a^3 + 4*(-a)^(2/3)*b^3*log(abs((b*x -
a)^(1/3) - (-a)^(1/3)))/a^3 - 3*(4*(b*x - a)^(5/3)*b^3 + 7*(b*x - a)^(2/3)*a*b^3)/(a^2*b^2*x^2))/b