\(\int \frac {x^3}{\sqrt [3]{-b x^2+a x^3} (-b^2+a^2 x^6)} \, dx\) [2141]

Optimal result

Integrand size = 35, antiderivative size = 156 \[ \int \frac {x^3}{\sqrt [3]{-b x^2+a x^3} \left (-b^2+a^2 x^6\right )} \, dx=\frac {\text {RootSum}\left [a^3-a b^2-3 a^2 \text {$\#$1}^3+3 a \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 a b}-\frac {\text {RootSum}\left [a^3+a b^2-3 a^2 \text {$\#$1}^3+3 a \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 a b} \]

[Out]

Unintegrable

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(3532\) vs. \(2(156)=312\).

Time = 4.76 (sec) , antiderivative size = 3532, normalized size of antiderivative = 22.64, number of steps used = 43, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {2081, 6857, 129, 490, 596, 544, 245, 384} \[ \int \frac {x^3}{\sqrt [3]{-b x^2+a x^3} \left (-b^2+a^2 x^6\right )} \, dx=-\frac {\left (3 a^{2/3}-2 b^{2/3}\right ) x (b-a x)}{18 a^{7/3} b^{4/3} \sqrt [3]{a x^3-b x^2}}-\frac {\sqrt [3]{-1} \left (3 \sqrt [3]{-1} a^{2/3}-2 b^{2/3}\right ) x (b-a x)}{18 a^{7/3} b^{4/3} \sqrt [3]{a x^3-b x^2}}-\frac {(-1)^{2/3} \left (3 (-1)^{2/3} a^{2/3}-2 b^{2/3}\right ) x (b-a x)}{18 a^{7/3} b^{4/3} \sqrt [3]{a x^3-b x^2}}-\frac {\left (3 a^{2/3}+2 b^{2/3}\right ) x (b-a x)}{18 a^{7/3} b^{4/3} \sqrt [3]{a x^3-b x^2}}-\frac {\sqrt [3]{-1} \left (3 \sqrt [3]{-1} a^{2/3}+2 b^{2/3}\right ) x (b-a x)}{18 a^{7/3} b^{4/3} \sqrt [3]{a x^3-b x^2}}-\frac {(-1)^{2/3} \left (3 (-1)^{2/3} a^{2/3}+2 b^{2/3}\right ) x (b-a x)}{18 a^{7/3} b^{4/3} \sqrt [3]{a x^3-b x^2}}-\frac {\left (9 a^{4/3}-3 b^{2/3} a^{2/3}+2 b^{4/3}\right ) x^{2/3} \sqrt [3]{a x-b} \arctan \left (\frac {\frac {2 \sqrt [3]{a} \sqrt [3]{x}}{\sqrt [3]{a x-b}}+1}{\sqrt {3}}\right )}{18 \sqrt {3} a^{8/3} b \sqrt [3]{a x^3-b x^2}}+\frac {\left (9 a^{4/3}+3 b^{2/3} a^{2/3}+2 b^{4/3}\right ) x^{2/3} \sqrt [3]{a x-b} \arctan \left (\frac {\frac {2 \sqrt [3]{a} \sqrt [3]{x}}{\sqrt [3]{a x-b}}+1}{\sqrt {3}}\right )}{18 \sqrt {3} a^{8/3} b \sqrt [3]{a x^3-b x^2}}-\frac {\left (9 a^{4/3}-3 (-1)^{2/3} b^{2/3} a^{2/3}-2 \sqrt [3]{-1} b^{4/3}\right ) x^{2/3} \sqrt [3]{a x-b} \arctan \left (\frac {\frac {2 \sqrt [3]{a} \sqrt [3]{x}}{\sqrt [3]{a x-b}}+1}{\sqrt {3}}\right )}{18 \sqrt {3} a^{8/3} b \sqrt [3]{a x^3-b x^2}}+\frac {\left (9 a^{4/3}+3 (-1)^{2/3} b^{2/3} a^{2/3}-2 \sqrt [3]{-1} b^{4/3}\right ) x^{2/3} \sqrt [3]{a x-b} \arctan \left (\frac {\frac {2 \sqrt [3]{a} \sqrt [3]{x}}{\sqrt [3]{a x-b}}+1}{\sqrt {3}}\right )}{18 \sqrt {3} a^{8/3} b \sqrt [3]{a x^3-b x^2}}+\frac {\left (9 a^{4/3}-3 \sqrt [3]{-1} b^{2/3} a^{2/3}+2 (-1)^{2/3} b^{4/3}\right ) x^{2/3} \sqrt [3]{a x-b} \arctan \left (\frac {\frac {2 \sqrt [3]{a} \sqrt [3]{x}}{\sqrt [3]{a x-b}}+1}{\sqrt {3}}\right )}{18 \sqrt {3} a^{8/3} b \sqrt [3]{a x^3-b x^2}}-\frac {\left (9 a^{4/3}+3 \sqrt [3]{-1} b^{2/3} a^{2/3}+2 (-1)^{2/3} b^{4/3}\right ) x^{2/3} \sqrt [3]{a x-b} \arctan \left (\frac {\frac {2 \sqrt [3]{a} \sqrt [3]{x}}{\sqrt [3]{a x-b}}+1}{\sqrt {3}}\right )}{18 \sqrt {3} a^{8/3} b \sqrt [3]{a x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a x-b} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{2/3}-b^{2/3}} \sqrt [3]{x}}{\sqrt [3]{a x-b}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}-b^{2/3}} b \sqrt [3]{a x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a x-b} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{2/3}+b^{2/3}} \sqrt [3]{x}}{\sqrt [3]{a x-b}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}+b^{2/3}} b \sqrt [3]{a x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a x-b} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [3]{x}}{\sqrt [3]{a x-b}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{a x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a x-b} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{2/3}+\sqrt [3]{-1} b^{2/3}} \sqrt [3]{x}}{\sqrt [3]{a x-b}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}+\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{a x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a x-b} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{2/3}-(-1)^{2/3} b^{2/3}} \sqrt [3]{x}}{\sqrt [3]{a x-b}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}-(-1)^{2/3} b^{2/3}} b \sqrt [3]{a x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a x-b} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [3]{x}}{\sqrt [3]{a x-b}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}+(-1)^{2/3} b^{2/3}} b \sqrt [3]{a x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{12 a^{10/9} \sqrt [3]{a^{2/3}-b^{2/3}} b \sqrt [3]{a x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (-\sqrt [3]{a} x-(-1)^{2/3} \sqrt [3]{b}\right )}{12 a^{10/9} \sqrt [3]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{a x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{12 a^{10/9} \sqrt [3]{a^{2/3}+b^{2/3}} b \sqrt [3]{a x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [3]{-1} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{12 a^{10/9} \sqrt [3]{a^{2/3}+\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{a x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right )}{12 a^{10/9} \sqrt [3]{a^{2/3}-(-1)^{2/3} b^{2/3}} b \sqrt [3]{a x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a x-b} \log \left ((-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{12 a^{10/9} \sqrt [3]{a^{2/3}+(-1)^{2/3} b^{2/3}} b \sqrt [3]{a x^3-b x^2}}+\frac {\left (9 a^{4/3}-3 b^{2/3} a^{2/3}+2 b^{4/3}\right ) x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [3]{a} \sqrt [3]{x}-\sqrt [3]{a x-b}\right )}{36 a^{8/3} b \sqrt [3]{a x^3-b x^2}}-\frac {\left (9 a^{4/3}+3 b^{2/3} a^{2/3}+2 b^{4/3}\right ) x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [3]{a} \sqrt [3]{x}-\sqrt [3]{a x-b}\right )}{36 a^{8/3} b \sqrt [3]{a x^3-b x^2}}+\frac {\left (9 a^{4/3}-3 (-1)^{2/3} b^{2/3} a^{2/3}-2 \sqrt [3]{-1} b^{4/3}\right ) x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [3]{a} \sqrt [3]{x}-\sqrt [3]{a x-b}\right )}{36 a^{8/3} b \sqrt [3]{a x^3-b x^2}}-\frac {\left (9 a^{4/3}+3 (-1)^{2/3} b^{2/3} a^{2/3}-2 \sqrt [3]{-1} b^{4/3}\right ) x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [3]{a} \sqrt [3]{x}-\sqrt [3]{a x-b}\right )}{36 a^{8/3} b \sqrt [3]{a x^3-b x^2}}-\frac {\left (9 a^{4/3}-3 \sqrt [3]{-1} b^{2/3} a^{2/3}+2 (-1)^{2/3} b^{4/3}\right ) x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [3]{a} \sqrt [3]{x}-\sqrt [3]{a x-b}\right )}{36 a^{8/3} b \sqrt [3]{a x^3-b x^2}}+\frac {\left (9 a^{4/3}+3 \sqrt [3]{-1} b^{2/3} a^{2/3}+2 (-1)^{2/3} b^{4/3}\right ) x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [3]{a} \sqrt [3]{x}-\sqrt [3]{a x-b}\right )}{36 a^{8/3} b \sqrt [3]{a x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [9]{a} \sqrt [3]{a^{2/3}-b^{2/3}} \sqrt [3]{x}-\sqrt [3]{a x-b}\right )}{4 a^{10/9} \sqrt [3]{a^{2/3}-b^{2/3}} b \sqrt [3]{a x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [9]{a} \sqrt [3]{a^{2/3}+b^{2/3}} \sqrt [3]{x}-\sqrt [3]{a x-b}\right )}{4 a^{10/9} \sqrt [3]{a^{2/3}+b^{2/3}} b \sqrt [3]{a x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [9]{a} \sqrt [3]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [3]{x}-\sqrt [3]{a x-b}\right )}{4 a^{10/9} \sqrt [3]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{a x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [9]{a} \sqrt [3]{a^{2/3}+\sqrt [3]{-1} b^{2/3}} \sqrt [3]{x}-\sqrt [3]{a x-b}\right )}{4 a^{10/9} \sqrt [3]{a^{2/3}+\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{a x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [9]{a} \sqrt [3]{a^{2/3}-(-1)^{2/3} b^{2/3}} \sqrt [3]{x}-\sqrt [3]{a x-b}\right )}{4 a^{10/9} \sqrt [3]{a^{2/3}-(-1)^{2/3} b^{2/3}} b \sqrt [3]{a x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [9]{a} \sqrt [3]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [3]{x}-\sqrt [3]{a x-b}\right )}{4 a^{10/9} \sqrt [3]{a^{2/3}+(-1)^{2/3} b^{2/3}} b \sqrt [3]{a x^3-b x^2}} \]

[In]

Int[x^3/((-(b*x^2) + a*x^3)^(1/3)*(-b^2 + a^2*x^6)),x]

[Out]

-1/18*((3*a^(2/3) - 2*b^(2/3))*x*(b - a*x))/(a^(7/3)*b^(4/3)*(-(b*x^2) + a*x^3)^(1/3)) - ((-1)^(1/3)*(3*(-1)^(
1/3)*a^(2/3) - 2*b^(2/3))*x*(b - a*x))/(18*a^(7/3)*b^(4/3)*(-(b*x^2) + a*x^3)^(1/3)) - ((-1)^(2/3)*(3*(-1)^(2/
3)*a^(2/3) - 2*b^(2/3))*x*(b - a*x))/(18*a^(7/3)*b^(4/3)*(-(b*x^2) + a*x^3)^(1/3)) - ((3*a^(2/3) + 2*b^(2/3))*
x*(b - a*x))/(18*a^(7/3)*b^(4/3)*(-(b*x^2) + a*x^3)^(1/3)) - ((-1)^(1/3)*(3*(-1)^(1/3)*a^(2/3) + 2*b^(2/3))*x*
(b - a*x))/(18*a^(7/3)*b^(4/3)*(-(b*x^2) + a*x^3)^(1/3)) - ((-1)^(2/3)*(3*(-1)^(2/3)*a^(2/3) + 2*b^(2/3))*x*(b
- a*x))/(18*a^(7/3)*b^(4/3)*(-(b*x^2) + a*x^3)^(1/3)) - ((9*a^(4/3) - 3*a^(2/3)*b^(2/3) + 2*b^(4/3))*x^(2/3)*
(-b + a*x)^(1/3)*ArcTan[(1 + (2*a^(1/3)*x^(1/3))/(-b + a*x)^(1/3))/Sqrt[3]])/(18*Sqrt[3]*a^(8/3)*b*(-(b*x^2) +
a*x^3)^(1/3)) + ((9*a^(4/3) + 3*a^(2/3)*b^(2/3) + 2*b^(4/3))*x^(2/3)*(-b + a*x)^(1/3)*ArcTan[(1 + (2*a^(1/3)*
x^(1/3))/(-b + a*x)^(1/3))/Sqrt[3]])/(18*Sqrt[3]*a^(8/3)*b*(-(b*x^2) + a*x^3)^(1/3)) - ((9*a^(4/3) - 3*(-1)^(2
/3)*a^(2/3)*b^(2/3) - 2*(-1)^(1/3)*b^(4/3))*x^(2/3)*(-b + a*x)^(1/3)*ArcTan[(1 + (2*a^(1/3)*x^(1/3))/(-b + a*x
)^(1/3))/Sqrt[3]])/(18*Sqrt[3]*a^(8/3)*b*(-(b*x^2) + a*x^3)^(1/3)) + ((9*a^(4/3) + 3*(-1)^(2/3)*a^(2/3)*b^(2/3
) - 2*(-1)^(1/3)*b^(4/3))*x^(2/3)*(-b + a*x)^(1/3)*ArcTan[(1 + (2*a^(1/3)*x^(1/3))/(-b + a*x)^(1/3))/Sqrt[3]])
/(18*Sqrt[3]*a^(8/3)*b*(-(b*x^2) + a*x^3)^(1/3)) + ((9*a^(4/3) - 3*(-1)^(1/3)*a^(2/3)*b^(2/3) + 2*(-1)^(2/3)*b
^(4/3))*x^(2/3)*(-b + a*x)^(1/3)*ArcTan[(1 + (2*a^(1/3)*x^(1/3))/(-b + a*x)^(1/3))/Sqrt[3]])/(18*Sqrt[3]*a^(8/
3)*b*(-(b*x^2) + a*x^3)^(1/3)) - ((9*a^(4/3) + 3*(-1)^(1/3)*a^(2/3)*b^(2/3) + 2*(-1)^(2/3)*b^(4/3))*x^(2/3)*(-
b + a*x)^(1/3)*ArcTan[(1 + (2*a^(1/3)*x^(1/3))/(-b + a*x)^(1/3))/Sqrt[3]])/(18*Sqrt[3]*a^(8/3)*b*(-(b*x^2) + a
*x^3)^(1/3)) - (x^(2/3)*(-b + a*x)^(1/3)*ArcTan[(1 + (2*a^(1/9)*(a^(2/3) - b^(2/3))^(1/3)*x^(1/3))/(-b + a*x)^
(1/3))/Sqrt[3]])/(2*Sqrt[3]*a^(10/9)*(a^(2/3) - b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) + (x^(2/3)*(-b + a*
x)^(1/3)*ArcTan[(1 + (2*a^(1/9)*(a^(2/3) + b^(2/3))^(1/3)*x^(1/3))/(-b + a*x)^(1/3))/Sqrt[3]])/(2*Sqrt[3]*a^(1
0/9)*(a^(2/3) + b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) + (x^(2/3)*(-b + a*x)^(1/3)*ArcTan[(1 + (2*a^(1/9)*
(a^(2/3) - (-1)^(1/3)*b^(2/3))^(1/3)*x^(1/3))/(-b + a*x)^(1/3))/Sqrt[3]])/(2*Sqrt[3]*a^(10/9)*(a^(2/3) - (-1)^
(1/3)*b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) - (x^(2/3)*(-b + a*x)^(1/3)*ArcTan[(1 + (2*a^(1/9)*(a^(2/3) +
(-1)^(1/3)*b^(2/3))^(1/3)*x^(1/3))/(-b + a*x)^(1/3))/Sqrt[3]])/(2*Sqrt[3]*a^(10/9)*(a^(2/3) + (-1)^(1/3)*b^(2
/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) - (x^(2/3)*(-b + a*x)^(1/3)*ArcTan[(1 + (2*a^(1/9)*(a^(2/3) - (-1)^(2/3
)*b^(2/3))^(1/3)*x^(1/3))/(-b + a*x)^(1/3))/Sqrt[3]])/(2*Sqrt[3]*a^(10/9)*(a^(2/3) - (-1)^(2/3)*b^(2/3))^(1/3)
*b*(-(b*x^2) + a*x^3)^(1/3)) + (x^(2/3)*(-b + a*x)^(1/3)*ArcTan[(1 + (2*a^(1/9)*(a^(2/3) + (-1)^(2/3)*b^(2/3))
^(1/3)*x^(1/3))/(-b + a*x)^(1/3))/Sqrt[3]])/(2*Sqrt[3]*a^(10/9)*(a^(2/3) + (-1)^(2/3)*b^(2/3))^(1/3)*b*(-(b*x^
2) + a*x^3)^(1/3)) - (x^(2/3)*(-b + a*x)^(1/3)*Log[b^(1/3) - a^(1/3)*x])/(12*a^(10/9)*(a^(2/3) - b^(2/3))^(1/3
)*b*(-(b*x^2) + a*x^3)^(1/3)) + (x^(2/3)*(-b + a*x)^(1/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*x])/(12*a^(10/9)
*(a^(2/3) - (-1)^(1/3)*b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) + (x^(2/3)*(-b + a*x)^(1/3)*Log[b^(1/3) + a^
(1/3)*x])/(12*a^(10/9)*(a^(2/3) + b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) - (x^(2/3)*(-b + a*x)^(1/3)*Log[b
^(1/3) + (-1)^(1/3)*a^(1/3)*x])/(12*a^(10/9)*(a^(2/3) + (-1)^(1/3)*b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3))
- (x^(2/3)*(-b + a*x)^(1/3)*Log[b^(1/3) - (-1)^(2/3)*a^(1/3)*x])/(12*a^(10/9)*(a^(2/3) - (-1)^(2/3)*b^(2/3))^(
1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) + (x^(2/3)*(-b + a*x)^(1/3)*Log[b^(1/3) + (-1)^(2/3)*a^(1/3)*x])/(12*a^(10/9)
*(a^(2/3) + (-1)^(2/3)*b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) + ((9*a^(4/3) - 3*a^(2/3)*b^(2/3) + 2*b^(4/3
))*x^(2/3)*(-b + a*x)^(1/3)*Log[a^(1/3)*x^(1/3) - (-b + a*x)^(1/3)])/(36*a^(8/3)*b*(-(b*x^2) + a*x^3)^(1/3)) -
((9*a^(4/3) + 3*a^(2/3)*b^(2/3) + 2*b^(4/3))*x^(2/3)*(-b + a*x)^(1/3)*Log[a^(1/3)*x^(1/3) - (-b + a*x)^(1/3)]
)/(36*a^(8/3)*b*(-(b*x^2) + a*x^3)^(1/3)) + ((9*a^(4/3) - 3*(-1)^(2/3)*a^(2/3)*b^(2/3) - 2*(-1)^(1/3)*b^(4/3))
*x^(2/3)*(-b + a*x)^(1/3)*Log[a^(1/3)*x^(1/3) - (-b + a*x)^(1/3)])/(36*a^(8/3)*b*(-(b*x^2) + a*x^3)^(1/3)) - (
(9*a^(4/3) + 3*(-1)^(2/3)*a^(2/3)*b^(2/3) - 2*(-1)^(1/3)*b^(4/3))*x^(2/3)*(-b + a*x)^(1/3)*Log[a^(1/3)*x^(1/3)
- (-b + a*x)^(1/3)])/(36*a^(8/3)*b*(-(b*x^2) + a*x^3)^(1/3)) - ((9*a^(4/3) - 3*(-1)^(1/3)*a^(2/3)*b^(2/3) + 2
*(-1)^(2/3)*b^(4/3))*x^(2/3)*(-b + a*x)^(1/3)*Log[a^(1/3)*x^(1/3) - (-b + a*x)^(1/3)])/(36*a^(8/3)*b*(-(b*x^2)
+ a*x^3)^(1/3)) + ((9*a^(4/3) + 3*(-1)^(1/3)*a^(2/3)*b^(2/3) + 2*(-1)^(2/3)*b^(4/3))*x^(2/3)*(-b + a*x)^(1/3)
*Log[a^(1/3)*x^(1/3) - (-b + a*x)^(1/3)])/(36*a^(8/3)*b*(-(b*x^2) + a*x^3)^(1/3)) + (x^(2/3)*(-b + a*x)^(1/3)*
Log[a^(1/9)*(a^(2/3) - b^(2/3))^(1/3)*x^(1/3) - (-b + a*x)^(1/3)])/(4*a^(10/9)*(a^(2/3) - b^(2/3))^(1/3)*b*(-(
b*x^2) + a*x^3)^(1/3)) - (x^(2/3)*(-b + a*x)^(1/3)*Log[a^(1/9)*(a^(2/3) + b^(2/3))^(1/3)*x^(1/3) - (-b + a*x)^
(1/3)])/(4*a^(10/9)*(a^(2/3) + b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) - (x^(2/3)*(-b + a*x)^(1/3)*Log[a^(1
/9)*(a^(2/3) - (-1)^(1/3)*b^(2/3))^(1/3)*x^(1/3) - (-b + a*x)^(1/3)])/(4*a^(10/9)*(a^(2/3) - (-1)^(1/3)*b^(2/3
))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) + (x^(2/3)*(-b + a*x)^(1/3)*Log[a^(1/9)*(a^(2/3) + (-1)^(1/3)*b^(2/3))^(1
/3)*x^(1/3) - (-b + a*x)^(1/3)])/(4*a^(10/9)*(a^(2/3) + (-1)^(1/3)*b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3))
+ (x^(2/3)*(-b + a*x)^(1/3)*Log[a^(1/9)*(a^(2/3) - (-1)^(2/3)*b^(2/3))^(1/3)*x^(1/3) - (-b + a*x)^(1/3)])/(4*a
^(10/9)*(a^(2/3) - (-1)^(2/3)*b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) - (x^(2/3)*(-b + a*x)^(1/3)*Log[a^(1/
9)*(a^(2/3) + (-1)^(2/3)*b^(2/3))^(1/3)*x^(1/3) - (-b + a*x)^(1/3)])/(4*a^(10/9)*(a^(2/3) + (-1)^(2/3)*b^(2/3)
)^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3))

Rule 129

Int[((e_.)*(x_))^(p_)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> With[{k = Denominator[p]
}, Dist[k/e, Subst[Int[x^(k*(p + 1) - 1)*(a + b*(x^k/e))^m*(c + d*(x^k/e))^n, x], x, (e*x)^(1/k)], x]] /; Free
Q[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && FractionQ[p] && IntegerQ[m]

Rule 245

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 490

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(2*n -
1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q) + 1))), x] - Dist[e^(2*n)
/(b*d*(m + n*(p + q) + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) + (a*d*(m +
n*(q - 1) + 1) + b*c*(m + n*(p - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d
, 0] && IGtQ[n, 0] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 544

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

Rule 596

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[f*g^(n - 1)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q +
1) + 1))), x] - Dist[g^n/(b*d*(m + n*(p + q + 1) + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*
f*c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*(f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x], x]
/; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 1]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-b+a x} \left (-b^2+a^2 x^6\right )} \, dx}{\sqrt [3]{-b x^2+a x^3}} \\ & = \frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \left (-\frac {x^{7/3}}{2 b \sqrt [3]{-b+a x} \left (b-a x^3\right )}-\frac {x^{7/3}}{2 b \sqrt [3]{-b+a x} \left (b+a x^3\right )}\right ) \, dx}{\sqrt [3]{-b x^2+a x^3}} \\ & = -\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-b+a x} \left (b-a x^3\right )} \, dx}{2 b \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-b+a x} \left (b+a x^3\right )} \, dx}{2 b \sqrt [3]{-b x^2+a x^3}} \\ & = -\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \left (-\frac {x^{7/3}}{3 b^{2/3} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}}-\frac {x^{7/3}}{3 b^{2/3} \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}}-\frac {x^{7/3}}{3 b^{2/3} \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}}\right ) \, dx}{2 b \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \left (\frac {x^{7/3}}{3 b^{2/3} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}}+\frac {x^{7/3}}{3 b^{2/3} \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}}+\frac {x^{7/3}}{3 b^{2/3} \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}}\right ) \, dx}{2 b \sqrt [3]{-b x^2+a x^3}} \\ & = \frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{7/3}}{\left (-\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 b^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{7/3}}{\left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 b^{5/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{7/3}}{\left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 b^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{7/3}}{\left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 b^{5/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{7/3}}{\left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 b^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{7/3}}{\left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 b^{5/3} \sqrt [3]{-b x^2+a x^3}} \\ & = \frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \text {Subst}\left (\int \frac {x^9}{\left (-\sqrt [3]{b}-\sqrt [3]{a} x^3\right ) \sqrt [3]{-b+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 b^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \text {Subst}\left (\int \frac {x^9}{\left (\sqrt [3]{b}-\sqrt [3]{a} x^3\right ) \sqrt [3]{-b+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 b^{5/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \text {Subst}\left (\int \frac {x^9}{\left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x^3\right ) \sqrt [3]{-b+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 b^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \text {Subst}\left (\int \frac {x^9}{\left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x^3\right ) \sqrt [3]{-b+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 b^{5/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \text {Subst}\left (\int \frac {x^9}{\left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x^3\right ) \sqrt [3]{-b+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 b^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \text {Subst}\left (\int \frac {x^9}{\left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x^3\right ) \sqrt [3]{-b+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 b^{5/3} \sqrt [3]{-b x^2+a x^3}} \\ & = \frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \text {Subst}\left (\int \frac {x^3 \left (4 b^{4/3}-2 \sqrt [3]{a} \left (3 a^{2/3}-2 b^{2/3}\right ) \sqrt [3]{b} x^3\right )}{\left (-\sqrt [3]{b}-\sqrt [3]{a} x^3\right ) \sqrt [3]{-b+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{12 a^{4/3} b^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \text {Subst}\left (\int \frac {x^3 \left (-4 b^{4/3}+2 \sqrt [3]{a} \left (3 a^{2/3}+2 b^{2/3}\right ) \sqrt [3]{b} x^3\right )}{\left (\sqrt [3]{b}-\sqrt [3]{a} x^3\right ) \sqrt [3]{-b+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{12 a^{4/3} b^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (\sqrt [3]{-1} x^{2/3} \sqrt [3]{-b+a x}\right ) \text {Subst}\left (\int \frac {x^3 \left (4 b^{4/3}-2 \sqrt [3]{a} \left (3 a^{2/3}-2 (-1)^{2/3} b^{2/3}\right ) \sqrt [3]{b} x^3\right )}{\left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x^3\right ) \sqrt [3]{-b+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{12 a^{4/3} b^{5/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (\sqrt [3]{-1} x^{2/3} \sqrt [3]{-b+a x}\right ) \text {Subst}\left (\int \frac {x^3 \left (-4 b^{4/3}+2 \sqrt [3]{a} \left (3 a^{2/3}+2 (-1)^{2/3} b^{2/3}\right ) \sqrt [3]{b} x^3\right )}{\left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x^3\right ) \sqrt [3]{-b+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{12 a^{4/3} b^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left ((-1)^{2/3} x^{2/3} \sqrt [3]{-b+a x}\right ) \text {Subst}\left (\int \frac {x^3 \left (-4 b^{4/3}+2 \sqrt [3]{a} \left (3 a^{2/3}-2 \sqrt [3]{-1} b^{2/3}\right ) \sqrt [3]{b} x^3\right )}{\left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x^3\right ) \sqrt [3]{-b+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{12 a^{4/3} b^{5/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left ((-1)^{2/3} x^{2/3} \sqrt [3]{-b+a x}\right ) \text {Subst}\left (\int \frac {x^3 \left (4 b^{4/3}-2 \sqrt [3]{a} \left (3 a^{2/3}+2 \sqrt [3]{-1} b^{2/3}\right ) \sqrt [3]{b} x^3\right )}{\left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x^3\right ) \sqrt [3]{-b+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{12 a^{4/3} b^{5/3} \sqrt [3]{-b x^2+a x^3}} \\ & = -\frac {\left (3 a^{2/3}-2 b^{2/3}\right ) x (b-a x)}{18 a^{7/3} b^{4/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\sqrt [3]{-1} \left (3 \sqrt [3]{-1} a^{2/3}-2 b^{2/3}\right ) x (b-a x)}{18 a^{7/3} b^{4/3} \sqrt [3]{-b x^2+a x^3}}-\frac {(-1)^{2/3} \left (3 (-1)^{2/3} a^{2/3}-2 b^{2/3}\right ) x (b-a x)}{18 a^{7/3} b^{4/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (3 a^{2/3}+2 b^{2/3}\right ) x (b-a x)}{18 a^{7/3} b^{4/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\sqrt [3]{-1} \left (3 \sqrt [3]{-1} a^{2/3}+2 b^{2/3}\right ) x (b-a x)}{18 a^{7/3} b^{4/3} \sqrt [3]{-b x^2+a x^3}}-\frac {(-1)^{2/3} \left (3 (-1)^{2/3} a^{2/3}+2 b^{2/3}\right ) x (b-a x)}{18 a^{7/3} b^{4/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \text {Subst}\left (\int \frac {-2 \sqrt [3]{a} \left (3 a^{2/3}-2 b^{2/3}\right ) b^{5/3}+2 a^{2/3} b^{2/3} \left (9 a^{4/3}-3 a^{2/3} b^{2/3}+2 b^{4/3}\right ) x^3}{\left (-\sqrt [3]{b}-\sqrt [3]{a} x^3\right ) \sqrt [3]{-b+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{36 a^{8/3} b^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \text {Subst}\left (\int \frac {-2 \sqrt [3]{a} \left (3 a^{2/3}+2 b^{2/3}\right ) b^{5/3}+2 a^{2/3} b^{2/3} \left (9 a^{4/3}+3 a^{2/3} b^{2/3}+2 b^{4/3}\right ) x^3}{\left (\sqrt [3]{b}-\sqrt [3]{a} x^3\right ) \sqrt [3]{-b+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{36 a^{8/3} b^{5/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (\sqrt [3]{-1} x^{2/3} \sqrt [3]{-b+a x}\right ) \text {Subst}\left (\int \frac {-2 \sqrt [3]{a} \left (3 a^{2/3}-2 \sqrt [3]{-1} b^{2/3}\right ) b^{5/3}+2 a^{2/3} b^{2/3} \left (9 a^{4/3}-3 \sqrt [3]{-1} a^{2/3} b^{2/3}+2 (-1)^{2/3} b^{4/3}\right ) x^3}{\left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x^3\right ) \sqrt [3]{-b+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{36 a^{8/3} b^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (\sqrt [3]{-1} x^{2/3} \sqrt [3]{-b+a x}\right ) \text {Subst}\left (\int \frac {-2 \sqrt [3]{a} \left (3 a^{2/3}+2 \sqrt [3]{-1} b^{2/3}\right ) b^{5/3}+2 a^{2/3} b^{2/3} \left (9 a^{4/3}+3 \sqrt [3]{-1} a^{2/3} b^{2/3}+2 (-1)^{2/3} b^{4/3}\right ) x^3}{\left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x^3\right ) \sqrt [3]{-b+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{36 a^{8/3} b^{5/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left ((-1)^{2/3} x^{2/3} \sqrt [3]{-b+a x}\right ) \text {Subst}\left (\int \frac {-2 \sqrt [3]{a} \left (3 a^{2/3}-2 (-1)^{2/3} b^{2/3}\right ) b^{5/3}+2 a^{2/3} b^{2/3} \left (9 a^{4/3}-3 (-1)^{2/3} a^{2/3} b^{2/3}-2 \sqrt [3]{-1} b^{4/3}\right ) x^3}{\left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x^3\right ) \sqrt [3]{-b+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{36 a^{8/3} b^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left ((-1)^{2/3} x^{2/3} \sqrt [3]{-b+a x}\right ) \text {Subst}\left (\int \frac {-2 \sqrt [3]{a} \left (3 a^{2/3}+2 (-1)^{2/3} b^{2/3}\right ) b^{5/3}+2 a^{2/3} b^{2/3} \left (9 a^{4/3}+3 (-1)^{2/3} a^{2/3} b^{2/3}-2 \sqrt [3]{-1} b^{4/3}\right ) x^3}{\left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x^3\right ) \sqrt [3]{-b+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{36 a^{8/3} b^{5/3} \sqrt [3]{-b x^2+a x^3}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.19 \[ \int \frac {x^3}{\sqrt [3]{-b x^2+a x^3} \left (-b^2+a^2 x^6\right )} \, dx=\frac {x^{2/3} \sqrt [3]{-b+a x} \left (\text {RootSum}\left [a^3-a b^2-3 a^2 \text {$\#$1}^3+3 a \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-b+a x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]-\text {RootSum}\left [a^3+a b^2-3 a^2 \text {$\#$1}^3+3 a \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-b+a x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{6 a b \sqrt [3]{x^2 (-b+a x)}} \]

[In]

Integrate[x^3/((-(b*x^2) + a*x^3)^(1/3)*(-b^2 + a^2*x^6)),x]

[Out]

(x^(2/3)*(-b + a*x)^(1/3)*(RootSum[a^3 - a*b^2 - 3*a^2*#1^3 + 3*a*#1^6 - #1^9 & , (-Log[x^(1/3)] + Log[(-b + a
*x)^(1/3) - x^(1/3)*#1])/#1 & ] - RootSum[a^3 + a*b^2 - 3*a^2*#1^3 + 3*a*#1^6 - #1^9 & , (-Log[x^(1/3)] + Log[
(-b + a*x)^(1/3) - x^(1/3)*#1])/#1 & ]))/(6*a*b*(x^2*(-b + a*x))^(1/3))

Maple [N/A] (verified)

Time = 0.66 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.84

method	result	size
pseudoelliptic	\(\frac {-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{9}-3 a \,\textit {\_Z}^{6}+3 a^{2} \textit {\_Z}^{3}-a^{3}-a \,b^{2}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a x -b \right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right )+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{9}-3 a \,\textit {\_Z}^{6}+3 a^{2} \textit {\_Z}^{3}-a^{3}+a \,b^{2}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a x -b \right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right )}{6 a b}\)	\(131\)

[In]

int(x^3/(a*x^3-b*x^2)^(1/3)/(a^2*x^6-b^2),x,method=_RETURNVERBOSE)

[Out]

1/6*(-sum(ln((-_R*x+(x^2*(a*x-b))^(1/3))/x)/_R,_R=RootOf(_Z^9-3*_Z^6*a+3*_Z^3*a^2-a^3-a*b^2))+sum(ln((-_R*x+(x
^2*(a*x-b))^(1/3))/x)/_R,_R=RootOf(_Z^9-3*_Z^6*a+3*_Z^3*a^2-a^3+a*b^2)))/a/b

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 1.36 (sec) , antiderivative size = 48833, normalized size of antiderivative = 313.03 \[ \int \frac {x^3}{\sqrt [3]{-b x^2+a x^3} \left (-b^2+a^2 x^6\right )} \, dx=\text {Too large to display} \]

[In]

integrate(x^3/(a*x^3-b*x^2)^(1/3)/(a^2*x^6-b^2),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {x^3}{\sqrt [3]{-b x^2+a x^3} \left (-b^2+a^2 x^6\right )} \, dx=\text {Timed out} \]

[In]

integrate(x**3/(a*x**3-b*x**2)**(1/3)/(a**2*x**6-b**2),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.22 \[ \int \frac {x^3}{\sqrt [3]{-b x^2+a x^3} \left (-b^2+a^2 x^6\right )} \, dx=\int { \frac {x^{3}}{{\left (a^{2} x^{6} - b^{2}\right )} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate(x^3/(a*x^3-b*x^2)^(1/3)/(a^2*x^6-b^2),x, algorithm="maxima")

[Out]

integrate(x^3/((a^2*x^6 - b^2)*(a*x^3 - b*x^2)^(1/3)), x)

Giac [N/A]

Not integrable

Time = 12.58 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.22 \[ \int \frac {x^3}{\sqrt [3]{-b x^2+a x^3} \left (-b^2+a^2 x^6\right )} \, dx=\int { \frac {x^{3}}{{\left (a^{2} x^{6} - b^{2}\right )} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate(x^3/(a*x^3-b*x^2)^(1/3)/(a^2*x^6-b^2),x, algorithm="giac")

[Out]

integrate(x^3/((a^2*x^6 - b^2)*(a*x^3 - b*x^2)^(1/3)), x)

Mupad [N/A]

Not integrable

Time = 5.77 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.23 \[ \int \frac {x^3}{\sqrt [3]{-b x^2+a x^3} \left (-b^2+a^2 x^6\right )} \, dx=-\int \frac {x^3}{\left (b^2-a^2\,x^6\right )\,{\left (a\,x^3-b\,x^2\right )}^{1/3}} \,d x \]

[In]

int(-x^3/((b^2 - a^2*x^6)*(a*x^3 - b*x^2)^(1/3)),x)

[Out]

-int(x^3/((b^2 - a^2*x^6)*(a*x^3 - b*x^2)^(1/3)), x)