Integrand size = 72, antiderivative size = 232 \[ \int \frac {2 a b^2 x-b (2 a+b) x^2+x^4}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{d^{2/3}}+\frac {\log \left (-\sqrt [3]{d} x+\sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}\right )}{d^{2/3}}-\frac {\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}+\left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}\right )}{2 d^{2/3}} \]
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\[ \int \frac {2 a b^2 x-b (2 a+b) x^2+x^4}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx=\int \frac {2 a b^2 x-b (2 a+b) x^2+x^4}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (2 a b^2-b (2 a+b) x+x^3\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx \\ & = \frac {\left (x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {\sqrt [3]{x} \left (2 a b^2-b (2 a+b) x+x^3\right )}{(-a+x)^{2/3} (-b+x)^{4/3} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}} \\ & = \frac {\left (x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {\sqrt [3]{x} \left (-2 a b+b x+x^2\right )}{(-a+x)^{2/3} \sqrt [3]{-b+x} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}} \\ & = \frac {\left (3 x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \frac {x^3 \left (-2 a b+b x^3+x^6\right )}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (-a b^2+b (2 a+b) x^3-(a+2 b+d) x^6+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3}} \\ & = \frac {\left (3 x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \left (\frac {1}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3}}+\frac {a b^2-b (4 a+b) x^3+(a+3 b+d) x^6}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (-a b^2+b (2 a+b) x^3-(a+2 b+d) x^6+x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3}} \\ & = \frac {\left (3 x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \frac {1}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3}} \, dx,x,\sqrt [3]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (3 x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \frac {a b^2-b (4 a+b) x^3+(a+3 b+d) x^6}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (-a b^2+b (2 a+b) x^3-(a+2 b+d) x^6+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3}} \\ & = \frac {\left (3 x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \left (\frac {b (4 a+b) x^3}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^3+a \left (1+\frac {2 b+d}{a}\right ) x^6-x^9\right )}+\frac {(-a-3 b-d) x^6}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^3+a \left (1+\frac {2 b+d}{a}\right ) x^6-x^9\right )}+\frac {a b^2}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (-a b^2+2 a b \left (1+\frac {b}{2 a}\right ) x^3-a \left (1+\frac {2 b+d}{a}\right ) x^6+x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (3 x^{2/3} (-b+x)^{4/3} \left (1-\frac {x}{a}\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-b+x^3} \left (1-\frac {x^3}{a}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3}} \\ & = \frac {\left (3 a b^2 x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \frac {1}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (-a b^2+2 a b \left (1+\frac {b}{2 a}\right ) x^3-a \left (1+\frac {2 b+d}{a}\right ) x^6+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (3 b (4 a+b) x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^3+a \left (1+\frac {2 b+d}{a}\right ) x^6-x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (3 (-a-3 b-d) x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \frac {x^6}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^3+a \left (1+\frac {2 b+d}{a}\right ) x^6-x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (3 x^{2/3} (-b+x) \left (1-\frac {x}{a}\right )^{2/3} \sqrt [3]{1-\frac {x}{b}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {x^3}{a}\right )^{2/3} \sqrt [3]{1-\frac {x^3}{b}}} \, dx,x,\sqrt [3]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3}} \\ & = -\frac {3 (b-x) x \left (1-\frac {x}{a}\right )^{2/3} \sqrt [3]{1-\frac {x}{b}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},\frac {1}{3},\frac {4}{3},\frac {x}{a},\frac {x}{b}\right )}{\left (-\left ((a-x) (b-x)^2 x\right )\right )^{2/3}}+\frac {\left (3 a b^2 x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \frac {1}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (-a b^2+2 a b \left (1+\frac {b}{2 a}\right ) x^3-a \left (1+\frac {2 b+d}{a}\right ) x^6+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (3 b (4 a+b) x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^3+a \left (1+\frac {2 b+d}{a}\right ) x^6-x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (3 (-a-3 b-d) x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \frac {x^6}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^3+a \left (1+\frac {2 b+d}{a}\right ) x^6-x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3}} \\ \end{align*}
\[ \int \frac {2 a b^2 x-b (2 a+b) x^2+x^4}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx=\int \frac {2 a b^2 x-b (2 a+b) x^2+x^4}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx \]
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Time = 0.76 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.58
method | result | size |
pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{3}} x +2 \left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{3}}\right )}{3 d^{\frac {1}{3}} x}\right )-\ln \left (\frac {d^{\frac {2}{3}} x^{2}+d^{\frac {1}{3}} \left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{3}} x +\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )+2 \ln \left (\frac {-d^{\frac {1}{3}} x +\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{3}}}{x}\right )}{2 d^{\frac {2}{3}}}\) | \(134\) |
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Timed out. \[ \int \frac {2 a b^2 x-b (2 a+b) x^2+x^4}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {2 a b^2 x-b (2 a+b) x^2+x^4}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {2 a b^2 x-b (2 a+b) x^2+x^4}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx=\int { -\frac {2 \, a b^{2} x - {\left (2 \, a + b\right )} b x^{2} + x^{4}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {2}{3}} {\left (a b^{2} - {\left (2 \, a + b\right )} b x + {\left (a + 2 \, b + d\right )} x^{2} - x^{3}\right )}} \,d x } \]
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\[ \int \frac {2 a b^2 x-b (2 a+b) x^2+x^4}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx=\int { -\frac {2 \, a b^{2} x - {\left (2 \, a + b\right )} b x^{2} + x^{4}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {2}{3}} {\left (a b^{2} - {\left (2 \, a + b\right )} b x + {\left (a + 2 \, b + d\right )} x^{2} - x^{3}\right )}} \,d x } \]
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Timed out. \[ \int \frac {2 a b^2 x-b (2 a+b) x^2+x^4}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx=\int -\frac {x^4-b\,x^2\,\left (2\,a+b\right )+2\,a\,b^2\,x}{{\left (-x\,\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{2/3}\,\left (x^2\,\left (a+2\,b+d\right )+a\,b^2-x^3-b\,x\,\left (2\,a+b\right )\right )} \,d x \]
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