Integrand size = 63, antiderivative size = 79 \[ \int \frac {-a b x+x^3}{(-a+x) (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx=\frac {2 \sqrt {a b x-a x^2-b x^2+x^3}}{(-a+x) (-b+x)}-2 \sqrt {d} \text {arctanh}\left (\frac {x}{\sqrt {d} \sqrt {a b x+(-a-b) x^2+x^3}}\right ) \]
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\[ \int \frac {-a b x+x^3}{(-a+x) (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx=\int \frac {-a b x+x^3}{(-a+x) (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (-a b+x^2\right )}{(-a+x) (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx \\ & = \int \frac {x^2 \left (-a b+x^2\right )}{(x (-a+x) (-b+x))^{3/2} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx \\ & = \frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {x} \left (-a b+x^2\right )}{(-a+x)^{3/2} (-b+x)^{3/2} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx}{\sqrt {x (-a+x) (-b+x)}} \\ & = \frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \left (\frac {\sqrt {x}}{d (-a+x)^{3/2} (-b+x)^{3/2}}-\frac {\sqrt {x} (2 a b d-(1+a d+b d) x)}{d (-a+x)^{3/2} (-b+x)^{3/2} \left (a b d+(-1-a d-b d) x+d x^2\right )}\right ) \, dx}{\sqrt {x (-a+x) (-b+x)}} \\ & = \frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {x}}{(-a+x)^{3/2} (-b+x)^{3/2}} \, dx}{d \sqrt {x (-a+x) (-b+x)}}-\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {x} (2 a b d-(1+a d+b d) x)}{(-a+x)^{3/2} (-b+x)^{3/2} \left (a b d+(-1-a d-b d) x+d x^2\right )} \, dx}{d \sqrt {x (-a+x) (-b+x)}} \\ & = -\frac {2 x}{(a-b) d \sqrt {(a-x) (b-x) x}}-\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \left (\frac {\left (-1-a d-b d-\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}\right ) \sqrt {x}}{(-a+x)^{3/2} (-b+x)^{3/2} \left (-1-a d-b d-\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )}+\frac {\left (-1-a d-b d+\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}\right ) \sqrt {x}}{(-a+x)^{3/2} (-b+x)^{3/2} \left (-1-a d-b d+\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )}\right ) \, dx}{d \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {-\frac {b}{2}-\frac {x}{2}}{\sqrt {x} \sqrt {-a+x} (-b+x)^{3/2}} \, dx}{(a-b) d \sqrt {x (-a+x) (-b+x)}} \\ & = -\frac {2 x}{(a-b) d \sqrt {(a-x) (b-x) x}}+\frac {4 (a-x) x}{(a-b)^2 d \sqrt {(a-x) (b-x) x}}+\frac {\left (4 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {-\frac {1}{4} b (a+b)+\frac {b x}{2}}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}} \, dx}{(a-b)^2 b d \sqrt {x (-a+x) (-b+x)}}-\frac {\left (\left (-1-a d-b d-\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {x}}{(-a+x)^{3/2} (-b+x)^{3/2} \left (-1-a d-b d-\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {x (-a+x) (-b+x)}}-\frac {\left (\left (-1-a d-b d+\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {x}}{(-a+x)^{3/2} (-b+x)^{3/2} \left (-1-a d-b d+\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {x (-a+x) (-b+x)}} \\ & = -\frac {2 x}{(a-b) d \sqrt {(a-x) (b-x) x}}+\frac {4 (a-x) x}{(a-b)^2 d \sqrt {(a-x) (b-x) x}}+\frac {\left (2 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-b+x}}{\sqrt {x} \sqrt {-a+x}} \, dx}{(a-b)^2 d \sqrt {x (-a+x) (-b+x)}}-\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {1}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}} \, dx}{(a-b) d \sqrt {x (-a+x) (-b+x)}}-\frac {\left (\left (-1-a d-b d-\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {x}}{(-a+x)^{3/2} (-b+x)^{3/2} \left (-1-a d-b d-\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {x (-a+x) (-b+x)}}-\frac {\left (\left (-1-a d-b d+\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {x}}{(-a+x)^{3/2} (-b+x)^{3/2} \left (-1-a d-b d+\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {x (-a+x) (-b+x)}} \\ & = -\frac {2 x}{(a-b) d \sqrt {(a-x) (b-x) x}}+\frac {4 (a-x) x}{(a-b)^2 d \sqrt {(a-x) (b-x) x}}-\frac {\left (\left (-1-a d-b d-\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {x}}{(-a+x)^{3/2} (-b+x)^{3/2} \left (-1-a d-b d-\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {x (-a+x) (-b+x)}}-\frac {\left (\left (-1-a d-b d+\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {x}}{(-a+x)^{3/2} (-b+x)^{3/2} \left (-1-a d-b d+\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 \sqrt {x} (-b+x) \sqrt {1-\frac {x}{a}}\right ) \int \frac {\sqrt {1-\frac {x}{b}}}{\sqrt {x} \sqrt {1-\frac {x}{a}}} \, dx}{(a-b)^2 d \sqrt {x (-a+x) (-b+x)} \sqrt {1-\frac {x}{b}}}-\frac {\left (\sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}} \, dx}{(a-b) d \sqrt {x (-a+x) (-b+x)}} \\ & = -\frac {2 x}{(a-b) d \sqrt {(a-x) (b-x) x}}+\frac {4 (a-x) x}{(a-b)^2 d \sqrt {(a-x) (b-x) x}}-\frac {4 \sqrt {a} (b-x) \sqrt {x} \sqrt {1-\frac {x}{a}} E\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{(a-b)^2 d \sqrt {(a-x) (b-x) x} \sqrt {1-\frac {x}{b}}}-\frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {a}}\right ),\frac {a}{b}\right )}{(a-b) d \sqrt {(a-x) (b-x) x}}-\frac {\left (\left (-1-a d-b d-\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {x}}{(-a+x)^{3/2} (-b+x)^{3/2} \left (-1-a d-b d-\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {x (-a+x) (-b+x)}}-\frac {\left (\left (-1-a d-b d+\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {x}}{(-a+x)^{3/2} (-b+x)^{3/2} \left (-1-a d-b d+\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {x (-a+x) (-b+x)}} \\ \end{align*}
Time = 15.39 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.65 \[ \int \frac {-a b x+x^3}{(-a+x) (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx=\frac {2 x}{\sqrt {x (-a+x) (-b+x)}}-2 \sqrt {d} \text {arctanh}\left (\frac {x}{\sqrt {d} \sqrt {x (-a+x) (-b+x)}}\right ) \]
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Time = 3.45 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.76
method | result | size |
default | \(\frac {-2 \sqrt {d}\, \operatorname {arctanh}\left (\frac {\sqrt {d}\, \sqrt {x \left (a -x \right ) \left (b -x \right )}}{x}\right ) \sqrt {x \left (a -x \right ) \left (b -x \right )}+2 x}{\sqrt {x \left (a -x \right ) \left (b -x \right )}}\) | \(60\) |
pseudoelliptic | \(\frac {-2 \sqrt {d}\, \operatorname {arctanh}\left (\frac {\sqrt {d}\, \sqrt {x \left (a -x \right ) \left (b -x \right )}}{x}\right ) \sqrt {x \left (a -x \right ) \left (b -x \right )}+2 x}{\sqrt {x \left (a -x \right ) \left (b -x \right )}}\) | \(60\) |
elliptic | \(\text {Expression too large to display}\) | \(4026\) |
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Time = 0.48 (sec) , antiderivative size = 395, normalized size of antiderivative = 5.00 \[ \int \frac {-a b x+x^3}{(-a+x) (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx=\left [\frac {{\left (a b - {\left (a + b\right )} x + x^{2}\right )} \sqrt {d} \log \left (\frac {a^{2} b^{2} d^{2} + d^{2} x^{4} - 2 \, {\left ({\left (a + b\right )} d^{2} - 3 \, d\right )} x^{3} + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d^{2} - 6 \, {\left (a + b\right )} d + 1\right )} x^{2} - 4 \, {\left (a b d + d x^{2} - {\left ({\left (a + b\right )} d - 1\right )} x\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {d} + 2 \, {\left (3 \, a b d - {\left (a^{2} b + a b^{2}\right )} d^{2}\right )} x}{a^{2} b^{2} d^{2} + d^{2} x^{4} - 2 \, {\left ({\left (a + b\right )} d^{2} + d\right )} x^{3} + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d^{2} + 2 \, {\left (a + b\right )} d + 1\right )} x^{2} - 2 \, {\left (a b d + {\left (a^{2} b + a b^{2}\right )} d^{2}\right )} x}\right ) + 4 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}}}{2 \, {\left (a b - {\left (a + b\right )} x + x^{2}\right )}}, \frac {{\left (a b - {\left (a + b\right )} x + x^{2}\right )} \sqrt {-d} \arctan \left (\frac {{\left (a b d + d x^{2} - {\left ({\left (a + b\right )} d - 1\right )} x\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {-d}}{2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}\right ) + 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}}}{a b - {\left (a + b\right )} x + x^{2}}\right ] \]
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Timed out. \[ \int \frac {-a b x+x^3}{(-a+x) (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {-a b x+x^3}{(-a+x) (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx=\int { -\frac {a b x - x^{3}}{{\left (a b d + d x^{2} - {\left (a d + b d + 1\right )} x\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x} {\left (a - x\right )} {\left (b - x\right )}} \,d x } \]
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\[ \int \frac {-a b x+x^3}{(-a+x) (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx=\int { -\frac {a b x - x^{3}}{{\left (a b d + d x^{2} - {\left (a d + b d + 1\right )} x\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x} {\left (a - x\right )} {\left (b - x\right )}} \,d x } \]
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Time = 6.05 (sec) , antiderivative size = 695, normalized size of antiderivative = 8.80 \[ \int \frac {-a b x+x^3}{(-a+x) (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx=-\frac {2\,a\,\sqrt {\frac {x}{a}}\,\left (\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {x}{a}}\right )\middle |\frac {a}{b}\right )-\frac {a\,\sin \left (2\,\mathrm {asin}\left (\sqrt {\frac {x}{a}}\right )\right )}{2\,b\,\sqrt {1-\frac {x}{b}}}\right )\,\sqrt {\frac {a-x}{a}}\,\sqrt {\frac {b-x}{b}}}{\left (\frac {a}{b}-1\right )\,\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}-\frac {b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (\frac {b}{b-\frac {a\,d+b\,d+\sqrt {a^2\,d^2-2\,a\,b\,d^2+2\,a\,d+b^2\,d^2+2\,b\,d+1}+1}{2\,d}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (a\,d+b\,d+\sqrt {a^2\,d^2-2\,a\,b\,d^2+2\,a\,d+b^2\,d^2+2\,b\,d+1}+1\right )}{d\,\left (b-\frac {a\,d+b\,d+\sqrt {a^2\,d^2-2\,a\,b\,d^2+2\,a\,d+b^2\,d^2+2\,b\,d+1}+1}{2\,d}\right )\,\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}-\frac {b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (\frac {b}{b-\frac {a\,d+b\,d-\sqrt {a^2\,d^2-2\,a\,b\,d^2+2\,a\,d+b^2\,d^2+2\,b\,d+1}+1}{2\,d}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (a\,d+b\,d-\sqrt {a^2\,d^2-2\,a\,b\,d^2+2\,a\,d+b^2\,d^2+2\,b\,d+1}+1\right )}{d\,\left (b-\frac {a\,d+b\,d-\sqrt {a^2\,d^2-2\,a\,b\,d^2+2\,a\,d+b^2\,d^2+2\,b\,d+1}+1}{2\,d}\right )\,\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}-\frac {2\,a\,b\,\left (\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )+\frac {b\,\sin \left (2\,\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\right )}{2\,\sqrt {\frac {b-x}{a-b}+1}\,\left (a-b\right )}\right )\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}}{\left (\frac {b}{a-b}+1\right )\,\left (a-b\right )\,\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}} \]
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