Optimal. Leaf size=127 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \left (x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3\right )^{3/4}}{(x-a) (b-x)}\right )}{d^{3/4}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \left (x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3\right )^{3/4}}{(x-a) (b-x)}\right )}{d^{3/4}} \]
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Rubi [C] time = 2.73, antiderivative size = 283, normalized size of antiderivative = 2.23, number of steps used = 8, number of rules used = 5, integrand size = 77, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {6688, 6719, 6728, 137, 136} \begin {gather*} -\frac {4 \left (1-\sqrt {-4 a d+4 b d+1}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )} F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};\frac {a-x}{a-b},-\frac {2 d (a-x)}{-2 a d+2 b d-\sqrt {-4 a d+4 b d+1}+1}\right )}{\left (-\sqrt {-4 a d+4 b d+1}-2 a d+2 b d+1\right ) \sqrt {-\frac {b-x}{a-b}}}-\frac {4 \left (\sqrt {-4 a d+4 b d+1}+1\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )} F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};\frac {a-x}{a-b},-\frac {2 d (a-x)}{-2 a d+2 b d+\sqrt {-4 a d+4 b d+1}+1}\right )}{\left (\sqrt {-4 a d+4 b d+1}-2 a d+2 b d+1\right ) \sqrt {-\frac {b-x}{a-b}}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 136
Rule 137
Rule 6688
Rule 6719
Rule 6728
Rubi steps
\begin {align*} \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^2 d-(1+2 b d) x+d x^2\right )} \, dx &=\int \frac {(2 a-b-x) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}{(a-x) \left (a+b^2 d-(1+2 b d) x+d x^2\right )} \, dx\\ &=\frac {\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \int \frac {(2 a-b-x) \sqrt {b-x}}{(a-x)^{3/4} \left (a+b^2 d-(1+2 b d) x+d x^2\right )} \, dx}{\sqrt [4]{a-x} \sqrt {b-x}}\\ &=\frac {\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \int \left (\frac {\left (-1-\sqrt {1-4 a d+4 b d}\right ) \sqrt {b-x}}{(a-x)^{3/4} \left (-1-2 b d-\sqrt {1-4 a d+4 b d}+2 d x\right )}+\frac {\left (-1+\sqrt {1-4 a d+4 b d}\right ) \sqrt {b-x}}{(a-x)^{3/4} \left (-1-2 b d+\sqrt {1-4 a d+4 b d}+2 d x\right )}\right ) \, dx}{\sqrt [4]{a-x} \sqrt {b-x}}\\ &=\frac {\left (\left (-1-\sqrt {1-4 a d+4 b d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}\right ) \int \frac {\sqrt {b-x}}{(a-x)^{3/4} \left (-1-2 b d-\sqrt {1-4 a d+4 b d}+2 d x\right )} \, dx}{\sqrt [4]{a-x} \sqrt {b-x}}+\frac {\left (\left (-1+\sqrt {1-4 a d+4 b d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}\right ) \int \frac {\sqrt {b-x}}{(a-x)^{3/4} \left (-1-2 b d+\sqrt {1-4 a d+4 b d}+2 d x\right )} \, dx}{\sqrt [4]{a-x} \sqrt {b-x}}\\ &=\frac {\left (\left (-1-\sqrt {1-4 a d+4 b d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}\right ) \int \frac {\sqrt {-\frac {b}{a-b}+\frac {x}{a-b}}}{(a-x)^{3/4} \left (-1-2 b d-\sqrt {1-4 a d+4 b d}+2 d x\right )} \, dx}{\sqrt [4]{a-x} \sqrt {-\frac {b-x}{a-b}}}+\frac {\left (\left (-1+\sqrt {1-4 a d+4 b d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}\right ) \int \frac {\sqrt {-\frac {b}{a-b}+\frac {x}{a-b}}}{(a-x)^{3/4} \left (-1-2 b d+\sqrt {1-4 a d+4 b d}+2 d x\right )} \, dx}{\sqrt [4]{a-x} \sqrt {-\frac {b-x}{a-b}}}\\ &=-\frac {4 \left (1-\sqrt {1-4 a d+4 b d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )} F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};\frac {a-x}{a-b},-\frac {2 d (a-x)}{1-2 a d+2 b d-\sqrt {1-4 a d+4 b d}}\right )}{\left (1-2 a d+2 b d-\sqrt {1-4 a d+4 b d}\right ) \sqrt {-\frac {b-x}{a-b}}}-\frac {4 \left (1+\sqrt {1-4 a d+4 b d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )} F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};\frac {a-x}{a-b},-\frac {2 d (a-x)}{1-2 a d+2 b d+\sqrt {1-4 a d+4 b d}}\right )}{\left (1-2 a d+2 b d+\sqrt {1-4 a d+4 b d}\right ) \sqrt {-\frac {b-x}{a-b}}}\\ \end {align*}
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Mathematica [C] time = 1.55, size = 488, normalized size = 3.84 \begin {gather*} \frac {2 \sqrt [4]{(x-a) (b-x)^2} \left (\sqrt [4]{a-b} \sqrt {\frac {x-b}{a-b}} \Pi \left (-\frac {\sqrt {2} \sqrt {a-b} \sqrt {d}}{\sqrt {2 a d-2 b d-\sqrt {-4 a d+4 b d+1}-1}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a-x}}{\sqrt [4]{a-b}}\right )\right |-1\right )+\sqrt [4]{a-b} \sqrt {\frac {x-b}{a-b}} \Pi \left (\frac {\sqrt {2} \sqrt {a-b} \sqrt {d}}{\sqrt {2 a d-2 b d-\sqrt {-4 a d+4 b d+1}-1}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a-x}}{\sqrt [4]{a-b}}\right )\right |-1\right )+\sqrt [4]{a-b} \sqrt {\frac {x-b}{a-b}} \Pi \left (-\frac {\sqrt {2} \sqrt {a-b} \sqrt {d}}{\sqrt {2 a d-2 b d+\sqrt {-4 a d+4 b d+1}-1}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a-x}}{\sqrt [4]{a-b}}\right )\right |-1\right )+\sqrt [4]{a-b} \sqrt {\frac {x-b}{a-b}} \Pi \left (\frac {\sqrt {2} \sqrt {a-b} \sqrt {d}}{\sqrt {2 a d-2 b d+\sqrt {-4 a d+4 b d+1}-1}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a-x}}{\sqrt [4]{a-b}}\right )\right |-1\right )+\frac {(x-b) \left (\frac {a-x}{a-b}\right )^{3/4} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};\frac {x-b}{a-b}\right )}{(a-x)^{3/4}}\right )}{d \sqrt [4]{a-x} (b-x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.51, size = 127, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{3/4}}{(b-x) (-a+x)}\right )}{d^{3/4}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{3/4}}{(b-x) (-a+x)}\right )}{d^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (2 \, a - b\right )} b^{2} - {\left (4 \, a - b\right )} b x + {\left (2 \, a + b\right )} x^{2} - x^{3}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {3}{4}} {\left (b^{2} d + d x^{2} - {\left (2 \, b d + 1\right )} x + a\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {-\left (2 a -b \right ) b^{2}+\left (4 a -b \right ) b x -\left (2 a +b \right ) x^{2}+x^{3}}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {3}{4}} \left (a +b^{2} d -\left (2 b d +1\right ) x +d \,x^{2}\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (2 \, a - b\right )} b^{2} - {\left (4 \, a - b\right )} b x + {\left (2 \, a + b\right )} x^{2} - x^{3}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {3}{4}} {\left (b^{2} d + d x^{2} - {\left (2 \, b d + 1\right )} x + a\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {b^2\,\left (2\,a-b\right )+x^2\,\left (2\,a+b\right )-x^3-b\,x\,\left (4\,a-b\right )}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{3/4}\,\left (a-x\,\left (2\,b\,d+1\right )+b^2\,d+d\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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