Optimal. Leaf size=165 \[ -2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3}}{b-x}\right )+2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3}}{b-x}\right )-\frac {4 \left (-a b^2+2 a b x-a x^2+b^2 x-2 b x^2+x^3\right )^{3/4}}{(x-a) (b-x)} \]
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Rubi [C] time = 3.34, antiderivative size = 325, normalized size of antiderivative = 1.97, number of steps used = 8, number of rules used = 5, integrand size = 81, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6688, 6719, 6728, 137, 136} \begin {gather*} \frac {4 (a-b) (b-x) \left (\sqrt {-4 a+4 b+d}+\sqrt {d}\right ) F_1\left (-\frac {1}{4};-\frac {3}{2},1;\frac {3}{4};\frac {a-x}{a-b},\frac {2 (a-x)}{2 a-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right )}{\sqrt {d} \left (-\sqrt {d} \sqrt {-4 a+4 b+d}+2 a-2 b-d\right ) \sqrt {-\frac {b-x}{a-b}} \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}+\frac {4 (a-b) (b-x) \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) F_1\left (-\frac {1}{4};-\frac {3}{2},1;\frac {3}{4};\frac {a-x}{a-b},\frac {2 (a-x)}{2 a-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right )}{\left (\sqrt {d} \sqrt {-4 a+4 b+d}+2 a-2 b-d\right ) \sqrt {-\frac {b-x}{a-b}} \sqrt [4]{-\left ((a-x) (b-x)^2\right )}} \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}{(-a+x) \sqrt [4]{(-a+x) (-b+x)^2} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx &=\int \frac {(b-x)^4 (-2 a+b+x)}{\left (-\left ((a-x) (b-x)^2\right )\right )^{5/4} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx\\ &=-\frac {\left (\sqrt [4]{a-x} \sqrt {b-x}\right ) \int \frac {(b-x)^{3/2} (-2 a+b+x)}{(a-x)^{5/4} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\\ &=-\frac {\left (\sqrt [4]{a-x} \sqrt {b-x}\right ) \int \left (\frac {\left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) (b-x)^{3/2}}{(a-x)^{5/4} \left (-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right )}+\frac {\left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) (b-x)^{3/2}}{(a-x)^{5/4} \left (-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right )}\right ) \, dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\\ &=-\frac {\left (\left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{a-x} \sqrt {b-x}\right ) \int \frac {(b-x)^{3/2}}{(a-x)^{5/4} \left (-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right )} \, dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}-\frac {\left (\left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{a-x} \sqrt {b-x}\right ) \int \frac {(b-x)^{3/2}}{(a-x)^{5/4} \left (-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right )} \, dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\\ &=\frac {\left ((a-b) \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{a-x} (b-x)\right ) \int \frac {\left (-\frac {b}{a-b}+\frac {x}{a-b}\right )^{3/2}}{(a-x)^{5/4} \left (-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right )} \, dx}{\sqrt {-\frac {b-x}{a-b}} \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}+\frac {\left ((a-b) \left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{a-x} (b-x)\right ) \int \frac {\left (-\frac {b}{a-b}+\frac {x}{a-b}\right )^{3/2}}{(a-x)^{5/4} \left (-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right )} \, dx}{\sqrt {-\frac {b-x}{a-b}} \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\\ &=\frac {4 (a-b) \left (\sqrt {d}+\sqrt {-4 a+4 b+d}\right ) (b-x) F_1\left (-\frac {1}{4};-\frac {3}{2},1;\frac {3}{4};\frac {a-x}{a-b},\frac {2 (a-x)}{2 a-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right )}{\sqrt {d} \left (2 a-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt {-\frac {b-x}{a-b}} \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}+\frac {4 (a-b) \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) (b-x) F_1\left (-\frac {1}{4};-\frac {3}{2},1;\frac {3}{4};\frac {a-x}{a-b},\frac {2 (a-x)}{2 a-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right )}{\left (2 a-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt {-\frac {b-x}{a-b}} \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\\ \end {align*}
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Mathematica [C] time = 18.73, size = 736, normalized size = 4.46 \begin {gather*} -\frac {4 (b-x)}{\sqrt [4]{(x-a) (b-x)^2}}-\frac {i \sqrt {2} d (a-x)^{11/4} (b-x)^3 \sqrt {\frac {b}{a-x}-\frac {a}{a-x}+1} \left (\left (\sqrt {d (-4 a+4 b+d)}+4 a-4 b-d\right ) \sqrt {-\frac {\sqrt {d (-4 a+4 b+d)}-2 a+2 b+d}{(a-b)^2}} \Pi \left (-\frac {\sqrt {2}}{\sqrt {a-b} \sqrt {\frac {2 a-2 b-d+\sqrt {d (-4 a+4 b+d)}}{(a-b)^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\sqrt {a-b}}}{\sqrt [4]{a-x}}\right )\right |-1\right )-\left (\sqrt {d (-4 a+4 b+d)}+4 a-4 b-d\right ) \sqrt {-\frac {\sqrt {d (-4 a+4 b+d)}-2 a+2 b+d}{(a-b)^2}} \Pi \left (\frac {\sqrt {2}}{\sqrt {a-b} \sqrt {\frac {2 a-2 b-d+\sqrt {d (-4 a+4 b+d)}}{(a-b)^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\sqrt {a-b}}}{\sqrt [4]{a-x}}\right )\right |-1\right )-\left (-\sqrt {d (-4 a+4 b+d)}+4 a-4 b-d\right ) \sqrt {\frac {\sqrt {d (-4 a+4 b+d)}+2 a-2 b-d}{(a-b)^2}} \left (\Pi \left (-\frac {\sqrt {2}}{\sqrt {a-b} \sqrt {-\frac {-2 a+2 b+d+\sqrt {d (-4 a+4 b+d)}}{(a-b)^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\sqrt {a-b}}}{\sqrt [4]{a-x}}\right )\right |-1\right )-\Pi \left (\frac {\sqrt {2}}{\sqrt {a-b} \sqrt {-\frac {-2 a+2 b+d+\sqrt {d (-4 a+4 b+d)}}{(a-b)^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\sqrt {a-b}}}{\sqrt [4]{a-x}}\right )\right |-1\right )\right )\right )}{\sqrt {-\sqrt {a-b}} (a-b) (x-a) (x-b) \sqrt {d (-4 a+4 b+d)} \sqrt {\frac {\sqrt {d (-4 a+4 b+d)}+2 a-2 b-d}{(a-b)^2}} \sqrt {-\frac {\sqrt {d (-4 a+4 b+d)}-2 a+2 b+d}{(a-b)^2}} \left (-\left ((a-x) (b-x)^2\right )\right )^{5/4}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.92, size = 165, normalized size = 1.00 \begin {gather*} -\frac {4 \left (-a b^2+2 a b x+b^2 x-a x^2-2 b x^2+x^3\right )^{3/4}}{(b-x) (-a+x)}-2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{b-x}\right )+2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{b-x}\right ) \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 {1}{4}} {\left (b^{2} + a d - {\left (2 \, b + d\right )} x + x^{2}\right )} {\left (a - x\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {-\left (2 a -b \right ) b^{2}+\left (4 a -b \right ) b x -\left (2 a +b \right ) x^{2}+x^{3}}{\left (-a +x \right ) \left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{4}} \left (b^{2}+a d -\left (2 b +d \right ) x +x^{2}\right )}\, dx\]
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 {1}{4}} {\left (b^{2} + a d - {\left (2 \, b + d\right )} x + x^{2}\right )} {\left (a - x\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 (a-x\right )\,{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/4}\,\left (a\,d-x\,\left (2\,b+d\right )+b^2+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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