Optimal. Leaf size=171 \[ -2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} \left (x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3\right )^{3/4}}{(x-a) (b-x)}\right )+2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \left (x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3\right )^{3/4}}{(x-a) (b-x)}\right )-\frac {4 \sqrt [4]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3}}{b-x} \]
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Rubi [C] time = 2.05, antiderivative size = 325, normalized size of antiderivative = 1.90, number of steps used = 7, number of rules used = 4, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {6719, 6728, 137, 136} \begin {gather*} \frac {4 (a-x)^2 (b-x) \left (1-\sqrt {-4 a d+4 b d+1}\right ) \sqrt {-\frac {b-x}{a-b}} F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{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 )}{5 (a-b) \left (-\sqrt {-4 a d+4 b d+1}-2 a d+2 b d+1\right ) \left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}+\frac {4 (a-x)^2 (b-x) \left (\sqrt {-4 a d+4 b d+1}+1\right ) \sqrt {-\frac {b-x}{a-b}} F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{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 )}{5 (a-b) \left (\sqrt {-4 a d+4 b d+1}-2 a d+2 b d+1\right ) \left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 136
Rule 137
Rule 6719
Rule 6728
Rubi steps
\begin {align*} \int \frac {(-a+x) (-2 a+b+x)}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^2 d-(1+2 b d) x+d x^2\right )} \, dx &=\frac {\left ((-a+x)^{3/4} (-b+x)^{3/2}\right ) \int \frac {\sqrt [4]{-a+x} (-2 a+b+x)}{(-b+x)^{3/2} \left (a+b^2 d-(1+2 b d) x+d x^2\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {\left ((-a+x)^{3/4} (-b+x)^{3/2}\right ) \int \left (\frac {\left (1+\sqrt {1-4 a d+4 b d}\right ) \sqrt [4]{-a+x}}{(-b+x)^{3/2} \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 [4]{-a+x}}{(-b+x)^{3/2} \left (-1-2 b d+\sqrt {1-4 a d+4 b d}+2 d x\right )}\right ) \, dx}{\left ((-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {\left (\left (1-\sqrt {1-4 a d+4 b d}\right ) (-a+x)^{3/4} (-b+x)^{3/2}\right ) \int \frac {\sqrt [4]{-a+x}}{(-b+x)^{3/2} \left (-1-2 b d+\sqrt {1-4 a d+4 b d}+2 d x\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{3/4}}+\frac {\left (\left (1+\sqrt {1-4 a d+4 b d}\right ) (-a+x)^{3/4} (-b+x)^{3/2}\right ) \int \frac {\sqrt [4]{-a+x}}{(-b+x)^{3/2} \left (-1-2 b d-\sqrt {1-4 a d+4 b d}+2 d x\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {\left (\left (1-\sqrt {1-4 a d+4 b d}\right ) (-a+x)^{3/4} (-b+x) \sqrt {\frac {-b+x}{a-b}}\right ) \int \frac {\sqrt [4]{-a+x}}{\left (-\frac {b}{a-b}+\frac {x}{a-b}\right )^{3/2} \left (-1-2 b d+\sqrt {1-4 a d+4 b d}+2 d x\right )} \, dx}{(a-b) \left ((-a+x) (-b+x)^2\right )^{3/4}}+\frac {\left (\left (1+\sqrt {1-4 a d+4 b d}\right ) (-a+x)^{3/4} (-b+x) \sqrt {\frac {-b+x}{a-b}}\right ) \int \frac {\sqrt [4]{-a+x}}{\left (-\frac {b}{a-b}+\frac {x}{a-b}\right )^{3/2} \left (-1-2 b d-\sqrt {1-4 a d+4 b d}+2 d x\right )} \, dx}{(a-b) \left ((-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {4 \left (1-\sqrt {1-4 a d+4 b d}\right ) (a-x)^2 (b-x) \sqrt {-\frac {b-x}{a-b}} F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{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 )}{5 (a-b) \left (1-2 a d+2 b d-\sqrt {1-4 a d+4 b d}\right ) \left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}+\frac {4 \left (1+\sqrt {1-4 a d+4 b d}\right ) (a-x)^2 (b-x) \sqrt {-\frac {b-x}{a-b}} F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{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 )}{5 (a-b) \left (1-2 a d+2 b d+\sqrt {1-4 a d+4 b d}\right ) \left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 1.66, size = 492, normalized size = 2.88 \begin {gather*} \frac {2 (b-x) \left ((x-a)^{3/4} \left (\sqrt [4]{b-a} \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]{x-a}}{\sqrt [4]{b-a}}\right )\right |-1\right )+\sqrt [4]{b-a} \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]{x-a}}{\sqrt [4]{b-a}}\right )\right |-1\right )+\sqrt [4]{b-a} \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]{x-a}}{\sqrt [4]{b-a}}\right )\right |-1\right )+\sqrt [4]{b-a} \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]{x-a}}{\sqrt [4]{b-a}}\right )\right |-1\right )+\frac {(b-x) \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 )}{(x-a)^{3/4}}\right )+2 (a-x)\right )}{\left ((x-a) (b-x)^2\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.07, size = 171, normalized size = 1.00 \begin {gather*} -\frac {4 \sqrt [4]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{b-x}-2 \sqrt [4]{d} \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 )+2 \sqrt [4]{d} \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 ) \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 - x\right )} {\left (a - x\right )}}{\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 = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (-a +x \right ) \left (-2 a +b +x \right )}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {3}{4}} \left (a +b^{2} d -\left (2 b d +1\right ) x +d \,x^{2}\right )}\, dx\]
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 - x\right )} {\left (a - x\right )}}{\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 {\left (a-x\right )\,\left (b-2\,a+x\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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