Optimal. Leaf size=133 \[ 2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} \left (x^3 (-a-b)+a b x^2+x^4\right )^{3/4}}{x^2 (x-b)}\right )-2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \left (x^3 (-a-b)+a b x^2+x^4\right )^{3/4}}{x^2 (x-b)}\right )-\frac {4 \sqrt [4]{x^3 (-a-b)+a b x^2+x^4}}{a-x} \]
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Rubi [F] time = 26.53, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x^3 (-b+x) (-2 a b+(3 a-b) x)}{(-a+x) \left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a^3 d+3 a^2 d x+(b-3 a d) x^2+(-1+d) x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^3 (-b+x) (-2 a b+(3 a-b) x)}{(-a+x) \left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a^3 d+3 a^2 d x+(b-3 a d) x^2+(-1+d) x^3\right )} \, dx &=\frac {\left (x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \int \frac {x^{3/2} \sqrt [4]{-b+x} (-2 a b+(3 a-b) x)}{(-a+x)^{7/4} \left (-a^3 d+3 a^2 d x+(b-3 a d) x^2+(-1+d) x^3\right )} \, dx}{\left (x^2 (-a+x) (-b+x)\right )^{3/4}}\\ &=\frac {\left (2 x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt [4]{-b+x^2} \left (-2 a b+(3 a-b) x^2\right )}{\left (-a+x^2\right )^{7/4} \left (-a^3 d+3 a^2 d x^2+(b-3 a d) x^4+(-1+d) x^6\right )} \, dx,x,\sqrt {x}\right )}{\left (x^2 (-a+x) (-b+x)\right )^{3/4}}\\ &=\frac {\left (2 x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (-\frac {(3 a-b) \sqrt [4]{-b+x^2}}{(1-d) \left (-a+x^2\right )^{7/4}}+\frac {\sqrt [4]{-b+x^2} \left (a^3 (3 a-b) d-3 a^2 (3 a-b) d x^2+\left (b^2+9 a^2 d-a (b+5 b d)\right ) x^4\right )}{(-1+d) \left (-a+x^2\right )^{7/4} \left (-a^3 d+3 a^2 d x^2+(b-3 a d) x^4+(-1+d) x^6\right )}\right ) \, dx,x,\sqrt {x}\right )}{\left (x^2 (-a+x) (-b+x)\right )^{3/4}}\\ &=-\frac {\left (2 (3 a-b) x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{-b+x^2}}{\left (-a+x^2\right )^{7/4}} \, dx,x,\sqrt {x}\right )}{(1-d) \left (x^2 (-a+x) (-b+x)\right )^{3/4}}+\frac {\left (2 x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{-b+x^2} \left (a^3 (3 a-b) d-3 a^2 (3 a-b) d x^2+\left (b^2+9 a^2 d-a (b+5 b d)\right ) x^4\right )}{\left (-a+x^2\right )^{7/4} \left (-a^3 d+3 a^2 d x^2+(b-3 a d) x^4+(-1+d) x^6\right )} \, dx,x,\sqrt {x}\right )}{(-1+d) \left (x^2 (-a+x) (-b+x)\right )^{3/4}}\\ &=-\frac {4 (3 a-b) (b-x) x^2}{3 a (1-d) \left ((a-x) (b-x) x^2\right )^{3/4}}-\frac {\left (2 (3 a-b) b x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a+x^2\right )^{3/4} \left (-b+x^2\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{3 a (1-d) \left (x^2 (-a+x) (-b+x)\right )^{3/4}}+\frac {\left (2 x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {a^3 (-3 a+b) d \sqrt [4]{-b+x^2}}{\left (-a+x^2\right )^{7/4} \left (a^3 d-3 a^2 d x^2-b \left (1-\frac {3 a d}{b}\right ) x^4+(1-d) x^6\right )}+\frac {3 a^2 (3 a-b) d x^2 \sqrt [4]{-b+x^2}}{\left (-a+x^2\right )^{7/4} \left (a^3 d-3 a^2 d x^2-b \left (1-\frac {3 a d}{b}\right ) x^4+(1-d) x^6\right )}+\frac {\left (-b^2-9 a^2 d+a (b+5 b d)\right ) x^4 \sqrt [4]{-b+x^2}}{\left (-a+x^2\right )^{7/4} \left (a^3 d-3 a^2 d x^2-b \left (1-\frac {3 a d}{b}\right ) x^4+(1-d) x^6\right )}\right ) \, dx,x,\sqrt {x}\right )}{(-1+d) \left (x^2 (-a+x) (-b+x)\right )^{3/4}}\\ &=-\frac {4 (3 a-b) (b-x) x^2}{3 a (1-d) \left ((a-x) (b-x) x^2\right )^{3/4}}-\frac {2 (3 a-b) \left (\frac {b (a-x)}{a (b-x)}\right )^{3/4} (b-x) x^2 \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};-\frac {(a-b) x}{a (b-x)}\right )}{3 a (1-d) \left ((a-x) (b-x) x^2\right )^{3/4}}+\frac {\left (6 a^2 (3 a-b) d x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{-b+x^2}}{\left (-a+x^2\right )^{7/4} \left (a^3 d-3 a^2 d x^2-b \left (1-\frac {3 a d}{b}\right ) x^4+(1-d) x^6\right )} \, dx,x,\sqrt {x}\right )}{(-1+d) \left (x^2 (-a+x) (-b+x)\right )^{3/4}}-\frac {\left (2 a^3 (3 a-b) d x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{-b+x^2}}{\left (-a+x^2\right )^{7/4} \left (a^3 d-3 a^2 d x^2-b \left (1-\frac {3 a d}{b}\right ) x^4+(1-d) x^6\right )} \, dx,x,\sqrt {x}\right )}{(-1+d) \left (x^2 (-a+x) (-b+x)\right )^{3/4}}+\frac {\left (2 \left (-b^2-9 a^2 d+a (b+5 b d)\right ) x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt [4]{-b+x^2}}{\left (-a+x^2\right )^{7/4} \left (a^3 d-3 a^2 d x^2-b \left (1-\frac {3 a d}{b}\right ) x^4+(1-d) x^6\right )} \, dx,x,\sqrt {x}\right )}{(-1+d) \left (x^2 (-a+x) (-b+x)\right )^{3/4}}\\ \end {align*}
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Mathematica [F] time = 5.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3 (-b+x) (-2 a b+(3 a-b) x)}{(-a+x) \left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a^3 d+3 a^2 d x+(b-3 a d) x^2+(-1+d) x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 6.79, size = 133, normalized size = 1.00 \begin {gather*} 2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} \left (x^3 (-a-b)+a b x^2+x^4\right )^{3/4}}{x^2 (x-b)}\right )-2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \left (x^3 (-a-b)+a b x^2+x^4\right )^{3/4}}{x^2 (x-b)}\right )-\frac {4 \sqrt [4]{x^3 (-a-b)+a b x^2+x^4}}{a-x} \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 - {\left (3 \, a - b\right )} x\right )} {\left (b - x\right )} x^{3}}{{\left (a^{3} d - 3 \, a^{2} d x - {\left (d - 1\right )} x^{3} + {\left (3 \, a d - b\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {3}{4}} {\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 = 0.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \left (-b +x \right ) \left (-2 a b +\left (3 a -b \right ) x \right )}{\left (-a +x \right ) \left (x^{2} \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {3}{4}} \left (-a^{3} d +3 a^{2} d x +\left (-3 a d +b \right ) x^{2}+\left (-1+d \right ) x^{3}\right )}\, dx \end {gather*}
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 - {\left (3 \, a - b\right )} x\right )} {\left (b - x\right )} x^{3}}{{\left (a^{3} d - 3 \, a^{2} d x - {\left (d - 1\right )} x^{3} + {\left (3 \, a d - b\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {3}{4}} {\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 {x^3\,\left (2\,a\,b-x\,\left (3\,a-b\right )\right )\,\left (b-x\right )}{\left (a-x\right )\,{\left (x^2\,\left (a-x\right )\,\left (b-x\right )\right )}^{3/4}\,\left (x^2\,\left (b-3\,a\,d\right )-a^3\,d+x^3\,\left (d-1\right )+3\,a^2\,d\,x\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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