Optimal. Leaf size=168 \[ 2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}}{x}\right )-2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}}{x}\right )+\frac {4 \left (-a b^2 x+2 a b x^2-a x^3+b^2 x^2-2 b x^3+x^4\right )^{3/4}}{(x-a) (x-b)^2} \]
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Rubi [F] time = 73.78, 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 (-3 a b+(a+2 b) x)}{(-a+x) (-b+x) \sqrt [4]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(a+2 b) 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 (-3 a b+(a+2 b) x)}{(-a+x) (-b+x) \sqrt [4]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(a+2 b) d x^2+(-1+d) x^3\right )} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {x^{11/4} (-3 a b+(a+2 b) x)}{(-a+x)^{5/4} (-b+x)^{3/2} \left (-a b^2 d+b (2 a+b) d x-(a+2 b) d x^2+(-1+d) x^3\right )} \, dx}{\sqrt [4]{x (-a+x) (-b+x)^2}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^{14} \left (-3 a b+(a+2 b) x^4\right )}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (-a b^2 d+b (2 a+b) d x^4-(a+2 b) d x^8+(-1+d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x) (-b+x)^2}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \left (\frac {\left (a^2 d+4 b^2 d+a b (3+d)\right ) x^2}{(1-d)^2 \left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2}}-\frac {(a+2 b) x^6}{(1-d) \left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2}}+\frac {x^2 \left (a b^2 d \left (a^2 d+4 b^2 d+a b (3+d)\right )-b d \left (2 a^3 d+4 b^3 d+a^2 b (7+2 d)+a b^2 (5+7 d)\right ) x^4+(a+2 b) d \left (a b (5-d)+a^2 d+b^2 (1+3 d)\right ) x^8\right )}{(-1+d)^2 \left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (-a b^2 d+b (2 a+b) d x^4-(a+2 b) d x^8+(-1+d) x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x) (-b+x)^2}}\\ &=-\frac {\left (4 (a+2 b) \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2}} \, dx,x,\sqrt [4]{x}\right )}{(1-d) \sqrt [4]{x (-a+x) (-b+x)^2}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (a b^2 d \left (a^2 d+4 b^2 d+a b (3+d)\right )-b d \left (2 a^3 d+4 b^3 d+a^2 b (7+2 d)+a b^2 (5+7 d)\right ) x^4+(a+2 b) d \left (a b (5-d)+a^2 d+b^2 (1+3 d)\right ) x^8\right )}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (-a b^2 d+b (2 a+b) d x^4-(a+2 b) d x^8+(-1+d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{(-1+d)^2 \sqrt [4]{x (-a+x) (-b+x)^2}}+\frac {\left (4 \left (a^2 d+4 b^2 d+a b (3+d)\right ) \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2}} \, dx,x,\sqrt [4]{x}\right )}{(1-d)^2 \sqrt [4]{x (-a+x) (-b+x)^2}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \left (\frac {a b^2 d \left (-a^2 d-4 b^2 d-a b (3+d)\right ) x^2}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (a b^2 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^4+a \left (1+\frac {2 b}{a}\right ) d x^8+(1-d) x^{12}\right )}+\frac {b d \left (2 a^3 d+4 b^3 d+a^2 b (7+2 d)+a b^2 (5+7 d)\right ) x^6}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (a b^2 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^4+a \left (1+\frac {2 b}{a}\right ) d x^8+(1-d) x^{12}\right )}+\frac {(a+2 b) d \left (-a b (5-d)-a^2 d-b^2 (1+3 d)\right ) x^{10}}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (a b^2 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^4+a \left (1+\frac {2 b}{a}\right ) d x^8+(1-d) x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{(-1+d)^2 \sqrt [4]{x (-a+x) (-b+x)^2}}+\frac {\left (4 (a+2 b) \sqrt [4]{x} \sqrt {-b+x} \sqrt [4]{1-\frac {x}{a}}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (-b+x^4\right )^{3/2} \left (1-\frac {x^4}{a}\right )^{5/4}} \, dx,x,\sqrt [4]{x}\right )}{a (1-d) \sqrt [4]{x (-a+x) (-b+x)^2}}-\frac {\left (4 \left (a^2 d+4 b^2 d+a b (3+d)\right ) \sqrt [4]{x} \sqrt {-b+x} \sqrt [4]{1-\frac {x}{a}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-b+x^4\right )^{3/2} \left (1-\frac {x^4}{a}\right )^{5/4}} \, dx,x,\sqrt [4]{x}\right )}{a (1-d)^2 \sqrt [4]{x (-a+x) (-b+x)^2}}\\ &=-\frac {\left (4 a b^2 d \left (a^2 d+4 b^2 d+a b (3+d)\right ) \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (a b^2 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^4+a \left (1+\frac {2 b}{a}\right ) d x^8+(1-d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{(-1+d)^2 \sqrt [4]{x (-a+x) (-b+x)^2}}-\frac {\left (4 (a+2 b) d \left (a b (5-d)+a^2 d+b^2 (1+3 d)\right ) \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (a b^2 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^4+a \left (1+\frac {2 b}{a}\right ) d x^8+(1-d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{(-1+d)^2 \sqrt [4]{x (-a+x) (-b+x)^2}}+\frac {\left (4 b d \left (2 a^3 d+4 b^3 d+a^2 b (7+2 d)+a b^2 (5+7 d)\right ) \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (a b^2 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^4+a \left (1+\frac {2 b}{a}\right ) d x^8+(1-d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{(-1+d)^2 \sqrt [4]{x (-a+x) (-b+x)^2}}-\frac {\left (4 (a+2 b) \sqrt [4]{x} \sqrt [4]{1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (1-\frac {x^4}{a}\right )^{5/4} \left (1-\frac {x^4}{b}\right )^{3/2}} \, dx,x,\sqrt [4]{x}\right )}{a b (1-d) \sqrt [4]{x (-a+x) (-b+x)^2}}+\frac {\left (4 \left (a^2 d+4 b^2 d+a b (3+d)\right ) \sqrt [4]{x} \sqrt [4]{1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1-\frac {x^4}{a}\right )^{5/4} \left (1-\frac {x^4}{b}\right )^{3/2}} \, dx,x,\sqrt [4]{x}\right )}{a b (1-d)^2 \sqrt [4]{x (-a+x) (-b+x)^2}}\\ &=-\frac {4 (a+2 b) x^2 \sqrt [4]{1-\frac {x}{a}} \, _2F_1\left (\frac {5}{4},\frac {7}{4};\frac {11}{4};\frac {\left (\frac {1}{a}-\frac {1}{b}\right ) b x}{b-x}\right )}{7 a b (1-d) \sqrt [4]{-\left ((a-x) (b-x)^2 x\right )} \left (1-\frac {x}{b}\right )^{5/4}}+\frac {16 \left (a^2 d+4 b^2 d+a b (3+d)\right ) x \Gamma \left (\frac {7}{4}\right ) \left (11 b (7 b-4 x) (a-x) \, _2F_1\left (1,\frac {5}{4};\frac {11}{4};\frac {(a-b) x}{b (a-x)}\right )+20 (a-b) (b-x) x \, _2F_1\left (2,\frac {9}{4};\frac {15}{4};\frac {(a-b) x}{b (a-x)}\right )\right )}{693 b^3 (1-d)^2 (a-x)^2 \sqrt [4]{-\left ((a-x) (b-x)^2 x\right )} \Gamma \left (\frac {3}{4}\right )}-\frac {\left (4 a b^2 d \left (a^2 d+4 b^2 d+a b (3+d)\right ) \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (a b^2 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^4+a \left (1+\frac {2 b}{a}\right ) d x^8+(1-d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{(-1+d)^2 \sqrt [4]{x (-a+x) (-b+x)^2}}-\frac {\left (4 (a+2 b) d \left (a b (5-d)+a^2 d+b^2 (1+3 d)\right ) \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (a b^2 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^4+a \left (1+\frac {2 b}{a}\right ) d x^8+(1-d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{(-1+d)^2 \sqrt [4]{x (-a+x) (-b+x)^2}}+\frac {\left (4 b d \left (2 a^3 d+4 b^3 d+a^2 b (7+2 d)+a b^2 (5+7 d)\right ) \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (a b^2 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^4+a \left (1+\frac {2 b}{a}\right ) d x^8+(1-d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{(-1+d)^2 \sqrt [4]{x (-a+x) (-b+x)^2}}\\ \end {align*}
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Mathematica [F] time = 7.31, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3 (-3 a b+(a+2 b) x)}{(-a+x) (-b+x) \sqrt [4]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(a+2 b) 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 = 1.34, size = 168, normalized size = 1.00 \begin {gather*} \frac {4 \left (-a b^2 x+2 a b x^2+b^2 x^2-a x^3-2 b x^3+x^4\right )^{3/4}}{(-a+x) (-b+x)^2}+2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{x}\right )-2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{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 (3 \, a b - {\left (a + 2 \, b\right )} x\right )} x^{3}}{{\left (a b^{2} d - {\left (2 \, a + b\right )} b d x + {\left (a + 2 \, b\right )} d x^{2} - {\left (d - 1\right )} x^{3}\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {1}{4}} {\left (a - x\right )} {\left (b - x\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \left (-3 a b +\left (a +2 b \right ) x \right )}{\left (-a +x \right ) \left (-b +x \right ) \left (x \left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{4}} \left (-a \,b^{2} d +b \left (2 a +b \right ) d x -\left (a +2 b \right ) d \,x^{2}+\left (-1+d \right ) x^{3}\right )}\, dx\]
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 (3 \, a b - {\left (a + 2 \, b\right )} x\right )} x^{3}}{{\left (a b^{2} d - {\left (2 \, a + b\right )} b d x + {\left (a + 2 \, b\right )} d x^{2} - {\left (d - 1\right )} x^{3}\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {1}{4}} {\left (a - x\right )} {\left (b - 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 (3\,a\,b-x\,\left (a+2\,b\right )\right )}{\left (a-x\right )\,\left (b-x\right )\,{\left (-x\,\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/4}\,\left (x^3\,\left (d-1\right )-d\,x^2\,\left (a+2\,b\right )-a\,b^2\,d+b\,d\,x\,\left (2\,a+b\right )\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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