Optimal. Leaf size=153 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b^3 x+x^3 \left (3 a b+3 b^2\right )+x^2 \left (-3 a b^2-b^3\right )+x^4 (-a-3 b)+x^5}}{a-x}\right )}{d^{3/4}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b^3 x+x^3 \left (3 a b+3 b^2\right )+x^2 \left (-3 a b^2-b^3\right )+x^4 (-a-3 b)+x^5}}{a-x}\right )}{d^{3/4}} \]
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Rubi [F] time = 19.22, 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 {a b^3-2 (3 a-b) b^2 x+3 (3 a-b) b x^2-4 a x^3+x^4}{\sqrt [4]{x (-a+x) (-b+x)^3} \left (a^3-\left (3 a^2+b^3 d\right ) x+3 \left (a+b^2 d\right ) x^2-(1+3 b d) x^3+d x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {a b^3-2 (3 a-b) b^2 x+3 (3 a-b) b x^2-4 a x^3+x^4}{\sqrt [4]{x (-a+x) (-b+x)^3} \left (a^3-\left (3 a^2+b^3 d\right ) x+3 \left (a+b^2 d\right ) x^2-(1+3 b d) x^3+d x^4\right )} \, dx &=\int \frac {(b-x)^2 \left (a b-2 (2 a-b) x+x^2\right )}{\sqrt [4]{(a-x) (b-x)^3 x} \left (a^3-\left (3 a^2+b^3 d\right ) x+3 \left (a+b^2 d\right ) x^2-(1+3 b d) x^3+d x^4\right )} \, dx\\ &=\frac {\left (\sqrt [4]{a-x} (b-x)^{3/4} \sqrt [4]{x}\right ) \int \frac {(b-x)^{5/4} \left (a b-2 (2 a-b) x+x^2\right )}{\sqrt [4]{a-x} \sqrt [4]{x} \left (a^3-\left (3 a^2+b^3 d\right ) x+3 \left (a+b^2 d\right ) x^2-(1+3 b d) x^3+d x^4\right )} \, dx}{\sqrt [4]{(a-x) (b-x)^3 x}}\\ &=\frac {\left (4 \sqrt [4]{a-x} (b-x)^{3/4} \sqrt [4]{x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (b-x^4\right )^{5/4} \left (a b-2 (2 a-b) x^4+x^8\right )}{\sqrt [4]{a-x^4} \left (a^3-\left (3 a^2+b^3 d\right ) x^4+3 \left (a+b^2 d\right ) x^8-(1+3 b d) x^{12}+d x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{(a-x) (b-x)^3 x}}\\ &=\frac {\left (4 \sqrt [4]{a-x} (b-x)^{3/4} \sqrt [4]{x}\right ) \operatorname {Subst}\left (\int \left (\frac {a b x^2 \left (b-x^4\right )^{5/4}}{\sqrt [4]{a-x^4} \left (a^3-3 a^2 \left (1+\frac {b^3 d}{3 a^2}\right ) x^4+3 a \left (1+\frac {b^2 d}{a}\right ) x^8-(1+3 b d) x^{12}+d x^{16}\right )}+\frac {2 (-2 a+b) x^6 \left (b-x^4\right )^{5/4}}{\sqrt [4]{a-x^4} \left (a^3-3 a^2 \left (1+\frac {b^3 d}{3 a^2}\right ) x^4+3 a \left (1+\frac {b^2 d}{a}\right ) x^8-(1+3 b d) x^{12}+d x^{16}\right )}+\frac {x^{10} \left (b-x^4\right )^{5/4}}{\sqrt [4]{a-x^4} \left (a^3-3 a^2 \left (1+\frac {b^3 d}{3 a^2}\right ) x^4+3 a \left (1+\frac {b^2 d}{a}\right ) x^8-(1+3 b d) x^{12}+d x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{(a-x) (b-x)^3 x}}\\ &=\frac {\left (4 \sqrt [4]{a-x} (b-x)^{3/4} \sqrt [4]{x}\right ) \operatorname {Subst}\left (\int \frac {x^{10} \left (b-x^4\right )^{5/4}}{\sqrt [4]{a-x^4} \left (a^3-3 a^2 \left (1+\frac {b^3 d}{3 a^2}\right ) x^4+3 a \left (1+\frac {b^2 d}{a}\right ) x^8-(1+3 b d) x^{12}+d x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{(a-x) (b-x)^3 x}}-\frac {\left (8 (2 a-b) \sqrt [4]{a-x} (b-x)^{3/4} \sqrt [4]{x}\right ) \operatorname {Subst}\left (\int \frac {x^6 \left (b-x^4\right )^{5/4}}{\sqrt [4]{a-x^4} \left (a^3-3 a^2 \left (1+\frac {b^3 d}{3 a^2}\right ) x^4+3 a \left (1+\frac {b^2 d}{a}\right ) x^8-(1+3 b d) x^{12}+d x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{(a-x) (b-x)^3 x}}+\frac {\left (4 a b \sqrt [4]{a-x} (b-x)^{3/4} \sqrt [4]{x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (b-x^4\right )^{5/4}}{\sqrt [4]{a-x^4} \left (a^3-3 a^2 \left (1+\frac {b^3 d}{3 a^2}\right ) x^4+3 a \left (1+\frac {b^2 d}{a}\right ) x^8-(1+3 b d) x^{12}+d x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{(a-x) (b-x)^3 x}}\\ \end {align*}
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Mathematica [F] time = 2.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a b^3-2 (3 a-b) b^2 x+3 (3 a-b) b x^2-4 a x^3+x^4}{\sqrt [4]{x (-a+x) (-b+x)^3} \left (a^3-\left (3 a^2+b^3 d\right ) x+3 \left (a+b^2 d\right ) x^2-(1+3 b d) x^3+d x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.48, size = 153, normalized size = 1.00 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b^3 x+x^3 \left (3 a b+3 b^2\right )+x^2 \left (-3 a b^2-b^3\right )+x^4 (-a-3 b)+x^5}}{a-x}\right )}{d^{3/4}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b^3 x+x^3 \left (3 a b+3 b^2\right )+x^2 \left (-3 a b^2-b^3\right )+x^4 (-a-3 b)+x^5}}{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 {a b^{3} - 2 \, {\left (3 \, a - b\right )} b^{2} x + 3 \, {\left (3 \, a - b\right )} b x^{2} - 4 \, a x^{3} + x^{4}}{\left ({\left (a - x\right )} {\left (b - x\right )}^{3} x\right )^{\frac {1}{4}} {\left (d x^{4} - {\left (3 \, b d + 1\right )} x^{3} + a^{3} + 3 \, {\left (b^{2} d + a\right )} x^{2} - {\left (b^{3} d + 3 \, a^{2}\right )} x\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a \,b^{3}-2 \left (3 a -b \right ) b^{2} x +3 \left (3 a -b \right ) b \,x^{2}-4 a \,x^{3}+x^{4}}{\left (x \left (-a +x \right ) \left (-b +x \right )^{3}\right )^{\frac {1}{4}} \left (a^{3}-\left (b^{3} d +3 a^{2}\right ) x +3 \left (b^{2} d +a \right ) x^{2}-\left (3 b d +1\right ) x^{3}+d \,x^{4}\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 {a b^{3} - 2 \, {\left (3 \, a - b\right )} b^{2} x + 3 \, {\left (3 \, a - b\right )} b x^{2} - 4 \, a x^{3} + x^{4}}{\left ({\left (a - x\right )} {\left (b - x\right )}^{3} x\right )^{\frac {1}{4}} {\left (d x^{4} - {\left (3 \, b d + 1\right )} x^{3} + a^{3} + 3 \, {\left (b^{2} d + a\right )} x^{2} - {\left (b^{3} d + 3 \, a^{2}\right )} 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 {a\,b^3-4\,a\,x^3+x^4+3\,b\,x^2\,\left (3\,a-b\right )-2\,b^2\,x\,\left (3\,a-b\right )}{{\left (x\,\left (a-x\right )\,{\left (b-x\right )}^3\right )}^{1/4}\,\left (3\,x^2\,\left (d\,b^2+a\right )+d\,x^4-x^3\,\left (3\,b\,d+1\right )+a^3-x\,\left (3\,a^2+d\,b^3\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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