Optimal. Leaf size=113 \[ -\frac {2 \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{a-x}\right )}{d^{3/4}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{a-x}\right )}{d^{3/4}} \]
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Rubi [F]
time = 4.95, 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 {-((2 a-3 b) b)+2 (a-2 b) x+x^2}{\sqrt [4]{(-a+x) (-b+x)^2} \left (-a^3-b^2 d+\left (3 a^2+2 b d\right ) x-(3 a+d) x^2+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-((2 a-3 b) b)+2 (a-2 b) x+x^2}{\sqrt [4]{(-a+x) (-b+x)^2} \left (-a^3-b^2 d+\left (3 a^2+2 b d\right ) x-(3 a+d) x^2+x^3\right )} \, dx &=\frac {\left (\sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {-((2 a-3 b) b)+2 (a-2 b) x+x^2}{\sqrt [4]{-a+x} \sqrt {-b+x} \left (-a^3-b^2 d+\left (3 a^2+2 b d\right ) x-(3 a+d) x^2+x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {(2 a-3 b+x) \sqrt {-b+x}}{\sqrt [4]{-a+x} \left (-a^3-b^2 d+\left (3 a^2+2 b d\right ) x-(3 a+d) x^2+x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \left (\frac {3 \left (1-\frac {2 a}{3 b}\right ) b \sqrt {-b+x}}{\sqrt [4]{-a+x} \left (a^3+b^2 d-\left (3 a^2+2 b d\right ) x+(3 a+d) x^2-x^3\right )}+\frac {x \sqrt {-b+x}}{\sqrt [4]{-a+x} \left (-a^3-b^2 d+\left (3 a^2+2 b d\right ) x-(3 a+d) x^2+x^3\right )}\right ) \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {x \sqrt {-b+x}}{\sqrt [4]{-a+x} \left (-a^3-b^2 d+\left (3 a^2+2 b d\right ) x-(3 a+d) x^2+x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left ((-2 a+3 b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-b+x}}{\sqrt [4]{-a+x} \left (a^3+b^2 d-\left (3 a^2+2 b d\right ) x+(3 a+d) x^2-x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (4 \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^2 \left (a+x^4\right ) \sqrt {a-b+x^4}}{-a^2 d-b^2 d+2 b d x^4-d x^8+x^{12}+2 a d \left (b-x^4\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (4 (-2 a+3 b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {a-b+x^4}}{a^2 d+b^2 d-2 b d x^4+x^8 \left (d-x^4\right )+2 a d \left (-b+x^4\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (4 \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^2 \left (-a-x^4\right ) \sqrt {a-b+x^4}}{a^2 \left (1+\frac {b^2}{a^2}\right ) d-2 b d x^4+d x^8-x^{12}-2 a d \left (b-x^4\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (4 (-2 a+3 b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {a-b+x^4}}{a^2 \left (1+\frac {b^2}{a^2}\right ) d-2 b d x^4+x^8 \left (d-x^4\right )+2 a d \left (-b+x^4\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (4 \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \left (\frac {a x^2 \sqrt {a-b+x^4}}{-a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d-2 a \left (1-\frac {b}{a}\right ) d x^4-d x^8+x^{12}}+\frac {x^6 \sqrt {a-b+x^4}}{-a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d-2 a \left (1-\frac {b}{a}\right ) d x^4-d x^8+x^{12}}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (4 (-2 a+3 b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {a-b+x^4}}{a^2 \left (1+\frac {b^2}{a^2}\right ) d-2 b d x^4+x^8 \left (d-x^4\right )+2 a d \left (-b+x^4\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (4 \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt {a-b+x^4}}{-a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d-2 a \left (1-\frac {b}{a}\right ) d x^4-d x^8+x^{12}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (4 a \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {a-b+x^4}}{-a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d-2 a \left (1-\frac {b}{a}\right ) d x^4-d x^8+x^{12}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (4 (-2 a+3 b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {a-b+x^4}}{a^2 \left (1+\frac {b^2}{a^2}\right ) d-2 b d x^4+x^8 \left (d-x^4\right )+2 a d \left (-b+x^4\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ \end {align*}
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Mathematica [A]
time = 2.87, size = 151, normalized size = 1.34 \begin {gather*} \frac {\sqrt {2} \sqrt [4]{a-x} \sqrt {b-x} \left (\text {ArcTan}\left (\frac {(a-x)^{3/2}+\sqrt {d} (-b+x)}{\sqrt {2} \sqrt [4]{d} (a-x)^{3/4} \sqrt {b-x}}\right )+\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} (a-x)^{3/4} \sqrt {b-x}}{(a-x)^{3/2}+\sqrt {d} (b-x)}\right )\right )}{d^{3/4} \sqrt [4]{(b-x)^2 (-a+x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {-\left (2 a -3 b \right ) b +2 \left (a -2 b \right ) x +x^{2}}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{4}} \left (-a^{3}-b^{2} d +\left (3 a^{2}+2 b d \right ) x -\left (3 a +d \right ) x^{2}+x^{3}\right )}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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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Sympy [F(-1)] Timed out
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*} \text {could not integrate} \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 {2\,x\,\left (a-2\,b\right )-b\,\left (2\,a-3\,b\right )+x^2}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/4}\,\left (b^2\,d-x\,\left (3\,a^2+2\,b\,d\right )+x^2\,\left (3\,a+d\right )+a^3-x^3\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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