Optimal. Leaf size=113 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3}}{(b-x)^2}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3}}{(b-x)^2}\right )}{d^{3/4}} \]
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Rubi [F] time = 28.01, 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 {(-b+x) (-6 a+b+5 x)}{\sqrt [4]{(-a+x) (-b+x)^2} \left (b^6+a d-\left (6 b^5+d\right ) x+15 b^4 x^2-20 b^3 x^3+15 b^2 x^4-6 b x^5+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(-b+x) (-6 a+b+5 x)}{\sqrt [4]{(-a+x) (-b+x)^2} \left (b^6+a d-\left (6 b^5+d\right ) x+15 b^4 x^2-20 b^3 x^3+15 b^2 x^4-6 b x^5+x^6\right )} \, dx &=\frac {\left (\sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-b+x} (-6 a+b+5 x)}{\sqrt [4]{-a+x} \left (b^6+a d-\left (6 b^5+d\right ) x+15 b^4 x^2-20 b^3 x^3+15 b^2 x^4-6 b x^5+x^6\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-b+x} (-6 a+b+5 x)}{\sqrt [4]{-a+x} \left (b^6 \left (1+\frac {a d}{b^6}\right )-\left (6 b^5+d\right ) x+15 b^4 x^2-20 b^3 x^3+15 b^2 x^4-6 b x^5+x^6\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \left (\frac {6 a \left (1-\frac {b}{6 a}\right ) \sqrt {-b+x}}{\sqrt [4]{-a+x} \left (-b^6 \left (1+\frac {a d}{b^6}\right )+6 b^5 \left (1+\frac {d}{6 b^5}\right ) x-15 b^4 x^2+20 b^3 x^3-15 b^2 x^4+6 b x^5-x^6\right )}+\frac {5 x \sqrt {-b+x}}{\sqrt [4]{-a+x} \left (b^6 \left (1+\frac {a d}{b^6}\right )-6 b^5 \left (1+\frac {d}{6 b^5}\right ) x+15 b^4 x^2-20 b^3 x^3+15 b^2 x^4-6 b x^5+x^6\right )}\right ) \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (5 \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {x \sqrt {-b+x}}{\sqrt [4]{-a+x} \left (b^6 \left (1+\frac {a d}{b^6}\right )-6 b^5 \left (1+\frac {d}{6 b^5}\right ) x+15 b^4 x^2-20 b^3 x^3+15 b^2 x^4-6 b x^5+x^6\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left ((6 a-b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-b+x}}{\sqrt [4]{-a+x} \left (-b^6 \left (1+\frac {a d}{b^6}\right )+6 b^5 \left (1+\frac {d}{6 b^5}\right ) x-15 b^4 x^2+20 b^3 x^3-15 b^2 x^4+6 b x^5-x^6\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (20 \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (a+x^4\right ) \sqrt {a-b+x^4}}{b^6+a d-\left (6 b^5+d\right ) \left (a+x^4\right )+15 b^4 \left (a+x^4\right )^2-20 b^3 \left (a+x^4\right )^3+15 b^2 \left (a+x^4\right )^4-6 b \left (a+x^4\right )^5+\left (a+x^4\right )^6} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (4 (6 a-b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {a-b+x^4}}{b^6+a d-\left (6 b^5+d\right ) \left (a+x^4\right )+15 b^4 \left (a+x^4\right )^2-20 b^3 \left (a+x^4\right )^3+15 b^2 \left (a+x^4\right )^4-6 b \left (a+x^4\right )^5+\left (a+x^4\right )^6} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (20 \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (a+x^4\right ) \sqrt {a-b+x^4}}{b^6 \left (1+\frac {a d}{b^6}\right )-\left (6 b^5+d\right ) \left (a+x^4\right )+15 b^4 \left (a+x^4\right )^2-20 b^3 \left (a+x^4\right )^3+15 b^2 \left (a+x^4\right )^4-6 b \left (a+x^4\right )^5+\left (a+x^4\right )^6} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (4 (6 a-b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {a-b+x^4}}{b^6 \left (1+\frac {a d}{b^6}\right )-\left (6 b^5+d\right ) \left (a+x^4\right )+15 b^4 \left (a+x^4\right )^2-20 b^3 \left (a+x^4\right )^3+15 b^2 \left (a+x^4\right )^4-6 b \left (a+x^4\right )^5+\left (a+x^4\right )^6} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (20 \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \left (\frac {a x^2 \sqrt {a-b+x^4}}{a^6 \left (1+\frac {b \left (-6 a^5+15 a^4 b-20 a^3 b^2+15 a^2 b^3-6 a b^4+b^5\right )}{a^6}\right )+6 a^5 \left (1-\frac {30 a^4 b-60 a^3 b^2+60 a^2 b^3-30 a b^4+6 b^5+d}{6 a^5}\right ) x^4+15 a^4 \left (1+\frac {b \left (-4 a^3+6 a^2 b-4 a b^2+b^3\right )}{a^4}\right ) x^8+20 a^3 \left (1-\frac {b \left (3 a^2-3 a b+b^2\right )}{a^3}\right ) x^{12}+15 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) x^{16}+6 a \left (1-\frac {b}{a}\right ) x^{20}+x^{24}}+\frac {x^6 \sqrt {a-b+x^4}}{a^6 \left (1+\frac {b \left (-6 a^5+15 a^4 b-20 a^3 b^2+15 a^2 b^3-6 a b^4+b^5\right )}{a^6}\right )+6 a^5 \left (1-\frac {30 a^4 b-60 a^3 b^2+60 a^2 b^3-30 a b^4+6 b^5+d}{6 a^5}\right ) x^4+15 a^4 \left (1+\frac {b \left (-4 a^3+6 a^2 b-4 a b^2+b^3\right )}{a^4}\right ) x^8+20 a^3 \left (1-\frac {b \left (3 a^2-3 a b+b^2\right )}{a^3}\right ) x^{12}+15 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) x^{16}+6 a \left (1-\frac {b}{a}\right ) x^{20}+x^{24}}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (4 (6 a-b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {a-b+x^4}}{b^6 \left (1+\frac {a d}{b^6}\right )-\left (6 b^5+d\right ) \left (a+x^4\right )+15 b^4 \left (a+x^4\right )^2-20 b^3 \left (a+x^4\right )^3+15 b^2 \left (a+x^4\right )^4-6 b \left (a+x^4\right )^5+\left (a+x^4\right )^6} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (20 \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt {a-b+x^4}}{a^6 \left (1+\frac {b \left (-6 a^5+15 a^4 b-20 a^3 b^2+15 a^2 b^3-6 a b^4+b^5\right )}{a^6}\right )+6 a^5 \left (1-\frac {30 a^4 b-60 a^3 b^2+60 a^2 b^3-30 a b^4+6 b^5+d}{6 a^5}\right ) x^4+15 a^4 \left (1+\frac {b \left (-4 a^3+6 a^2 b-4 a b^2+b^3\right )}{a^4}\right ) x^8+20 a^3 \left (1-\frac {b \left (3 a^2-3 a b+b^2\right )}{a^3}\right ) x^{12}+15 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) x^{16}+6 a \left (1-\frac {b}{a}\right ) x^{20}+x^{24}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (20 a \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {a-b+x^4}}{a^6 \left (1+\frac {b \left (-6 a^5+15 a^4 b-20 a^3 b^2+15 a^2 b^3-6 a b^4+b^5\right )}{a^6}\right )+6 a^5 \left (1-\frac {30 a^4 b-60 a^3 b^2+60 a^2 b^3-30 a b^4+6 b^5+d}{6 a^5}\right ) x^4+15 a^4 \left (1+\frac {b \left (-4 a^3+6 a^2 b-4 a b^2+b^3\right )}{a^4}\right ) x^8+20 a^3 \left (1-\frac {b \left (3 a^2-3 a b+b^2\right )}{a^3}\right ) x^{12}+15 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) x^{16}+6 a \left (1-\frac {b}{a}\right ) x^{20}+x^{24}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (4 (6 a-b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {a-b+x^4}}{b^6 \left (1+\frac {a d}{b^6}\right )-\left (6 b^5+d\right ) \left (a+x^4\right )+15 b^4 \left (a+x^4\right )^2-20 b^3 \left (a+x^4\right )^3+15 b^2 \left (a+x^4\right )^4-6 b \left (a+x^4\right )^5+\left (a+x^4\right )^6} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (20 \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt {a-b+x^4}}{a^6 \left (1+\frac {b^6}{a^6}\right )-6 b^5 \left (1+\frac {d}{6 b^5}\right ) x^4+15 b^4 x^8-20 b^3 x^{12}+15 b^2 x^{16}-6 b x^{20}+x^{24}-6 a^5 \left (b-x^4\right )+15 a^4 \left (b-x^4\right )^2-20 a^3 \left (b-x^4\right )^3+15 a^2 \left (b-x^4\right )^4-6 a \left (b-x^4\right )^5} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (20 a \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {a-b+x^4}}{a^6 \left (1+\frac {b^6}{a^6}\right )-6 b^5 \left (1+\frac {d}{6 b^5}\right ) x^4+15 b^4 x^8-20 b^3 x^{12}+15 b^2 x^{16}-6 b x^{20}+x^{24}-6 a^5 \left (b-x^4\right )+15 a^4 \left (b-x^4\right )^2-20 a^3 \left (b-x^4\right )^3+15 a^2 \left (b-x^4\right )^4-6 a \left (b-x^4\right )^5} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (4 (6 a-b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {a-b+x^4}}{b^6 \left (1+\frac {a d}{b^6}\right )-\left (6 b^5+d\right ) \left (a+x^4\right )+15 b^4 \left (a+x^4\right )^2-20 b^3 \left (a+x^4\right )^3+15 b^2 \left (a+x^4\right )^4-6 b \left (a+x^4\right )^5+\left (a+x^4\right )^6} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ \end {align*}
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Mathematica [C] time = 22.59, size = 333050, normalized size = 2947.35 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.46, size = 113, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3}}{(b-x)^2}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3}}{(b-x)^2}\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 {{\left (6 \, a - b - 5 \, x\right )} {\left (b - x\right )}}{{\left (b^{6} + 15 \, b^{4} x^{2} - 20 \, b^{3} x^{3} + 15 \, b^{2} x^{4} - 6 \, b x^{5} + x^{6} + a d - {\left (6 \, b^{5} + d\right )} x\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-b +x \right ) \left (-6 a +b +5 x \right )}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{4}} \left (b^{6}+a d -\left (6 b^{5}+d \right ) x +15 b^{4} x^{2}-20 b^{3} x^{3}+15 b^{2} x^{4}-6 b \,x^{5}+x^{6}\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 (6 \, a - b - 5 \, x\right )} {\left (b - x\right )}}{{\left (b^{6} + 15 \, b^{4} x^{2} - 20 \, b^{3} x^{3} + 15 \, b^{2} x^{4} - 6 \, b x^{5} + x^{6} + a d - {\left (6 \, b^{5} + d\right )} x\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{4}}}\,{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 (b-x\right )\,\left (b-6\,a+5\,x\right )}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/4}\,\left (a\,d-6\,b\,x^5-x\,\left (6\,b^5+d\right )+b^6+x^6+15\,b^2\,x^4-20\,b^3\,x^3+15\,b^4\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- b + x\right ) \left (- 6 a + b + 5 x\right )}{\sqrt [4]{\left (- a + x\right ) \left (- b + x\right )^{2}} \left (a d + b^{6} - 6 b^{5} x + 15 b^{4} x^{2} - 20 b^{3} x^{3} + 15 b^{2} x^{4} - 6 b x^{5} - d x + x^{6}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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