Optimal. Leaf size=141 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b^3 x+x^3 \left (3 a b+3 b^2\right )+x^2 \left (-3 a b^2-b^3\right )+x^4 (-a-3 b)+x^5}}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b^3 x+x^3 \left (3 a b+3 b^2\right )+x^2 \left (-3 a b^2-b^3\right )+x^4 (-a-3 b)+x^5}}\right )}{d^{3/4}} \]
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Rubi [F] time = 29.43, 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^2 \left (-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4} \left (a b^3-b^2 (3 a+b) x+3 b (a+b) x^2-(a+3 b+d) x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^2 \left (-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4} \left (a b^3-b^2 (3 a+b) x+3 b (a+b) x^2-(a+3 b+d) x^3+x^4\right )} \, dx &=\frac {\left (x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \int \frac {x^{5/4} \left (-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4\right )}{(-a+x)^{3/4} (-b+x)^{9/4} \left (a b^3-b^2 (3 a+b) x+3 b (a+b) x^2-(a+3 b+d) x^3+x^4\right )} \, dx}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}\\ &=\frac {\left (x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \int \frac {x^{5/4} \left (3 a b^2+\left (-3 a b-2 b^2\right ) x+b x^2+x^3\right )}{(-a+x)^{3/4} (-b+x)^{5/4} \left (a b^3-b^2 (3 a+b) x+3 b (a+b) x^2-(a+3 b+d) x^3+x^4\right )} \, dx}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}\\ &=\frac {\left (x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \int \frac {x^{5/4} \left (-3 a b+2 b x+x^2\right )}{(-a+x)^{3/4} \sqrt [4]{-b+x} \left (a b^3-b^2 (3 a+b) x+3 b (a+b) x^2-(a+3 b+d) x^3+x^4\right )} \, dx}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}\\ &=\frac {\left (4 x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {x^8 \left (-3 a b+2 b x^4+x^8\right )}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-b^2 (3 a+b) x^4+3 b (a+b) x^8-(a+3 b+d) x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}\\ &=\frac {\left (4 x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4}}-\frac {a b^3-b^2 (3 a+b) x^4+3 b (2 a+b) x^8-(a+5 b+d) x^{12}}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-b^2 (3 a+b) x^4+3 b (a+b) x^8-(a+3 b+d) x^{12}+x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}\\ &=\frac {\left (4 x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4}} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}-\frac {\left (4 x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {a b^3-b^2 (3 a+b) x^4+3 b (2 a+b) x^8-(a+5 b+d) x^{12}}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-b^2 (3 a+b) x^4+3 b (a+b) x^8-(a+3 b+d) x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}\\ &=-\frac {\left (4 x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \left (\frac {a b^3}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )}+\frac {(-3 a-b) b^2 x^4}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )}+\frac {3 b (2 a+b) x^8}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )}+\frac {(-a-5 b-d) x^{12}}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}+\frac {\left (4 x^{3/4} (-b+x)^{9/4} \left (1-\frac {x}{a}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-b+x^4} \left (1-\frac {x^4}{a}\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}\\ &=-\frac {\left (4 a b^3 x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}-\frac {\left (12 b (2 a+b) x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}+\frac {\left (4 b^2 (3 a+b) x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}-\frac {\left (4 (-a-5 b-d) x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}+\frac {\left (4 x^{3/4} (-b+x)^2 \left (1-\frac {x}{a}\right )^{3/4} \sqrt [4]{1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {x^4}{a}\right )^{3/4} \sqrt [4]{1-\frac {x^4}{b}}} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}\\ &=\frac {4 (b-x)^2 x \left (1-\frac {x}{a}\right )^{3/4} \sqrt [4]{1-\frac {x}{b}} F_1\left (\frac {1}{4};\frac {3}{4},\frac {1}{4};\frac {5}{4};\frac {x}{a},\frac {x}{b}\right )}{\left ((a-x) (b-x)^3 x\right )^{3/4}}-\frac {\left (4 a b^3 x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}-\frac {\left (12 b (2 a+b) x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}+\frac {\left (4 b^2 (3 a+b) x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}-\frac {\left (4 (-a-5 b-d) x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}\\ \end {align*}
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Mathematica [F] time = 4.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \left (-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4} \left (a b^3-b^2 (3 a+b) x+3 b (a+b) x^2-(a+3 b+d) x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 3.93, size = 141, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b^3 x+\left (-3 a b^2-b^3\right ) x^2+\left (3 a b+3 b^2\right ) x^3+(-a-3 b) x^4+x^5}}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b^3 x+\left (-3 a b^2-b^3\right ) x^2+\left (3 a b+3 b^2\right ) x^3+(-a-3 b) x^4+x^5}}\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 (3 \, a b^{3} - 2 \, {\left (3 \, a + b\right )} b^{2} x + 3 \, {\left (a + b\right )} b x^{2} - x^{4}\right )} x^{2}}{\left ({\left (a - x\right )} {\left (b - x\right )}^{3} x\right )^{\frac {3}{4}} {\left (a b^{3} - {\left (3 \, a + b\right )} b^{2} x + 3 \, {\left (a + b\right )} b x^{2} - {\left (a + 3 \, b + d\right )} x^{3} + x^{4}\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (-3 a \,b^{3}+2 b^{2} \left (3 a +b \right ) x -3 b \left (a +b \right ) x^{2}+x^{4}\right )}{\left (x \left (-a +x \right ) \left (-b +x \right )^{3}\right )^{\frac {3}{4}} \left (a \,b^{3}-b^{2} \left (3 a +b \right ) x +3 b \left (a +b \right ) x^{2}-\left (a +3 b +d \right ) x^{3}+x^{4}\right )}\, dx\]
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^{3} - 2 \, {\left (3 \, a + b\right )} b^{2} x + 3 \, {\left (a + b\right )} b x^{2} - x^{4}\right )} x^{2}}{\left ({\left (a - x\right )} {\left (b - x\right )}^{3} x\right )^{\frac {3}{4}} {\left (a b^{3} - {\left (3 \, a + b\right )} b^{2} x + 3 \, {\left (a + b\right )} b x^{2} - {\left (a + 3 \, b + d\right )} x^{3} + x^{4}\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^2\,\left (3\,a\,b^3-x^4-2\,b^2\,x\,\left (3\,a+b\right )+3\,b\,x^2\,\left (a+b\right )\right )}{{\left (x\,\left (a-x\right )\,{\left (b-x\right )}^3\right )}^{3/4}\,\left (a\,b^3-x^3\,\left (a+3\,b+d\right )+x^4-b^2\,x\,\left (3\,a+b\right )+3\,b\,x^2\,\left (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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