Optimal. Leaf size=33 \[ \frac {4 \left ((x-1)^3\right )^{3/4} \left (135 x^3+245 x^2+158 x+47\right )}{585 (x-1)^2} \]
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Rubi [B] time = 0.19, antiderivative size = 70, normalized size of antiderivative = 2.12, number of steps used = 17, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {6742, 2067, 15, 30, 2081, 43} \begin {gather*} \frac {12 (1-x)^4}{13 \sqrt [4]{(x-1)^3}}+\frac {36 (1-x)^2}{5 \sqrt [4]{(x-1)^3}}-\frac {4 (1-x)}{\sqrt [4]{(x-1)^3}}+\frac {40}{9} \left ((x-1)^3\right )^{3/4} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rule 43
Rule 2067
Rule 2081
Rule 6742
Rubi steps
\begin {align*} \int \frac {-1-2 x+x^2+3 x^3}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx &=\int \left (-\frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3}}-\frac {2 x}{\sqrt [4]{-1+3 x-3 x^2+x^3}}+\frac {x^2}{\sqrt [4]{-1+3 x-3 x^2+x^3}}+\frac {3 x^3}{\sqrt [4]{-1+3 x-3 x^2+x^3}}\right ) \, dx\\ &=-\left (2 \int \frac {x}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx\right )+3 \int \frac {x^3}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx-\int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx+\int \frac {x^2}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1+x}{\sqrt [4]{x^3}} \, dx,x,-1+x\right )\right )+3 \operatorname {Subst}\left (\int \frac {(1+x)^3}{\sqrt [4]{x^3}} \, dx,x,-1+x\right )-\operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{x^3}} \, dx,x,-1+x\right )+\operatorname {Subst}\left (\int \frac {(1+x)^2}{\sqrt [4]{x^3}} \, dx,x,-1+x\right )\\ &=-\frac {(-1+x)^{3/4} \operatorname {Subst}\left (\int \frac {1}{x^{3/4}} \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}+\frac {(-1+x)^{3/4} \operatorname {Subst}\left (\int \frac {(1+x)^2}{x^{3/4}} \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}-\frac {\left (2 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1+x}{x^{3/4}} \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}+\frac {\left (3 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {(1+x)^3}{x^{3/4}} \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}\\ &=\frac {4 (1-x)}{\sqrt [4]{(-1+x)^3}}+\frac {(-1+x)^{3/4} \operatorname {Subst}\left (\int \left (\frac {1}{x^{3/4}}+2 \sqrt [4]{x}+x^{5/4}\right ) \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}-\frac {\left (2 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^{3/4}}+\sqrt [4]{x}\right ) \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}+\frac {\left (3 (-1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^{3/4}}+3 \sqrt [4]{x}+3 x^{5/4}+x^{9/4}\right ) \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}\\ &=-\frac {4 (1-x)}{\sqrt [4]{(-1+x)^3}}+\frac {36 (1-x)^2}{5 \sqrt [4]{(-1+x)^3}}+\frac {12 (1-x)^4}{13 \sqrt [4]{(-1+x)^3}}+\frac {40}{9} \left ((-1+x)^3\right )^{3/4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 31, normalized size = 0.94 \begin {gather*} \frac {4 (x-1) \left (135 x^3+245 x^2+158 x+47\right )}{585 \sqrt [4]{(x-1)^3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.92, size = 57, normalized size = 1.73 \begin {gather*} \frac {4 \left (585 \sqrt [4]{-1+x}+1053 (-1+x)^{5/4}+650 (-1+x)^{9/4}+135 (-1+x)^{13/4}\right ) \left ((-1+x)^3\right )^{3/4}}{585 (-1+x)^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 42, normalized size = 1.27 \begin {gather*} \frac {4 \, {\left (135 \, x^{3} + 245 \, x^{2} + 158 \, x + 47\right )} {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}^{\frac {3}{4}}}{585 \, {\left (x^{2} - 2 \, x + 1\right )}} \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 {3 \, x^{3} + x^{2} - 2 \, x - 1}{{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 28, normalized size = 0.85
method | result | size |
risch | \(\frac {4 \left (-1+x \right ) \left (135 x^{3}+245 x^{2}+158 x +47\right )}{585 \left (\left (-1+x \right )^{3}\right )^{\frac {1}{4}}}\) | \(28\) |
gosper | \(\frac {4 \left (-1+x \right ) \left (135 x^{3}+245 x^{2}+158 x +47\right )}{585 \left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}}}\) | \(36\) |
trager | \(\frac {4 \left (135 x^{3}+245 x^{2}+158 x +47\right ) \left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {3}{4}}}{585 \left (-1+x \right )^{2}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 72, normalized size = 2.18 \begin {gather*} \frac {4 \, {\left (15 \, x^{4} + 5 \, x^{3} + 12 \, x^{2} + 96 \, x - 128\right )}}{65 \, {\left (x - 1\right )}^{\frac {3}{4}}} + \frac {4 \, {\left (5 \, x^{3} + 3 \, x^{2} + 24 \, x - 32\right )}}{45 \, {\left (x - 1\right )}^{\frac {3}{4}}} - \frac {8 \, {\left (x^{2} + 3 \, x - 4\right )}}{5 \, {\left (x - 1\right )}^{\frac {3}{4}}} - 4 \, {\left (x - 1\right )}^{\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 41, normalized size = 1.24 \begin {gather*} \frac {{\left (x^3-3\,x^2+3\,x-1\right )}^{3/4}\,\left (\frac {12\,x^3}{13}+\frac {196\,x^2}{117}+\frac {632\,x}{585}+\frac {188}{585}\right )}{x^2-2\,x+1} \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 {3 x^{3} + x^{2} - 2 x - 1}{\sqrt [4]{\left (x - 1\right )^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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