Optimal. Leaf size=250 \[ -\frac {2467 \sqrt [4]{\frac {1}{a x}+1}}{192 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {475 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{64 a^4}+\frac {475 \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{64 a^4}+\frac {521 x \sqrt [4]{\frac {1}{a x}+1}}{192 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {113 x^2 \sqrt [4]{\frac {1}{a x}+1}}{96 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {17 x^3 \sqrt [4]{\frac {1}{a x}+1}}{24 a \sqrt [4]{1-\frac {1}{a x}}} \]
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Rubi [A] time = 0.14, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6171, 98, 151, 155, 12, 93, 212, 206, 203} \[ \frac {113 x^2 \sqrt [4]{\frac {1}{a x}+1}}{96 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {521 x \sqrt [4]{\frac {1}{a x}+1}}{192 a^3 \sqrt [4]{1-\frac {1}{a x}}}-\frac {2467 \sqrt [4]{\frac {1}{a x}+1}}{192 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {475 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{64 a^4}+\frac {475 \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{64 a^4}+\frac {x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {17 x^3 \sqrt [4]{\frac {1}{a x}+1}}{24 a \sqrt [4]{1-\frac {1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 98
Rule 151
Rule 155
Rule 203
Rule 206
Rule 212
Rule 6171
Rubi steps
\begin {align*} \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^3 \, dx &=-\operatorname {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{5/4}}{x^5 \left (1-\frac {x}{a}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\sqrt [4]{1+\frac {1}{a x}} x^4}{4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {-\frac {17}{2 a}-\frac {8 x}{a^2}}{x^4 \left (1-\frac {x}{a}\right )^{5/4} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {17 \sqrt [4]{1+\frac {1}{a x}} x^3}{24 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^4}{4 \sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{12} \operatorname {Subst}\left (\int \frac {\frac {113}{4 a^2}+\frac {51 x}{2 a^3}}{x^3 \left (1-\frac {x}{a}\right )^{5/4} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {113 \sqrt [4]{1+\frac {1}{a x}} x^2}{96 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {17 \sqrt [4]{1+\frac {1}{a x}} x^3}{24 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^4}{4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1}{24} \operatorname {Subst}\left (\int \frac {-\frac {521}{8 a^3}-\frac {113 x}{2 a^4}}{x^2 \left (1-\frac {x}{a}\right )^{5/4} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {521 \sqrt [4]{1+\frac {1}{a x}} x}{192 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {113 \sqrt [4]{1+\frac {1}{a x}} x^2}{96 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {17 \sqrt [4]{1+\frac {1}{a x}} x^3}{24 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^4}{4 \sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{24} \operatorname {Subst}\left (\int \frac {\frac {1425}{16 a^4}+\frac {521 x}{8 a^5}}{x \left (1-\frac {x}{a}\right )^{5/4} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2467 \sqrt [4]{1+\frac {1}{a x}}}{192 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {521 \sqrt [4]{1+\frac {1}{a x}} x}{192 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {113 \sqrt [4]{1+\frac {1}{a x}} x^2}{96 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {17 \sqrt [4]{1+\frac {1}{a x}} x^3}{24 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^4}{4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1}{12} a \operatorname {Subst}\left (\int -\frac {1425}{32 a^5 x \sqrt [4]{1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2467 \sqrt [4]{1+\frac {1}{a x}}}{192 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {521 \sqrt [4]{1+\frac {1}{a x}} x}{192 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {113 \sqrt [4]{1+\frac {1}{a x}} x^2}{96 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {17 \sqrt [4]{1+\frac {1}{a x}} x^3}{24 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^4}{4 \sqrt [4]{1-\frac {1}{a x}}}-\frac {475 \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{128 a^4}\\ &=-\frac {2467 \sqrt [4]{1+\frac {1}{a x}}}{192 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {521 \sqrt [4]{1+\frac {1}{a x}} x}{192 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {113 \sqrt [4]{1+\frac {1}{a x}} x^2}{96 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {17 \sqrt [4]{1+\frac {1}{a x}} x^3}{24 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^4}{4 \sqrt [4]{1-\frac {1}{a x}}}-\frac {475 \operatorname {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{32 a^4}\\ &=-\frac {2467 \sqrt [4]{1+\frac {1}{a x}}}{192 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {521 \sqrt [4]{1+\frac {1}{a x}} x}{192 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {113 \sqrt [4]{1+\frac {1}{a x}} x^2}{96 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {17 \sqrt [4]{1+\frac {1}{a x}} x^3}{24 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^4}{4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {475 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{64 a^4}+\frac {475 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{64 a^4}\\ &=-\frac {2467 \sqrt [4]{1+\frac {1}{a x}}}{192 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {521 \sqrt [4]{1+\frac {1}{a x}} x}{192 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {113 \sqrt [4]{1+\frac {1}{a x}} x^2}{96 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {17 \sqrt [4]{1+\frac {1}{a x}} x^3}{24 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^4}{4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {475 \tan ^{-1}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{64 a^4}+\frac {475 \tanh ^{-1}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{64 a^4}\\ \end {align*}
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Mathematica [A] time = 5.25, size = 161, normalized size = 0.64 \[ \frac {-3072 e^{\frac {1}{2} \coth ^{-1}(a x)}+\frac {6292 e^{\frac {1}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}+\frac {7376 e^{\frac {1}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^2}+\frac {5248 e^{\frac {1}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^3}+\frac {1536 e^{\frac {1}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^4}-1425 \log \left (1-e^{\frac {1}{2} \coth ^{-1}(a x)}\right )+1425 \log \left (e^{\frac {1}{2} \coth ^{-1}(a x)}+1\right )+2850 \tan ^{-1}\left (e^{\frac {1}{2} \coth ^{-1}(a x)}\right )}{384 a^4} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.56, size = 144, normalized size = 0.58 \[ -\frac {2850 \, {\left (a x - 1\right )} \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) - 1425 \, {\left (a x - 1\right )} \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + 1425 \, {\left (a x - 1\right )} \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right ) - 2 \, {\left (48 \, a^{5} x^{5} + 184 \, a^{4} x^{4} + 362 \, a^{3} x^{3} + 747 \, a^{2} x^{2} - 1946 \, a x - 2467\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{384 \, {\left (a^{5} x - a^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 223, normalized size = 0.89 \[ -\frac {1}{384} \, a {\left (\frac {2850 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{5}} - \frac {1425 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{5}} + \frac {1425 \, \log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1 \right |}\right )}{a^{5}} + \frac {3072}{a^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} + \frac {4 \, {\left (\frac {2875 \, {\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{a x + 1} - \frac {2343 \, {\left (a x - 1\right )}^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{{\left (a x + 1\right )}^{2}} + \frac {657 \, {\left (a x - 1\right )}^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{{\left (a x + 1\right )}^{3}} - 1573 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\right )}}{a^{5} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 238, normalized size = 0.95 \[ \frac {1}{384} \, a {\left (\frac {4 \, {\left (\frac {4645 \, {\left (a x - 1\right )}}{a x + 1} - \frac {7483 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {5415 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac {1425 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} - 768\right )}}{a^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {17}{4}} - 4 \, a^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{4}} + 6 \, a^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{4}} - 4 \, a^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} + a^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} - \frac {2850 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{5}} + \frac {1425 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{5}} - \frac {1425 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{a^{5}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 211, normalized size = 0.84 \[ \frac {475\,\mathrm {atanh}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{64\,a^4}-\frac {475\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{64\,a^4}-\frac {\frac {7483\,{\left (a\,x-1\right )}^2}{96\,{\left (a\,x+1\right )}^2}-\frac {1805\,{\left (a\,x-1\right )}^3}{32\,{\left (a\,x+1\right )}^3}+\frac {475\,{\left (a\,x-1\right )}^4}{32\,{\left (a\,x+1\right )}^4}-\frac {4645\,\left (a\,x-1\right )}{96\,\left (a\,x+1\right )}+8}{a^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}-4\,a^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}+6\,a^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/4}-4\,a^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/4}+a^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{17/4}} \]
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 \[ \int \frac {x^{3}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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