Optimal. Leaf size=317 \[ \frac {(1-a x)^{3/4} (a x+1)^{5/4} (452 a x+521)}{96 a^4}+\frac {475 (1-a x)^{3/4} \sqrt [4]{a x+1}}{64 a^4}+\frac {475 \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt {2} a^4}-\frac {475 \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt {2} a^4}-\frac {475 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{64 \sqrt {2} a^4}+\frac {475 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{64 \sqrt {2} a^4}+\frac {17 x^2 (1-a x)^{3/4} (a x+1)^{5/4}}{4 a^2}+\frac {4 x^3 (a x+1)^{5/4}}{a \sqrt [4]{1-a x}} \]
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Rubi [A] time = 0.24, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {6126, 97, 153, 147, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ \frac {17 x^2 (1-a x)^{3/4} (a x+1)^{5/4}}{4 a^2}+\frac {(1-a x)^{3/4} (a x+1)^{5/4} (452 a x+521)}{96 a^4}+\frac {475 (1-a x)^{3/4} \sqrt [4]{a x+1}}{64 a^4}+\frac {475 \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt {2} a^4}-\frac {475 \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt {2} a^4}-\frac {475 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{64 \sqrt {2} a^4}+\frac {475 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{64 \sqrt {2} a^4}+\frac {4 x^3 (a x+1)^{5/4}}{a \sqrt [4]{1-a x}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 97
Rule 147
Rule 153
Rule 204
Rule 297
Rule 331
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 6126
Rubi steps
\begin {align*} \int e^{\frac {5}{2} \tanh ^{-1}(a x)} x^3 \, dx &=\int \frac {x^3 (1+a x)^{5/4}}{(1-a x)^{5/4}} \, dx\\ &=\frac {4 x^3 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}-\frac {4 \int \frac {x^2 \sqrt [4]{1+a x} \left (3+\frac {17 a x}{4}\right )}{\sqrt [4]{1-a x}} \, dx}{a}\\ &=\frac {4 x^3 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}+\frac {17 x^2 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^2}+\frac {\int \frac {x \sqrt [4]{1+a x} \left (-\frac {17 a}{2}-\frac {113 a^2 x}{8}\right )}{\sqrt [4]{1-a x}} \, dx}{a^3}\\ &=\frac {4 x^3 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}+\frac {17 x^2 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^2}+\frac {(1-a x)^{3/4} (1+a x)^{5/4} (521+452 a x)}{96 a^4}-\frac {475 \int \frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}} \, dx}{64 a^3}\\ &=\frac {475 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}+\frac {4 x^3 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}+\frac {17 x^2 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^2}+\frac {(1-a x)^{3/4} (1+a x)^{5/4} (521+452 a x)}{96 a^4}-\frac {475 \int \frac {1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx}{128 a^3}\\ &=\frac {475 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}+\frac {4 x^3 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}+\frac {17 x^2 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^2}+\frac {(1-a x)^{3/4} (1+a x)^{5/4} (521+452 a x)}{96 a^4}+\frac {475 \operatorname {Subst}\left (\int \frac {x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-a x}\right )}{32 a^4}\\ &=\frac {475 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}+\frac {4 x^3 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}+\frac {17 x^2 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^2}+\frac {(1-a x)^{3/4} (1+a x)^{5/4} (521+452 a x)}{96 a^4}+\frac {475 \operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{32 a^4}\\ &=\frac {475 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}+\frac {4 x^3 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}+\frac {17 x^2 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^2}+\frac {(1-a x)^{3/4} (1+a x)^{5/4} (521+452 a x)}{96 a^4}-\frac {475 \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 a^4}+\frac {475 \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 a^4}\\ &=\frac {475 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}+\frac {4 x^3 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}+\frac {17 x^2 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^2}+\frac {(1-a x)^{3/4} (1+a x)^{5/4} (521+452 a x)}{96 a^4}+\frac {475 \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 a^4}+\frac {475 \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 a^4}+\frac {475 \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a^4}+\frac {475 \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a^4}\\ &=\frac {475 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}+\frac {4 x^3 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}+\frac {17 x^2 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^2}+\frac {(1-a x)^{3/4} (1+a x)^{5/4} (521+452 a x)}{96 a^4}+\frac {475 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a^4}-\frac {475 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a^4}+\frac {475 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt {2} a^4}-\frac {475 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt {2} a^4}\\ &=\frac {475 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}+\frac {4 x^3 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}+\frac {17 x^2 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^2}+\frac {(1-a x)^{3/4} (1+a x)^{5/4} (521+452 a x)}{96 a^4}-\frac {475 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt {2} a^4}+\frac {475 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt {2} a^4}+\frac {475 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a^4}-\frac {475 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a^4}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 74, normalized size = 0.23 \[ \frac {(a x+1)^{9/4} \left (-6 a^2 x^2-5 a x+59\right )-380 \sqrt [4]{2} (a x-1) \, _2F_1\left (-\frac {5}{4},\frac {3}{4};\frac {7}{4};\frac {1}{2} (1-a x)\right )}{24 a^4 \sqrt [4]{1-a x}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.61, size = 553, normalized size = 1.74 \[ \frac {5700 \, \sqrt {2} a^{4} \frac {1}{a^{16}}^{\frac {1}{4}} \arctan \left (\sqrt {2} a^{12} \sqrt {\frac {\sqrt {2} {\left (a^{5} x - a^{4}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {1}{4}} + {\left (a^{9} x - a^{8}\right )} \sqrt {\frac {1}{a^{16}}} - \sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {3}{4}} - \sqrt {2} a^{12} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {3}{4}} - 1\right ) + 5700 \, \sqrt {2} a^{4} \frac {1}{a^{16}}^{\frac {1}{4}} \arctan \left (\sqrt {2} a^{12} \sqrt {-\frac {\sqrt {2} {\left (a^{5} x - a^{4}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {1}{4}} - {\left (a^{9} x - a^{8}\right )} \sqrt {\frac {1}{a^{16}}} + \sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {3}{4}} - \sqrt {2} a^{12} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {3}{4}} + 1\right ) - 1425 \, \sqrt {2} a^{4} \frac {1}{a^{16}}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} {\left (a^{5} x - a^{4}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {1}{4}} + {\left (a^{9} x - a^{8}\right )} \sqrt {\frac {1}{a^{16}}} - \sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right ) + 1425 \, \sqrt {2} a^{4} \frac {1}{a^{16}}^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} {\left (a^{5} x - a^{4}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {1}{4}} - {\left (a^{9} x - a^{8}\right )} \sqrt {\frac {1}{a^{16}}} + \sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right ) - 4 \, {\left (48 \, a^{4} x^{4} + 136 \, a^{3} x^{3} + 226 \, a^{2} x^{2} + 521 \, a x - 2467\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{768 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )^{\frac {5}{2}} x^{3}\, dx \]
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 \[ \int x^{3} \left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{5/2} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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