Optimal. Leaf size=136 \[ \frac {25 a^2 \sqrt [4]{a x+1}}{2 \sqrt [4]{1-a x}}-\frac {25}{4} a^2 \tan ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {25}{4} a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {(a x+1)^{9/4}}{2 x^2 \sqrt [4]{1-a x}}-\frac {5 a (a x+1)^{5/4}}{4 x \sqrt [4]{1-a x}} \]
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Rubi [A] time = 0.05, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6126, 96, 94, 93, 212, 206, 203} \[ \frac {25 a^2 \sqrt [4]{a x+1}}{2 \sqrt [4]{1-a x}}-\frac {25}{4} a^2 \tan ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {25}{4} a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {(a x+1)^{9/4}}{2 x^2 \sqrt [4]{1-a x}}-\frac {5 a (a x+1)^{5/4}}{4 x \sqrt [4]{1-a x}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 203
Rule 206
Rule 212
Rule 6126
Rubi steps
\begin {align*} \int \frac {e^{\frac {5}{2} \tanh ^{-1}(a x)}}{x^3} \, dx &=\int \frac {(1+a x)^{5/4}}{x^3 (1-a x)^{5/4}} \, dx\\ &=-\frac {(1+a x)^{9/4}}{2 x^2 \sqrt [4]{1-a x}}+\frac {1}{4} (5 a) \int \frac {(1+a x)^{5/4}}{x^2 (1-a x)^{5/4}} \, dx\\ &=-\frac {5 a (1+a x)^{5/4}}{4 x \sqrt [4]{1-a x}}-\frac {(1+a x)^{9/4}}{2 x^2 \sqrt [4]{1-a x}}+\frac {1}{8} \left (25 a^2\right ) \int \frac {\sqrt [4]{1+a x}}{x (1-a x)^{5/4}} \, dx\\ &=\frac {25 a^2 \sqrt [4]{1+a x}}{2 \sqrt [4]{1-a x}}-\frac {5 a (1+a x)^{5/4}}{4 x \sqrt [4]{1-a x}}-\frac {(1+a x)^{9/4}}{2 x^2 \sqrt [4]{1-a x}}+\frac {1}{8} \left (25 a^2\right ) \int \frac {1}{x \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=\frac {25 a^2 \sqrt [4]{1+a x}}{2 \sqrt [4]{1-a x}}-\frac {5 a (1+a x)^{5/4}}{4 x \sqrt [4]{1-a x}}-\frac {(1+a x)^{9/4}}{2 x^2 \sqrt [4]{1-a x}}+\frac {1}{2} \left (25 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=\frac {25 a^2 \sqrt [4]{1+a x}}{2 \sqrt [4]{1-a x}}-\frac {5 a (1+a x)^{5/4}}{4 x \sqrt [4]{1-a x}}-\frac {(1+a x)^{9/4}}{2 x^2 \sqrt [4]{1-a x}}-\frac {1}{4} \left (25 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac {1}{4} \left (25 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=\frac {25 a^2 \sqrt [4]{1+a x}}{2 \sqrt [4]{1-a x}}-\frac {5 a (1+a x)^{5/4}}{4 x \sqrt [4]{1-a x}}-\frac {(1+a x)^{9/4}}{2 x^2 \sqrt [4]{1-a x}}-\frac {25}{4} a^2 \tan ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac {25}{4} a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ \end {align*}
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Mathematica [C] time = 0.03, size = 86, normalized size = 0.63 \[ \frac {50 a^2 x^2 (a x-1) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {1-a x}{a x+1}\right )+3 \left (43 a^3 x^3+34 a^2 x^2-11 a x-2\right )}{12 x^2 \sqrt [4]{1-a x} (a x+1)^{3/4}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.57, size = 145, normalized size = 1.07 \[ -\frac {50 \, a^{2} x^{2} \arctan \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) + 25 \, a^{2} x^{2} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) - 25 \, a^{2} x^{2} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) - 2 \, {\left (43 \, a^{2} x^{2} - 9 \, a x - 2\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{8 \, x^{2}} \]
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 \[ \int \frac {\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )^{\frac {5}{2}}}{x^{3}}\, dx \]
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 \[ \int \frac {\left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{5/2}}{x^3} \,d x \]
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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