Optimal. Leaf size=98 \[ \frac {5 (d x)^{5/2} e^{-\frac {5 a}{2 b n}} \left (c x^n\right )^{\left .-\frac {5}{2}\right /n} \text {Ei}\left (\frac {5 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac {(d x)^{5/2}}{b d n \left (a+b \log \left (c x^n\right )\right )} \]
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Rubi [A] time = 0.10, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2306, 2310, 2178} \[ \frac {5 (d x)^{5/2} e^{-\frac {5 a}{2 b n}} \left (c x^n\right )^{\left .-\frac {5}{2}\right /n} \text {Ei}\left (\frac {5 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac {(d x)^{5/2}}{b d n \left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2306
Rule 2310
Rubi steps
\begin {align*} \int \frac {(d x)^{3/2}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx &=-\frac {(d x)^{5/2}}{b d n \left (a+b \log \left (c x^n\right )\right )}+\frac {5 \int \frac {(d x)^{3/2}}{a+b \log \left (c x^n\right )} \, dx}{2 b n}\\ &=-\frac {(d x)^{5/2}}{b d n \left (a+b \log \left (c x^n\right )\right )}+\frac {\left (5 (d x)^{5/2} \left (c x^n\right )^{\left .-\frac {5}{2}\right /n}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {5 x}{2 n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 b d n^2}\\ &=\frac {5 e^{-\frac {5 a}{2 b n}} (d x)^{5/2} \left (c x^n\right )^{\left .-\frac {5}{2}\right /n} \text {Ei}\left (\frac {5 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac {(d x)^{5/2}}{b d n \left (a+b \log \left (c x^n\right )\right )}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 84, normalized size = 0.86 \[ \frac {x (d x)^{3/2} \left (5 e^{-\frac {5 a}{2 b n}} \left (c x^n\right )^{\left .-\frac {5}{2}\right /n} \text {Ei}\left (\frac {5 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )-\frac {2 b n}{a+b \log \left (c x^n\right )}\right )}{2 b^2 n^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x} d x}{b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}}, x\right ) \]
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 \[ \int \frac {\left (d x\right )^{\frac {3}{2}}}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 5.22, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x \right )^{\frac {3}{2}}}{\left (b \ln \left (c \,x^{n}\right )+a \right )^{2}}\, dx \]
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 \[ 4 \, b d^{\frac {3}{2}} n \int \frac {x^{\frac {3}{2}}}{5 \, {\left (b^{3} \log \relax (c)^{3} + b^{3} \log \left (x^{n}\right )^{3} + 3 \, a b^{2} \log \relax (c)^{2} + 3 \, a^{2} b \log \relax (c) + a^{3} + 3 \, {\left (b^{3} \log \relax (c) + a b^{2}\right )} \log \left (x^{n}\right )^{2} + 3 \, {\left (b^{3} \log \relax (c)^{2} + 2 \, a b^{2} \log \relax (c) + a^{2} b\right )} \log \left (x^{n}\right )\right )}}\,{d x} + \frac {2 \, d^{\frac {3}{2}} x^{\frac {5}{2}}}{5 \, {\left (b^{2} \log \relax (c)^{2} + b^{2} \log \left (x^{n}\right )^{2} + 2 \, a b \log \relax (c) + a^{2} + 2 \, {\left (b^{2} \log \relax (c) + a b\right )} \log \left (x^{n}\right )\right )}} \]
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 \[ \int \frac {{\left (d\,x\right )}^{3/2}}{{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]
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 \[ \int \frac {\left (d x\right )^{\frac {3}{2}}}{\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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