Optimal. Leaf size=98 \[ -\frac {3 e^{\frac {3 a}{2 b n}} \left (c x^n\right )^{\left .\frac {3}{2}\right /n} \text {Ei}\left (-\frac {3 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{2 b^2 d n^2 (d x)^{3/2}}-\frac {1}{b d n (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )} \]
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Rubi [A] time = 0.09, 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} \[ -\frac {3 e^{\frac {3 a}{2 b n}} \left (c x^n\right )^{\left .\frac {3}{2}\right /n} \text {Ei}\left (-\frac {3 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{2 b^2 d n^2 (d x)^{3/2}}-\frac {1}{b d n (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2306
Rule 2310
Rubi steps
\begin {align*} \int \frac {1}{(d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx &=-\frac {1}{b d n (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}-\frac {3 \int \frac {1}{(d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )} \, dx}{2 b n}\\ &=-\frac {1}{b d n (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}-\frac {\left (3 \left (c x^n\right )^{\left .\frac {3}{2}\right /n}\right ) \operatorname {Subst}\left (\int \frac {e^{-\frac {3 x}{2 n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 b d n^2 (d x)^{3/2}}\\ &=-\frac {3 e^{\frac {3 a}{2 b n}} \left (c x^n\right )^{\left .\frac {3}{2}\right /n} \text {Ei}\left (-\frac {3 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{2 b^2 d n^2 (d x)^{3/2}}-\frac {1}{b d n (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 94, normalized size = 0.96 \[ -\frac {x \left (3 e^{\frac {3 a}{2 b n}} \left (c x^n\right )^{\left .\frac {3}{2}\right /n} \left (a+b \log \left (c x^n\right )\right ) \text {Ei}\left (-\frac {3 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )+2 b n\right )}{2 b^2 n^2 (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x}}{b^{2} d^{3} x^{3} \log \left (c x^{n}\right )^{2} + 2 \, a b d^{3} x^{3} \log \left (c x^{n}\right ) + a^{2} d^{3} x^{3}}, x\right ) \]
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 \[ \int \frac {1}{\left (d x\right )^{\frac {5}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 5.16, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x \right )^{\frac {5}{2}} \left (b \ln \left (c \,x^{n}\right )+a \right )^{2}}\, dx \]
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 \[ -4 \, b n \int \frac {1}{3 \, {\left (b^{3} d^{\frac {5}{2}} \log \relax (c)^{3} + b^{3} d^{\frac {5}{2}} \log \left (x^{n}\right )^{3} + 3 \, a b^{2} d^{\frac {5}{2}} \log \relax (c)^{2} + 3 \, a^{2} b d^{\frac {5}{2}} \log \relax (c) + a^{3} d^{\frac {5}{2}} + 3 \, {\left (b^{3} d^{\frac {5}{2}} \log \relax (c) + a b^{2} d^{\frac {5}{2}}\right )} \log \left (x^{n}\right )^{2} + 3 \, {\left (b^{3} d^{\frac {5}{2}} \log \relax (c)^{2} + 2 \, a b^{2} d^{\frac {5}{2}} \log \relax (c) + a^{2} b d^{\frac {5}{2}}\right )} \log \left (x^{n}\right )\right )} x^{\frac {5}{2}}}\,{d x} - \frac {2}{3 \, {\left (b^{2} d^{\frac {5}{2}} \log \relax (c)^{2} + b^{2} d^{\frac {5}{2}} \log \left (x^{n}\right )^{2} + 2 \, a b d^{\frac {5}{2}} \log \relax (c) + a^{2} d^{\frac {5}{2}} + 2 \, {\left (b^{2} d^{\frac {5}{2}} \log \relax (c) + a b d^{\frac {5}{2}}\right )} \log \left (x^{n}\right )\right )} x^{\frac {3}{2}}} \]
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 \[ \int \frac {1}{{\left (d\,x\right )}^{5/2}\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]
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 \[ \int \frac {1}{\left (d x\right )^{\frac {5}{2}} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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