Optimal. Leaf size=101 \[ -\frac {e^{\frac {a}{2 b n}} \left (c x^n\right )^{\left .\frac {1}{2}\right /n} \text {Ei}\left (\frac {-a-b \log \left (c x^n\right )}{2 b n}\right )}{2 b^2 d n^2 \sqrt {d x}}-\frac {1}{b d n \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \]
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Rubi [A] time = 0.09, antiderivative size = 98, normalized size of antiderivative = 0.97, 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} \[ -\frac {e^{\frac {a}{2 b n}} \left (c x^n\right )^{\left .\frac {1}{2}\right /n} \text {Ei}\left (-\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{2 b^2 d n^2 \sqrt {d x}}-\frac {1}{b d n \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2306
Rule 2310
Rubi steps
\begin {align*} \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx &=-\frac {1}{b d n \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}-\frac {\int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )} \, dx}{2 b n}\\ &=-\frac {1}{b d n \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}-\frac {\left (c x^n\right )^{\left .\frac {1}{2}\right /n} \operatorname {Subst}\left (\int \frac {e^{-\frac {x}{2 n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 b d n^2 \sqrt {d x}}\\ &=-\frac {e^{\frac {a}{2 b n}} \left (c x^n\right )^{\left .\frac {1}{2}\right /n} \text {Ei}\left (-\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{2 b^2 d n^2 \sqrt {d x}}-\frac {1}{b d n \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 93, normalized size = 0.92 \[ -\frac {x \left (e^{\frac {a}{2 b n}} \left (c x^n\right )^{\left .\frac {1}{2}\right /n} \left (a+b \log \left (c x^n\right )\right ) \text {Ei}\left (-\frac {a+b \log \left (c x^n\right )}{2 b n}\right )+2 b n\right )}{2 b^2 n^2 (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x}}{b^{2} d^{2} x^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b d^{2} x^{2} \log \left (c x^{n}\right ) + a^{2} d^{2} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.43, size = 293, normalized size = 2.90 \[ -\frac {\frac {b c^{\frac {1}{2 \, n}} n {\rm Ei}\left (-\frac {\log \relax (c)}{2 \, n} - \frac {a}{2 \, b n} - \frac {1}{2} \, \log \relax (x)\right ) e^{\left (\frac {a}{2 \, b n}\right )} \log \relax (x)}{b^{3} \sqrt {d} n^{3} \log \relax (x) + b^{3} \sqrt {d} n^{2} \log \relax (c) + a b^{2} \sqrt {d} n^{2}} + \frac {b c^{\frac {1}{2 \, n}} {\rm Ei}\left (-\frac {\log \relax (c)}{2 \, n} - \frac {a}{2 \, b n} - \frac {1}{2} \, \log \relax (x)\right ) e^{\left (\frac {a}{2 \, b n}\right )} \log \relax (c)}{b^{3} \sqrt {d} n^{3} \log \relax (x) + b^{3} \sqrt {d} n^{2} \log \relax (c) + a b^{2} \sqrt {d} n^{2}} + \frac {a c^{\frac {1}{2 \, n}} {\rm Ei}\left (-\frac {\log \relax (c)}{2 \, n} - \frac {a}{2 \, b n} - \frac {1}{2} \, \log \relax (x)\right ) e^{\left (\frac {a}{2 \, b n}\right )}}{b^{3} \sqrt {d} n^{3} \log \relax (x) + b^{3} \sqrt {d} n^{2} \log \relax (c) + a b^{2} \sqrt {d} n^{2}} + \frac {2 \, b n}{{\left (b^{3} \sqrt {d} n^{3} \log \relax (x) + b^{3} \sqrt {d} n^{2} \log \relax (c) + a b^{2} \sqrt {d} n^{2}\right )} \sqrt {x}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 5.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x \right )^{\frac {3}{2}} \left (b \ln \left (c \,x^{n}\right )+a \right )^{2}}\, dx \]
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 \[ -4 \, b n \int \frac {1}{{\left (b^{3} d^{\frac {3}{2}} \log \relax (c)^{3} + b^{3} d^{\frac {3}{2}} \log \left (x^{n}\right )^{3} + 3 \, a b^{2} d^{\frac {3}{2}} \log \relax (c)^{2} + 3 \, a^{2} b d^{\frac {3}{2}} \log \relax (c) + a^{3} d^{\frac {3}{2}} + 3 \, {\left (b^{3} d^{\frac {3}{2}} \log \relax (c) + a b^{2} d^{\frac {3}{2}}\right )} \log \left (x^{n}\right )^{2} + 3 \, {\left (b^{3} d^{\frac {3}{2}} \log \relax (c)^{2} + 2 \, a b^{2} d^{\frac {3}{2}} \log \relax (c) + a^{2} b d^{\frac {3}{2}}\right )} \log \left (x^{n}\right )\right )} x^{\frac {3}{2}}}\,{d x} - \frac {2}{{\left (b^{2} d^{\frac {3}{2}} \log \relax (c)^{2} + b^{2} d^{\frac {3}{2}} \log \left (x^{n}\right )^{2} + 2 \, a b d^{\frac {3}{2}} \log \relax (c) + a^{2} d^{\frac {3}{2}} + 2 \, {\left (b^{2} d^{\frac {3}{2}} \log \relax (c) + a b d^{\frac {3}{2}}\right )} \log \left (x^{n}\right )\right )} \sqrt {x}} \]
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 \[ \int \frac {1}{{\left (d\,x\right )}^{3/2}\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]
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 \[ \int \frac {1}{\left (d x\right )^{\frac {3}{2}} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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