Optimal. Leaf size=195 \[ \frac{b n \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{e^2}-\frac{\left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^2}+\frac{2 d \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^2}-\frac{4 a b d n}{e \sqrt{x}}-\frac{4 b^2 d n \left (d+\frac{e}{\sqrt{x}}\right ) \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )}{e^2}-\frac{b^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^2}{2 e^2}+\frac{4 b^2 d n^2}{e \sqrt{x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.1977, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304} \[ \frac{b n \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{e^2}-\frac{\left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^2}+\frac{2 d \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^2}-\frac{4 a b d n}{e \sqrt{x}}-\frac{4 b^2 d n \left (d+\frac{e}{\sqrt{x}}\right ) \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )}{e^2}-\frac{b^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^2}{2 e^2}+\frac{4 b^2 d n^2}{e \sqrt{x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2454
Rule 2401
Rule 2389
Rule 2296
Rule 2295
Rule 2390
Rule 2305
Rule 2304
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{x^2} \, dx &=-\left (2 \operatorname{Subst}\left (\int x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \left (-\frac{d \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}\right ) \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=-\frac{2 \operatorname{Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\frac{1}{\sqrt{x}}\right )}{e}+\frac{(2 d) \operatorname{Subst}\left (\int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\frac{1}{\sqrt{x}}\right )}{e}\\ &=-\frac{2 \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^2}+\frac{(2 d) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^2}\\ &=\frac{2 d \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^2}-\frac{\left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^2}+\frac{(2 b n) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^2}-\frac{(4 b d n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^2}\\ &=-\frac{b^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^2}{2 e^2}-\frac{4 a b d n}{e \sqrt{x}}+\frac{b n \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{e^2}+\frac{2 d \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^2}-\frac{\left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^2}-\frac{\left (4 b^2 d n\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^2}\\ &=-\frac{b^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^2}{2 e^2}-\frac{4 a b d n}{e \sqrt{x}}+\frac{4 b^2 d n^2}{e \sqrt{x}}-\frac{4 b^2 d n \left (d+\frac{e}{\sqrt{x}}\right ) \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )}{e^2}+\frac{b n \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{e^2}+\frac{2 d \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^2}-\frac{\left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^2}\\ \end{align*}
Mathematica [C] time = 0.29817, size = 298, normalized size = 1.53 \[ -\frac{\frac{b n \left (-4 b d^2 n x \text{PolyLog}\left (2,\frac{e}{d \sqrt{x}}+1\right )+2 b d^2 n x \left (\log \left (d \sqrt{x}+e\right ) \left (\log \left (d \sqrt{x}+e\right )-2 \log \left (-\frac{d \sqrt{x}}{e}\right )\right )-2 \text{PolyLog}\left (2,\frac{d \sqrt{x}}{e}+1\right )\right )-4 d^2 x \log \left (d \sqrt{x}+e\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )-4 d^2 x \log \left (-\frac{e}{d \sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )-2 e^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )+4 a d e \sqrt{x}+4 b d \sqrt{x} \left (d \sqrt{x}+e\right ) \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )+b n \left (2 d^2 x \log \left (d+\frac{e}{\sqrt{x}}\right )+e \left (e-2 d \sqrt{x}\right )\right )-4 b d e n \sqrt{x}\right )}{e^2}+2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.331, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+b\ln \left ( c \left ( d+{e{\frac{1}{\sqrt{x}}}} \right ) ^{n} \right ) \right ) ^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.08132, size = 335, normalized size = 1.72 \begin{align*} a b e n{\left (\frac{2 \, d^{2} \log \left (d \sqrt{x} + e\right )}{e^{3}} - \frac{d^{2} \log \left (x\right )}{e^{3}} - \frac{2 \, d \sqrt{x} - e}{e^{2} x}\right )} + \frac{1}{4} \,{\left (4 \, e n{\left (\frac{2 \, d^{2} \log \left (d \sqrt{x} + e\right )}{e^{3}} - \frac{d^{2} \log \left (x\right )}{e^{3}} - \frac{2 \, d \sqrt{x} - e}{e^{2} x}\right )} \log \left (c{\left (d + \frac{e}{\sqrt{x}}\right )}^{n}\right ) - \frac{{\left (4 \, d^{2} x \log \left (d \sqrt{x} + e\right )^{2} + d^{2} x \log \left (x\right )^{2} - 6 \, d^{2} x \log \left (x\right ) - 12 \, d e \sqrt{x} + 2 \, e^{2} - 4 \,{\left (d^{2} x \log \left (x\right ) - 3 \, d^{2} x\right )} \log \left (d \sqrt{x} + e\right )\right )} n^{2}}{e^{2} x}\right )} b^{2} - \frac{b^{2} \log \left (c{\left (d + \frac{e}{\sqrt{x}}\right )}^{n}\right )^{2}}{x} - \frac{2 \, a b \log \left (c{\left (d + \frac{e}{\sqrt{x}}\right )}^{n}\right )}{x} - \frac{a^{2}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.73133, size = 521, normalized size = 2.67 \begin{align*} -\frac{b^{2} e^{2} n^{2} + 2 \, b^{2} e^{2} \log \left (c\right )^{2} - 2 \, a b e^{2} n + 2 \, a^{2} e^{2} - 2 \,{\left (b^{2} d^{2} n^{2} x - b^{2} e^{2} n^{2}\right )} \log \left (\frac{d x + e \sqrt{x}}{x}\right )^{2} - 2 \,{\left (b^{2} e^{2} n - 2 \, a b e^{2}\right )} \log \left (c\right ) + 2 \,{\left (2 \, b^{2} d e n^{2} \sqrt{x} - b^{2} e^{2} n^{2} + 2 \, a b e^{2} n +{\left (3 \, b^{2} d^{2} n^{2} - 2 \, a b d^{2} n\right )} x - 2 \,{\left (b^{2} d^{2} n x - b^{2} e^{2} n\right )} \log \left (c\right )\right )} \log \left (\frac{d x + e \sqrt{x}}{x}\right ) - 2 \,{\left (3 \, b^{2} d e n^{2} - 2 \, b^{2} d e n \log \left (c\right ) - 2 \, a b d e n\right )} \sqrt{x}}{2 \, e^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c{\left (d + \frac{e}{\sqrt{x}}\right )}^{n}\right ) + a\right )}^{2}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]