Optimal. Leaf size=78 \[ \frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f}-2 a b p q x-\frac{2 b^2 p q (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}+2 b^2 p^2 q^2 x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0963122, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2389, 2296, 2295, 2445} \[ \frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f}-2 a b p q x-\frac{2 b^2 p q (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}+2 b^2 p^2 q^2 x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2389
Rule 2296
Rule 2295
Rule 2445
Rubi steps
\begin{align*} \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx &=\operatorname{Subst}\left (\int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2 \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname{Subst}\left (\frac{\operatorname{Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right )^2 \, dx,x,e+f x\right )}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f}-\operatorname{Subst}\left (\frac{(2 b p q) \operatorname{Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right ) \, dx,x,e+f x\right )}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-2 a b p q x+\frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f}-\operatorname{Subst}\left (\frac{\left (2 b^2 p q\right ) \operatorname{Subst}\left (\int \log \left (c d^q x^{p q}\right ) \, dx,x,e+f x\right )}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-2 a b p q x+2 b^2 p^2 q^2 x-\frac{2 b^2 p q (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}+\frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f}\\ \end{align*}
Mathematica [A] time = 0.0156952, size = 69, normalized size = 0.88 \[ \frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f}-2 b p q \left (a x+\frac{b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}-b p q x\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.279, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.14768, size = 200, normalized size = 2.56 \begin{align*} -2 \, a b f p q{\left (\frac{x}{f} - \frac{e \log \left (f x + e\right )}{f^{2}}\right )} + b^{2} x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + 2 \, a b x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) -{\left (2 \, f p q{\left (\frac{x}{f} - \frac{e \log \left (f x + e\right )}{f^{2}}\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + \frac{{\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} p^{2} q^{2}}{f}\right )} b^{2} + a^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.90963, size = 516, normalized size = 6.62 \begin{align*} \frac{b^{2} f q^{2} x \log \left (d\right )^{2} + b^{2} f x \log \left (c\right )^{2} +{\left (b^{2} f p^{2} q^{2} x + b^{2} e p^{2} q^{2}\right )} \log \left (f x + e\right )^{2} - 2 \,{\left (b^{2} f p q - a b f\right )} x \log \left (c\right ) +{\left (2 \, b^{2} f p^{2} q^{2} - 2 \, a b f p q + a^{2} f\right )} x - 2 \,{\left (b^{2} e p^{2} q^{2} - a b e p q +{\left (b^{2} f p^{2} q^{2} - a b f p q\right )} x -{\left (b^{2} f p q x + b^{2} e p q\right )} \log \left (c\right ) -{\left (b^{2} f p q^{2} x + b^{2} e p q^{2}\right )} \log \left (d\right )\right )} \log \left (f x + e\right ) + 2 \,{\left (b^{2} f q x \log \left (c\right ) -{\left (b^{2} f p q^{2} - a b f q\right )} x\right )} \log \left (d\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 3.27804, size = 343, normalized size = 4.4 \begin{align*} \begin{cases} a^{2} x + \frac{2 a b e p q \log{\left (e + f x \right )}}{f} + 2 a b p q x \log{\left (e + f x \right )} - 2 a b p q x + 2 a b q x \log{\left (d \right )} + 2 a b x \log{\left (c \right )} + \frac{b^{2} e p^{2} q^{2} \log{\left (e + f x \right )}^{2}}{f} - \frac{2 b^{2} e p^{2} q^{2} \log{\left (e + f x \right )}}{f} + \frac{2 b^{2} e p q^{2} \log{\left (d \right )} \log{\left (e + f x \right )}}{f} + \frac{2 b^{2} e p q \log{\left (c \right )} \log{\left (e + f x \right )}}{f} + b^{2} p^{2} q^{2} x \log{\left (e + f x \right )}^{2} - 2 b^{2} p^{2} q^{2} x \log{\left (e + f x \right )} + 2 b^{2} p^{2} q^{2} x + 2 b^{2} p q^{2} x \log{\left (d \right )} \log{\left (e + f x \right )} - 2 b^{2} p q^{2} x \log{\left (d \right )} + 2 b^{2} p q x \log{\left (c \right )} \log{\left (e + f x \right )} - 2 b^{2} p q x \log{\left (c \right )} + b^{2} q^{2} x \log{\left (d \right )}^{2} + 2 b^{2} q x \log{\left (c \right )} \log{\left (d \right )} + b^{2} x \log{\left (c \right )}^{2} & \text{for}\: f \neq 0 \\x \left (a + b \log{\left (c \left (d e^{p}\right )^{q} \right )}\right )^{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.26098, size = 409, normalized size = 5.24 \begin{align*} \frac{{\left (f x + e\right )} b^{2} p^{2} q^{2} \log \left (f x + e\right )^{2}}{f} - \frac{2 \,{\left (f x + e\right )} b^{2} p^{2} q^{2} \log \left (f x + e\right )}{f} + \frac{2 \,{\left (f x + e\right )} b^{2} p q^{2} \log \left (f x + e\right ) \log \left (d\right )}{f} + \frac{2 \,{\left (f x + e\right )} b^{2} p^{2} q^{2}}{f} + \frac{2 \,{\left (f x + e\right )} b^{2} p q \log \left (f x + e\right ) \log \left (c\right )}{f} - \frac{2 \,{\left (f x + e\right )} b^{2} p q^{2} \log \left (d\right )}{f} + \frac{{\left (f x + e\right )} b^{2} q^{2} \log \left (d\right )^{2}}{f} + \frac{2 \,{\left (f x + e\right )} a b p q \log \left (f x + e\right )}{f} - \frac{2 \,{\left (f x + e\right )} b^{2} p q \log \left (c\right )}{f} + \frac{2 \,{\left (f x + e\right )} b^{2} q \log \left (c\right ) \log \left (d\right )}{f} - \frac{2 \,{\left (f x + e\right )} a b p q}{f} + \frac{{\left (f x + e\right )} b^{2} \log \left (c\right )^{2}}{f} + \frac{2 \,{\left (f x + e\right )} a b q \log \left (d\right )}{f} + \frac{2 \,{\left (f x + e\right )} a b \log \left (c\right )}{f} + \frac{{\left (f x + e\right )} a^{2}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]