Optimal. Leaf size=98 \[ \frac{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 h}-\frac{b p q (f g-e h)^2 \log (e+f x)}{2 f^2 h}-\frac{b p q x (f g-e h)}{2 f}-\frac{b p q (g+h x)^2}{4 h} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0813518, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2395, 43, 2445} \[ \frac{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 h}-\frac{b p q (f g-e h)^2 \log (e+f x)}{2 f^2 h}-\frac{b p q x (f g-e h)}{2 f}-\frac{b p q (g+h x)^2}{4 h} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2395
Rule 43
Rule 2445
Rubi steps
\begin{align*} \int (g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx &=\operatorname{Subst}\left (\int (g+h x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 h}-\operatorname{Subst}\left (\frac{(b f p q) \int \frac{(g+h x)^2}{e+f x} \, dx}{2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 h}-\operatorname{Subst}\left (\frac{(b f p q) \int \left (\frac{h (f g-e h)}{f^2}+\frac{(f g-e h)^2}{f^2 (e+f x)}+\frac{h (g+h x)}{f}\right ) \, dx}{2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{b (f g-e h) p q x}{2 f}-\frac{b p q (g+h x)^2}{4 h}-\frac{b (f g-e h)^2 p q \log (e+f x)}{2 f^2 h}+\frac{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 h}\\ \end{align*}
Mathematica [A] time = 0.0591751, size = 113, normalized size = 1.15 \[ a g x+\frac{1}{2} a h x^2+\frac{b g (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}+\frac{1}{2} b h x^2 \log \left (c \left (d (e+f x)^p\right )^q\right )-\frac{b e^2 h p q \log (e+f x)}{2 f^2}+\frac{b e h p q x}{2 f}-b g p q x-\frac{1}{4} b h p q x^2 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.278, size = 0, normalized size = 0. \begin{align*} \int \left ( hx+g \right ) \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.04245, size = 151, normalized size = 1.54 \begin{align*} -b f g p q{\left (\frac{x}{f} - \frac{e \log \left (f x + e\right )}{f^{2}}\right )} - \frac{1}{4} \, b f h p q{\left (\frac{2 \, e^{2} \log \left (f x + e\right )}{f^{3}} + \frac{f x^{2} - 2 \, e x}{f^{2}}\right )} + \frac{1}{2} \, b h x^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + \frac{1}{2} \, a h x^{2} + b g x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a g x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.11382, size = 339, normalized size = 3.46 \begin{align*} -\frac{{\left (b f^{2} h p q - 2 \, a f^{2} h\right )} x^{2} - 2 \,{\left (2 \, a f^{2} g -{\left (2 \, b f^{2} g - b e f h\right )} p q\right )} x - 2 \,{\left (b f^{2} h p q x^{2} + 2 \, b f^{2} g p q x +{\left (2 \, b e f g - b e^{2} h\right )} p q\right )} \log \left (f x + e\right ) - 2 \,{\left (b f^{2} h x^{2} + 2 \, b f^{2} g x\right )} \log \left (c\right ) - 2 \,{\left (b f^{2} h q x^{2} + 2 \, b f^{2} g q x\right )} \log \left (d\right )}{4 \, f^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 3.47904, size = 187, normalized size = 1.91 \begin{align*} \begin{cases} a g x + \frac{a h x^{2}}{2} - \frac{b e^{2} h p q \log{\left (e + f x \right )}}{2 f^{2}} + \frac{b e g p q \log{\left (e + f x \right )}}{f} + \frac{b e h p q x}{2 f} + b g p q x \log{\left (e + f x \right )} - b g p q x + b g q x \log{\left (d \right )} + b g x \log{\left (c \right )} + \frac{b h p q x^{2} \log{\left (e + f x \right )}}{2} - \frac{b h p q x^{2}}{4} + \frac{b h q x^{2} \log{\left (d \right )}}{2} + \frac{b h x^{2} \log{\left (c \right )}}{2} & \text{for}\: f \neq 0 \\\left (a + b \log{\left (c \left (d e^{p}\right )^{q} \right )}\right ) \left (g x + \frac{h x^{2}}{2}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.29359, size = 350, normalized size = 3.57 \begin{align*} \frac{{\left (f x + e\right )} b g p q \log \left (f x + e\right )}{f} + \frac{{\left (f x + e\right )}^{2} b h p q \log \left (f x + e\right )}{2 \, f^{2}} - \frac{{\left (f x + e\right )} b h p q e \log \left (f x + e\right )}{f^{2}} - \frac{{\left (f x + e\right )} b g p q}{f} - \frac{{\left (f x + e\right )}^{2} b h p q}{4 \, f^{2}} + \frac{{\left (f x + e\right )} b h p q e}{f^{2}} + \frac{{\left (f x + e\right )} b g q \log \left (d\right )}{f} + \frac{{\left (f x + e\right )}^{2} b h q \log \left (d\right )}{2 \, f^{2}} - \frac{{\left (f x + e\right )} b h q e \log \left (d\right )}{f^{2}} + \frac{{\left (f x + e\right )} b g \log \left (c\right )}{f} + \frac{{\left (f x + e\right )}^{2} b h \log \left (c\right )}{2 \, f^{2}} - \frac{{\left (f x + e\right )} b h e \log \left (c\right )}{f^{2}} + \frac{{\left (f x + e\right )} a g}{f} + \frac{{\left (f x + e\right )}^{2} a h}{2 \, f^{2}} - \frac{{\left (f x + e\right )} a h e}{f^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]