3.187 \(\int \frac{(a+b \log (c (e+f x)))^2}{d e+d f x} \, dx\)

Optimal. Leaf size=27 \[ \frac{(a+b \log (c (e+f x)))^3}{3 b d f} \]

[Out]

(a + b*Log[c*(e + f*x)])^3/(3*b*d*f)

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Rubi [A]  time = 0.0595308, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2390, 12, 2302, 30} \[ \frac{(a+b \log (c (e+f x)))^3}{3 b d f} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(e + f*x)])^2/(d*e + d*f*x),x]

[Out]

(a + b*Log[c*(e + f*x)])^3/(3*b*d*f)

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{(a+b \log (c (e+f x)))^2}{d e+d f x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{d x} \, dx,x,e+f x\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f}\\ &=\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log (c (e+f x))\right )}{b d f}\\ &=\frac{(a+b \log (c (e+f x)))^3}{3 b d f}\\ \end{align*}

Mathematica [A]  time = 0.004426, size = 27, normalized size = 1. \[ \frac{(a+b \log (c (e+f x)))^3}{3 b d f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(e + f*x)])^2/(d*e + d*f*x),x]

[Out]

(a + b*Log[c*(e + f*x)])^3/(3*b*d*f)

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Maple [B]  time = 0.06, size = 63, normalized size = 2.3 \begin{align*}{\frac{{a}^{2}\ln \left ( cfx+ce \right ) }{df}}+{\frac{ab \left ( \ln \left ( cfx+ce \right ) \right ) ^{2}}{df}}+{\frac{{b}^{2} \left ( \ln \left ( cfx+ce \right ) \right ) ^{3}}{3\,df}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(f*x+e)))^2/(d*f*x+d*e),x)

[Out]

1/f/d*a^2*ln(c*f*x+c*e)+1/f/d*a*b*ln(c*f*x+c*e)^2+1/3/f/d*b^2*ln(c*f*x+c*e)^3

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Maxima [B]  time = 1.16093, size = 173, normalized size = 6.41 \begin{align*} -a b{\left (\frac{2 \, \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} - \frac{\log \left (f x + e\right )^{2} + 2 \, \log \left (f x + e\right ) \log \left (c\right )}{d f}\right )} + \frac{b^{2} \log \left (c f x + c e\right )^{3}}{3 \, d f} + \frac{2 \, a b \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} + \frac{a^{2} \log \left (d f x + d e\right )}{d f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(f*x+e)))^2/(d*f*x+d*e),x, algorithm="maxima")

[Out]

-a*b*(2*log(c*f*x + c*e)*log(d*f*x + d*e)/(d*f) - (log(f*x + e)^2 + 2*log(f*x + e)*log(c))/(d*f)) + 1/3*b^2*lo
g(c*f*x + c*e)^3/(d*f) + 2*a*b*log(c*f*x + c*e)*log(d*f*x + d*e)/(d*f) + a^2*log(d*f*x + d*e)/(d*f)

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Fricas [B]  time = 1.7152, size = 119, normalized size = 4.41 \begin{align*} \frac{b^{2} \log \left (c f x + c e\right )^{3} + 3 \, a b \log \left (c f x + c e\right )^{2} + 3 \, a^{2} \log \left (c f x + c e\right )}{3 \, d f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(f*x+e)))^2/(d*f*x+d*e),x, algorithm="fricas")

[Out]

1/3*(b^2*log(c*f*x + c*e)^3 + 3*a*b*log(c*f*x + c*e)^2 + 3*a^2*log(c*f*x + c*e))/(d*f)

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Sympy [B]  time = 0.485953, size = 51, normalized size = 1.89 \begin{align*} \frac{a^{2} \log{\left (d e + d f x \right )}}{d f} + \frac{a b \log{\left (c \left (e + f x\right ) \right )}^{2}}{d f} + \frac{b^{2} \log{\left (c \left (e + f x\right ) \right )}^{3}}{3 d f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(f*x+e)))**2/(d*f*x+d*e),x)

[Out]

a**2*log(d*e + d*f*x)/(d*f) + a*b*log(c*(e + f*x))**2/(d*f) + b**2*log(c*(e + f*x))**3/(3*d*f)

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Giac [B]  time = 1.2072, size = 72, normalized size = 2.67 \begin{align*} \frac{b^{2} \log \left ({\left (f x + e\right )} c\right )^{3} + 3 \, a b \log \left ({\left (f x + e\right )} c\right )^{2} + 3 \, a^{2} \log \left ({\left (f x + e\right )} c\right )}{3 \, d f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(f*x+e)))^2/(d*f*x+d*e),x, algorithm="giac")

[Out]

1/3*(b^2*log((f*x + e)*c)^3 + 3*a*b*log((f*x + e)*c)^2 + 3*a^2*log((f*x + e)*c))/(d*f)