3.406 \(\int (a+b \log (c (d (e+f x)^m)^n))^2 \, dx\)

Optimal. Leaf size=78 \[ \frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}-2 a b m n x-\frac{2 b^2 m n (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}+2 b^2 m^2 n^2 x \]

[Out]

-2*a*b*m*n*x + 2*b^2*m^2*n^2*x - (2*b^2*m*n*(e + f*x)*Log[c*(d*(e + f*x)^m)^n])/f + ((e + f*x)*(a + b*Log[c*(d
*(e + f*x)^m)^n])^2)/f

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Rubi [A]  time = 0.0944507, 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} \[ \frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}-2 a b m n x-\frac{2 b^2 m n (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}+2 b^2 m^2 n^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*a*b*m*n*x + 2*b^2*m^2*n^2*x - (2*b^2*m*n*(e + f*x)*Log[c*(d*(e + f*x)^m)^n])/f + ((e + f*x)*(a + b*Log[c*(d
*(e + f*x)^m)^n])^2)/f

Rule 2389

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

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2445

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

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

Mathematica [A]  time = 0.0096565, size = 69, normalized size = 0.88 \[ \frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}-2 b m n \left (a x+\frac{b (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}-b m n x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((e + f*x)*(a + b*Log[c*(d*(e + f*x)^m)^n])^2)/f - 2*b*m*n*(a*x - b*m*n*x + (b*(e + f*x)*Log[c*(d*(e + f*x)^m)
^n])/f)

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{m} \right ) ^{n} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.10344, size = 200, normalized size = 2.56 \begin{align*} -2 \, a b f m n{\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 )}^{m} d\right )^{n} c\right )^{2} + 2 \, a b x \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) -{\left (2 \, f m n{\left (\frac{x}{f} - \frac{e \log \left (f x + e\right )}{f^{2}}\right )} \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + \frac{{\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} m^{2} n^{2}}{f}\right )} b^{2} + a^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*b*f*m*n*(x/f - e*log(f*x + e)/f^2) + b^2*x*log(((f*x + e)^m*d)^n*c)^2 + 2*a*b*x*log(((f*x + e)^m*d)^n*c)
- (2*f*m*n*(x/f - e*log(f*x + e)/f^2)*log(((f*x + e)^m*d)^n*c) + (e*log(f*x + e)^2 - 2*f*x + 2*e*log(f*x + e))
*m^2*n^2/f)*b^2 + a^2*x

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Fricas [B]  time = 2.29019, size = 516, normalized size = 6.62 \begin{align*} \frac{b^{2} f n^{2} x \log \left (d\right )^{2} + b^{2} f x \log \left (c\right )^{2} +{\left (b^{2} f m^{2} n^{2} x + b^{2} e m^{2} n^{2}\right )} \log \left (f x + e\right )^{2} - 2 \,{\left (b^{2} f m n - a b f\right )} x \log \left (c\right ) +{\left (2 \, b^{2} f m^{2} n^{2} - 2 \, a b f m n + a^{2} f\right )} x - 2 \,{\left (b^{2} e m^{2} n^{2} - a b e m n +{\left (b^{2} f m^{2} n^{2} - a b f m n\right )} x -{\left (b^{2} f m n x + b^{2} e m n\right )} \log \left (c\right ) -{\left (b^{2} f m n^{2} x + b^{2} e m n^{2}\right )} \log \left (d\right )\right )} \log \left (f x + e\right ) + 2 \,{\left (b^{2} f n x \log \left (c\right ) -{\left (b^{2} f m n^{2} - a b f n\right )} x\right )} \log \left (d\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^2*f*n^2*x*log(d)^2 + b^2*f*x*log(c)^2 + (b^2*f*m^2*n^2*x + b^2*e*m^2*n^2)*log(f*x + e)^2 - 2*(b^2*f*m*n - a
*b*f)*x*log(c) + (2*b^2*f*m^2*n^2 - 2*a*b*f*m*n + a^2*f)*x - 2*(b^2*e*m^2*n^2 - a*b*e*m*n + (b^2*f*m^2*n^2 - a
*b*f*m*n)*x - (b^2*f*m*n*x + b^2*e*m*n)*log(c) - (b^2*f*m*n^2*x + b^2*e*m*n^2)*log(d))*log(f*x + e) + 2*(b^2*f
*n*x*log(c) - (b^2*f*m*n^2 - a*b*f*n)*x)*log(d))/f

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Sympy [A]  time = 3.27153, size = 343, normalized size = 4.4 \begin{align*} \begin{cases} a^{2} x + \frac{2 a b e m n \log{\left (e + f x \right )}}{f} + 2 a b m n x \log{\left (e + f x \right )} - 2 a b m n x + 2 a b n x \log{\left (d \right )} + 2 a b x \log{\left (c \right )} + \frac{b^{2} e m^{2} n^{2} \log{\left (e + f x \right )}^{2}}{f} - \frac{2 b^{2} e m^{2} n^{2} \log{\left (e + f x \right )}}{f} + \frac{2 b^{2} e m n^{2} \log{\left (d \right )} \log{\left (e + f x \right )}}{f} + \frac{2 b^{2} e m n \log{\left (c \right )} \log{\left (e + f x \right )}}{f} + b^{2} m^{2} n^{2} x \log{\left (e + f x \right )}^{2} - 2 b^{2} m^{2} n^{2} x \log{\left (e + f x \right )} + 2 b^{2} m^{2} n^{2} x + 2 b^{2} m n^{2} x \log{\left (d \right )} \log{\left (e + f x \right )} - 2 b^{2} m n^{2} x \log{\left (d \right )} + 2 b^{2} m n x \log{\left (c \right )} \log{\left (e + f x \right )} - 2 b^{2} m n x \log{\left (c \right )} + b^{2} n^{2} x \log{\left (d \right )}^{2} + 2 b^{2} n 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^{m}\right )^{n} \right )}\right )^{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*x + 2*a*b*e*m*n*log(e + f*x)/f + 2*a*b*m*n*x*log(e + f*x) - 2*a*b*m*n*x + 2*a*b*n*x*log(d) + 2
*a*b*x*log(c) + b**2*e*m**2*n**2*log(e + f*x)**2/f - 2*b**2*e*m**2*n**2*log(e + f*x)/f + 2*b**2*e*m*n**2*log(d
)*log(e + f*x)/f + 2*b**2*e*m*n*log(c)*log(e + f*x)/f + b**2*m**2*n**2*x*log(e + f*x)**2 - 2*b**2*m**2*n**2*x*
log(e + f*x) + 2*b**2*m**2*n**2*x + 2*b**2*m*n**2*x*log(d)*log(e + f*x) - 2*b**2*m*n**2*x*log(d) + 2*b**2*m*n*
x*log(c)*log(e + f*x) - 2*b**2*m*n*x*log(c) + b**2*n**2*x*log(d)**2 + 2*b**2*n*x*log(c)*log(d) + b**2*x*log(c)
**2, Ne(f, 0)), (x*(a + b*log(c*(d*e**m)**n))**2, True))

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Giac [B]  time = 1.22781, size = 409, normalized size = 5.24 \begin{align*} \frac{{\left (f x + e\right )} b^{2} m^{2} n^{2} \log \left (f x + e\right )^{2}}{f} - \frac{2 \,{\left (f x + e\right )} b^{2} m^{2} n^{2} \log \left (f x + e\right )}{f} + \frac{2 \,{\left (f x + e\right )} b^{2} m n^{2} \log \left (f x + e\right ) \log \left (d\right )}{f} + \frac{2 \,{\left (f x + e\right )} b^{2} m^{2} n^{2}}{f} + \frac{2 \,{\left (f x + e\right )} b^{2} m n \log \left (f x + e\right ) \log \left (c\right )}{f} - \frac{2 \,{\left (f x + e\right )} b^{2} m n^{2} \log \left (d\right )}{f} + \frac{{\left (f x + e\right )} b^{2} n^{2} \log \left (d\right )^{2}}{f} + \frac{2 \,{\left (f x + e\right )} a b m n \log \left (f x + e\right )}{f} - \frac{2 \,{\left (f x + e\right )} b^{2} m n \log \left (c\right )}{f} + \frac{2 \,{\left (f x + e\right )} b^{2} n \log \left (c\right ) \log \left (d\right )}{f} - \frac{2 \,{\left (f x + e\right )} a b m n}{f} + \frac{{\left (f x + e\right )} b^{2} \log \left (c\right )^{2}}{f} + \frac{2 \,{\left (f x + e\right )} a b n \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]

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

[Out]

(f*x + e)*b^2*m^2*n^2*log(f*x + e)^2/f - 2*(f*x + e)*b^2*m^2*n^2*log(f*x + e)/f + 2*(f*x + e)*b^2*m*n^2*log(f*
x + e)*log(d)/f + 2*(f*x + e)*b^2*m^2*n^2/f + 2*(f*x + e)*b^2*m*n*log(f*x + e)*log(c)/f - 2*(f*x + e)*b^2*m*n^
2*log(d)/f + (f*x + e)*b^2*n^2*log(d)^2/f + 2*(f*x + e)*a*b*m*n*log(f*x + e)/f - 2*(f*x + e)*b^2*m*n*log(c)/f
+ 2*(f*x + e)*b^2*n*log(c)*log(d)/f - 2*(f*x + e)*a*b*m*n/f + (f*x + e)*b^2*log(c)^2/f + 2*(f*x + e)*a*b*n*log
(d)/f + 2*(f*x + e)*a*b*log(c)/f + (f*x + e)*a^2/f