3.1.47 \(\int (a+b \log (c (d+e x)^n))^2 \, dx\) [47]

Optimal. Leaf size=65 \[ -2 a b n x+2 b^2 n^2 x-\frac {2 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e} \]

[Out]

-2*a*b*n*x+2*b^2*n^2*x-2*b^2*n*(e*x+d)*ln(c*(e*x+d)^n)/e+(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/e

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Rubi [A]
time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2436, 2333, 2332} \begin {gather*} \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-2 a b n x-\frac {2 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e}+2 b^2 n^2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x)^n])^2,x]

[Out]

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

Rule 2332

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

Rule 2333

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 2436

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]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 59, normalized size = 0.91 \begin {gather*} \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-2 b n \left (a x-b n x+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.08, size = 130, normalized size = 2.00

method	result	size
norman	\(\left (2 b^{2} n^{2}-2 b a n +a^{2}\right ) x +b^{2} x \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )^{2}+\left (-2 b^{2} n +2 b a \right ) x \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )+\frac {b^{2} d \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )^{2}}{e}+\frac {n \left (-2 b^{2} d n +2 a d b \right ) \ln \left (e x +d \right )}{e}\)	\(111\)
default	\(a^{2} x +b^{2} x \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )^{2}+\frac {b^{2} d \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )^{2}}{e}+2 b^{2} n^{2} x -2 b^{2} n x \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )-\frac {2 n^{2} b^{2} d \ln \left (e x +d \right )}{e}+2 b a \ln \left (c \left (e x +d \right )^{n}\right ) x -2 a b n x +\frac {2 b a n d \ln \left (e x +d \right )}{e}\)	\(130\)
risch	\(\text {Expression too large to display}\)	\(1125\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(e*x+d)^n))^2,x,method=_RETURNVERBOSE)

[Out]

a^2*x+b^2*x*ln(c*exp(n*ln(e*x+d)))^2+b^2*d/e*ln(c*exp(n*ln(e*x+d)))^2+2*b^2*n^2*x-2*b^2*n*x*ln(c*exp(n*ln(e*x+
d)))-2*n^2*b^2*d/e*ln(e*x+d)+2*b*a*ln(c*(e*x+d)^n)*x-2*a*b*n*x+2*b*a/e*n*d*ln(e*x+d)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (67) = 134\).
time = 0.28, size = 136, normalized size = 2.09 \begin {gather*} 2 \, {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} a b n e + b^{2} x \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + 2 \, a b x \log \left ({\left (x e + d\right )}^{n} c\right ) - {\left ({\left (d \log \left (x e + d\right )^{2} - 2 \, x e + 2 \, d \log \left (x e + d\right )\right )} n^{2} e^{\left (-1\right )} - 2 \, {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} n e \log \left ({\left (x e + d\right )}^{n} c\right )\right )} b^{2} + a^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(d*e^(-2)*log(x*e + d) - x*e^(-1))*a*b*n*e + b^2*x*log((x*e + d)^n*c)^2 + 2*a*b*x*log((x*e + d)^n*c) - ((d*l
og(x*e + d)^2 - 2*x*e + 2*d*log(x*e + d))*n^2*e^(-1) - 2*(d*e^(-2)*log(x*e + d) - x*e^(-1))*n*e*log((x*e + d)^
n*c))*b^2 + a^2*x

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (67) = 134\).
time = 0.35, size = 142, normalized size = 2.18 \begin {gather*} {\left (b^{2} x e \log \left (c\right )^{2} - 2 \, {\left (b^{2} n - a b\right )} x e \log \left (c\right ) + {\left (2 \, b^{2} n^{2} - 2 \, a b n + a^{2}\right )} x e + {\left (b^{2} n^{2} x e + b^{2} d n^{2}\right )} \log \left (x e + d\right )^{2} - 2 \, {\left (b^{2} d n^{2} - a b d n + {\left (b^{2} n^{2} - a b n\right )} x e - {\left (b^{2} n x e + b^{2} d n\right )} \log \left (c\right )\right )} \log \left (x e + d\right )\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^2*x*e*log(c)^2 - 2*(b^2*n - a*b)*x*e*log(c) + (2*b^2*n^2 - 2*a*b*n + a^2)*x*e + (b^2*n^2*x*e + b^2*d*n^2)*l
og(x*e + d)^2 - 2*(b^2*d*n^2 - a*b*d*n + (b^2*n^2 - a*b*n)*x*e - (b^2*n*x*e + b^2*d*n)*log(c))*log(x*e + d))*e
^(-1)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (63) = 126\).
time = 0.37, size = 146, normalized size = 2.25 \begin {gather*} \begin {cases} a^{2} x + \frac {2 a b d \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - 2 a b n x + 2 a b x \log {\left (c \left (d + e x\right )^{n} \right )} - \frac {2 b^{2} d n \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {b^{2} d \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} + 2 b^{2} n^{2} x - 2 b^{2} n x \log {\left (c \left (d + e x\right )^{n} \right )} + b^{2} x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} & \text {for}\: e \neq 0 \\x \left (a + b \log {\left (c d^{n} \right )}\right )^{2} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (67) = 134\).
time = 5.25, size = 178, normalized size = 2.74 \begin {gather*} {\left (x e + d\right )} b^{2} n^{2} e^{\left (-1\right )} \log \left (x e + d\right )^{2} - 2 \, {\left (x e + d\right )} b^{2} n^{2} e^{\left (-1\right )} \log \left (x e + d\right ) + 2 \, {\left (x e + d\right )} b^{2} n e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right ) + 2 \, {\left (x e + d\right )} b^{2} n^{2} e^{\left (-1\right )} + 2 \, {\left (x e + d\right )} a b n e^{\left (-1\right )} \log \left (x e + d\right ) - 2 \, {\left (x e + d\right )} b^{2} n e^{\left (-1\right )} \log \left (c\right ) + {\left (x e + d\right )} b^{2} e^{\left (-1\right )} \log \left (c\right )^{2} - 2 \, {\left (x e + d\right )} a b n e^{\left (-1\right )} + 2 \, {\left (x e + d\right )} a b e^{\left (-1\right )} \log \left (c\right ) + {\left (x e + d\right )} a^{2} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*e + d)*b^2*n^2*e^(-1)*log(x*e + d)^2 - 2*(x*e + d)*b^2*n^2*e^(-1)*log(x*e + d) + 2*(x*e + d)*b^2*n*e^(-1)*l
og(x*e + d)*log(c) + 2*(x*e + d)*b^2*n^2*e^(-1) + 2*(x*e + d)*a*b*n*e^(-1)*log(x*e + d) - 2*(x*e + d)*b^2*n*e^
(-1)*log(c) + (x*e + d)*b^2*e^(-1)*log(c)^2 - 2*(x*e + d)*a*b*n*e^(-1) + 2*(x*e + d)*a*b*e^(-1)*log(c) + (x*e
+ d)*a^2*e^(-1)

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Mupad [B]
time = 0.00, size = 94, normalized size = 1.45 \begin {gather*} x\,\left (a^2-2\,a\,b\,n+2\,b^2\,n^2\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2\,\left (b^2\,x+\frac {b^2\,d}{e}\right )-\frac {\ln \left (d+e\,x\right )\,\left (2\,b^2\,d\,n^2-2\,a\,b\,d\,n\right )}{e}+2\,b\,x\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (a-b\,n\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*(d + e*x)^n))^2,x)

[Out]

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

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