3.4.3 \(\int (f+\frac {g}{x}) x (a+b \log (c (d+e x)^n)) \, dx\) [303]

Optimal. Leaf size=91 \[ \frac {b (d f-e g) n x}{2 e}-\frac {b n (g+f x)^2}{4 f}-\frac {b (d f-e g)^2 n \log (d+e x)}{2 e^2 f}+\frac {(g+f x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f} \]

[Out]

1/2*b*(d*f-e*g)*n*x/e-1/4*b*n*(f*x+g)^2/f-1/2*b*(d*f-e*g)^2*n*ln(e*x+d)/e^2/f+1/2*(f*x+g)^2*(a+b*ln(c*(e*x+d)^
n))/f

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Rubi [A]
time = 0.05, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2459, 2442, 45} \begin {gather*} \frac {(f x+g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f}-\frac {b n (d f-e g)^2 \log (d+e x)}{2 e^2 f}+\frac {b n x (d f-e g)}{2 e}-\frac {b n (f x+g)^2}{4 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(d*f - e*g)*n*x)/(2*e) - (b*n*(g + f*x)^2)/(4*f) - (b*(d*f - e*g)^2*n*Log[d + e*x])/(2*e^2*f) + ((g + f*x)^
2*(a + b*Log[c*(d + e*x)^n]))/(2*f)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2442

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

Rule 2459

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

Rubi steps

\begin {align*} \int \left (f+\frac {g}{x}\right ) x \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx &=\int (g+f x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx\\ &=\frac {(g+f x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f}-\frac {(b e n) \int \frac {(g+f x)^2}{d+e x} \, dx}{2 f}\\ &=\frac {(g+f x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f}-\frac {(b e n) \int \left (\frac {f (-d f+e g)}{e^2}+\frac {(-d f+e g)^2}{e^2 (d+e x)}+\frac {f (g+f x)}{e}\right ) \, dx}{2 f}\\ &=\frac {b (d f-e g) n x}{2 e}-\frac {b n (g+f x)^2}{4 f}-\frac {b (d f-e g)^2 n \log (d+e x)}{2 e^2 f}+\frac {(g+f x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 101, normalized size = 1.11 \begin {gather*} a g x+\frac {b d f n x}{2 e}-b g n x+\frac {1}{2} a f x^2-\frac {1}{4} b f n x^2-\frac {b d^2 f n \log (d+e x)}{2 e^2}+\frac {1}{2} b f x^2 \log \left (c (d+e x)^n\right )+\frac {b g (d+e x) \log \left (c (d+e x)^n\right )}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a*g*x + (b*d*f*n*x)/(2*e) - b*g*n*x + (a*f*x^2)/2 - (b*f*n*x^2)/4 - (b*d^2*f*n*Log[d + e*x])/(2*e^2) + (b*f*x^
2*Log[c*(d + e*x)^n])/2 + (b*g*(d + e*x)*Log[c*(d + e*x)^n])/e

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Maple [A]
time = 0.35, size = 101, normalized size = 1.11

method	result	size
default	\(a g x +\frac {a f \,x^{2}}{2}+b g \ln \left (c \left (e x +d \right )^{n}\right ) x -b g n x +\frac {b g n d \ln \left (e x +d \right )}{e}+\frac {b f \,x^{2} \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )}{2}-\frac {n b f \,x^{2}}{4}-\frac {n b \,d^{2} f \ln \left (e x +d \right )}{2 e^{2}}+\frac {b d f n x}{2 e}\)	\(101\)
risch	\(\frac {b x \left (f x +2 g \right ) \ln \left (\left (e x +d \right )^{n}\right )}{2}+\frac {i \pi b g x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2}+\frac {i \pi b f \,x^{2} \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{4}-\frac {i \pi b f \,x^{2} \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{4}+\frac {i \pi b f \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{4}-\frac {i \pi b g x \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{2}-\frac {i \pi b g x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )}{2}+\frac {i \pi b g x \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2}-\frac {i \pi b f \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )}{4}+\frac {\ln \left (c \right ) b f \,x^{2}}{2}-\frac {n b f \,x^{2}}{4}-\frac {n b \,d^{2} f \ln \left (e x +d \right )}{2 e^{2}}+\frac {b g n d \ln \left (e x +d \right )}{e}+\ln \left (c \right ) b g x +\frac {a f \,x^{2}}{2}+\frac {b d f n x}{2 e}-b g n x +a g x\)	\(338\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*g*x+1/2*a*f*x^2+b*g*ln(c*(e*x+d)^n)*x-b*g*n*x+b*g/e*n*d*ln(e*x+d)+1/2*b*f*x^2*ln(c*exp(n*ln(e*x+d)))-1/4*n*b
*f*x^2-1/2*n*b*d^2*f/e^2*ln(e*x+d)+1/2*b*d*f*n*x/e

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Maxima [A]
time = 0.26, size = 104, normalized size = 1.14 \begin {gather*} -\frac {1}{4} \, {\left (2 \, d^{2} e^{\left (-3\right )} \log \left (x e + d\right ) + {\left (x^{2} e - 2 \, d x\right )} e^{\left (-2\right )}\right )} b f n e + {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} b g n e + \frac {1}{2} \, b f x^{2} \log \left ({\left (x e + d\right )}^{n} c\right ) + \frac {1}{2} \, a f x^{2} + b g x \log \left ({\left (x e + d\right )}^{n} c\right ) + a g x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*d^2*e^(-3)*log(x*e + d) + (x^2*e - 2*d*x)*e^(-2))*b*f*n*e + (d*e^(-2)*log(x*e + d) - x*e^(-1))*b*g*n*e
+ 1/2*b*f*x^2*log((x*e + d)^n*c) + 1/2*a*f*x^2 + b*g*x*log((x*e + d)^n*c) + a*g*x

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Fricas [A]
time = 0.35, size = 105, normalized size = 1.15 \begin {gather*} \frac {1}{4} \, {\left (2 \, b d f n x e + 2 \, {\left (b f x^{2} + 2 \, b g x\right )} e^{2} \log \left (c\right ) - {\left ({\left (b f n - 2 \, a f\right )} x^{2} + 4 \, {\left (b g n - a g\right )} x\right )} e^{2} - 2 \, {\left (b d^{2} f n - 2 \, b d g n e - {\left (b f n x^{2} + 2 \, b g n x\right )} e^{2}\right )} \log \left (x e + d\right )\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*b*d*f*n*x*e + 2*(b*f*x^2 + 2*b*g*x)*e^2*log(c) - ((b*f*n - 2*a*f)*x^2 + 4*(b*g*n - a*g)*x)*e^2 - 2*(b*d
^2*f*n - 2*b*d*g*n*e - (b*f*n*x^2 + 2*b*g*n*x)*e^2)*log(x*e + d))*e^(-2)

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Sympy [A]
time = 0.69, size = 134, normalized size = 1.47 \begin {gather*} \begin {cases} \frac {a f x^{2}}{2} + a g x - \frac {b d^{2} f \log {\left (c \left (d + e x\right )^{n} \right )}}{2 e^{2}} + \frac {b d f n x}{2 e} + \frac {b d g \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - \frac {b f n x^{2}}{4} + \frac {b f x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{2} - b g n x + b g x \log {\left (c \left (d + e x\right )^{n} \right )} & \text {for}\: e \neq 0 \\\left (a + b \log {\left (c d^{n} \right )}\right ) \left (\frac {f x^{2}}{2} + g x\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*f*x**2/2 + a*g*x - b*d**2*f*log(c*(d + e*x)**n)/(2*e**2) + b*d*f*n*x/(2*e) + b*d*g*log(c*(d + e*x
)**n)/e - b*f*n*x**2/4 + b*f*x**2*log(c*(d + e*x)**n)/2 - b*g*n*x + b*g*x*log(c*(d + e*x)**n), Ne(e, 0)), ((a
+ b*log(c*d**n))*(f*x**2/2 + g*x), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (85) = 170\).
time = 4.36, size = 186, normalized size = 2.04 \begin {gather*} \frac {1}{2} \, {\left (x e + d\right )}^{2} b f n e^{\left (-2\right )} \log \left (x e + d\right ) - {\left (x e + d\right )} b d f n e^{\left (-2\right )} \log \left (x e + d\right ) - \frac {1}{4} \, {\left (x e + d\right )}^{2} b f n e^{\left (-2\right )} + {\left (x e + d\right )} b d f n e^{\left (-2\right )} + {\left (x e + d\right )} b g n e^{\left (-1\right )} \log \left (x e + d\right ) + \frac {1}{2} \, {\left (x e + d\right )}^{2} b f e^{\left (-2\right )} \log \left (c\right ) - {\left (x e + d\right )} b d f e^{\left (-2\right )} \log \left (c\right ) - {\left (x e + d\right )} b g n e^{\left (-1\right )} + \frac {1}{2} \, {\left (x e + d\right )}^{2} a f e^{\left (-2\right )} - {\left (x e + d\right )} a d f e^{\left (-2\right )} + {\left (x e + d\right )} b g e^{\left (-1\right )} \log \left (c\right ) + {\left (x e + d\right )} a g e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x*e + d)^2*b*f*n*e^(-2)*log(x*e + d) - (x*e + d)*b*d*f*n*e^(-2)*log(x*e + d) - 1/4*(x*e + d)^2*b*f*n*e^(-
2) + (x*e + d)*b*d*f*n*e^(-2) + (x*e + d)*b*g*n*e^(-1)*log(x*e + d) + 1/2*(x*e + d)^2*b*f*e^(-2)*log(c) - (x*e
+ d)*b*d*f*e^(-2)*log(c) - (x*e + d)*b*g*n*e^(-1) + 1/2*(x*e + d)^2*a*f*e^(-2) - (x*e + d)*a*d*f*e^(-2) + (x*
e + d)*b*g*e^(-1)*log(c) + (x*e + d)*a*g*e^(-1)

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Mupad [B]
time = 0.27, size = 104, normalized size = 1.14 \begin {gather*} x\,\left (\frac {2\,a\,d\,f+2\,a\,e\,g-2\,b\,e\,g\,n}{2\,e}-\frac {d\,f\,\left (2\,a-b\,n\right )}{2\,e}\right )+\ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (\frac {b\,f\,x^2}{2}+b\,g\,x\right )-\frac {\ln \left (d+e\,x\right )\,\left (b\,d^2\,f\,n-2\,b\,d\,e\,g\,n\right )}{2\,e^2}+\frac {f\,x^2\,\left (2\,a-b\,n\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(f + g/x)*(a + b*log(c*(d + e*x)^n)),x)

[Out]

x*((2*a*d*f + 2*a*e*g - 2*b*e*g*n)/(2*e) - (d*f*(2*a - b*n))/(2*e)) + log(c*(d + e*x)^n)*(b*g*x + (b*f*x^2)/2)
- (log(d + e*x)*(b*d^2*f*n - 2*b*d*e*g*n))/(2*e^2) + (f*x^2*(2*a - b*n))/4

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