3.4 \(\int (d+e x) (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=48 \[ d x \left (a+b \log \left (c x^n\right )\right )+\frac{1}{2} e x^2 \left (a+b \log \left (c x^n\right )\right )-b d n x-\frac{1}{4} b e n x^2 \]

[Out]

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

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Rubi [A]  time = 0.0163162, antiderivative size = 41, normalized size of antiderivative = 0.85, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {2313} \[ \frac{1}{2} \left (2 d x+e x^2\right ) \left (a+b \log \left (c x^n\right )\right )-b d n x-\frac{1}{4} b e n x^2 \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*(a + b*Log[c*x^n]),x]

[Out]

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

Rule 2313

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(d +
e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a,
b, c, d, e, n, r}, x] && IGtQ[q, 0]

Rubi steps

\begin{align*} \int (d+e x) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{2} \left (2 d x+e x^2\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (d+\frac{e x}{2}\right ) \, dx\\ &=-b d n x-\frac{1}{4} b e n x^2+\frac{1}{2} \left (2 d x+e x^2\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0017839, size = 55, normalized size = 1.15 \[ a d x+\frac{1}{2} a e x^2+b d x \log \left (c x^n\right )+\frac{1}{2} b e x^2 \log \left (c x^n\right )-b d n x-\frac{1}{4} b e n x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.054, size = 52, normalized size = 1.1 \begin{align*} axd+{\frac{ae{x}^{2}}{2}}+xb\ln \left ( c{x}^{n} \right ) d-bdnx+{\frac{be{x}^{2}\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{2}}-{\frac{ben{x}^{2}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.19946, size = 66, normalized size = 1.38 \begin{align*} -\frac{1}{4} \, b e n x^{2} + \frac{1}{2} \, b e x^{2} \log \left (c x^{n}\right ) - b d n x + \frac{1}{2} \, a e x^{2} + b d x \log \left (c x^{n}\right ) + a d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.9753, size = 154, normalized size = 3.21 \begin{align*} -\frac{1}{4} \,{\left (b e n - 2 \, a e\right )} x^{2} -{\left (b d n - a d\right )} x + \frac{1}{2} \,{\left (b e x^{2} + 2 \, b d x\right )} \log \left (c\right ) + \frac{1}{2} \,{\left (b e n x^{2} + 2 \, b d n x\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.823084, size = 73, normalized size = 1.52 \begin{align*} a d x + \frac{a e x^{2}}{2} + b d n x \log{\left (x \right )} - b d n x + b d x \log{\left (c \right )} + \frac{b e n x^{2} \log{\left (x \right )}}{2} - \frac{b e n x^{2}}{4} + \frac{b e x^{2} \log{\left (c \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d*x + a*e*x**2/2 + b*d*n*x*log(x) - b*d*n*x + b*d*x*log(c) + b*e*n*x**2*log(x)/2 - b*e*n*x**2/4 + b*e*x**2*l
og(c)/2

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Giac [A]  time = 1.30296, size = 84, normalized size = 1.75 \begin{align*} \frac{1}{2} \, b n x^{2} e \log \left (x\right ) - \frac{1}{4} \, b n x^{2} e + \frac{1}{2} \, b x^{2} e \log \left (c\right ) + b d n x \log \left (x\right ) - b d n x + \frac{1}{2} \, a x^{2} e + b d x \log \left (c\right ) + a d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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