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

Optimal. Leaf size=48 \[ d x \left (a+b \log \left (c x^n\right )\right )+\frac{1}{3} e x^3 \left (a+b \log \left (c x^n\right )\right )-b d n x-\frac{1}{9} b e n x^3 \]

[Out]

-(b*d*n*x) - (b*e*n*x^3)/9 + d*x*(a + b*Log[c*x^n]) + (e*x^3*(a + b*Log[c*x^n]))/3

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Rubi [A]  time = 0.0176313, antiderivative size = 41, normalized size of antiderivative = 0.85, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {2313} \[ \frac{1}{3} \left (3 d x+e x^3\right ) \left (a+b \log \left (c x^n\right )\right )-b d n x-\frac{1}{9} b e n x^3 \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)*(a + b*Log[c*x^n]),x]

[Out]

-(b*d*n*x) - (b*e*n*x^3)/9 + ((3*d*x + e*x^3)*(a + b*Log[c*x^n]))/3

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 \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{3} \left (3 d x+e x^3\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (d+\frac{e x^2}{3}\right ) \, dx\\ &=-b d n x-\frac{1}{9} b e n x^3+\frac{1}{3} \left (3 d x+e x^3\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0014115, size = 55, normalized size = 1.15 \[ a d x+\frac{1}{3} a e x^3+b d x \log \left (c x^n\right )+\frac{1}{3} b e x^3 \log \left (c x^n\right )-b d n x-\frac{1}{9} b e n x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*d*x - b*d*n*x + (a*e*x^3)/3 - (b*e*n*x^3)/9 + b*d*x*Log[c*x^n] + (b*e*x^3*Log[c*x^n])/3

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Maple [C]  time = 0.187, size = 247, normalized size = 5.2 \begin{align*}{\frac{bx \left ( e{x}^{2}+3\,d \right ) \ln \left ({x}^{n} \right ) }{3}}+{\frac{i}{6}}\pi \,be{x}^{3}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{6}}\pi \,be{x}^{3}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{6}}\pi \,be{x}^{3} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{6}}\pi \,be{x}^{3} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{i}{2}}\pi \,bd{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}x-{\frac{i}{2}}\pi \,bd{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) x-{\frac{i}{2}}\pi \,bd \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}x+{\frac{i}{2}}\pi \,bd \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) x+{\frac{\ln \left ( c \right ) be{x}^{3}}{3}}-{\frac{ben{x}^{3}}{9}}+{\frac{ae{x}^{3}}{3}}+\ln \left ( c \right ) bdx-bdnx+axd \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b*x*(e*x^2+3*d)*ln(x^n)+1/6*I*Pi*b*e*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2-1/6*I*Pi*b*e*x^3*csgn(I*x^n)*csgn(I*c
*x^n)*csgn(I*c)-1/6*I*Pi*b*e*x^3*csgn(I*c*x^n)^3+1/6*I*Pi*b*e*x^3*csgn(I*c*x^n)^2*csgn(I*c)+1/2*I*Pi*b*d*csgn(
I*x^n)*csgn(I*c*x^n)^2*x-1/2*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x-1/2*I*Pi*b*d*csgn(I*c*x^n)^3*x+1/2
*I*Pi*b*d*csgn(I*c*x^n)^2*csgn(I*c)*x+1/3*ln(c)*b*e*x^3-1/9*b*e*n*x^3+1/3*a*e*x^3+ln(c)*b*d*x-b*d*n*x+a*x*d

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Maxima [A]  time = 1.04568, size = 66, normalized size = 1.38 \begin{align*} -\frac{1}{9} \, b e n x^{3} + \frac{1}{3} \, b e x^{3} \log \left (c x^{n}\right ) + \frac{1}{3} \, a e x^{3} - b d n x + 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^2+d)*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

-1/9*b*e*n*x^3 + 1/3*b*e*x^3*log(c*x^n) + 1/3*a*e*x^3 - b*d*n*x + b*d*x*log(c*x^n) + a*d*x

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Fricas [A]  time = 1.23191, size = 154, normalized size = 3.21 \begin{align*} -\frac{1}{9} \,{\left (b e n - 3 \, a e\right )} x^{3} -{\left (b d n - a d\right )} x + \frac{1}{3} \,{\left (b e x^{3} + 3 \, b d x\right )} \log \left (c\right ) + \frac{1}{3} \,{\left (b e n x^{3} + 3 \, 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^2+d)*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

-1/9*(b*e*n - 3*a*e)*x^3 - (b*d*n - a*d)*x + 1/3*(b*e*x^3 + 3*b*d*x)*log(c) + 1/3*(b*e*n*x^3 + 3*b*d*n*x)*log(
x)

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Sympy [A]  time = 1.14463, size = 73, normalized size = 1.52 \begin{align*} a d x + \frac{a e x^{3}}{3} + b d n x \log{\left (x \right )} - b d n x + b d x \log{\left (c \right )} + \frac{b e n x^{3} \log{\left (x \right )}}{3} - \frac{b e n x^{3}}{9} + \frac{b e x^{3} \log{\left (c \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d*x + a*e*x**3/3 + b*d*n*x*log(x) - b*d*n*x + b*d*x*log(c) + b*e*n*x**3*log(x)/3 - b*e*n*x**3/9 + b*e*x**3*l
og(c)/3

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Giac [A]  time = 1.34424, size = 84, normalized size = 1.75 \begin{align*} \frac{1}{3} \, b n x^{3} e \log \left (x\right ) - \frac{1}{9} \, b n x^{3} e + \frac{1}{3} \, b x^{3} e \log \left (c\right ) + \frac{1}{3} \, a x^{3} e + b d n x \log \left (x\right ) - b d n x + 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^2+d)*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

1/3*b*n*x^3*e*log(x) - 1/9*b*n*x^3*e + 1/3*b*x^3*e*log(c) + 1/3*a*x^3*e + b*d*n*x*log(x) - b*d*n*x + b*d*x*log
(c) + a*d*x