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

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Rubi [A]  time = 0.02, 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*}

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Mathematica [A]  time = 0.00, 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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fricas [A]  time = 0.53, size = 61, normalized size = 1.27 \[ -\frac {1}{9} \, {\left (b e n - 3 \, a e\right )} x^{3} - {\left (b d n - a d\right )} x + \frac {1}{3} \, {\left (b e x^{3} + 3 \, b d x\right )} \log \relax (c) + \frac {1}{3} \, {\left (b e n x^{3} + 3 \, b d n x\right )} \log \relax (x) \]

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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giac [A]  time = 0.24, size = 62, normalized size = 1.29 \[ \frac {1}{3} \, b n x^{3} e \log \relax (x) - \frac {1}{9} \, b n x^{3} e + \frac {1}{3} \, b x^{3} e \log \relax (c) + \frac {1}{3} \, a x^{3} e + b d n x \log \relax (x) - b d n x + b d x \log \relax (c) + a d x \]

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

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maple [C]  time = 0.21, size = 247, normalized size = 5.15 \[ -\frac {i \pi b e \,x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{6}+\frac {i \pi b e \,x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{6}+\frac {i \pi b e \,x^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{6}-\frac {i \pi b e \,x^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{6}-\frac {i \pi b d x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i \pi b d x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i \pi b d x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i \pi b d x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}-\frac {b e n \,x^{3}}{9}+\frac {b e \,x^{3} \ln \relax (c )}{3}+\frac {a e \,x^{3}}{3}-b d n x +b d x \ln \relax (c )+a d x +\frac {\left (e \,x^{2}+3 d \right ) b x \ln \left (x^{n}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)*(b*ln(c*x^n)+a),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*b*e*x^3*ln(c)-1/9*b*e*n*x^3+1/3*a*e*x^3+b*d*x*ln(c)-b*d*n*x+a*d*x

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maxima [A]  time = 0.48, size = 49, normalized size = 1.02 \[ -\frac {1}{9} \, b e n x^{3} + \frac {1}{3} \, b e x^{3} \log \left (c x^{n}\right ) + \frac {1}{3} \, a e x^{3} - b d n x + b d x \log \left (c x^{n}\right ) + a d x \]

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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mupad [B]  time = 3.31, size = 43, normalized size = 0.90 \[ \ln \left (c\,x^n\right )\,\left (\frac {b\,e\,x^3}{3}+b\,d\,x\right )+d\,x\,\left (a-b\,n\right )+\frac {e\,x^3\,\left (3\,a-b\,n\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x^2)*(a + b*log(c*x^n)),x)

[Out]

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

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sympy [A]  time = 0.86, size = 73, normalized size = 1.52 \[ a d x + \frac {a e x^{3}}{3} + b d n x \log {\relax (x )} - b d n x + b d x \log {\relax (c )} + \frac {b e n x^{3} \log {\relax (x )}}{3} - \frac {b e n x^{3}}{9} + \frac {b e x^{3} \log {\relax (c )}}{3} \]

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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