3.180 \(\int \frac {(d+e x^2) (a+b \log (c x^n))}{x^2} \, dx\)

Optimal. Leaf size=44 \[ -\frac {d \left (a+b \log \left (c x^n\right )\right )}{x}+e x \left (a+b \log \left (c x^n\right )\right )-\frac {b d n}{x}-b e n x \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 37, normalized size of antiderivative = 0.84, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {14, 2334} \[ -\left (\frac {d}{x}-e x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b d n}{x}-b e n x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(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] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 49, normalized size = 1.11 \[ -\frac {a d}{x}+a e x-\frac {b d \log \left (c x^n\right )}{x}+b e x \log \left (c x^n\right )-\frac {b d n}{x}-b e n x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 58, normalized size = 1.32 \[ -\frac {b d n + {\left (b e n - a e\right )} x^{2} + a d - {\left (b e x^{2} - b d\right )} \log \relax (c) - {\left (b e n x^{2} - b d n\right )} \log \relax (x)}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.35, size = 62, normalized size = 1.41 \[ \frac {b n x^{2} e \log \relax (x) - b n x^{2} e + b x^{2} e \log \relax (c) + a x^{2} e - b d n \log \relax (x) - b d n - b d \log \relax (c) - a d}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.22, size = 249, normalized size = 5.66 \[ -\frac {\left (-e \,x^{2}+d \right ) b \ln \left (x^{n}\right )}{x}-\frac {i \pi b e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-i \pi b e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b e \,x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b e \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 b e n \,x^{2}-2 b e \,x^{2} \ln \relax (c )-2 a e \,x^{2}+2 b d n +2 b d \ln \relax (c )+2 a d}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.47, size = 49, normalized size = 1.11 \[ -b e n x + b e x \log \left (c x^{n}\right ) + a e x - \frac {b d n}{x} - \frac {b d \log \left (c x^{n}\right )}{x} - \frac {a d}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.34, size = 51, normalized size = 1.16 \[ e\,x\,\left (a-b\,n\right )-\ln \left (c\,x^n\right )\,\left (\frac {b\,e\,x^2+b\,d}{x}-2\,b\,e\,x\right )-\frac {a\,d+b\,d\,n}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.89, size = 60, normalized size = 1.36 \[ - \frac {a d}{x} + a e x - \frac {b d n \log {\relax (x )}}{x} - \frac {b d n}{x} - \frac {b d \log {\relax (c )}}{x} + b e n x \log {\relax (x )} - b e n x + b e x \log {\relax (c )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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