3.4.75 \(\int (d+e x^r) (a+b \log (c x^n)) \, dx\) [375]

Optimal. Leaf size=57 \[ -b d n x-\frac {b e n x^{1+r}}{(1+r)^2}+d x \left (a+b \log \left (c x^n\right )\right )+\frac {e x^{1+r} \left (a+b \log \left (c x^n\right )\right )}{1+r} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2350, 12} \begin {gather*} d x \left (a+b \log \left (c x^n\right )\right )+\frac {e x^{r+1} \left (a+b \log \left (c x^n\right )\right )}{r+1}-b d n x-\frac {b e n x^{r+1}}{(r+1)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x^r)*(a + b*Log[c*x^n]),x]

[Out]

-(b*d*n*x) - (b*e*n*x^(1 + r))/(1 + r)^2 + d*x*(a + b*Log[c*x^n]) + (e*x^(1 + r)*(a + b*Log[c*x^n]))/(1 + r)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2350

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(d +
e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, 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^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\left (d x+\frac {e x^{1+r}}{1+r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {d+d r+e x^r}{1+r} \, dx\\ &=\left (d x+\frac {e x^{1+r}}{1+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(b n) \int \left (d+d r+e x^r\right ) \, dx}{1+r}\\ &=-b d n x-\frac {b e n x^{1+r}}{(1+r)^2}+\left (d x+\frac {e x^{1+r}}{1+r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 61, normalized size = 1.07 \begin {gather*} \frac {x \left (a (1+r) \left (d+d r+e x^r\right )-b n \left (d (1+r)^2+e x^r\right )+b (1+r) \left (d+d r+e x^r\right ) \log \left (c x^n\right )\right )}{(1+r)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a*(1 + r)*(d + d*r + e*x^r) - b*n*(d*(1 + r)^2 + e*x^r) + b*(1 + r)*(d + d*r + e*x^r)*Log[c*x^n]))/(1 + r)
^2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.08, size = 606, normalized size = 10.63

method	result	size
risch	\(\frac {b x \left (d r +e \,x^{r}+d \right ) \ln \left (x^{n}\right )}{1+r}-\frac {x \left (-2 x^{r} a e +2 b d n +2 x^{r} b e n -2 x^{r} a e r -2 a d +4 b d n r -4 \ln \left (c \right ) b d r -2 \ln \left (c \right ) b d \,r^{2}-2 a d \,r^{2}-2 d b \ln \left (c \right )-2 \ln \left (c \right ) b e \,x^{r} r -4 a d r -i \pi b d \,r^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r}-i \pi b e \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r}-i \pi b d \,r^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-2 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} r -2 i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} r +i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 b d n \,r^{2}+i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x^{r}-i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r -i \pi b e \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r +i \pi b d \,r^{2} \mathrm {csgn}\left (i c \right ) \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 \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) r -2 \ln \left (c \right ) b e \,x^{r}+i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x^{r} r +2 i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3} r +i \pi b d \,r^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+i \pi b e \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{r}+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 e \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{r} r \right )}{2 \left (1+r \right )^{2}}\)	\(606\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x*(d*r+e*x^r+d)/(1+r)*ln(x^n)-1/2*x*(-2*x^r*a*e+2*b*d*n+I*x^r*csgn(I*c*x^n)^3*e*b*Pi+2*x^r*b*e*n-2*x^r*a*e*r
-2*a*d+4*b*d*n*r+I*Pi*b*e*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+2*I*Pi*b*d*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*
x^n)-4*ln(c)*b*d*r-2*ln(c)*b*d*r^2-2*a*d*r^2-2*d*b*ln(c)-2*ln(c)*b*e*x^r*r+I*Pi*b*d*csgn(I*c*x^n)^3-4*a*d*r+2*
b*d*n*r^2-2*ln(c)*b*e*x^r+2*I*Pi*b*d*csgn(I*c*x^n)^3*r-I*Pi*b*e*csgn(I*c)*csgn(I*c*x^n)^2*x^r*r-I*Pi*b*e*csgn(
I*x^n)*csgn(I*c*x^n)^2*x^r*r+I*Pi*b*d*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*Pi*b*e*csgn(I*c)*csgn(I*x^n)*c
sgn(I*c*x^n)*x^r*r+I*Pi*b*d*r^2*csgn(I*c*x^n)^3-I*x^r*csgn(I*c*x^n)^2*csgn(I*x^n)*e*b*Pi+I*Pi*b*e*csgn(I*c*x^n
)^3*x^r*r-I*x^r*csgn(I*c*x^n)^2*csgn(I*c)*e*b*Pi-I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*d*csgn(I*c)*csgn(
I*c*x^n)^2-I*Pi*b*d*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*d*r^2*csgn(I*c)*csgn(I*c*x^n)^2-2*I*Pi*b*d*csgn(I*x
^n)*csgn(I*c*x^n)^2*r-2*I*Pi*b*d*csgn(I*c)*csgn(I*c*x^n)^2*r+I*Pi*b*d*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n))/(1+
r)^2

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Maxima [A]
time = 0.29, size = 68, normalized size = 1.19 \begin {gather*} -b d n x + b d x \log \left (c x^{n}\right ) + a d x + \frac {b e x^{r + 1} \log \left (c x^{n}\right )}{r + 1} - \frac {b e n x^{r + 1}}{{\left (r + 1\right )}^{2}} + \frac {a e x^{r + 1}}{r + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (59) = 118\).
time = 0.39, size = 136, normalized size = 2.39 \begin {gather*} \frac {{\left (b d r^{2} + 2 \, b d r + b d\right )} x \log \left (c\right ) + {\left (b d n r^{2} + 2 \, b d n r + b d n\right )} x \log \left (x\right ) + {\left ({\left (b r + b\right )} x e \log \left (c\right ) + {\left (b n r + b n\right )} x e \log \left (x\right ) - {\left (b n - a r - a\right )} x e\right )} x^{r} - {\left (b d n + {\left (b d n - a d\right )} r^{2} - a d + 2 \, {\left (b d n - a d\right )} r\right )} x}{r^{2} + 2 \, r + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (54) = 108\).
time = 0.48, size = 323, normalized size = 5.67 \begin {gather*} \begin {cases} \frac {a d r^{2} x}{r^{2} + 2 r + 1} + \frac {2 a d r x}{r^{2} + 2 r + 1} + \frac {a d x}{r^{2} + 2 r + 1} + \frac {a e r x x^{r}}{r^{2} + 2 r + 1} + \frac {a e x x^{r}}{r^{2} + 2 r + 1} - \frac {b d n r^{2} x}{r^{2} + 2 r + 1} - \frac {2 b d n r x}{r^{2} + 2 r + 1} - \frac {b d n x}{r^{2} + 2 r + 1} + \frac {b d r^{2} x \log {\left (c x^{n} \right )}}{r^{2} + 2 r + 1} + \frac {2 b d r x \log {\left (c x^{n} \right )}}{r^{2} + 2 r + 1} + \frac {b d x \log {\left (c x^{n} \right )}}{r^{2} + 2 r + 1} - \frac {b e n x x^{r}}{r^{2} + 2 r + 1} + \frac {b e r x x^{r} \log {\left (c x^{n} \right )}}{r^{2} + 2 r + 1} + \frac {b e x x^{r} \log {\left (c x^{n} \right )}}{r^{2} + 2 r + 1} & \text {for}\: r \neq -1 \\a d x + \frac {a e \log {\left (c x^{n} \right )}}{n} - b d n x + b d x \log {\left (c x^{n} \right )} + \frac {b e \log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d*r**2*x/(r**2 + 2*r + 1) + 2*a*d*r*x/(r**2 + 2*r + 1) + a*d*x/(r**2 + 2*r + 1) + a*e*r*x*x**r/(r
**2 + 2*r + 1) + a*e*x*x**r/(r**2 + 2*r + 1) - b*d*n*r**2*x/(r**2 + 2*r + 1) - 2*b*d*n*r*x/(r**2 + 2*r + 1) -
b*d*n*x/(r**2 + 2*r + 1) + b*d*r**2*x*log(c*x**n)/(r**2 + 2*r + 1) + 2*b*d*r*x*log(c*x**n)/(r**2 + 2*r + 1) +
b*d*x*log(c*x**n)/(r**2 + 2*r + 1) - b*e*n*x*x**r/(r**2 + 2*r + 1) + b*e*r*x*x**r*log(c*x**n)/(r**2 + 2*r + 1)
+ b*e*x*x**r*log(c*x**n)/(r**2 + 2*r + 1), Ne(r, -1)), (a*d*x + a*e*log(c*x**n)/n - b*d*n*x + b*d*x*log(c*x**
n) + b*e*log(c*x**n)**2/(2*n), True))

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Giac [A]
time = 2.45, size = 115, normalized size = 2.02 \begin {gather*} \frac {b n r x x^{r} e \log \left (x\right )}{r^{2} + 2 \, r + 1} + b d n x \log \left (x\right ) + \frac {b n x x^{r} e \log \left (x\right )}{r^{2} + 2 \, r + 1} - b d n x - \frac {b n x x^{r} e}{r^{2} + 2 \, r + 1} + b d x \log \left (c\right ) + \frac {b x x^{r} e \log \left (c\right )}{r + 1} + a d x + \frac {a x x^{r} e}{r + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x^r)*(a + b*log(c*x^n)),x)

[Out]

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

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