3.1.22 \(\int (d+e x)^3 (a+b \log (c x^n)) \, dx\) [22]

Optimal. Leaf size=85 \[ -b d^3 n x-\frac {3}{4} b d^2 e n x^2-\frac {1}{3} b d e^2 n x^3-\frac {1}{16} b e^3 n x^4-\frac {b d^4 n \log (x)}{4 e}+\frac {(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 e} \]

[Out]

-b*d^3*n*x-3/4*b*d^2*e*n*x^2-1/3*b*d*e^2*n*x^3-1/16*b*e^3*n*x^4-1/4*b*d^4*n*ln(x)/e+1/4*(e*x+d)^4*(a+b*ln(c*x^
n))/e

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Rubi [A]
time = 0.03, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {32, 2350, 12, 45} \begin {gather*} \frac {(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac {b d^4 n \log (x)}{4 e}-b d^3 n x-\frac {3}{4} b d^2 e n x^2-\frac {1}{3} b d e^2 n x^3-\frac {1}{16} b e^3 n x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(a + b*Log[c*x^n]),x]

[Out]

-(b*d^3*n*x) - (3*b*d^2*e*n*x^2)/4 - (b*d*e^2*n*x^3)/3 - (b*e^3*n*x^4)/16 - (b*d^4*n*Log[x])/(4*e) + ((d + e*x
)^4*(a + b*Log[c*x^n]))/(4*e)

Rule 12

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 (d+e x)^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 e}-(b n) \int \frac {(d+e x)^4}{4 e x} \, dx\\ &=\frac {(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac {(b n) \int \frac {(d+e x)^4}{x} \, dx}{4 e}\\ &=\frac {(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac {(b n) \int \left (4 d^3 e+\frac {d^4}{x}+6 d^2 e^2 x+4 d e^3 x^2+e^4 x^3\right ) \, dx}{4 e}\\ &=-b d^3 n x-\frac {3}{4} b d^2 e n x^2-\frac {1}{3} b d e^2 n x^3-\frac {1}{16} b e^3 n x^4-\frac {b d^4 n \log (x)}{4 e}+\frac {(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 e}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 110, normalized size = 1.29 \begin {gather*} \frac {1}{48} x \left (12 a \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )-b n \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )+12 b \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right ) \log \left (c x^n\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*(a + b*Log[c*x^n]),x]

[Out]

(x*(12*a*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) - b*n*(48*d^3 + 36*d^2*e*x + 16*d*e^2*x^2 + 3*e^3*x^3) +
12*b*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)*Log[c*x^n]))/48

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.15, size = 571, normalized size = 6.72

method	result	size
risch	\(\frac {a \,e^{3} x^{4}}{4}+x a \,d^{3}+a d \,e^{2} x^{3}+\frac {3 a \,d^{2} e \,x^{2}}{2}+\frac {3 \ln \left (c \right ) b \,d^{2} e \,x^{2}}{2}+\ln \left (c \right ) b d \,e^{2} x^{3}-b \,d^{3} n x +\frac {\left (e x +d \right )^{4} b \ln \left (x^{n}\right )}{4 e}+\ln \left (c \right ) b \,d^{3} x +\frac {\ln \left (c \right ) b \,e^{3} x^{4}}{4}-\frac {i e^{2} \pi b d \,x^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}-\frac {3 i e \pi b \,d^{2} x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{4}-\frac {b \,e^{3} n \,x^{4}}{16}-\frac {i e^{2} \pi b d \,x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}-\frac {3 b \,d^{2} e n \,x^{2}}{4}-\frac {b d \,e^{2} n \,x^{3}}{3}-\frac {b \,d^{4} n \ln \left (x \right )}{4 e}-\frac {3 i e \pi b \,d^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{4}-\frac {i e^{3} \pi b \,x^{4} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{8}-\frac {i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x}{2}+\frac {i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x}{2}+\frac {i \pi b \,d^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x}{2}+\frac {i e^{3} \pi b \,x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8}+\frac {i e^{3} \pi b \,x^{4} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8}-\frac {i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x}{2}-\frac {i e^{3} \pi b \,x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{8}+\frac {i e^{2} \pi b d \,x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i e^{2} \pi b d \,x^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {3 i e \pi b \,d^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4}+\frac {3 i e \pi b \,d^{2} x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4}\)	\(571\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(a+b*ln(c*x^n)),x,method=_RETURNVERBOSE)

[Out]

1/4*a*e^3*x^4+x*a*d^3+a*d*e^2*x^3+3/2*a*d^2*e*x^2+3/2*ln(c)*b*d^2*e*x^2+ln(c)*b*d*e^2*x^3-1/8*I*e^3*Pi*b*x^4*c
sgn(I*c*x^n)^3-1/2*I*Pi*b*d^3*csgn(I*c*x^n)^3*x-b*d^3*n*x+1/4*(e*x+d)^4*b/e*ln(x^n)+ln(c)*b*d^3*x+1/4*ln(c)*b*
e^3*x^4-1/2*I*Pi*b*d^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x-1/8*I*e^3*Pi*b*x^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c
*x^n)-1/16*b*e^3*n*x^4-1/2*I*e^2*Pi*b*d*x^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-3/4*I*e*Pi*b*d^2*x^2*csgn(I*c)
*csgn(I*x^n)*csgn(I*c*x^n)-3/4*b*d^2*e*n*x^2-1/3*b*d*e^2*n*x^3-1/4*b*d^4*n*ln(x)/e+1/2*I*Pi*b*d^3*csgn(I*c)*cs
gn(I*c*x^n)^2*x+1/2*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2*x+1/8*I*e^3*Pi*b*x^4*csgn(I*c)*csgn(I*c*x^n)^2+1/8*
I*e^3*Pi*b*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*e^2*Pi*b*d*x^3*csgn(I*c*x^n)^3-3/4*I*e*Pi*b*d^2*x^2*csgn(I*c*
x^n)^3+1/2*I*e^2*Pi*b*d*x^3*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I*e^2*Pi*b*d*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2+3/4*I*e
*Pi*b*d^2*x^2*csgn(I*c)*csgn(I*c*x^n)^2+3/4*I*e*Pi*b*d^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2

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Maxima [A]
time = 0.26, size = 130, normalized size = 1.53 \begin {gather*} -\frac {1}{16} \, b n x^{4} e^{3} - \frac {1}{3} \, b d n x^{3} e^{2} - \frac {3}{4} \, b d^{2} n x^{2} e + \frac {1}{4} \, b x^{4} e^{3} \log \left (c x^{n}\right ) + b d x^{3} e^{2} \log \left (c x^{n}\right ) + \frac {3}{2} \, b d^{2} x^{2} e \log \left (c x^{n}\right ) - b d^{3} n x + \frac {1}{4} \, a x^{4} e^{3} + a d x^{3} e^{2} + \frac {3}{2} \, a d^{2} x^{2} e + b d^{3} x \log \left (c x^{n}\right ) + a d^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

-1/16*b*n*x^4*e^3 - 1/3*b*d*n*x^3*e^2 - 3/4*b*d^2*n*x^2*e + 1/4*b*x^4*e^3*log(c*x^n) + b*d*x^3*e^2*log(c*x^n)
+ 3/2*b*d^2*x^2*e*log(c*x^n) - b*d^3*n*x + 1/4*a*x^4*e^3 + a*d*x^3*e^2 + 3/2*a*d^2*x^2*e + b*d^3*x*log(c*x^n)
+ a*d^3*x

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (73) = 146\).
time = 0.34, size = 149, normalized size = 1.75 \begin {gather*} -\frac {1}{16} \, {\left (b n - 4 \, a\right )} x^{4} e^{3} - \frac {1}{3} \, {\left (b d n - 3 \, a d\right )} x^{3} e^{2} - \frac {3}{4} \, {\left (b d^{2} n - 2 \, a d^{2}\right )} x^{2} e - {\left (b d^{3} n - a d^{3}\right )} x + \frac {1}{4} \, {\left (b x^{4} e^{3} + 4 \, b d x^{3} e^{2} + 6 \, b d^{2} x^{2} e + 4 \, b d^{3} x\right )} \log \left (c\right ) + \frac {1}{4} \, {\left (b n x^{4} e^{3} + 4 \, b d n x^{3} e^{2} + 6 \, b d^{2} n x^{2} e + 4 \, b d^{3} n x\right )} \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

-1/16*(b*n - 4*a)*x^4*e^3 - 1/3*(b*d*n - 3*a*d)*x^3*e^2 - 3/4*(b*d^2*n - 2*a*d^2)*x^2*e - (b*d^3*n - a*d^3)*x
+ 1/4*(b*x^4*e^3 + 4*b*d*x^3*e^2 + 6*b*d^2*x^2*e + 4*b*d^3*x)*log(c) + 1/4*(b*n*x^4*e^3 + 4*b*d*n*x^3*e^2 + 6*
b*d^2*n*x^2*e + 4*b*d^3*n*x)*log(x)

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Sympy [A]
time = 0.29, size = 156, normalized size = 1.84 \begin {gather*} a d^{3} x + \frac {3 a d^{2} e x^{2}}{2} + a d e^{2} x^{3} + \frac {a e^{3} x^{4}}{4} - b d^{3} n x + b d^{3} x \log {\left (c x^{n} \right )} - \frac {3 b d^{2} e n x^{2}}{4} + \frac {3 b d^{2} e x^{2} \log {\left (c x^{n} \right )}}{2} - \frac {b d e^{2} n x^{3}}{3} + b d e^{2} x^{3} \log {\left (c x^{n} \right )} - \frac {b e^{3} n x^{4}}{16} + \frac {b e^{3} x^{4} \log {\left (c x^{n} \right )}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(a+b*ln(c*x**n)),x)

[Out]

a*d**3*x + 3*a*d**2*e*x**2/2 + a*d*e**2*x**3 + a*e**3*x**4/4 - b*d**3*n*x + b*d**3*x*log(c*x**n) - 3*b*d**2*e*
n*x**2/4 + 3*b*d**2*e*x**2*log(c*x**n)/2 - b*d*e**2*n*x**3/3 + b*d*e**2*x**3*log(c*x**n) - b*e**3*n*x**4/16 +
b*e**3*x**4*log(c*x**n)/4

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (73) = 146\).
time = 1.91, size = 159, normalized size = 1.87 \begin {gather*} \frac {1}{4} \, b n x^{4} e^{3} \log \left (x\right ) + b d n x^{3} e^{2} \log \left (x\right ) + \frac {3}{2} \, b d^{2} n x^{2} e \log \left (x\right ) - \frac {1}{16} \, b n x^{4} e^{3} - \frac {1}{3} \, b d n x^{3} e^{2} - \frac {3}{4} \, b d^{2} n x^{2} e + \frac {1}{4} \, b x^{4} e^{3} \log \left (c\right ) + b d x^{3} e^{2} \log \left (c\right ) + \frac {3}{2} \, b d^{2} x^{2} e \log \left (c\right ) + b d^{3} n x \log \left (x\right ) - b d^{3} n x + \frac {1}{4} \, a x^{4} e^{3} + a d x^{3} e^{2} + \frac {3}{2} \, a d^{2} x^{2} e + b d^{3} x \log \left (c\right ) + a d^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

1/4*b*n*x^4*e^3*log(x) + b*d*n*x^3*e^2*log(x) + 3/2*b*d^2*n*x^2*e*log(x) - 1/16*b*n*x^4*e^3 - 1/3*b*d*n*x^3*e^
2 - 3/4*b*d^2*n*x^2*e + 1/4*b*x^4*e^3*log(c) + b*d*x^3*e^2*log(c) + 3/2*b*d^2*x^2*e*log(c) + b*d^3*n*x*log(x)
- b*d^3*n*x + 1/4*a*x^4*e^3 + a*d*x^3*e^2 + 3/2*a*d^2*x^2*e + b*d^3*x*log(c) + a*d^3*x

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Mupad [B]
time = 3.57, size = 104, normalized size = 1.22 \begin {gather*} \ln \left (c\,x^n\right )\,\left (b\,d^3\,x+\frac {3\,b\,d^2\,e\,x^2}{2}+b\,d\,e^2\,x^3+\frac {b\,e^3\,x^4}{4}\right )+\frac {e^3\,x^4\,\left (4\,a-b\,n\right )}{16}+d^3\,x\,\left (a-b\,n\right )+\frac {3\,d^2\,e\,x^2\,\left (2\,a-b\,n\right )}{4}+\frac {d\,e^2\,x^3\,\left (3\,a-b\,n\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))*(d + e*x)^3,x)

[Out]

log(c*x^n)*((b*e^3*x^4)/4 + b*d^3*x + (3*b*d^2*e*x^2)/2 + b*d*e^2*x^3) + (e^3*x^4*(4*a - b*n))/16 + d^3*x*(a -
b*n) + (3*d^2*e*x^2*(2*a - b*n))/4 + (d*e^2*x^3*(3*a - b*n))/3

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