3.24 \(\int \frac{(d+e x)^3 (a+b \log (c x^n))}{x^2} \, dx\)

Optimal. Leaf size=119 \[ 3 d^2 e \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{x}+3 d e^2 x \left (a+b \log \left (c x^n\right )\right )+\frac{1}{2} e^3 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{3}{2} b d^2 e n \log ^2(x)-\frac{b d^3 n}{x}-3 b d e^2 n x-\frac{1}{4} b e^3 n x^2 \]

[Out]

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

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Rubi [A]  time = 0.0877142, antiderivative size = 92, normalized size of antiderivative = 0.77, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {43, 2334, 2301} \[ -\frac{1}{2} \left (-6 d^2 e \log (x)+\frac{2 d^3}{x}-6 d e^2 x-e^3 x^2\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3}{2} b d^2 e n \log ^2(x)-\frac{b d^3 n}{x}-3 b d e^2 n x-\frac{1}{4} b e^3 n x^2 \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^3*(a + b*Log[c*x^n]))/x^2,x]

[Out]

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

Rule 43

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 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])

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

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

Mathematica [A]  time = 0.0802664, size = 118, normalized size = 0.99 \[ \frac{3 d^2 e \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{1}{2} e^3 x^2 \left (a+b \log \left (c x^n\right )\right )+3 a d e^2 x+3 b d e^2 x \log \left (c x^n\right )-\frac{b d^3 n}{x}-3 b d e^2 n x-\frac{1}{4} b e^3 n x^2 \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^3*(a + b*Log[c*x^n]))/x^2,x]

[Out]

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

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Maple [C]  time = 0.276, size = 588, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b*(-e^3*x^3-6*d^2*e*ln(x)*x-6*d*e^2*x^2+2*d^3)/x*ln(x^n)-1/4*(4*a*d^3-2*ln(c)*b*e^3*x^3-12*a*d*e^2*x^2+4*
ln(c)*b*d^3-12*ln(x)*ln(c)*b*d^2*e*x+6*b*d^2*e*n*ln(x)^2*x+I*Pi*b*e^3*x^3*csgn(I*c*x^n)^3+6*I*ln(x)*Pi*b*d^2*e
*csgn(I*c*x^n)^3*x+6*I*ln(x)*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x-6*I*Pi*b*d*e^2*x^2*csgn(I*x^n)*c
sgn(I*c*x^n)^2+I*Pi*b*e^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-6*I*Pi*b*d*e^2*x^2*csgn(I*c*x^n)^2*csgn(I*c)
+2*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2+2*I*Pi*b*d^3*csgn(I*c*x^n)^2*csgn(I*c)+6*I*Pi*b*d*e^2*x^2*csgn(I*x^n
)*csgn(I*c*x^n)*csgn(I*c)-6*I*ln(x)*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x-12*ln(c)*b*d*e^2*x^2-2*a*e^3*x^3+
4*b*d^3*n-2*I*Pi*b*d^3*csgn(I*c*x^n)^3-6*I*ln(x)*Pi*b*d^2*e*csgn(I*c*x^n)^2*csgn(I*c)*x-12*ln(x)*a*d^2*e*x+6*I
*Pi*b*d*e^2*x^2*csgn(I*c*x^n)^3+b*e^3*n*x^3+12*b*d*e^2*n*x^2-I*Pi*b*e^3*x^3*csgn(I*c*x^n)^2*csgn(I*c)-I*Pi*b*e
^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2-2*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c))/x

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Maxima [A]  time = 1.11752, size = 171, normalized size = 1.44 \begin{align*} -\frac{1}{4} \, b e^{3} n x^{2} + \frac{1}{2} \, b e^{3} x^{2} \log \left (c x^{n}\right ) - 3 \, b d e^{2} n x + \frac{1}{2} \, a e^{3} x^{2} + 3 \, b d e^{2} x \log \left (c x^{n}\right ) + 3 \, a d e^{2} x + \frac{3 \, b d^{2} e \log \left (c x^{n}\right )^{2}}{2 \, n} + 3 \, a d^{2} e \log \left (x\right ) - \frac{b d^{3} n}{x} - \frac{b d^{3} \log \left (c x^{n}\right )}{x} - \frac{a d^{3}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a+b*log(c*x^n))/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.03515, size = 338, normalized size = 2.84 \begin{align*} \frac{6 \, b d^{2} e n x \log \left (x\right )^{2} - 4 \, b d^{3} n - 4 \, a d^{3} -{\left (b e^{3} n - 2 \, a e^{3}\right )} x^{3} - 12 \,{\left (b d e^{2} n - a d e^{2}\right )} x^{2} + 2 \,{\left (b e^{3} x^{3} + 6 \, b d e^{2} x^{2} - 2 \, b d^{3}\right )} \log \left (c\right ) + 2 \,{\left (b e^{3} n x^{3} + 6 \, b d e^{2} n x^{2} + 6 \, b d^{2} e x \log \left (c\right ) - 2 \, b d^{3} n + 6 \, a d^{2} e x\right )} \log \left (x\right )}{4 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a+b*log(c*x^n))/x^2,x, algorithm="fricas")

[Out]

1/4*(6*b*d^2*e*n*x*log(x)^2 - 4*b*d^3*n - 4*a*d^3 - (b*e^3*n - 2*a*e^3)*x^3 - 12*(b*d*e^2*n - a*d*e^2)*x^2 + 2
*(b*e^3*x^3 + 6*b*d*e^2*x^2 - 2*b*d^3)*log(c) + 2*(b*e^3*n*x^3 + 6*b*d*e^2*n*x^2 + 6*b*d^2*e*x*log(c) - 2*b*d^
3*n + 6*a*d^2*e*x)*log(x))/x

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Sympy [A]  time = 4.76585, size = 182, normalized size = 1.53 \begin{align*} - \frac{a d^{3}}{x} + 3 a d^{2} e \log{\left (x \right )} + 3 a d e^{2} x + \frac{a e^{3} x^{2}}{2} - \frac{b d^{3} n \log{\left (x \right )}}{x} - \frac{b d^{3} n}{x} - \frac{b d^{3} \log{\left (c \right )}}{x} + \frac{3 b d^{2} e n \log{\left (x \right )}^{2}}{2} + 3 b d^{2} e \log{\left (c \right )} \log{\left (x \right )} + 3 b d e^{2} n x \log{\left (x \right )} - 3 b d e^{2} n x + 3 b d e^{2} x \log{\left (c \right )} + \frac{b e^{3} n x^{2} \log{\left (x \right )}}{2} - \frac{b e^{3} n x^{2}}{4} + \frac{b e^{3} x^{2} \log{\left (c \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(a+b*ln(c*x**n))/x**2,x)

[Out]

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

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Giac [A]  time = 1.31787, size = 208, normalized size = 1.75 \begin{align*} \frac{6 \, b d^{2} n x e \log \left (x\right )^{2} + 2 \, b n x^{3} e^{3} \log \left (x\right ) + 12 \, b d n x^{2} e^{2} \log \left (x\right ) + 12 \, b d^{2} x e \log \left (c\right ) \log \left (x\right ) - b n x^{3} e^{3} - 12 \, b d n x^{2} e^{2} + 2 \, b x^{3} e^{3} \log \left (c\right ) + 12 \, b d x^{2} e^{2} \log \left (c\right ) - 4 \, b d^{3} n \log \left (x\right ) + 12 \, a d^{2} x e \log \left (x\right ) - 4 \, b d^{3} n + 2 \, a x^{3} e^{3} + 12 \, a d x^{2} e^{2} - 4 \, b d^{3} \log \left (c\right ) - 4 \, a d^{3}}{4 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a+b*log(c*x^n))/x^2,x, algorithm="giac")

[Out]

1/4*(6*b*d^2*n*x*e*log(x)^2 + 2*b*n*x^3*e^3*log(x) + 12*b*d*n*x^2*e^2*log(x) + 12*b*d^2*x*e*log(c)*log(x) - b*
n*x^3*e^3 - 12*b*d*n*x^2*e^2 + 2*b*x^3*e^3*log(c) + 12*b*d*x^2*e^2*log(c) - 4*b*d^3*n*log(x) + 12*a*d^2*x*e*lo
g(x) - 4*b*d^3*n + 2*a*x^3*e^3 + 12*a*d*x^2*e^2 - 4*b*d^3*log(c) - 4*a*d^3)/x