3.69 \(\int \frac{\log (d (b x+c x^2)^n)}{x^5} \, dx\)

Optimal. Leaf size=100 \[ \frac{c^2 n}{8 b^2 x^2}-\frac{c^3 n}{4 b^3 x}-\frac{c^4 n \log (x)}{4 b^4}+\frac{c^4 n \log (b+c x)}{4 b^4}-\frac{\log \left (d \left (b x+c x^2\right )^n\right )}{4 x^4}-\frac{c n}{12 b x^3}-\frac{n}{16 x^4} \]

[Out]

-n/(16*x^4) - (c*n)/(12*b*x^3) + (c^2*n)/(8*b^2*x^2) - (c^3*n)/(4*b^3*x) - (c^4*n*Log[x])/(4*b^4) + (c^4*n*Log
[b + c*x])/(4*b^4) - Log[d*(b*x + c*x^2)^n]/(4*x^4)

________________________________________________________________________________________

Rubi [A]  time = 0.0632197, antiderivative size = 100, normalized size of antiderivative = 1., 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 = {2525, 77} \[ \frac{c^2 n}{8 b^2 x^2}-\frac{c^3 n}{4 b^3 x}-\frac{c^4 n \log (x)}{4 b^4}+\frac{c^4 n \log (b+c x)}{4 b^4}-\frac{\log \left (d \left (b x+c x^2\right )^n\right )}{4 x^4}-\frac{c n}{12 b x^3}-\frac{n}{16 x^4} \]

Antiderivative was successfully verified.

[In]

Int[Log[d*(b*x + c*x^2)^n]/x^5,x]

[Out]

-n/(16*x^4) - (c*n)/(12*b*x^3) + (c^2*n)/(8*b^2*x^2) - (c^3*n)/(4*b^3*x) - (c^4*n*Log[x])/(4*b^4) + (c^4*n*Log
[b + c*x])/(4*b^4) - Log[d*(b*x + c*x^2)^n]/(4*x^4)

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{\log \left (d \left (b x+c x^2\right )^n\right )}{x^5} \, dx &=-\frac{\log \left (d \left (b x+c x^2\right )^n\right )}{4 x^4}+\frac{1}{4} n \int \frac{b+2 c x}{x^5 (b+c x)} \, dx\\ &=-\frac{\log \left (d \left (b x+c x^2\right )^n\right )}{4 x^4}+\frac{1}{4} n \int \left (\frac{1}{x^5}+\frac{c}{b x^4}-\frac{c^2}{b^2 x^3}+\frac{c^3}{b^3 x^2}-\frac{c^4}{b^4 x}+\frac{c^5}{b^4 (b+c x)}\right ) \, dx\\ &=-\frac{n}{16 x^4}-\frac{c n}{12 b x^3}+\frac{c^2 n}{8 b^2 x^2}-\frac{c^3 n}{4 b^3 x}-\frac{c^4 n \log (x)}{4 b^4}+\frac{c^4 n \log (b+c x)}{4 b^4}-\frac{\log \left (d \left (b x+c x^2\right )^n\right )}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.0462707, size = 87, normalized size = 0.87 \[ -\frac{b n \left (4 b^2 c x+3 b^3-6 b c^2 x^2+12 c^3 x^3\right )+12 b^4 \log \left (d (x (b+c x))^n\right )-12 c^4 n x^4 \log (b+c x)+12 c^4 n x^4 \log (x)}{48 b^4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[d*(b*x + c*x^2)^n]/x^5,x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( d \left ( c{x}^{2}+bx \right ) ^{n} \right ) }{{x}^{5}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(d*(c*x^2+b*x)^n)/x^5,x)

[Out]

int(ln(d*(c*x^2+b*x)^n)/x^5,x)

________________________________________________________________________________________

Maxima [A]  time = 1.02596, size = 116, normalized size = 1.16 \begin{align*} \frac{1}{48} \, n{\left (\frac{12 \, c^{4} \log \left (c x + b\right )}{b^{4}} - \frac{12 \, c^{4} \log \left (x\right )}{b^{4}} - \frac{12 \, c^{3} x^{3} - 6 \, b c^{2} x^{2} + 4 \, b^{2} c x + 3 \, b^{3}}{b^{3} x^{4}}\right )} - \frac{\log \left ({\left (c x^{2} + b x\right )}^{n} d\right )}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(d*(c*x^2+b*x)^n)/x^5,x, algorithm="maxima")

[Out]

1/48*n*(12*c^4*log(c*x + b)/b^4 - 12*c^4*log(x)/b^4 - (12*c^3*x^3 - 6*b*c^2*x^2 + 4*b^2*c*x + 3*b^3)/(b^3*x^4)
) - 1/4*log((c*x^2 + b*x)^n*d)/x^4

________________________________________________________________________________________

Fricas [A]  time = 1.92054, size = 228, normalized size = 2.28 \begin{align*} \frac{12 \, c^{4} n x^{4} \log \left (c x + b\right ) - 12 \, c^{4} n x^{4} \log \left (x\right ) - 12 \, b c^{3} n x^{3} + 6 \, b^{2} c^{2} n x^{2} - 4 \, b^{3} c n x - 12 \, b^{4} n \log \left (c x^{2} + b x\right ) - 3 \, b^{4} n - 12 \, b^{4} \log \left (d\right )}{48 \, b^{4} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(d*(c*x^2+b*x)^n)/x^5,x, algorithm="fricas")

[Out]

1/48*(12*c^4*n*x^4*log(c*x + b) - 12*c^4*n*x^4*log(x) - 12*b*c^3*n*x^3 + 6*b^2*c^2*n*x^2 - 4*b^3*c*n*x - 12*b^
4*n*log(c*x^2 + b*x) - 3*b^4*n - 12*b^4*log(d))/(b^4*x^4)

________________________________________________________________________________________

Sympy [A]  time = 16.6428, size = 141, normalized size = 1.41 \begin{align*} \begin{cases} - \frac{n \log{\left (b x + c x^{2} \right )}}{4 x^{4}} - \frac{n}{16 x^{4}} - \frac{\log{\left (d \right )}}{4 x^{4}} - \frac{c n}{12 b x^{3}} + \frac{c^{2} n}{8 b^{2} x^{2}} - \frac{c^{3} n}{4 b^{3} x} + \frac{c^{4} n \log{\left (b + c x \right )}}{2 b^{4}} - \frac{c^{4} n \log{\left (b x + c x^{2} \right )}}{4 b^{4}} & \text{for}\: b \neq 0 \\- \frac{n \log{\left (c \right )}}{4 x^{4}} - \frac{n \log{\left (x \right )}}{2 x^{4}} - \frac{n}{8 x^{4}} - \frac{\log{\left (d \right )}}{4 x^{4}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(d*(c*x**2+b*x)**n)/x**5,x)

[Out]

Piecewise((-n*log(b*x + c*x**2)/(4*x**4) - n/(16*x**4) - log(d)/(4*x**4) - c*n/(12*b*x**3) + c**2*n/(8*b**2*x*
*2) - c**3*n/(4*b**3*x) + c**4*n*log(b + c*x)/(2*b**4) - c**4*n*log(b*x + c*x**2)/(4*b**4), Ne(b, 0)), (-n*log
(c)/(4*x**4) - n*log(x)/(2*x**4) - n/(8*x**4) - log(d)/(4*x**4), True))

________________________________________________________________________________________

Giac [A]  time = 1.16691, size = 124, normalized size = 1.24 \begin{align*} \frac{c^{4} n \log \left (c x + b\right )}{4 \, b^{4}} - \frac{c^{4} n \log \left (x\right )}{4 \, b^{4}} - \frac{n \log \left (c x^{2} + b x\right )}{4 \, x^{4}} - \frac{12 \, c^{3} n x^{3} - 6 \, b c^{2} n x^{2} + 4 \, b^{2} c n x + 3 \, b^{3} n + 12 \, b^{3} \log \left (d\right )}{48 \, b^{3} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(d*(c*x^2+b*x)^n)/x^5,x, algorithm="giac")

[Out]

1/4*c^4*n*log(c*x + b)/b^4 - 1/4*c^4*n*log(x)/b^4 - 1/4*n*log(c*x^2 + b*x)/x^4 - 1/48*(12*c^3*n*x^3 - 6*b*c^2*
n*x^2 + 4*b^2*c*n*x + 3*b^3*n + 12*b^3*log(d))/(b^3*x^4)