3.68 \(\int \frac{\log (d (b x+c x^2)^n)}{x^4} \, dx\)

Optimal. Leaf size=86 \[ \frac{c^2 n}{3 b^2 x}+\frac{c^3 n \log (x)}{3 b^3}-\frac{c^3 n \log (b+c x)}{3 b^3}-\frac{\log \left (d \left (b x+c x^2\right )^n\right )}{3 x^3}-\frac{c n}{6 b x^2}-\frac{n}{9 x^3} \]

[Out]

-n/(9*x^3) - (c*n)/(6*b*x^2) + (c^2*n)/(3*b^2*x) + (c^3*n*Log[x])/(3*b^3) - (c^3*n*Log[b + c*x])/(3*b^3) - Log
[d*(b*x + c*x^2)^n]/(3*x^3)

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Rubi [A]  time = 0.0571364, antiderivative size = 86, 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} \[ \frac{c^2 n}{3 b^2 x}+\frac{c^3 n \log (x)}{3 b^3}-\frac{c^3 n \log (b+c x)}{3 b^3}-\frac{\log \left (d \left (b x+c x^2\right )^n\right )}{3 x^3}-\frac{c n}{6 b x^2}-\frac{n}{9 x^3} \]

Antiderivative was successfully verified.

[In]

Int[Log[d*(b*x + c*x^2)^n]/x^4,x]

[Out]

-n/(9*x^3) - (c*n)/(6*b*x^2) + (c^2*n)/(3*b^2*x) + (c^3*n*Log[x])/(3*b^3) - (c^3*n*Log[b + c*x])/(3*b^3) - Log
[d*(b*x + c*x^2)^n]/(3*x^3)

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^4} \, dx &=-\frac{\log \left (d \left (b x+c x^2\right )^n\right )}{3 x^3}+\frac{1}{3} n \int \frac{b+2 c x}{x^4 (b+c x)} \, dx\\ &=-\frac{\log \left (d \left (b x+c x^2\right )^n\right )}{3 x^3}+\frac{1}{3} n \int \left (\frac{1}{x^4}+\frac{c}{b x^3}-\frac{c^2}{b^2 x^2}+\frac{c^3}{b^3 x}-\frac{c^4}{b^3 (b+c x)}\right ) \, dx\\ &=-\frac{n}{9 x^3}-\frac{c n}{6 b x^2}+\frac{c^2 n}{3 b^2 x}+\frac{c^3 n \log (x)}{3 b^3}-\frac{c^3 n \log (b+c x)}{3 b^3}-\frac{\log \left (d \left (b x+c x^2\right )^n\right )}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0352008, size = 77, normalized size = 0.9 \[ \frac{1}{3} n \left (\frac{c^2}{b^2 x}+\frac{c^3 \log (x)}{b^3}-\frac{c^3 \log (b+c x)}{b^3}-\frac{c}{2 b x^2}-\frac{1}{3 x^3}\right )-\frac{\log \left (d (x (b+c x))^n\right )}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[d*(b*x + c*x^2)^n]/x^4,x]

[Out]

(n*(-1/(3*x^3) - c/(2*b*x^2) + c^2/(b^2*x) + (c^3*Log[x])/b^3 - (c^3*Log[b + c*x])/b^3))/3 - Log[d*(x*(b + c*x
))^n]/(3*x^3)

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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}^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(d*(c*x^2+b*x)^n)/x^4,x)

[Out]

int(ln(d*(c*x^2+b*x)^n)/x^4,x)

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Maxima [A]  time = 1.0167, size = 101, normalized size = 1.17 \begin{align*} -\frac{1}{18} \, n{\left (\frac{6 \, c^{3} \log \left (c x + b\right )}{b^{3}} - \frac{6 \, c^{3} \log \left (x\right )}{b^{3}} - \frac{6 \, c^{2} x^{2} - 3 \, b c x - 2 \, b^{2}}{b^{2} x^{3}}\right )} - \frac{\log \left ({\left (c x^{2} + b x\right )}^{n} d\right )}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(d*(c*x^2+b*x)^n)/x^4,x, algorithm="maxima")

[Out]

-1/18*n*(6*c^3*log(c*x + b)/b^3 - 6*c^3*log(x)/b^3 - (6*c^2*x^2 - 3*b*c*x - 2*b^2)/(b^2*x^3)) - 1/3*log((c*x^2
+ b*x)^n*d)/x^3

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Fricas [A]  time = 1.87479, size = 198, normalized size = 2.3 \begin{align*} -\frac{6 \, c^{3} n x^{3} \log \left (c x + b\right ) - 6 \, c^{3} n x^{3} \log \left (x\right ) - 6 \, b c^{2} n x^{2} + 3 \, b^{2} c n x + 6 \, b^{3} n \log \left (c x^{2} + b x\right ) + 2 \, b^{3} n + 6 \, b^{3} \log \left (d\right )}{18 \, b^{3} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(d*(c*x^2+b*x)^n)/x^4,x, algorithm="fricas")

[Out]

-1/18*(6*c^3*n*x^3*log(c*x + b) - 6*c^3*n*x^3*log(x) - 6*b*c^2*n*x^2 + 3*b^2*c*n*x + 6*b^3*n*log(c*x^2 + b*x)
+ 2*b^3*n + 6*b^3*log(d))/(b^3*x^3)

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Sympy [A]  time = 8.75475, size = 133, normalized size = 1.55 \begin{align*} \begin{cases} - \frac{n \log{\left (b x + c x^{2} \right )}}{3 x^{3}} - \frac{n}{9 x^{3}} - \frac{\log{\left (d \right )}}{3 x^{3}} - \frac{c n}{6 b x^{2}} + \frac{c^{2} n}{3 b^{2} x} - \frac{2 c^{3} n \log{\left (b + c x \right )}}{3 b^{3}} + \frac{c^{3} n \log{\left (b x + c x^{2} \right )}}{3 b^{3}} & \text{for}\: b \neq 0 \\- \frac{n \log{\left (c \right )}}{3 x^{3}} - \frac{2 n \log{\left (x \right )}}{3 x^{3}} - \frac{2 n}{9 x^{3}} - \frac{\log{\left (d \right )}}{3 x^{3}} & \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**4,x)

[Out]

Piecewise((-n*log(b*x + c*x**2)/(3*x**3) - n/(9*x**3) - log(d)/(3*x**3) - c*n/(6*b*x**2) + c**2*n/(3*b**2*x) -
2*c**3*n*log(b + c*x)/(3*b**3) + c**3*n*log(b*x + c*x**2)/(3*b**3), Ne(b, 0)), (-n*log(c)/(3*x**3) - 2*n*log(
x)/(3*x**3) - 2*n/(9*x**3) - log(d)/(3*x**3), True))

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Giac [A]  time = 1.16039, size = 108, normalized size = 1.26 \begin{align*} -\frac{c^{3} n \log \left (c x + b\right )}{3 \, b^{3}} + \frac{c^{3} n \log \left (x\right )}{3 \, b^{3}} - \frac{n \log \left (c x^{2} + b x\right )}{3 \, x^{3}} + \frac{6 \, c^{2} n x^{2} - 3 \, b c n x - 2 \, b^{2} n - 6 \, b^{2} \log \left (d\right )}{18 \, b^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(d*(c*x^2+b*x)^n)/x^4,x, algorithm="giac")

[Out]

-1/3*c^3*n*log(c*x + b)/b^3 + 1/3*c^3*n*log(x)/b^3 - 1/3*n*log(c*x^2 + b*x)/x^3 + 1/18*(6*c^2*n*x^2 - 3*b*c*n*
x - 2*b^2*n - 6*b^2*log(d))/(b^2*x^3)