3.219 \(\int \frac{x^3 \log (c (a+b x)^p)}{d+e x} \, dx\)

Optimal. Leaf size=250 \[ -\frac{d^3 p \text{PolyLog}\left (2,-\frac{e (a+b x)}{b d-a e}\right )}{e^4}+\frac{a^2 d p \log (a+b x)}{2 b^2 e^2}-\frac{a^2 p x}{3 b^2 e}+\frac{a^3 p \log (a+b x)}{3 b^3 e}+\frac{d^2 (a+b x) \log \left (c (a+b x)^p\right )}{b e^3}-\frac{d^3 \log \left (c (a+b x)^p\right ) \log \left (\frac{b (d+e x)}{b d-a e}\right )}{e^4}-\frac{d x^2 \log \left (c (a+b x)^p\right )}{2 e^2}+\frac{x^3 \log \left (c (a+b x)^p\right )}{3 e}-\frac{a d p x}{2 b e^2}+\frac{a p x^2}{6 b e}-\frac{d^2 p x}{e^3}+\frac{d p x^2}{4 e^2}-\frac{p x^3}{9 e} \]

[Out]

-((d^2*p*x)/e^3) - (a*d*p*x)/(2*b*e^2) - (a^2*p*x)/(3*b^2*e) + (d*p*x^2)/(4*e^2) + (a*p*x^2)/(6*b*e) - (p*x^3)
/(9*e) + (a^2*d*p*Log[a + b*x])/(2*b^2*e^2) + (a^3*p*Log[a + b*x])/(3*b^3*e) - (d*x^2*Log[c*(a + b*x)^p])/(2*e
^2) + (x^3*Log[c*(a + b*x)^p])/(3*e) + (d^2*(a + b*x)*Log[c*(a + b*x)^p])/(b*e^3) - (d^3*Log[c*(a + b*x)^p]*Lo
g[(b*(d + e*x))/(b*d - a*e)])/e^4 - (d^3*p*PolyLog[2, -((e*(a + b*x))/(b*d - a*e))])/e^4

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Rubi [A]  time = 0.245857, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {43, 2416, 2389, 2295, 2395, 2394, 2393, 2391} \[ -\frac{d^3 p \text{PolyLog}\left (2,-\frac{e (a+b x)}{b d-a e}\right )}{e^4}+\frac{a^2 d p \log (a+b x)}{2 b^2 e^2}-\frac{a^2 p x}{3 b^2 e}+\frac{a^3 p \log (a+b x)}{3 b^3 e}+\frac{d^2 (a+b x) \log \left (c (a+b x)^p\right )}{b e^3}-\frac{d^3 \log \left (c (a+b x)^p\right ) \log \left (\frac{b (d+e x)}{b d-a e}\right )}{e^4}-\frac{d x^2 \log \left (c (a+b x)^p\right )}{2 e^2}+\frac{x^3 \log \left (c (a+b x)^p\right )}{3 e}-\frac{a d p x}{2 b e^2}+\frac{a p x^2}{6 b e}-\frac{d^2 p x}{e^3}+\frac{d p x^2}{4 e^2}-\frac{p x^3}{9 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((d^2*p*x)/e^3) - (a*d*p*x)/(2*b*e^2) - (a^2*p*x)/(3*b^2*e) + (d*p*x^2)/(4*e^2) + (a*p*x^2)/(6*b*e) - (p*x^3)
/(9*e) + (a^2*d*p*Log[a + b*x])/(2*b^2*e^2) + (a^3*p*Log[a + b*x])/(3*b^3*e) - (d*x^2*Log[c*(a + b*x)^p])/(2*e
^2) + (x^3*Log[c*(a + b*x)^p])/(3*e) + (d^2*(a + b*x)*Log[c*(a + b*x)^p])/(b*e^3) - (d^3*Log[c*(a + b*x)^p]*Lo
g[(b*(d + e*x))/(b*d - a*e)])/e^4 - (d^3*p*PolyLog[2, -((e*(a + b*x))/(b*d - a*e))])/e^4

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 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{x^3 \log \left (c (a+b x)^p\right )}{d+e x} \, dx &=\int \left (\frac{d^2 \log \left (c (a+b x)^p\right )}{e^3}-\frac{d x \log \left (c (a+b x)^p\right )}{e^2}+\frac{x^2 \log \left (c (a+b x)^p\right )}{e}-\frac{d^3 \log \left (c (a+b x)^p\right )}{e^3 (d+e x)}\right ) \, dx\\ &=\frac{d^2 \int \log \left (c (a+b x)^p\right ) \, dx}{e^3}-\frac{d^3 \int \frac{\log \left (c (a+b x)^p\right )}{d+e x} \, dx}{e^3}-\frac{d \int x \log \left (c (a+b x)^p\right ) \, dx}{e^2}+\frac{\int x^2 \log \left (c (a+b x)^p\right ) \, dx}{e}\\ &=-\frac{d x^2 \log \left (c (a+b x)^p\right )}{2 e^2}+\frac{x^3 \log \left (c (a+b x)^p\right )}{3 e}-\frac{d^3 \log \left (c (a+b x)^p\right ) \log \left (\frac{b (d+e x)}{b d-a e}\right )}{e^4}+\frac{d^2 \operatorname{Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+b x\right )}{b e^3}+\frac{\left (b d^3 p\right ) \int \frac{\log \left (\frac{b (d+e x)}{b d-a e}\right )}{a+b x} \, dx}{e^4}+\frac{(b d p) \int \frac{x^2}{a+b x} \, dx}{2 e^2}-\frac{(b p) \int \frac{x^3}{a+b x} \, dx}{3 e}\\ &=-\frac{d^2 p x}{e^3}-\frac{d x^2 \log \left (c (a+b x)^p\right )}{2 e^2}+\frac{x^3 \log \left (c (a+b x)^p\right )}{3 e}+\frac{d^2 (a+b x) \log \left (c (a+b x)^p\right )}{b e^3}-\frac{d^3 \log \left (c (a+b x)^p\right ) \log \left (\frac{b (d+e x)}{b d-a e}\right )}{e^4}+\frac{\left (d^3 p\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{e x}{b d-a e}\right )}{x} \, dx,x,a+b x\right )}{e^4}+\frac{(b d p) \int \left (-\frac{a}{b^2}+\frac{x}{b}+\frac{a^2}{b^2 (a+b x)}\right ) \, dx}{2 e^2}-\frac{(b p) \int \left (\frac{a^2}{b^3}-\frac{a x}{b^2}+\frac{x^2}{b}-\frac{a^3}{b^3 (a+b x)}\right ) \, dx}{3 e}\\ &=-\frac{d^2 p x}{e^3}-\frac{a d p x}{2 b e^2}-\frac{a^2 p x}{3 b^2 e}+\frac{d p x^2}{4 e^2}+\frac{a p x^2}{6 b e}-\frac{p x^3}{9 e}+\frac{a^2 d p \log (a+b x)}{2 b^2 e^2}+\frac{a^3 p \log (a+b x)}{3 b^3 e}-\frac{d x^2 \log \left (c (a+b x)^p\right )}{2 e^2}+\frac{x^3 \log \left (c (a+b x)^p\right )}{3 e}+\frac{d^2 (a+b x) \log \left (c (a+b x)^p\right )}{b e^3}-\frac{d^3 \log \left (c (a+b x)^p\right ) \log \left (\frac{b (d+e x)}{b d-a e}\right )}{e^4}-\frac{d^3 p \text{Li}_2\left (-\frac{e (a+b x)}{b d-a e}\right )}{e^4}\\ \end{align*}

Mathematica [A]  time = 0.18436, size = 183, normalized size = 0.73 \[ \frac{-36 b^3 d^3 p \text{PolyLog}\left (2,\frac{e (a+b x)}{a e-b d}\right )+b \left (6 b \log \left (c (a+b x)^p\right ) \left (-6 b d^3 \log \left (\frac{b (d+e x)}{b d-a e}\right )+6 a d^2 e+b e x \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-e p x \left (12 a^2 e^2-6 a b e (e x-3 d)+b^2 \left (36 d^2-9 d e x+4 e^2 x^2\right )\right )\right )+6 a^2 e^2 p (2 a e+3 b d) \log (a+b x)}{36 b^3 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*a^2*e^2*(3*b*d + 2*a*e)*p*Log[a + b*x] + b*(-(e*p*x*(12*a^2*e^2 - 6*a*b*e*(-3*d + e*x) + b^2*(36*d^2 - 9*d*
e*x + 4*e^2*x^2))) + 6*b*Log[c*(a + b*x)^p]*(6*a*d^2*e + b*e*x*(6*d^2 - 3*d*e*x + 2*e^2*x^2) - 6*b*d^3*Log[(b*
(d + e*x))/(b*d - a*e)])) - 36*b^3*d^3*p*PolyLog[2, (e*(a + b*x))/(-(b*d) + a*e)])/(36*b^3*e^4)

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Maple [C]  time = 0.66, size = 919, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*d*p*x^2/e^2-1/2*I*Pi*csgn(I*c)*csgn(I*(b*x+a)^p)*csgn(I*c*(b*x+a)^p)/e^3*x*d^2+1/4*I*Pi*csgn(I*c)*csgn(I*(
b*x+a)^p)*csgn(I*c*(b*x+a)^p)/e^2*x^2*d+1/2*I*Pi*csgn(I*c)*csgn(I*(b*x+a)^p)*csgn(I*c*(b*x+a)^p)*d^3/e^4*ln(e*
x+d)+1/3*ln((b*x+a)^p)/e*x^3-ln((b*x+a)^p)*d^3/e^4*ln(e*x+d)-1/2*ln((b*x+a)^p)/e^2*x^2*d+ln((b*x+a)^p)/e^3*x*d
^2+1/3*ln(c)/e*x^3+1/2*I*Pi*csgn(I*c)*csgn(I*c*(b*x+a)^p)^2/e^3*x*d^2-ln(c)*d^3/e^4*ln(e*x+d)-1/2*ln(c)/e^2*x^
2*d+ln(c)/e^3*x*d^2-49/36*p/e^4*d^3+p/e^4*d^3*dilog((b*(e*x+d)+a*e-b*d)/(a*e-b*d))-1/6*I*Pi*csgn(I*c)*csgn(I*(
b*x+a)^p)*csgn(I*c*(b*x+a)^p)/e*x^3+1/4*I*Pi*csgn(I*c*(b*x+a)^p)^3/e^2*x^2*d-1/2*I*Pi*csgn(I*c*(b*x+a)^p)^3/e^
3*x*d^2+1/6*I*Pi*csgn(I*c)*csgn(I*c*(b*x+a)^p)^2/e*x^3+1/2*I*Pi*csgn(I*c*(b*x+a)^p)^3*d^3/e^4*ln(e*x+d)+1/6*I*
Pi*csgn(I*(b*x+a)^p)*csgn(I*c*(b*x+a)^p)^2/e*x^3+1/b*p/e^3*a*ln(b*(e*x+d)+a*e-b*d)*d^2+1/2/b^2*p/e^2*a^2*ln(b*
(e*x+d)+a*e-b*d)*d-1/6*I*Pi*csgn(I*c*(b*x+a)^p)^3/e*x^3+1/3/b^3*p/e*a^3*ln(b*(e*x+d)+a*e-b*d)+p/e^4*d^3*ln(e*x
+d)*ln((b*(e*x+d)+a*e-b*d)/(a*e-b*d))-2/3/b*p/e^3*a*d^2-1/3/b^2*p/e^2*a^2*d-1/2*I*Pi*csgn(I*c)*csgn(I*c*(b*x+a
)^p)^2*d^3/e^4*ln(e*x+d)-1/4*I*Pi*csgn(I*c)*csgn(I*c*(b*x+a)^p)^2/e^2*x^2*d+1/2*I*Pi*csgn(I*(b*x+a)^p)*csgn(I*
c*(b*x+a)^p)^2/e^3*x*d^2-1/3*a^2*p*x/b^2/e+1/6*a*p*x^2/b/e-1/9*p*x^3/e-d^2*p*x/e^3-1/2*I*Pi*csgn(I*(b*x+a)^p)*
csgn(I*c*(b*x+a)^p)^2*d^3/e^4*ln(e*x+d)-1/4*I*Pi*csgn(I*(b*x+a)^p)*csgn(I*c*(b*x+a)^p)^2/e^2*x^2*d-1/2*a*d*p*x
/b/e^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \log \left ({\left (b x + a\right )}^{p} c\right )}{e x + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3*log((b*x + a)^p*c)/(e*x + d), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{3} \log \left ({\left (b x + a\right )}^{p} c\right )}{e x + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^3*log((b*x + a)^p*c)/(e*x + d), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \log \left ({\left (b x + a\right )}^{p} c\right )}{e x + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3*log((b*x + a)^p*c)/(e*x + d), x)