3.294 \(\int \frac{x^3 \log (c+d x)}{a+b x^4} \, dx\)

Optimal. Leaf size=401 \[ \frac{\text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 b}+\frac{\text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 b}+\frac{\text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b}+\frac{\text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b}+\frac{\log (c+d x) \log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 b}+\frac{\log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b}+\frac{\log (c+d x) \log \left (-\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 b}+\frac{\log (c+d x) \log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b} \]

[Out]

(Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/(4*b) + (Log[(d*((-a)^(1
/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*b) + (Log[-((d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(
b^(1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Log[c + d*x])/(4*b) + (Log[-((d*((-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(
1/4)*d))]*Log[c + d*x])/(4*b) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)]/(4*b) + PolyLo
g[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]/(4*b) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c -
(-a)^(1/4)*d)]/(4*b) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)]/(4*b)

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Rubi [A]  time = 0.460136, antiderivative size = 401, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {260, 2416, 2394, 2393, 2391} \[ \frac{\text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 b}+\frac{\text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 b}+\frac{\text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b}+\frac{\text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b}+\frac{\log (c+d x) \log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 b}+\frac{\log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b}+\frac{\log (c+d x) \log \left (-\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 b}+\frac{\log (c+d x) \log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*Log[c + d*x])/(a + b*x^4),x]

[Out]

(Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/(4*b) + (Log[(d*((-a)^(1
/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*b) + (Log[-((d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(
b^(1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Log[c + d*x])/(4*b) + (Log[-((d*((-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(
1/4)*d))]*Log[c + d*x])/(4*b) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)]/(4*b) + PolyLo
g[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]/(4*b) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c -
(-a)^(1/4)*d)]/(4*b) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)]/(4*b)

Rule 260

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

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 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 (c+d x)}{a+b x^4} \, dx &=\int \left (\frac{x \log (c+d x)}{2 \left (-\sqrt{-a} \sqrt{b}+b x^2\right )}+\frac{x \log (c+d x)}{2 \left (\sqrt{-a} \sqrt{b}+b x^2\right )}\right ) \, dx\\ &=\frac{1}{2} \int \frac{x \log (c+d x)}{-\sqrt{-a} \sqrt{b}+b x^2} \, dx+\frac{1}{2} \int \frac{x \log (c+d x)}{\sqrt{-a} \sqrt{b}+b x^2} \, dx\\ &=\frac{1}{2} \int \left (-\frac{\log (c+d x)}{2 b^{3/4} \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}+\frac{\log (c+d x)}{2 b^{3/4} \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}\right ) \, dx+\frac{1}{2} \int \left (-\frac{\log (c+d x)}{2 b^{3/4} \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}+\frac{\log (c+d x)}{2 b^{3/4} \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}\right ) \, dx\\ &=-\frac{\int \frac{\log (c+d x)}{\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x} \, dx}{4 b^{3/4}}-\frac{\int \frac{\log (c+d x)}{\sqrt [4]{-a}-\sqrt [4]{b} x} \, dx}{4 b^{3/4}}+\frac{\int \frac{\log (c+d x)}{\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x} \, dx}{4 b^{3/4}}+\frac{\int \frac{\log (c+d x)}{\sqrt [4]{-a}+\sqrt [4]{b} x} \, dx}{4 b^{3/4}}\\ &=\frac{\log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right ) \log (c+d x)}{4 b}+\frac{\log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b}+\frac{\log \left (-\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right ) \log (c+d x)}{4 b}+\frac{\log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b}-\frac{d \int \frac{\log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right )}{c+d x} \, dx}{4 b}-\frac{d \int \frac{\log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{c+d x} \, dx}{4 b}-\frac{d \int \frac{\log \left (\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{-\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right )}{c+d x} \, dx}{4 b}-\frac{d \int \frac{\log \left (\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{-\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{c+d x} \, dx}{4 b}\\ &=\frac{\log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right ) \log (c+d x)}{4 b}+\frac{\log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b}+\frac{\log \left (-\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right ) \log (c+d x)}{4 b}+\frac{\log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b}-\frac{\operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [4]{b} x}{-\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right )}{x} \, dx,x,c+d x\right )}{4 b}-\frac{\operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt [4]{b} x}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right )}{x} \, dx,x,c+d x\right )}{4 b}-\frac{\operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [4]{b} x}{-\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{x} \, dx,x,c+d x\right )}{4 b}-\frac{\operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt [4]{b} x}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{x} \, dx,x,c+d x\right )}{4 b}\\ &=\frac{\log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right ) \log (c+d x)}{4 b}+\frac{\log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b}+\frac{\log \left (-\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right ) \log (c+d x)}{4 b}+\frac{\log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b}+\frac{\text{Li}_2\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 b}+\frac{\text{Li}_2\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right )}{4 b}+\frac{\text{Li}_2\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b}+\frac{\text{Li}_2\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b}\\ \end{align*}

Mathematica [C]  time = 0.0613948, size = 383, normalized size = 0.96 \[ \frac{\text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b}+\frac{\text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )}{4 b}+\frac{\text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )}{4 b}+\frac{\text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b}+\frac{\log (c+d x) \log \left (\frac{d \left (-\sqrt [4]{b} x+i \sqrt [4]{-a}\right )}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )}{4 b}+\frac{\log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b}+\frac{\log (c+d x) \log \left (-\frac{d \left (\sqrt [4]{b} x+i \sqrt [4]{-a}\right )}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )}{4 b}+\frac{\log (c+d x) \log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*Log[c + d*x])/(a + b*x^4),x]

[Out]

(Log[(d*(I*(-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + I*(-a)^(1/4)*d)]*Log[c + d*x])/(4*b) + (Log[(d*((-a)^(1/4) -
b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*b) + (Log[-((d*(I*(-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c
- I*(-a)^(1/4)*d))]*Log[c + d*x])/(4*b) + (Log[-((d*((-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Lo
g[c + d*x])/(4*b) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)]/(4*b) + PolyLog[2, (b^(1/4)*(c
+ d*x))/(b^(1/4)*c - I*(-a)^(1/4)*d)]/(4*b) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + I*(-a)^(1/4)*d)]/(4*
b) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)]/(4*b)

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Maple [C]  time = 0.378, size = 85, normalized size = 0.2 \begin{align*}{\frac{1}{4\,b}\sum _{{\it \_R1}={\it RootOf} \left ( b{{\it \_Z}}^{4}-4\,{{\it \_Z}}^{3}bc+6\,{{\it \_Z}}^{2}b{c}^{2}-4\,{\it \_Z}\,b{c}^{3}+a{d}^{4}+b{c}^{4} \right ) }\ln \left ( dx+c \right ) \ln \left ({\frac{-dx+{\it \_R1}-c}{{\it \_R1}}} \right ) +{\it dilog} \left ({\frac{-dx+{\it \_R1}-c}{{\it \_R1}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*ln(d*x+c)/(b*x^4+a),x)

[Out]

1/4/b*sum(ln(d*x+c)*ln((-d*x+_R1-c)/_R1)+dilog((-d*x+_R1-c)/_R1),_R1=RootOf(_Z^4*b-4*_Z^3*b*c+6*_Z^2*b*c^2-4*_
Z*b*c^3+a*d^4+b*c^4))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \log \left (d x + c\right )}{b x^{4} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*log(d*x+c)/(b*x^4+a),x, algorithm="maxima")

[Out]

integrate(x^3*log(d*x + c)/(b*x^4 + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{3} \log \left (d x + c\right )}{b x^{4} + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*log(d*x+c)/(b*x^4+a),x, algorithm="fricas")

[Out]

integral(x^3*log(d*x + c)/(b*x^4 + a), 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(d*x+c)/(b*x**4+a),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 (d x + c\right )}{b x^{4} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*log(d*x+c)/(b*x^4+a),x, algorithm="giac")

[Out]

integrate(x^3*log(d*x + c)/(b*x^4 + a), x)