3.518 \(\int \frac{a+b \log (c (d (e+f x)^p)^q)}{g+h x^2} \, dx\)

Optimal. Leaf size=249 \[ -\frac{b p q \text{PolyLog}\left (2,-\frac{\sqrt{h} (e+f x)}{f \sqrt{-g}-e \sqrt{h}}\right )}{2 \sqrt{-g} \sqrt{h}}+\frac{b p q \text{PolyLog}\left (2,\frac{\sqrt{h} (e+f x)}{e \sqrt{h}+f \sqrt{-g}}\right )}{2 \sqrt{-g} \sqrt{h}}+\frac{\log \left (\frac{f \left (\sqrt{-g}-\sqrt{h} x\right )}{e \sqrt{h}+f \sqrt{-g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 \sqrt{-g} \sqrt{h}}-\frac{\log \left (\frac{f \left (\sqrt{-g}+\sqrt{h} x\right )}{f \sqrt{-g}-e \sqrt{h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 \sqrt{-g} \sqrt{h}} \]

[Out]

((a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(Sqrt[-g] - Sqrt[h]*x))/(f*Sqrt[-g] + e*Sqrt[h])])/(2*Sqrt[-g]*Sqrt[h
]) - ((a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(Sqrt[-g] + Sqrt[h]*x))/(f*Sqrt[-g] - e*Sqrt[h])])/(2*Sqrt[-g]*S
qrt[h]) - (b*p*q*PolyLog[2, -((Sqrt[h]*(e + f*x))/(f*Sqrt[-g] - e*Sqrt[h]))])/(2*Sqrt[-g]*Sqrt[h]) + (b*p*q*Po
lyLog[2, (Sqrt[h]*(e + f*x))/(f*Sqrt[-g] + e*Sqrt[h])])/(2*Sqrt[-g]*Sqrt[h])

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Rubi [A]  time = 0.503994, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {2409, 2394, 2393, 2391, 2445} \[ -\frac{b p q \text{PolyLog}\left (2,-\frac{\sqrt{h} (e+f x)}{f \sqrt{-g}-e \sqrt{h}}\right )}{2 \sqrt{-g} \sqrt{h}}+\frac{b p q \text{PolyLog}\left (2,\frac{\sqrt{h} (e+f x)}{e \sqrt{h}+f \sqrt{-g}}\right )}{2 \sqrt{-g} \sqrt{h}}+\frac{\log \left (\frac{f \left (\sqrt{-g}-\sqrt{h} x\right )}{e \sqrt{h}+f \sqrt{-g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 \sqrt{-g} \sqrt{h}}-\frac{\log \left (\frac{f \left (\sqrt{-g}+\sqrt{h} x\right )}{f \sqrt{-g}-e \sqrt{h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 \sqrt{-g} \sqrt{h}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d*(e + f*x)^p)^q])/(g + h*x^2),x]

[Out]

((a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(Sqrt[-g] - Sqrt[h]*x))/(f*Sqrt[-g] + e*Sqrt[h])])/(2*Sqrt[-g]*Sqrt[h
]) - ((a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(Sqrt[-g] + Sqrt[h]*x))/(f*Sqrt[-g] - e*Sqrt[h])])/(2*Sqrt[-g]*S
qrt[h]) - (b*p*q*PolyLog[2, -((Sqrt[h]*(e + f*x))/(f*Sqrt[-g] - e*Sqrt[h]))])/(2*Sqrt[-g]*Sqrt[h]) + (b*p*q*Po
lyLog[2, (Sqrt[h]*(e + f*x))/(f*Sqrt[-g] + e*Sqrt[h])])/(2*Sqrt[-g]*Sqrt[h])

Rule 2409

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 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]

Rule 2445

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

\begin{align*} \int \frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{g+h x^2} \, dx &=\operatorname{Subst}\left (\int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{g+h x^2} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{\sqrt{-g} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{2 g \left (\sqrt{-g}-\sqrt{h} x\right )}+\frac{\sqrt{-g} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{2 g \left (\sqrt{-g}+\sqrt{h} x\right )}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\operatorname{Subst}\left (\frac{\int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{\sqrt{-g}-\sqrt{h} x} \, dx}{2 \sqrt{-g}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{\sqrt{-g}+\sqrt{h} x} \, dx}{2 \sqrt{-g}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f \left (\sqrt{-g}-\sqrt{h} x\right )}{f \sqrt{-g}+e \sqrt{h}}\right )}{2 \sqrt{-g} \sqrt{h}}-\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f \left (\sqrt{-g}+\sqrt{h} x\right )}{f \sqrt{-g}-e \sqrt{h}}\right )}{2 \sqrt{-g} \sqrt{h}}-\operatorname{Subst}\left (\frac{(b f p q) \int \frac{\log \left (\frac{f \left (\sqrt{-g}-\sqrt{h} x\right )}{f \sqrt{-g}+e \sqrt{h}}\right )}{e+f x} \, dx}{2 \sqrt{-g} \sqrt{h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(b f p q) \int \frac{\log \left (\frac{f \left (\sqrt{-g}+\sqrt{h} x\right )}{f \sqrt{-g}-e \sqrt{h}}\right )}{e+f x} \, dx}{2 \sqrt{-g} \sqrt{h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f \left (\sqrt{-g}-\sqrt{h} x\right )}{f \sqrt{-g}+e \sqrt{h}}\right )}{2 \sqrt{-g} \sqrt{h}}-\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f \left (\sqrt{-g}+\sqrt{h} x\right )}{f \sqrt{-g}-e \sqrt{h}}\right )}{2 \sqrt{-g} \sqrt{h}}+\operatorname{Subst}\left (\frac{(b p q) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{h} x}{f \sqrt{-g}-e \sqrt{h}}\right )}{x} \, dx,x,e+f x\right )}{2 \sqrt{-g} \sqrt{h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(b p q) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{h} x}{f \sqrt{-g}+e \sqrt{h}}\right )}{x} \, dx,x,e+f x\right )}{2 \sqrt{-g} \sqrt{h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f \left (\sqrt{-g}-\sqrt{h} x\right )}{f \sqrt{-g}+e \sqrt{h}}\right )}{2 \sqrt{-g} \sqrt{h}}-\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f \left (\sqrt{-g}+\sqrt{h} x\right )}{f \sqrt{-g}-e \sqrt{h}}\right )}{2 \sqrt{-g} \sqrt{h}}-\frac{b p q \text{Li}_2\left (-\frac{\sqrt{h} (e+f x)}{f \sqrt{-g}-e \sqrt{h}}\right )}{2 \sqrt{-g} \sqrt{h}}+\frac{b p q \text{Li}_2\left (\frac{\sqrt{h} (e+f x)}{f \sqrt{-g}+e \sqrt{h}}\right )}{2 \sqrt{-g} \sqrt{h}}\\ \end{align*}

Mathematica [A]  time = 0.137004, size = 190, normalized size = 0.76 \[ \frac{-b p q \text{PolyLog}\left (2,-\frac{\sqrt{h} (e+f x)}{f \sqrt{-g}-e \sqrt{h}}\right )+b p q \text{PolyLog}\left (2,\frac{\sqrt{h} (e+f x)}{e \sqrt{h}+f \sqrt{-g}}\right )+\left (\log \left (\frac{f \left (\sqrt{-g}-\sqrt{h} x\right )}{e \sqrt{h}+f \sqrt{-g}}\right )-\log \left (\frac{f \left (\sqrt{-g}+\sqrt{h} x\right )}{f \sqrt{-g}-e \sqrt{h}}\right )\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 \sqrt{-g} \sqrt{h}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d*(e + f*x)^p)^q])/(g + h*x^2),x]

[Out]

((a + b*Log[c*(d*(e + f*x)^p)^q])*(Log[(f*(Sqrt[-g] - Sqrt[h]*x))/(f*Sqrt[-g] + e*Sqrt[h])] - Log[(f*(Sqrt[-g]
+ Sqrt[h]*x))/(f*Sqrt[-g] - e*Sqrt[h])]) - b*p*q*PolyLog[2, -((Sqrt[h]*(e + f*x))/(f*Sqrt[-g] - e*Sqrt[h]))]
+ b*p*q*PolyLog[2, (Sqrt[h]*(e + f*x))/(f*Sqrt[-g] + e*Sqrt[h])])/(2*Sqrt[-g]*Sqrt[h])

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Maple [F]  time = 0.731, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) }{h{x}^{2}+g}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d*(f*x+e)^p)^q))/(h*x^2+g),x)

[Out]

int((a+b*ln(c*(d*(f*x+e)^p)^q))/(h*x^2+g),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(h*x^2+g),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{h x^{2} + g}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(h*x^2+g),x, algorithm="fricas")

[Out]

integral((b*log(((f*x + e)^p*d)^q*c) + a)/(h*x^2 + g), 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((a+b*ln(c*(d*(f*x+e)**p)**q))/(h*x**2+g),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{h x^{2} + g}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(h*x^2+g),x, algorithm="giac")

[Out]

integrate((b*log(((f*x + e)^p*d)^q*c) + a)/(h*x^2 + g), x)