3.6.18 \(\int \frac {a+b \log (c (d (e+f x)^p)^q)}{g+h x^2} \, dx\) [518]

Optimal. Leaf size=249 \[ \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f \left (\sqrt {-g}-\sqrt {h} x\right )}{f \sqrt {-g}+e \sqrt {h}}\right )}{2 \sqrt {-g} \sqrt {h}}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f \left (\sqrt {-g}+\sqrt {h} x\right )}{f \sqrt {-g}-e \sqrt {h}}\right )}{2 \sqrt {-g} \sqrt {h}}-\frac {b p q \text {Li}_2\left (-\frac {\sqrt {h} (e+f x)}{f \sqrt {-g}-e \sqrt {h}}\right )}{2 \sqrt {-g} \sqrt {h}}+\frac {b p q \text {Li}_2\left (\frac {\sqrt {h} (e+f x)}{f \sqrt {-g}+e \sqrt {h}}\right )}{2 \sqrt {-g} \sqrt {h}} \]

[Out]

1/2*(a+b*ln(c*(d*(f*x+e)^p)^q))*ln(f*((-g)^(1/2)-x*h^(1/2))/(f*(-g)^(1/2)+e*h^(1/2)))/(-g)^(1/2)/h^(1/2)-1/2*(
a+b*ln(c*(d*(f*x+e)^p)^q))*ln(f*((-g)^(1/2)+x*h^(1/2))/(f*(-g)^(1/2)-e*h^(1/2)))/(-g)^(1/2)/h^(1/2)-1/2*b*p*q*
polylog(2,-(f*x+e)*h^(1/2)/(f*(-g)^(1/2)-e*h^(1/2)))/(-g)^(1/2)/h^(1/2)+1/2*b*p*q*polylog(2,(f*x+e)*h^(1/2)/(f
*(-g)^(1/2)+e*h^(1/2)))/(-g)^(1/2)/h^(1/2)

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Rubi [A]
time = 0.34, antiderivative size = 249, normalized size of antiderivative = 1.00, 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 = {2456, 2441, 2440, 2438, 2495} \begin {gather*} -\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}} \end {gather*}

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 2438

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 2440

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 2441

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 2456

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 2495

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 &=\text {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 )\\ &=\text {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 )\\ &=-\text {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 )-\text {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}}-\text {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 )+\text {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}}+\text {Subst}\left (\frac {(b p q) \text {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 )-\text {Subst}\left (\frac {(b p q) \text {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*}

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Mathematica [A]
time = 0.11, size = 190, normalized size = 0.76 \begin {gather*} \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \left (\log \left (\frac {f \left (\sqrt {-g}-\sqrt {h} x\right )}{f \sqrt {-g}+e \sqrt {h}}\right )-\log \left (\frac {f \left (\sqrt {-g}+\sqrt {h} x\right )}{f \sqrt {-g}-e \sqrt {h}}\right )\right )-b p q \text {Li}_2\left (-\frac {\sqrt {h} (e+f x)}{f \sqrt {-g}-e \sqrt {h}}\right )+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 {gather*}

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 = 0.23, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{h \,x^{2}+g}\, dx\]

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{h\,x^2+g} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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