3.3.66 \(\int \frac {\log (c (d+e \sqrt {x})^p)}{f+g x^2} \, dx\) [266]

Optimal. Leaf size=541 \[ -\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \text {Li}_2\left (-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {Li}_2\left (-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \text {Li}_2\left (\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {Li}_2\left (\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}} \]

[Out]

1/2*ln(c*(d+e*x^(1/2))^p)*ln(e*((-f)^(1/4)-g^(1/4)*x^(1/2))/(e*(-f)^(1/4)+d*g^(1/4)))/(-f)^(1/2)/g^(1/2)+1/2*l
n(c*(d+e*x^(1/2))^p)*ln(e*((-f)^(1/4)+g^(1/4)*x^(1/2))/(e*(-f)^(1/4)-d*g^(1/4)))/(-f)^(1/2)/g^(1/2)-1/2*ln(c*(
d+e*x^(1/2))^p)*ln(e*(g^(1/4)*x^(1/2)+(-(-f)^(1/2))^(1/2))/(-d*g^(1/4)+e*(-(-f)^(1/2))^(1/2)))/(-f)^(1/2)/g^(1
/2)-1/2*ln(c*(d+e*x^(1/2))^p)*ln(e*(-g^(1/4)*x^(1/2)+(-(-f)^(1/2))^(1/2))/(d*g^(1/4)+e*(-(-f)^(1/2))^(1/2)))/(
-f)^(1/2)/g^(1/2)+1/2*p*polylog(2,-g^(1/4)*(d+e*x^(1/2))/(e*(-f)^(1/4)-d*g^(1/4)))/(-f)^(1/2)/g^(1/2)+1/2*p*po
lylog(2,g^(1/4)*(d+e*x^(1/2))/(e*(-f)^(1/4)+d*g^(1/4)))/(-f)^(1/2)/g^(1/2)-1/2*p*polylog(2,-g^(1/4)*(d+e*x^(1/
2))/(-d*g^(1/4)+e*(-(-f)^(1/2))^(1/2)))/(-f)^(1/2)/g^(1/2)-1/2*p*polylog(2,g^(1/4)*(d+e*x^(1/2))/(d*g^(1/4)+e*
(-(-f)^(1/2))^(1/2)))/(-f)^(1/2)/g^(1/2)

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Rubi [A]
time = 0.57, antiderivative size = 541, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2522, 281, 211, 2463, 266, 2441, 2440, 2438} \begin {gather*} -\frac {p \text {PolyLog}\left (2,-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {PolyLog}\left (2,-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \text {PolyLog}\left (2,\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{d \sqrt [4]{g}+e \sqrt {-\sqrt {-f}}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {PolyLog}\left (2,\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{d \sqrt [4]{g}+e \sqrt [4]{-f}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}-\sqrt [4]{g} \sqrt {x}\right )}{d \sqrt [4]{g}+e \sqrt {-\sqrt {-f}}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right )}{d \sqrt [4]{g}+e \sqrt [4]{-f}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*Sqrt[x])^p]/(f + g*x^2),x]

[Out]

-1/2*(Log[c*(d + e*Sqrt[x])^p]*Log[(e*(Sqrt[-Sqrt[-f]] - g^(1/4)*Sqrt[x]))/(e*Sqrt[-Sqrt[-f]] + d*g^(1/4))])/(
Sqrt[-f]*Sqrt[g]) + (Log[c*(d + e*Sqrt[x])^p]*Log[(e*((-f)^(1/4) - g^(1/4)*Sqrt[x]))/(e*(-f)^(1/4) + d*g^(1/4)
)])/(2*Sqrt[-f]*Sqrt[g]) - (Log[c*(d + e*Sqrt[x])^p]*Log[(e*(Sqrt[-Sqrt[-f]] + g^(1/4)*Sqrt[x]))/(e*Sqrt[-Sqrt
[-f]] - d*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (Log[c*(d + e*Sqrt[x])^p]*Log[(e*((-f)^(1/4) + g^(1/4)*Sqrt[x]))/(
e*(-f)^(1/4) - d*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, -((g^(1/4)*(d + e*Sqrt[x]))/(e*Sqrt[-Sqrt[-f]
] - d*g^(1/4)))])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, -((g^(1/4)*(d + e*Sqrt[x]))/(e*(-f)^(1/4) - d*g^(1/4)))
])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (g^(1/4)*(d + e*Sqrt[x]))/(e*Sqrt[-Sqrt[-f]] + d*g^(1/4))])/(2*Sqrt[-f
]*Sqrt[g]) + (p*PolyLog[2, (g^(1/4)*(d + e*Sqrt[x]))/(e*(-f)^(1/4) + d*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 266

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 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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 2463

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 2522

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol]
:> With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k - 1)*(f + g*x^(k*s))^r*(a + b*Log[c*(d + e*x^(k*n))^p])^q
, x], x, x^(1/k)], x] /; IntegerQ[k*s]] /; FreeQ[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && FractionQ[n]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.20, size = 422, normalized size = 0.78 \begin {gather*} \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )-\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-i \sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}+i d \sqrt [4]{g}}\right )-\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+i \sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-i d \sqrt [4]{g}}\right )+\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )+p \text {Li}_2\left (-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )-p \text {Li}_2\left (\frac {i \sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}+i d \sqrt [4]{g}}\right )-p \text {Li}_2\left (\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{i e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )+p \text {Li}_2\left (\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e*Sqrt[x])^p]/(f + g*x^2),x]

[Out]

(Log[c*(d + e*Sqrt[x])^p]*Log[(e*((-f)^(1/4) - g^(1/4)*Sqrt[x]))/(e*(-f)^(1/4) + d*g^(1/4))] - Log[c*(d + e*Sq
rt[x])^p]*Log[(e*((-f)^(1/4) - I*g^(1/4)*Sqrt[x]))/(e*(-f)^(1/4) + I*d*g^(1/4))] - Log[c*(d + e*Sqrt[x])^p]*Lo
g[(e*((-f)^(1/4) + I*g^(1/4)*Sqrt[x]))/(e*(-f)^(1/4) - I*d*g^(1/4))] + Log[c*(d + e*Sqrt[x])^p]*Log[(e*((-f)^(
1/4) + g^(1/4)*Sqrt[x]))/(e*(-f)^(1/4) - d*g^(1/4))] + p*PolyLog[2, -((g^(1/4)*(d + e*Sqrt[x]))/(e*(-f)^(1/4)
- d*g^(1/4)))] - p*PolyLog[2, (I*g^(1/4)*(d + e*Sqrt[x]))/(e*(-f)^(1/4) + I*d*g^(1/4))] - p*PolyLog[2, (g^(1/4
)*(d + e*Sqrt[x]))/(I*e*(-f)^(1/4) + d*g^(1/4))] + p*PolyLog[2, (g^(1/4)*(d + e*Sqrt[x]))/(e*(-f)^(1/4) + d*g^
(1/4))])/(2*Sqrt[-f]*Sqrt[g])

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Maple [F]
time = 0.27, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (d +e \sqrt {x}\right )^{p}\right )}{g \,x^{2}+f}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(d+e*x^(1/2))^p)/(g*x^2+f),x)

[Out]

int(ln(c*(d+e*x^(1/2))^p)/(g*x^2+f),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(log(c*(d+e*x^(1/2))^p)/(g*x^2+f),x, algorithm="maxima")

[Out]

integrate(log((sqrt(x)*e + d)^p*c)/(g*x^2 + f), x)

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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(log(c*(d+e*x^(1/2))^p)/(g*x^2+f),x, algorithm="fricas")

[Out]

integral(log((sqrt(x)*e + d)^p*c)/(g*x^2 + f), 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(ln(c*(d+e*x**(1/2))**p)/(g*x**2+f),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(log(c*(d+e*x^(1/2))^p)/(g*x^2+f),x, algorithm="giac")

[Out]

integrate(log((sqrt(x)*e + d)^p*c)/(g*x^2 + f), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(d + e*x^(1/2))^p)/(f + g*x^2),x)

[Out]

int(log(c*(d + e*x^(1/2))^p)/(f + g*x^2), x)

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