Integrand size = 31, antiderivative size = 1361 \[ \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {h x} (f+g x)} \, dx =\text {Too large to display} \]
2*arctan(g^(1/2)*(h*x)^(1/2)/f^(1/2)/h^(1/2))*(a+b*ln(c*(e*x^2+d)^p))/f^(1 /2)/g^(1/2)/h^(1/2)+8*b*p*arctan(g^(1/2)*(h*x)^(1/2)/f^(1/2)/h^(1/2))*ln(2 *f^(1/2)*h^(1/2)/(f^(1/2)*h^(1/2)-I*g^(1/2)*(h*x)^(1/2)))/f^(1/2)/g^(1/2)/ h^(1/2)-2*b*p*arctan(g^(1/2)*(h*x)^(1/2)/f^(1/2)/h^(1/2))*ln(2*f^(1/2)*g^( 1/2)*h^(1/2)*((-d)^(1/4)*(-h)^(1/2)-e^(1/4)*(h*x)^(1/2))/((-d)^(1/4)*g^(1/ 2)*(-h)^(1/2)-I*e^(1/4)*f^(1/2)*h^(1/2))/(f^(1/2)*h^(1/2)-I*g^(1/2)*(h*x)^ (1/2)))/f^(1/2)/g^(1/2)/h^(1/2)-2*b*p*arctan(g^(1/2)*(h*x)^(1/2)/f^(1/2)/h ^(1/2))*ln(-2*f^(1/2)*g^(1/2)*((-d)^(1/4)*h^(1/2)-e^(1/4)*(h*x)^(1/2))/(I* e^(1/4)*f^(1/2)-(-d)^(1/4)*g^(1/2))/(f^(1/2)*h^(1/2)-I*g^(1/2)*(h*x)^(1/2) ))/f^(1/2)/g^(1/2)/h^(1/2)-2*b*p*arctan(g^(1/2)*(h*x)^(1/2)/f^(1/2)/h^(1/2 ))*ln(2*f^(1/2)*g^(1/2)*h^(1/2)*((-d)^(1/4)*(-h)^(1/2)+e^(1/4)*(h*x)^(1/2) )/((-d)^(1/4)*g^(1/2)*(-h)^(1/2)+I*e^(1/4)*f^(1/2)*h^(1/2))/(f^(1/2)*h^(1/ 2)-I*g^(1/2)*(h*x)^(1/2)))/f^(1/2)/g^(1/2)/h^(1/2)-2*b*p*arctan(g^(1/2)*(h *x)^(1/2)/f^(1/2)/h^(1/2))*ln(2*f^(1/2)*g^(1/2)*((-d)^(1/4)*h^(1/2)+e^(1/4 )*(h*x)^(1/2))/(I*e^(1/4)*f^(1/2)+(-d)^(1/4)*g^(1/2))/(f^(1/2)*h^(1/2)-I*g ^(1/2)*(h*x)^(1/2)))/f^(1/2)/g^(1/2)/h^(1/2)-4*I*b*p*polylog(2,1-2*f^(1/2) *h^(1/2)/(f^(1/2)*h^(1/2)-I*g^(1/2)*(h*x)^(1/2)))/f^(1/2)/g^(1/2)/h^(1/2)+ I*b*p*polylog(2,1-2*f^(1/2)*g^(1/2)*h^(1/2)*((-d)^(1/4)*(-h)^(1/2)-e^(1/4) *(h*x)^(1/2))/((-d)^(1/4)*g^(1/2)*(-h)^(1/2)-I*e^(1/4)*f^(1/2)*h^(1/2))/(f ^(1/2)*h^(1/2)-I*g^(1/2)*(h*x)^(1/2)))/f^(1/2)/g^(1/2)/h^(1/2)+I*b*p*po...
Time = 0.30 (sec) , antiderivative size = 1297, normalized size of antiderivative = 0.95 \[ \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {h x} (f+g x)} \, dx=\frac {\sqrt {x} \left (a \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )-b p \log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}-\sqrt [4]{e} \sqrt {x}\right )}{-\sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )-b p \log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}+i \sqrt [4]{e} \sqrt {x}\right )}{i \sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )-b p \log \left (\frac {\sqrt {g} \left (i \sqrt [4]{-d}+\sqrt [4]{e} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+i \sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )-b p \log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}+\sqrt [4]{e} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )-a \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )+b p \log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}-\sqrt [4]{e} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )+b p \log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}-i \sqrt [4]{e} \sqrt {x}\right )}{i \sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )+b p \log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}+i \sqrt [4]{e} \sqrt {x}\right )}{-i \sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )+b p \log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}+\sqrt [4]{e} \sqrt {x}\right )}{-\sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )+b \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right ) \log \left (c \left (d+e x^2\right )^p\right )-b \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right ) \log \left (c \left (d+e x^2\right )^p\right )-b p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{e} \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}-\sqrt [4]{-d} \sqrt {g}}\right )-b p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{e} \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}-i \sqrt [4]{-d} \sqrt {g}}\right )-b p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{e} \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+i \sqrt [4]{-d} \sqrt {g}}\right )-b p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{e} \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right )+b p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{e} \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}-\sqrt [4]{-d} \sqrt {g}}\right )+b p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{e} \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}-i \sqrt [4]{-d} \sqrt {g}}\right )+b p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{e} \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+i \sqrt [4]{-d} \sqrt {g}}\right )+b p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{e} \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right )\right )}{\sqrt {-f} \sqrt {g} \sqrt {h x}} \]
(Sqrt[x]*(a*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]] - b*p*Log[(Sqrt[g]*((-d)^(1/4) - e^(1/4)*Sqrt[x]))/(-(e^(1/4)*Sqrt[-f]) + (-d)^(1/4)*Sqrt[g])]*Log[Sqrt[ -f] - Sqrt[g]*Sqrt[x]] - b*p*Log[(Sqrt[g]*((-d)^(1/4) + I*e^(1/4)*Sqrt[x]) )/(I*e^(1/4)*Sqrt[-f] + (-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x ]] - b*p*Log[(Sqrt[g]*(I*(-d)^(1/4) + e^(1/4)*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + I*(-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]] - b*p*Log[(Sqrt[g ]*((-d)^(1/4) + e^(1/4)*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + (-d)^(1/4)*Sqrt[g])] *Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]] - a*Log[Sqrt[-f] + Sqrt[g]*Sqrt[x]] + b*p *Log[(Sqrt[g]*((-d)^(1/4) - e^(1/4)*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + (-d)^(1/ 4)*Sqrt[g])]*Log[Sqrt[-f] + Sqrt[g]*Sqrt[x]] + b*p*Log[(Sqrt[g]*((-d)^(1/4 ) - I*e^(1/4)*Sqrt[x]))/(I*e^(1/4)*Sqrt[-f] + (-d)^(1/4)*Sqrt[g])]*Log[Sqr t[-f] + Sqrt[g]*Sqrt[x]] + b*p*Log[(Sqrt[g]*((-d)^(1/4) + I*e^(1/4)*Sqrt[x ]))/((-I)*e^(1/4)*Sqrt[-f] + (-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] + Sqrt[g]*S qrt[x]] + b*p*Log[(Sqrt[g]*((-d)^(1/4) + e^(1/4)*Sqrt[x]))/(-(e^(1/4)*Sqrt [-f]) + (-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] + Sqrt[g]*Sqrt[x]] + b*Log[Sqrt[ -f] - Sqrt[g]*Sqrt[x]]*Log[c*(d + e*x^2)^p] - b*Log[Sqrt[-f] + Sqrt[g]*Sqr t[x]]*Log[c*(d + e*x^2)^p] - b*p*PolyLog[2, (e^(1/4)*(Sqrt[-f] - Sqrt[g]*S qrt[x]))/(e^(1/4)*Sqrt[-f] - (-d)^(1/4)*Sqrt[g])] - b*p*PolyLog[2, (e^(1/4 )*(Sqrt[-f] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] - I*(-d)^(1/4)*Sqrt[g])] - b*p*PolyLog[2, (e^(1/4)*(Sqrt[-f] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[...
Time = 1.66 (sec) , antiderivative size = 1261, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2917, 27, 2920, 27, 7276, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {h x} (f+g x)} \, dx\) |
\(\Big \downarrow \) 2917 |
\(\displaystyle \frac {2 \int \frac {h \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{f h+g x h}d\sqrt {h x}}{h}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {a+b \log \left (c \left (e x^2+d\right )^p\right )}{f h+g x h}d\sqrt {h x}\) |
\(\Big \downarrow \) 2920 |
\(\displaystyle 2 \left (\frac {\arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {4 b e p \int \frac {h^{3/2} (h x)^{3/2} \arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right )}{\sqrt {f} \sqrt {g} \left (e x^2 h^2+d h^2\right )}d\sqrt {h x}}{h^2}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \left (\frac {\arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {4 b e p \int \frac {(h x)^{3/2} \arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right )}{e x^2 h^2+d h^2}d\sqrt {h x}}{\sqrt {f} \sqrt {g} \sqrt {h}}\right )\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle 2 \left (\frac {\arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {4 b e p \int \left (\frac {\sqrt {h x} \arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right )}{2 \left (e h x-\sqrt {-d} \sqrt {e} h\right )}+\frac {\sqrt {h x} \arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right )}{2 \left (e x h+\sqrt {-d} \sqrt {e} h\right )}\right )d\sqrt {h x}}{\sqrt {f} \sqrt {g} \sqrt {h}}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {\arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {4 b e p \left (-\frac {\arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{e}+\frac {\arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{4 e}+\frac {\arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{4 e}+\frac {\arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{4 e}+\frac {\arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{4 e}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{2 e}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{8 e}-\frac {i \operatorname {PolyLog}\left (2,\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}+1\right )}{8 e}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{8 e}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{8 e}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}\right )\) |
2*((ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*(a + b*Log[c*(d + e*x^2) ^p]))/(Sqrt[f]*Sqrt[g]*Sqrt[h]) - (4*b*e*p*(-((ArcTan[(Sqrt[g]*Sqrt[h*x])/ (Sqrt[f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqrt[h])/(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sq rt[h*x])])/e) + (ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(2*Sqrt [f]*Sqrt[g]*Sqrt[h]*((-d)^(1/4)*Sqrt[-h] - e^(1/4)*Sqrt[h*x]))/(((-d)^(1/4 )*Sqrt[g]*Sqrt[-h] - I*e^(1/4)*Sqrt[f]*Sqrt[h])*(Sqrt[f]*Sqrt[h] - I*Sqrt[ g]*Sqrt[h*x]))])/(4*e) + (ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Lo g[(-2*Sqrt[f]*Sqrt[g]*((-d)^(1/4)*Sqrt[h] - e^(1/4)*Sqrt[h*x]))/((I*e^(1/4 )*Sqrt[f] - (-d)^(1/4)*Sqrt[g])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))]) /(4*e) + (ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqr t[g]*Sqrt[h]*((-d)^(1/4)*Sqrt[-h] + e^(1/4)*Sqrt[h*x]))/(((-d)^(1/4)*Sqrt[ g]*Sqrt[-h] + I*e^(1/4)*Sqrt[f]*Sqrt[h])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt [h*x]))])/(4*e) + (ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(2*Sq rt[f]*Sqrt[g]*((-d)^(1/4)*Sqrt[h] + e^(1/4)*Sqrt[h*x]))/((I*e^(1/4)*Sqrt[f ] + (-d)^(1/4)*Sqrt[g])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(4*e) + ((I/2)*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[h])/(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sq rt[h*x])])/e - ((I/8)*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*Sqrt[h]*((-d)^(1/4 )*Sqrt[-h] - e^(1/4)*Sqrt[h*x]))/(((-d)^(1/4)*Sqrt[g]*Sqrt[-h] - I*e^(1/4) *Sqrt[f]*Sqrt[h])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/e - ((I/8)*Po lyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*((-d)^(1/4)*Sqrt[h] - e^(1/4)*Sqrt[h*x]...
3.7.17.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))^(p_.)]*(b_.))^(q_.)*((h_.) *(x_))^(m_)*((f_.) + (g_.)*(x_))^(r_.), x_Symbol] :> With[{k = Denominator[ m]}, Simp[k/h Subst[Int[x^(k*(m + 1) - 1)*(f + g*(x^k/h))^r*(a + b*Log[c* (d + e*(x^(k*n)/h^n))^p])^q, x], x, (h*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, g, h, p, r}, x] && FractionQ[m] && IntegerQ[n] && IntegerQ[r]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.) *(x_)^2), x_Symbol] :> With[{u = IntHide[1/(f + g*x^2), x]}, Simp[u*(a + b* Log[c*(d + e*x^n)^p]), x] - Simp[b*e*n*p Int[u*(x^(n - 1)/(d + e*x^n)), x ], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
\[\int \frac {a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{\sqrt {h x}\, \left (g x +f \right )}d x\]
\[ \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {h x} (f+g x)} \, dx=\int { \frac {b \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) + a}{{\left (g x + f\right )} \sqrt {h x}} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {h x} (f+g x)} \, dx=\text {Timed out} \]
\[ \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {h x} (f+g x)} \, dx=\int { \frac {b \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) + a}{{\left (g x + f\right )} \sqrt {h x}} \,d x } \]
b*integrate((sqrt(h)*log((e*x^2 + d)^p) + sqrt(h)*log(c))/(g*h*x^(3/2) + f *h*sqrt(x)), x) + 2*a*arctan(sqrt(h*x)*g/sqrt(f*g*h))/sqrt(f*g*h)
\[ \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {h x} (f+g x)} \, dx=\int { \frac {b \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) + a}{{\left (g x + f\right )} \sqrt {h x}} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {h x} (f+g x)} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{\left (f+g\,x\right )\,\sqrt {h\,x}} \,d x \]