Integrand size = 31, antiderivative size = 1680 \[ \int \frac {\sqrt {h x} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f+g x} \, dx =\text {Too large to display} \]
-2*b*d^(1/4)*p*arctan(1-e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/ 2)*h^(1/2)/e^(1/4)/g+2*b*d^(1/4)*p*arctan(1+e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^ (1/4)/h^(1/2))*2^(1/2)*h^(1/2)/e^(1/4)/g-b*d^(1/4)*p*ln(d^(1/2)*h^(1/2)+x* e^(1/2)*h^(1/2)-d^(1/4)*e^(1/4)*2^(1/2)*(h*x)^(1/2))*2^(1/2)*h^(1/2)/e^(1/ 4)/g+b*d^(1/4)*p*ln(d^(1/2)*h^(1/2)+x*e^(1/2)*h^(1/2)+d^(1/4)*e^(1/4)*2^(1 /2)*(h*x)^(1/2))*2^(1/2)*h^(1/2)/e^(1/4)/g-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)*h^(1/2)/g^(3/2)-8*b*p*arcta n(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)*h^(1/2)/g^(3/2)+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)*h^(1/2)/g^(3/2) +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)*h^(1/2)/g^(3/2)+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)*h^(1/2)/g^(3/2)+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^...
Time = 0.91 (sec) , antiderivative size = 1506, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {h x} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f+g x} \, dx =\text {Too large to display} \]
(Sqrt[h*x]*(2*a*Sqrt[g]*Sqrt[x] - 8*b*Sqrt[g]*p*Sqrt[x] - (2*Sqrt[2]*b*d^( 1/4)*Sqrt[g]*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)])/e^(1/4) + (2 *Sqrt[2]*b*d^(1/4)*Sqrt[g]*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)] )/e^(1/4) - (Sqrt[2]*b*d^(1/4)*Sqrt[g]*p*Log[Sqrt[d] - Sqrt[2]*d^(1/4)*e^( 1/4)*Sqrt[x] + Sqrt[e]*x])/e^(1/4) + (Sqrt[2]*b*d^(1/4)*Sqrt[g]*p*Log[Sqrt [d] + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x])/e^(1/4) + 2*b*Sqrt[g]* Sqrt[x]*Log[c*(d + e*x^2)^p] + Sqrt[-f]*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]]*(a + b*Log[c*(d + e*x^2)^p]) - Sqrt[-f]*Log[Sqrt[-f] + Sqrt[g]*Sqrt[x]]*(a + b*Log[c*(d + e*x^2)^p]) - b*Sqrt[-f]*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] - Sqr t[g]*Sqrt[x]] + 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]] + Log[(Sqr t[g]*(I*(-d)^(1/4) + e^(1/4)*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + I*(-d)^(1/4)*Sq rt[g])]*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]] + 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]] + PolyLog[2, (e^(1/4)*(Sqrt[-f] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*S qrt[-f] - (-d)^(1/4)*Sqrt[g])] + PolyLog[2, (e^(1/4)*(Sqrt[-f] - Sqrt[g]*S qrt[x]))/(e^(1/4)*Sqrt[-f] - I*(-d)^(1/4)*Sqrt[g])] + PolyLog[2, (e^(1/4)* (Sqrt[-f] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + I*(-d)^(1/4)*Sqrt[g])] + PolyLog[2, (e^(1/4)*(Sqrt[-f] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + ...
Time = 2.52 (sec) , antiderivative size = 1677, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2917, 27, 2926, 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.
\(\displaystyle \int \frac {\sqrt {h x} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f+g x} \, dx\) |
\(\Big \downarrow \) 2917 |
\(\displaystyle \frac {2 \int \frac {h^2 x \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{f h+g x h}d\sqrt {h x}}{h}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {h x \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{f h+g x h}d\sqrt {h x}\) |
\(\Big \downarrow \) 2926 |
\(\displaystyle 2 \int \left (\frac {a+b \log \left (c \left (e x^2+d\right )^p\right )}{g}-\frac {f h \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{g (f h+g x h)}\right )d\sqrt {h x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {\sqrt {h x} a}{g}-\frac {\sqrt {2} b \sqrt [4]{d} \sqrt {h} p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} g}+\frac {\sqrt {2} b \sqrt [4]{d} \sqrt {h} p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{\sqrt [4]{e} g}+\frac {b \sqrt {h x} \log \left (c \left (e x^2+d\right )^p\right )}{g}-\frac {\sqrt {f} \sqrt {h} \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 )}{g^{3/2}}-\frac {4 b \sqrt {f} \sqrt {h} p \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 )}{g^{3/2}}+\frac {b \sqrt {f} \sqrt {h} p \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 )}{g^{3/2}}+\frac {b \sqrt {f} \sqrt {h} p \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 )}{g^{3/2}}+\frac {b \sqrt {f} \sqrt {h} p \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 )}{g^{3/2}}+\frac {b \sqrt {f} \sqrt {h} p \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 )}{g^{3/2}}-\frac {b \sqrt [4]{d} \sqrt {h} p \log \left (\sqrt {e} x h+\sqrt {d} h-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x} \sqrt {h}\right )}{\sqrt {2} \sqrt [4]{e} g}+\frac {b \sqrt [4]{d} \sqrt {h} p \log \left (\sqrt {e} x h+\sqrt {d} h+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x} \sqrt {h}\right )}{\sqrt {2} \sqrt [4]{e} g}+\frac {2 i b \sqrt {f} \sqrt {h} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} p \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 )}{2 g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} p \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 )}{2 g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} p \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 )}{2 g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} p \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 )}{2 g^{3/2}}-\frac {4 b p \sqrt {h x}}{g}\right )\) |
2*((a*Sqrt[h*x])/g - (4*b*p*Sqrt[h*x])/g - (Sqrt[2]*b*d^(1/4)*Sqrt[h]*p*Ar cTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(e^(1/4)*g) + (Sq rt[2]*b*d^(1/4)*Sqrt[h]*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)* Sqrt[h])])/(e^(1/4)*g) + (b*Sqrt[h*x]*Log[c*(d + e*x^2)^p])/g - (Sqrt[f]*S qrt[h]*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*(a + b*Log[c*(d + e*x ^2)^p]))/g^(3/2) - (4*b*Sqrt[f]*Sqrt[h]*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt [f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqrt[h])/(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h* x])])/g^(3/2) + (b*Sqrt[f]*Sqrt[h]*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*S qrt[h])]*Log[(2*Sqrt[f]*Sqrt[g]*Sqrt[h]*((-d)^(1/4)*Sqrt[-h] - e^(1/4)*Sqr t[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]))])/g^(3/2) + (b*Sqrt[f]*Sqrt[h]*p*ArcTan [(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(-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]))])/g^(3/2) + (b*Sqrt[f]*Sqrt[h]*p* 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]))])/ g^(3/2) + (b*Sqrt[f]*Sqrt[h]*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h] )]*Log[(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...
3.7.16.3.1 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_.))^(q_.)*(x_)^(m _.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Int[ExpandIntegrand[(a + b *Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c, d, e , f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] & & IntegerQ[s]
\[\int \frac {\sqrt {h x}\, \left (a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )\right )}{g x +f}d x\]
\[ \int \frac {\sqrt {h x} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f+g x} \, dx=\int { \frac {\sqrt {h x} {\left (b \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) + a\right )}}{g x + f} \,d x } \]
Timed out. \[ \int \frac {\sqrt {h x} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f+g x} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {h x} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f+g x} \, dx=\int { \frac {\sqrt {h x} {\left (b \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) + a\right )}}{g x + f} \,d x } \]
b*integrate((sqrt(h)*sqrt(x)*log((e*x^2 + d)^p) + sqrt(h)*sqrt(x)*log(c))/ (g*x + f), x) - 2*(f*h^2*arctan(sqrt(h*x)*g/sqrt(f*g*h))/(sqrt(f*g*h)*g) - sqrt(h*x)*h/g)*a/h
\[ \int \frac {\sqrt {h x} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f+g x} \, dx=\int { \frac {\sqrt {h x} {\left (b \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) + a\right )}}{g x + f} \,d x } \]
Timed out. \[ \int \frac {\sqrt {h x} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f+g x} \, dx=\int \frac {\sqrt {h\,x}\,\left (a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\right )}{f+g\,x} \,d x \]