Integrand size = 31, antiderivative size = 1659 \[ \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{(h x)^{3/2} (f+g x)} \, dx =\text {Too large to display} \]
-2*b*e^(1/4)*p*arctan(1-e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/ 2)/d^(1/4)/f/h^(3/2)+2*b*e^(1/4)*p*arctan(1+e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^ (1/4)/h^(1/2))*2^(1/2)/d^(1/4)/f/h^(3/2)+b*e^(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)/d^(1/4)/f/h^( 3/2)-b*e^(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)/d^(1/4)/f/h^(3/2)-2*arctan(g^(1/2)*(h*x)^(1/2)/f^ (1/2)/h^(1/2))*(a+b*ln(c*(e*x^2+d)^p))*g^(1/2)/f^(3/2)/h^(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)))*g^(1/2)/f^(3/2)/h^(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)))*g^(1/2)/f^(3/2)/h^(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)))*g^(1/2)/f^(3/2)/h^(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)))*g^(1 /2)/f^(3/2)/h^(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 = 1.06 (sec) , antiderivative size = 1336, normalized size of antiderivative = 0.81 \[ \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{(h x)^{3/2} (f+g x)} \, dx =\text {Too large to display} \]
(x^(3/2)*((4*b*e^(1/4)*p*(ArcTan[(e^(1/4)*Sqrt[x])/(-d)^(1/4)] + ArcTanh[( d*e^(1/4)*Sqrt[x])/(-d)^(5/4)]))/(-d)^(1/4) - (2*(a + b*Log[c*(d + e*x^2)^ p]))/Sqrt[x] + (f*Sqrt[g]*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]]*(a + b*Log[c*(d + e*x^2)^p]))/(-f)^(3/2) + (Sqrt[g]*Log[Sqrt[-f] + Sqrt[g]*Sqrt[x]]*(a + b *Log[c*(d + e*x^2)^p]))/Sqrt[-f] + (b*Sqrt[g]*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]] + 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[(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]] + 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)*Sqrt[-f] - (-d)^(1/4)*Sqrt[g])] + PolyLog[2, (e^(1/4)*(Sqrt[-f] - S qrt[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] + I*(-d)^(1/4)*Sqr t[g])] + PolyLog[2, (e^(1/4)*(Sqrt[-f] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[- f] + (-d)^(1/4)*Sqrt[g])]))/Sqrt[-f] + (b*f*Sqrt[g]*p*(Log[(Sqrt[g]*((-d)^ (1/4) - e^(1/4)*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + (-d)^(1/4)*Sqrt[g])]*Log[Sqr t[-f] + Sqrt[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[...
Time = 2.26 (sec) , antiderivative size = 1658, 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.
\(\displaystyle \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{(h x)^{3/2} (f+g x)} \, dx\) |
\(\Big \downarrow \) 2917 |
\(\displaystyle \frac {2 \int \frac {a+b \log \left (c \left (e x^2+d\right )^p\right )}{x (f h+g x h)}d\sqrt {h x}}{h}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {a+b \log \left (c \left (e x^2+d\right )^p\right )}{h x (f h+g x h)}d\sqrt {h x}\) |
\(\Big \downarrow \) 2926 |
\(\displaystyle 2 \int \left (\frac {a+b \log \left (c \left (e x^2+d\right )^p\right )}{f h^2 x}-\frac {g \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{f h (f h+g x h)}\right )d\sqrt {h x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {\sqrt {2} b \sqrt [4]{e} p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} f h^{3/2}}+\frac {\sqrt {2} b \sqrt [4]{e} p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{\sqrt [4]{d} f h^{3/2}}-\frac {\sqrt {g} \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 )}{f^{3/2} h^{3/2}}-\frac {a+b \log \left (c \left (e x^2+d\right )^p\right )}{f h \sqrt {h x}}-\frac {4 b \sqrt {g} 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 )}{f^{3/2} h^{3/2}}+\frac {b \sqrt {g} 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 )}{f^{3/2} h^{3/2}}+\frac {b \sqrt {g} 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 )}{f^{3/2} h^{3/2}}+\frac {b \sqrt {g} 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 )}{f^{3/2} h^{3/2}}+\frac {b \sqrt {g} 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 )}{f^{3/2} h^{3/2}}+\frac {b \sqrt [4]{e} 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]{d} f h^{3/2}}-\frac {b \sqrt [4]{e} 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]{d} f h^{3/2}}+\frac {2 i b \sqrt {g} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{f^{3/2} h^{3/2}}-\frac {i b \sqrt {g} 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 f^{3/2} h^{3/2}}-\frac {i b \sqrt {g} 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 f^{3/2} h^{3/2}}-\frac {i b \sqrt {g} 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 f^{3/2} h^{3/2}}-\frac {i b \sqrt {g} 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 f^{3/2} h^{3/2}}\right )\) |
2*(-((Sqrt[2]*b*e^(1/4)*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)* Sqrt[h])])/(d^(1/4)*f*h^(3/2))) + (Sqrt[2]*b*e^(1/4)*p*ArcTan[1 + (Sqrt[2] *e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(d^(1/4)*f*h^(3/2)) - (a + b*Log[c *(d + e*x^2)^p])/(f*h*Sqrt[h*x]) - (Sqrt[g]*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sq rt[f]*Sqrt[h])]*(a + b*Log[c*(d + e*x^2)^p]))/(f^(3/2)*h^(3/2)) - (4*b*Sqr t[g]*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])])/(f^(3/2)*h^(3/2)) + (b*Sqrt[g ]*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqrt[g]*S qrt[h]*((-d)^(1/4)*Sqrt[-h] - e^(1/4)*Sqrt[h*x]))/(((-d)^(1/4)*Sqrt[g]*Sqr t[-h] - I*e^(1/4)*Sqrt[f]*Sqrt[h])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]) )])/(f^(3/2)*h^(3/2)) + (b*Sqrt[g]*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*S qrt[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]*Sq rt[h*x]))])/(f^(3/2)*h^(3/2)) + (b*Sqrt[g]*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(S qrt[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]))])/(f^(3/2)*h^(3/2)) + (b*Sqrt[g ]*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]))])/(f^(3/2)*h^(3/2)) +...
3.7.18.3.1 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]
\[\int \frac {a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{\left (h x \right )^{\frac {3}{2}} \left (g x +f \right )}d x\]
\[ \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{(h x)^{3/2} (f+g x)} \, dx=\int { \frac {b \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) + a}{{\left (g x + f\right )} \left (h x\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{(h x)^{3/2} (f+g x)} \, dx=\text {Timed out} \]
\[ \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{(h x)^{3/2} (f+g x)} \, dx=\int { \frac {b \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) + a}{{\left (g x + f\right )} \left (h x\right )^{\frac {3}{2}}} \,d x } \]
b*integrate((sqrt(h)*log((e*x^2 + d)^p) + sqrt(h)*log(c))/(g*h^2*x^(5/2) + f*h^2*x^(3/2)), x) - 2*a*(g*arctan(sqrt(h*x)*g/sqrt(f*g*h))/(sqrt(f*g*h)* f) + 1/(sqrt(h*x)*f))/h
\[ \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{(h x)^{3/2} (f+g x)} \, dx=\int { \frac {b \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) + a}{{\left (g x + f\right )} \left (h x\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{(h x)^{3/2} (f+g x)} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{\left (f+g\,x\right )\,{\left (h\,x\right )}^{3/2}} \,d x \]