Integrand size = 29, antiderivative size = 603 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=\frac {2 a g \sqrt {h x}}{h^2}-\frac {8 b g p \sqrt {h x}}{h^2}-\frac {2 \sqrt {2} b \sqrt [4]{e} f p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}-\frac {2 \sqrt {2} b \sqrt [4]{d} g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} f p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{d} g p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} h^{3/2}}+\frac {2 b g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}-\frac {2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {\sqrt {2} b \sqrt [4]{e} f p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{d} g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{e} f p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {\sqrt {2} b \sqrt [4]{d} g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} h^{3/2}} \]
-2*b*e^(1/4)*f*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)/h^(3/2)-2*b*d^(1/4)*g*p*arctan(1-e^(1/4)*2^(1/2)*(h*x)^(1/2)/ d^(1/4)/h^(1/2))*2^(1/2)/e^(1/4)/h^(3/2)+2*b*e^(1/4)*f*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)/h^(3/2)+2*b*d^(1/4)*g *p*arctan(1+e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/2)/e^(1/4)/h ^(3/2)+b*e^(1/4)*f*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)/h^(3/2)-b*d^(1/4)*g*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)/e^(1/4) /h^(3/2)-b*e^(1/4)*f*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)/h^(3/2)+b*d^(1/4)*g*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)/e^(1/ 4)/h^(3/2)-2*f*(a+b*ln(c*(e*x^2+d)^p))/h/(h*x)^(1/2)+2*a*g*(h*x)^(1/2)/h^2 -8*b*g*p*(h*x)^(1/2)/h^2+2*b*g*ln(c*(e*x^2+d)^p)*(h*x)^(1/2)/h^2
Time = 0.32 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.55 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=\frac {2 x^{3/2} \left (a g \sqrt {x}-4 b g p \sqrt {x}-\frac {\sqrt {2} b \sqrt [4]{d} g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )}{\sqrt [4]{e}}+\frac {\sqrt {2} b \sqrt [4]{d} g p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )}{\sqrt [4]{e}}+\frac {2 b \sqrt [4]{e} f p \left (\arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )+\text {arctanh}\left (\frac {d \sqrt [4]{e} \sqrt {x}}{(-d)^{5/4}}\right )\right )}{\sqrt [4]{-d}}-\frac {b \sqrt [4]{d} g p \log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )}{\sqrt {2} \sqrt [4]{e}}+\frac {b \sqrt [4]{d} g p \log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )}{\sqrt {2} \sqrt [4]{e}}+b g \sqrt {x} \log \left (c \left (d+e x^2\right )^p\right )-\frac {f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {x}}\right )}{(h x)^{3/2}} \]
(2*x^(3/2)*(a*g*Sqrt[x] - 4*b*g*p*Sqrt[x] - (Sqrt[2]*b*d^(1/4)*g*p*ArcTan[ 1 - (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)])/e^(1/4) + (Sqrt[2]*b*d^(1/4)*g*p*A rcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)])/e^(1/4) + (2*b*e^(1/4)*f*p*( ArcTan[(e^(1/4)*Sqrt[x])/(-d)^(1/4)] + ArcTanh[(d*e^(1/4)*Sqrt[x])/(-d)^(5 /4)]))/(-d)^(1/4) - (b*d^(1/4)*g*p*Log[Sqrt[d] - Sqrt[2]*d^(1/4)*e^(1/4)*S qrt[x] + Sqrt[e]*x])/(Sqrt[2]*e^(1/4)) + (b*d^(1/4)*g*p*Log[Sqrt[d] + Sqrt [2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x])/(Sqrt[2]*e^(1/4)) + b*g*Sqrt[x]* Log[c*(d + e*x^2)^p] - (f*(a + b*Log[c*(d + e*x^2)^p]))/Sqrt[x]))/(h*x)^(3 /2)
Time = 0.85 (sec) , antiderivative size = 581, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, 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.
| \(\displaystyle \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2917 |
| \(\displaystyle \frac {2 \int \frac {(f h+g x h) \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{h^2 x}d\sqrt {h x}}{h}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \int \frac {(f h+g x h) \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{h x}d\sqrt {h x}}{h^2}\) |
| \(\Big \downarrow \) 2926 |
| \(\displaystyle \frac {2 \int \left (g \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )+\frac {f \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{x}\right )d\sqrt {h x}}{h^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (-\frac {f h \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}}+a g \sqrt {h x}-\frac {\sqrt {2} b \sqrt [4]{e} f \sqrt {h} p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d}}+\frac {\sqrt {2} b \sqrt [4]{e} f \sqrt {h} p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{\sqrt [4]{d}}-\frac {\sqrt {2} b \sqrt [4]{d} g \sqrt {h} p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e}}+\frac {\sqrt {2} b \sqrt [4]{d} g \sqrt {h} p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{\sqrt [4]{e}}+b g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )+\frac {b \sqrt [4]{e} f \sqrt {h} p \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h} \sqrt {h x}+\sqrt {d} h+\sqrt {e} h x\right )}{\sqrt {2} \sqrt [4]{d}}-\frac {b \sqrt [4]{e} f \sqrt {h} p \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h} \sqrt {h x}+\sqrt {d} h+\sqrt {e} h x\right )}{\sqrt {2} \sqrt [4]{d}}-\frac {b \sqrt [4]{d} g \sqrt {h} p \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h} \sqrt {h x}+\sqrt {d} h+\sqrt {e} h x\right )}{\sqrt {2} \sqrt [4]{e}}+\frac {b \sqrt [4]{d} g \sqrt {h} p \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h} \sqrt {h x}+\sqrt {d} h+\sqrt {e} h x\right )}{\sqrt {2} \sqrt [4]{e}}-4 b g p \sqrt {h x}\right )}{h^2}\) |
(2*(a*g*Sqrt[h*x] - 4*b*g*p*Sqrt[h*x] - (Sqrt[2]*b*e^(1/4)*f*Sqrt[h]*p*Arc Tan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/d^(1/4) - (Sqrt[2] *b*d^(1/4)*g*Sqrt[h]*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqr t[h])])/e^(1/4) + (Sqrt[2]*b*e^(1/4)*f*Sqrt[h]*p*ArcTan[1 + (Sqrt[2]*e^(1/ 4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/d^(1/4) + (Sqrt[2]*b*d^(1/4)*g*Sqrt[h]*p *ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/e^(1/4) + b*g* Sqrt[h*x]*Log[c*(d + e*x^2)^p] - (f*h*(a + b*Log[c*(d + e*x^2)^p]))/Sqrt[h *x] + (b*e^(1/4)*f*Sqrt[h]*p*Log[Sqrt[d]*h + Sqrt[e]*h*x - Sqrt[2]*d^(1/4) *e^(1/4)*Sqrt[h]*Sqrt[h*x]])/(Sqrt[2]*d^(1/4)) - (b*d^(1/4)*g*Sqrt[h]*p*Lo g[Sqrt[d]*h + Sqrt[e]*h*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h]*Sqrt[h*x]])/(S qrt[2]*e^(1/4)) - (b*e^(1/4)*f*Sqrt[h]*p*Log[Sqrt[d]*h + Sqrt[e]*h*x + Sqr t[2]*d^(1/4)*e^(1/4)*Sqrt[h]*Sqrt[h*x]])/(Sqrt[2]*d^(1/4)) + (b*d^(1/4)*g* Sqrt[h]*p*Log[Sqrt[d]*h + Sqrt[e]*h*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h]*Sq rt[h*x]])/(Sqrt[2]*e^(1/4))))/h^2
3.7.7.3.1 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]
\[\int \frac {\left (g x +f \right ) \left (a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )\right )}{\left (h x \right )^{\frac {3}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 1162 vs. \(2 (427) = 854\).
Time = 0.37 (sec) , antiderivative size = 1162, normalized size of antiderivative = 1.93 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=\frac {2 \, {\left (h^{2} x \sqrt {-\frac {2 \, b^{2} f g p^{2} + h^{3} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}}}{h^{3}}} \log \left (-32 \, {\left (b^{3} e^{2} f^{4} - b^{3} d^{2} g^{4}\right )} \sqrt {h x} p^{3} + 32 \, {\left (d e f h^{5} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}} - {\left (b^{2} d e f^{2} g - b^{2} d^{2} g^{3}\right )} h^{2} p^{2}\right )} \sqrt {-\frac {2 \, b^{2} f g p^{2} + h^{3} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}}}{h^{3}}}\right ) - h^{2} x \sqrt {-\frac {2 \, b^{2} f g p^{2} + h^{3} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}}}{h^{3}}} \log \left (-32 \, {\left (b^{3} e^{2} f^{4} - b^{3} d^{2} g^{4}\right )} \sqrt {h x} p^{3} - 32 \, {\left (d e f h^{5} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}} - {\left (b^{2} d e f^{2} g - b^{2} d^{2} g^{3}\right )} h^{2} p^{2}\right )} \sqrt {-\frac {2 \, b^{2} f g p^{2} + h^{3} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}}}{h^{3}}}\right ) - h^{2} x \sqrt {-\frac {2 \, b^{2} f g p^{2} - h^{3} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}}}{h^{3}}} \log \left (-32 \, {\left (b^{3} e^{2} f^{4} - b^{3} d^{2} g^{4}\right )} \sqrt {h x} p^{3} + 32 \, {\left (d e f h^{5} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}} + {\left (b^{2} d e f^{2} g - b^{2} d^{2} g^{3}\right )} h^{2} p^{2}\right )} \sqrt {-\frac {2 \, b^{2} f g p^{2} - h^{3} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}}}{h^{3}}}\right ) + h^{2} x \sqrt {-\frac {2 \, b^{2} f g p^{2} - h^{3} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}}}{h^{3}}} \log \left (-32 \, {\left (b^{3} e^{2} f^{4} - b^{3} d^{2} g^{4}\right )} \sqrt {h x} p^{3} - 32 \, {\left (d e f h^{5} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}} + {\left (b^{2} d e f^{2} g - b^{2} d^{2} g^{3}\right )} h^{2} p^{2}\right )} \sqrt {-\frac {2 \, b^{2} f g p^{2} - h^{3} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}}}{h^{3}}}\right ) - {\left (a f + {\left (4 \, b g p - a g\right )} x - {\left (b g p x - b f p\right )} \log \left (e x^{2} + d\right ) - {\left (b g x - b f\right )} \log \left (c\right )\right )} \sqrt {h x}\right )}}{h^{2} x} \]
2*(h^2*x*sqrt(-(2*b^2*f*g*p^2 + h^3*sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p^4/(d*e*h^6)))/h^3)*log(-32*(b^3*e^2*f^4 - b^3*d^2*g^4)*s qrt(h*x)*p^3 + 32*(d*e*f*h^5*sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^4* d^2*g^4)*p^4/(d*e*h^6)) - (b^2*d*e*f^2*g - b^2*d^2*g^3)*h^2*p^2)*sqrt(-(2* b^2*f*g*p^2 + h^3*sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p^ 4/(d*e*h^6)))/h^3)) - h^2*x*sqrt(-(2*b^2*f*g*p^2 + h^3*sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p^4/(d*e*h^6)))/h^3)*log(-32*(b^3*e^2*f ^4 - b^3*d^2*g^4)*sqrt(h*x)*p^3 - 32*(d*e*f*h^5*sqrt(-(b^4*e^2*f^4 - 2*b^4 *d*e*f^2*g^2 + b^4*d^2*g^4)*p^4/(d*e*h^6)) - (b^2*d*e*f^2*g - b^2*d^2*g^3) *h^2*p^2)*sqrt(-(2*b^2*f*g*p^2 + h^3*sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^ 2 + b^4*d^2*g^4)*p^4/(d*e*h^6)))/h^3)) - h^2*x*sqrt(-(2*b^2*f*g*p^2 - h^3* sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p^4/(d*e*h^6)))/h^3) *log(-32*(b^3*e^2*f^4 - b^3*d^2*g^4)*sqrt(h*x)*p^3 + 32*(d*e*f*h^5*sqrt(-( b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p^4/(d*e*h^6)) + (b^2*d*e*f ^2*g - b^2*d^2*g^3)*h^2*p^2)*sqrt(-(2*b^2*f*g*p^2 - h^3*sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p^4/(d*e*h^6)))/h^3)) + h^2*x*sqrt(-(2 *b^2*f*g*p^2 - h^3*sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p ^4/(d*e*h^6)))/h^3)*log(-32*(b^3*e^2*f^4 - b^3*d^2*g^4)*sqrt(h*x)*p^3 - 32 *(d*e*f*h^5*sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p^4/(d*e *h^6)) + (b^2*d*e*f^2*g - b^2*d^2*g^3)*h^2*p^2)*sqrt(-(2*b^2*f*g*p^2 - ...
Exception generated. \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Time = 0.29 (sec) , antiderivative size = 731, normalized size of antiderivative = 1.21 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=-\frac {b e f p {\left (\frac {\sqrt {2} \log \left (\sqrt {e} h x + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\frac {1}{4}} + \sqrt {d} h\right )}{\left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {3}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {e} h x - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\frac {1}{4}} + \sqrt {d} h\right )}{\left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {3}{4}}} - \frac {\sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} - 2 \, \sqrt {h x} \sqrt {e}}{\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} + 2 \, \sqrt {h x} \sqrt {e}}\right )}{\sqrt {-\sqrt {d} \sqrt {e} h} \sqrt {e}} - \frac {\sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} - 2 \, \sqrt {h x} \sqrt {e}}{\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} + 2 \, \sqrt {h x} \sqrt {e}}\right )}{\sqrt {-\sqrt {d} \sqrt {e} h} \sqrt {e}}\right )}}{h} + \frac {2 \, b g x^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{\left (h x\right )^{\frac {3}{2}}} + \frac {2 \, a g x^{2}}{\left (h x\right )^{\frac {3}{2}}} - \frac {2 \, b f \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{\sqrt {h x} h} - \frac {{\left (\frac {8 \, \sqrt {h x} h^{2}}{e} - \frac {{\left (\frac {\sqrt {2} h^{4} \log \left (\sqrt {e} h x + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\frac {1}{4}} + \sqrt {d} h\right )}{\left (d h^{2}\right )^{\frac {3}{4}} e^{\frac {1}{4}}} - \frac {\sqrt {2} h^{4} \log \left (\sqrt {e} h x - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\frac {1}{4}} + \sqrt {d} h\right )}{\left (d h^{2}\right )^{\frac {3}{4}} e^{\frac {1}{4}}} + \frac {\sqrt {2} h^{3} \log \left (-\frac {\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} - 2 \, \sqrt {h x} \sqrt {e}}{\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} + 2 \, \sqrt {h x} \sqrt {e}}\right )}{\sqrt {-\sqrt {d} \sqrt {e} h} \sqrt {d}} + \frac {\sqrt {2} h^{3} \log \left (-\frac {\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} - 2 \, \sqrt {h x} \sqrt {e}}{\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} + 2 \, \sqrt {h x} \sqrt {e}}\right )}{\sqrt {-\sqrt {d} \sqrt {e} h} \sqrt {d}}\right )} d}{e}\right )} b e g p}{h^{4}} - \frac {2 \, a f}{\sqrt {h x} h} \]
-b*e*f*p*(sqrt(2)*log(sqrt(e)*h*x + sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4 ) + sqrt(d)*h)/((d*h^2)^(1/4)*e^(3/4)) - sqrt(2)*log(sqrt(e)*h*x - sqrt(2) *(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/((d*h^2)^(1/4)*e^(3/4)) - sq rt(2)*log(-(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) + sqrt(2)*(d*h^2)^(1/4)*e^(1/ 4) - 2*sqrt(h*x)*sqrt(e))/(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) - sqrt(2)*(d*h ^2)^(1/4)*e^(1/4) + 2*sqrt(h*x)*sqrt(e)))/(sqrt(-sqrt(d)*sqrt(e)*h)*sqrt(e )) - sqrt(2)*log(-(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) - sqrt(2)*(d*h^2)^(1/4 )*e^(1/4) - 2*sqrt(h*x)*sqrt(e))/(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) + sqrt( 2)*(d*h^2)^(1/4)*e^(1/4) + 2*sqrt(h*x)*sqrt(e)))/(sqrt(-sqrt(d)*sqrt(e)*h) *sqrt(e)))/h + 2*b*g*x^2*log((e*x^2 + d)^p*c)/(h*x)^(3/2) + 2*a*g*x^2/(h*x )^(3/2) - 2*b*f*log((e*x^2 + d)^p*c)/(sqrt(h*x)*h) - (8*sqrt(h*x)*h^2/e - (sqrt(2)*h^4*log(sqrt(e)*h*x + sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + s qrt(d)*h)/((d*h^2)^(3/4)*e^(1/4)) - sqrt(2)*h^4*log(sqrt(e)*h*x - sqrt(2)* (d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/((d*h^2)^(3/4)*e^(1/4)) + sqr t(2)*h^3*log(-(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) + sqrt(2)*(d*h^2)^(1/4)*e^ (1/4) - 2*sqrt(h*x)*sqrt(e))/(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) - sqrt(2)*( d*h^2)^(1/4)*e^(1/4) + 2*sqrt(h*x)*sqrt(e)))/(sqrt(-sqrt(d)*sqrt(e)*h)*sqr t(d)) + sqrt(2)*h^3*log(-(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) - sqrt(2)*(d*h^ 2)^(1/4)*e^(1/4) - 2*sqrt(h*x)*sqrt(e))/(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) + sqrt(2)*(d*h^2)^(1/4)*e^(1/4) + 2*sqrt(h*x)*sqrt(e)))/(sqrt(-sqrt(d)*...
Time = 0.38 (sec) , antiderivative size = 448, normalized size of antiderivative = 0.74 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=-\frac {2 \, {\left (\frac {b f p}{\sqrt {h x}} - \frac {\sqrt {h x} b g p}{h}\right )} \log \left (e h^{2} x^{2} + d h^{2}\right ) - \frac {2 \, {\left (b f p \log \left (h^{2}\right ) - b f \log \left (c\right ) - a f\right )}}{\sqrt {h x}} + \frac {2 \, {\left (b g p \log \left (h^{2}\right ) + 4 \, b g p - b g \log \left (c\right ) - a g\right )} \sqrt {h x}}{h} - \frac {2 \, {\left (\sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e g h p + \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b f p\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} + 2 \, \sqrt {h x}\right )}}{2 \, \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}}}\right )}{d e^{2} h^{2}} - \frac {2 \, {\left (\sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e g h p + \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b f p\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} - 2 \, \sqrt {h x}\right )}}{2 \, \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}}}\right )}{d e^{2} h^{2}} - \frac {{\left (\sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e g h p - \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b f p\right )} \log \left (h x + \sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} \sqrt {h x} + \sqrt {\frac {d h^{2}}{e}}\right )}{d e^{2} h^{2}} + \frac {{\left (\sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e g h p - \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b f p\right )} \log \left (h x - \sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} \sqrt {h x} + \sqrt {\frac {d h^{2}}{e}}\right )}{d e^{2} h^{2}}}{h} \]
-(2*(b*f*p/sqrt(h*x) - sqrt(h*x)*b*g*p/h)*log(e*h^2*x^2 + d*h^2) - 2*(b*f* p*log(h^2) - b*f*log(c) - a*f)/sqrt(h*x) + 2*(b*g*p*log(h^2) + 4*b*g*p - b *g*log(c) - a*g)*sqrt(h*x)/h - 2*(sqrt(2)*(d*e^3*h^2)^(1/4)*b*d*e*g*h*p + sqrt(2)*(d*e^3*h^2)^(3/4)*b*f*p)*arctan(1/2*sqrt(2)*(sqrt(2)*(d*h^2/e)^(1/ 4) + 2*sqrt(h*x))/(d*h^2/e)^(1/4))/(d*e^2*h^2) - 2*(sqrt(2)*(d*e^3*h^2)^(1 /4)*b*d*e*g*h*p + sqrt(2)*(d*e^3*h^2)^(3/4)*b*f*p)*arctan(-1/2*sqrt(2)*(sq rt(2)*(d*h^2/e)^(1/4) - 2*sqrt(h*x))/(d*h^2/e)^(1/4))/(d*e^2*h^2) - (sqrt( 2)*(d*e^3*h^2)^(1/4)*b*d*e*g*h*p - sqrt(2)*(d*e^3*h^2)^(3/4)*b*f*p)*log(h* x + sqrt(2)*(d*h^2/e)^(1/4)*sqrt(h*x) + sqrt(d*h^2/e))/(d*e^2*h^2) + (sqrt (2)*(d*e^3*h^2)^(1/4)*b*d*e*g*h*p - sqrt(2)*(d*e^3*h^2)^(3/4)*b*f*p)*log(h *x - sqrt(2)*(d*h^2/e)^(1/4)*sqrt(h*x) + sqrt(d*h^2/e))/(d*e^2*h^2))/h
Timed out. \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=\int \frac {\left (f+g\,x\right )\,\left (a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\right )}{{\left (h\,x\right )}^{3/2}} \,d x \]