Integrand size = 31, antiderivative size = 968 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=-\frac {8 b e f^2 p}{21 d h^3 (h x)^{3/2}}-\frac {16 b e f g p}{5 d h^4 \sqrt {h x}}+\frac {2 \sqrt {2} b e^{7/4} f^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{7 d^{7/4} h^{9/2}}+\frac {4 \sqrt {2} b e^{5/4} f g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{9/2}}-\frac {2 \sqrt {2} b e^{3/4} g^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{9/2}}-\frac {2 \sqrt {2} b e^{7/4} f^2 p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{7 d^{7/4} h^{9/2}}-\frac {4 \sqrt {2} b e^{5/4} f g p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{9/2}}+\frac {2 \sqrt {2} b e^{3/4} g^2 p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{9/2}}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac {4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}}-\frac {2 g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3 (h x)^{3/2}}+\frac {\sqrt {2} b e^{7/4} f^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{7 d^{7/4} h^{9/2}}-\frac {2 \sqrt {2} b e^{5/4} f 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 )}{5 d^{5/4} h^{9/2}}-\frac {\sqrt {2} b e^{3/4} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 d^{3/4} h^{9/2}}-\frac {\sqrt {2} b e^{7/4} f^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{7 d^{7/4} h^{9/2}}+\frac {2 \sqrt {2} b e^{5/4} f 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 )}{5 d^{5/4} h^{9/2}}+\frac {\sqrt {2} b e^{3/4} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 d^{3/4} h^{9/2}} \]
-8/21*b*e*f^2*p/d/h^3/(h*x)^(3/2)-2/7*f^2*(a+b*ln(c*(e*x^2+d)^p))/h/(h*x)^ (7/2)-4/5*f*g*(a+b*ln(c*(e*x^2+d)^p))/h^2/(h*x)^(5/2)-2/3*g^2*(a+b*ln(c*(e *x^2+d)^p))/h^3/(h*x)^(3/2)+2/7*b*e^(7/4)*f^2*p*arctan(1-e^(1/4)*2^(1/2)*( h*x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/2)/d^(7/4)/h^(9/2)+4/5*b*e^(5/4)*f*g*p*ar ctan(1-e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/2)/d^(5/4)/h^(9/2 )-2/3*b*e^(3/4)*g^2*p*arctan(1-e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2) )*2^(1/2)/d^(3/4)/h^(9/2)-2/7*b*e^(7/4)*f^2*p*arctan(1+e^(1/4)*2^(1/2)*(h* x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/2)/d^(7/4)/h^(9/2)-4/5*b*e^(5/4)*f*g*p*arct an(1+e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/2)/d^(5/4)/h^(9/2)+ 2/3*b*e^(3/4)*g^2*p*arctan(1+e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))* 2^(1/2)/d^(3/4)/h^(9/2)+1/7*b*e^(7/4)*f^2*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^(7/4)/h^(9/2)-2/5*b* e^(5/4)*f*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)/d^(5/4)/h^(9/2)-1/3*b*e^(3/4)*g^2*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^(3/4)/ h^(9/2)-1/7*b*e^(7/4)*f^2*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^(7/4)/h^(9/2)+2/5*b*e^(5/4)*f*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)/d^(5/4)/h^(9/2)+1/3*b*e^(3/4)*g^2*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^(3/4)/h^(9/2)-16/5*...
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.18 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.30 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=\frac {x \left (-40 b e f^2 p x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},-\frac {e x^2}{d}\right )-336 b e f g p x^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\frac {e x^2}{d}\right )-35 \sqrt {2} b \sqrt [4]{d} e^{3/4} g^2 p x^{7/2} \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )+\log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )-\log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )\right )-30 d f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )-84 d f g x \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )-70 d g^2 x^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )\right )}{105 d (h x)^{9/2}} \]
(x*(-40*b*e*f^2*p*x^2*Hypergeometric2F1[-3/4, 1, 1/4, -((e*x^2)/d)] - 336* b*e*f*g*p*x^3*Hypergeometric2F1[-1/4, 1, 3/4, -((e*x^2)/d)] - 35*Sqrt[2]*b *d^(1/4)*e^(3/4)*g^2*p*x^(7/2)*(2*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[x])/d^( 1/4)] - 2*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)] + Log[Sqrt[d] - Sq rt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x] - Log[Sqrt[d] + Sqrt[2]*d^(1/4) *e^(1/4)*Sqrt[x] + Sqrt[e]*x]) - 30*d*f^2*(a + b*Log[c*(d + e*x^2)^p]) - 8 4*d*f*g*x*(a + b*Log[c*(d + e*x^2)^p]) - 70*d*g^2*x^2*(a + b*Log[c*(d + e* x^2)^p])))/(105*d*(h*x)^(9/2))
Time = 1.15 (sec) , antiderivative size = 947, normalized size of antiderivative = 0.98, 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.
| \(\displaystyle \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2917 |
| \(\displaystyle \frac {2 \int \frac {(f h+g x h)^2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{h^6 x^4}d\sqrt {h x}}{h}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \int \frac {(f h+g x h)^2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{h^4 x^4}d\sqrt {h x}}{h^3}\) |
| \(\Big \downarrow \) 2926 |
| \(\displaystyle \frac {2 \int \left (\frac {\left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f^2}{h^2 x^4}+\frac {2 g \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f}{h^2 x^3}+\frac {g^2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{h^2 x^2}\right )d\sqrt {h x}}{h^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (\frac {\sqrt {2} b e^{7/4} p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right ) f^2}{7 d^{7/4} h^{3/2}}-\frac {\sqrt {2} b e^{7/4} p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right ) f^2}{7 d^{7/4} h^{3/2}}-\frac {h^2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f^2}{7 (h x)^{7/2}}+\frac {b e^{7/4} p \log \left (\sqrt {e} x h+\sqrt {d} h-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x} \sqrt {h}\right ) f^2}{7 \sqrt {2} d^{7/4} h^{3/2}}-\frac {b e^{7/4} p \log \left (\sqrt {e} x h+\sqrt {d} h+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x} \sqrt {h}\right ) f^2}{7 \sqrt {2} d^{7/4} h^{3/2}}-\frac {4 b e p f^2}{21 d (h x)^{3/2}}+\frac {2 \sqrt {2} b e^{5/4} g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right ) f}{5 d^{5/4} h^{3/2}}-\frac {2 \sqrt {2} b e^{5/4} g p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right ) f}{5 d^{5/4} h^{3/2}}-\frac {2 g h \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f}{5 (h x)^{5/2}}-\frac {\sqrt {2} b e^{5/4} g p \log \left (\sqrt {e} x h+\sqrt {d} h-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x} \sqrt {h}\right ) f}{5 d^{5/4} h^{3/2}}+\frac {\sqrt {2} b e^{5/4} g p \log \left (\sqrt {e} x h+\sqrt {d} h+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x} \sqrt {h}\right ) f}{5 d^{5/4} h^{3/2}}-\frac {8 b e g p f}{5 d h \sqrt {h x}}-\frac {\sqrt {2} b e^{3/4} g^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{3/2}}+\frac {\sqrt {2} b e^{3/4} g^2 p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{3 d^{3/4} h^{3/2}}-\frac {g^2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{3 (h x)^{3/2}}-\frac {b e^{3/4} g^2 p \log \left (\sqrt {e} x h+\sqrt {d} h-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x} \sqrt {h}\right )}{3 \sqrt {2} d^{3/4} h^{3/2}}+\frac {b e^{3/4} g^2 p \log \left (\sqrt {e} x h+\sqrt {d} h+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x} \sqrt {h}\right )}{3 \sqrt {2} d^{3/4} h^{3/2}}\right )}{h^3}\) |
(2*((-4*b*e*f^2*p)/(21*d*(h*x)^(3/2)) - (8*b*e*f*g*p)/(5*d*h*Sqrt[h*x]) + (Sqrt[2]*b*e^(7/4)*f^2*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*S qrt[h])])/(7*d^(7/4)*h^(3/2)) + (2*Sqrt[2]*b*e^(5/4)*f*g*p*ArcTan[1 - (Sqr t[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*d^(5/4)*h^(3/2)) - (Sqrt[2] *b*e^(3/4)*g^2*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])] )/(3*d^(3/4)*h^(3/2)) - (Sqrt[2]*b*e^(7/4)*f^2*p*ArcTan[1 + (Sqrt[2]*e^(1/ 4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(7*d^(7/4)*h^(3/2)) - (2*Sqrt[2]*b*e^(5/ 4)*f*g*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*d^( 5/4)*h^(3/2)) + (Sqrt[2]*b*e^(3/4)*g^2*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[ h*x])/(d^(1/4)*Sqrt[h])])/(3*d^(3/4)*h^(3/2)) - (f^2*h^2*(a + b*Log[c*(d + e*x^2)^p]))/(7*(h*x)^(7/2)) - (2*f*g*h*(a + b*Log[c*(d + e*x^2)^p]))/(5*( h*x)^(5/2)) - (g^2*(a + b*Log[c*(d + e*x^2)^p]))/(3*(h*x)^(3/2)) + (b*e^(7 /4)*f^2*p*Log[Sqrt[d]*h + Sqrt[e]*h*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h]*Sq rt[h*x]])/(7*Sqrt[2]*d^(7/4)*h^(3/2)) - (Sqrt[2]*b*e^(5/4)*f*g*p*Log[Sqrt[ d]*h + Sqrt[e]*h*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h]*Sqrt[h*x]])/(5*d^(5/4 )*h^(3/2)) - (b*e^(3/4)*g^2*p*Log[Sqrt[d]*h + Sqrt[e]*h*x - Sqrt[2]*d^(1/4 )*e^(1/4)*Sqrt[h]*Sqrt[h*x]])/(3*Sqrt[2]*d^(3/4)*h^(3/2)) - (b*e^(7/4)*f^2 *p*Log[Sqrt[d]*h + Sqrt[e]*h*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h]*Sqrt[h*x] ])/(7*Sqrt[2]*d^(7/4)*h^(3/2)) + (Sqrt[2]*b*e^(5/4)*f*g*p*Log[Sqrt[d]*h + Sqrt[e]*h*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h]*Sqrt[h*x]])/(5*d^(5/4)*h^...
3.7.15.3.1 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]
\[\int \frac {\left (g x +f \right )^{2} \left (a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )\right )}{\left (h x \right )^{\frac {9}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 2283 vs. \(2 (672) = 1344\).
Time = 0.43 (sec) , antiderivative size = 2283, normalized size of antiderivative = 2.36 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=\text {Too large to display} \]
-2/105*(d*h^5*x^4*sqrt(-(d^3*h^9*sqrt(-(50625*b^4*e^7*f^8 - 1266300*b^4*d* e^6*f^6*g^2 + 8469846*b^4*d^2*e^5*f^4*g^4 - 6894300*b^4*d^3*e^4*f^2*g^6 + 1500625*b^4*d^4*e^3*g^8)*p^4/(d^7*h^18)) + 420*(3*b^2*e^3*f^3*g - 7*b^2*d* e^2*f*g^3)*p^2)/(d^3*h^9))*log(16*(50625*b^3*e^6*f^8 - 472500*b^3*d*e^5*f^ 6*g^2 - 1457946*b^3*d^2*e^4*f^4*g^4 - 2572500*b^3*d^3*e^3*f^2*g^6 + 150062 5*b^3*d^4*e^2*g^8)*sqrt(h*x)*p^3 + 16*(42*d^6*f*g*h^14*sqrt(-(50625*b^4*e^ 7*f^8 - 1266300*b^4*d*e^6*f^6*g^2 + 8469846*b^4*d^2*e^5*f^4*g^4 - 6894300* b^4*d^3*e^4*f^2*g^6 + 1500625*b^4*d^4*e^3*g^8)*p^4/(d^7*h^18)) + 5*(675*b^ 2*d^2*e^4*f^6 - 10017*b^2*d^3*e^3*f^4*g^2 + 23373*b^2*d^4*e^2*f^2*g^4 - 85 75*b^2*d^5*e*g^6)*h^5*p^2)*sqrt(-(d^3*h^9*sqrt(-(50625*b^4*e^7*f^8 - 12663 00*b^4*d*e^6*f^6*g^2 + 8469846*b^4*d^2*e^5*f^4*g^4 - 6894300*b^4*d^3*e^4*f ^2*g^6 + 1500625*b^4*d^4*e^3*g^8)*p^4/(d^7*h^18)) + 420*(3*b^2*e^3*f^3*g - 7*b^2*d*e^2*f*g^3)*p^2)/(d^3*h^9))) - d*h^5*x^4*sqrt(-(d^3*h^9*sqrt(-(506 25*b^4*e^7*f^8 - 1266300*b^4*d*e^6*f^6*g^2 + 8469846*b^4*d^2*e^5*f^4*g^4 - 6894300*b^4*d^3*e^4*f^2*g^6 + 1500625*b^4*d^4*e^3*g^8)*p^4/(d^7*h^18)) + 420*(3*b^2*e^3*f^3*g - 7*b^2*d*e^2*f*g^3)*p^2)/(d^3*h^9))*log(16*(50625*b^ 3*e^6*f^8 - 472500*b^3*d*e^5*f^6*g^2 - 1457946*b^3*d^2*e^4*f^4*g^4 - 25725 00*b^3*d^3*e^3*f^2*g^6 + 1500625*b^3*d^4*e^2*g^8)*sqrt(h*x)*p^3 - 16*(42*d ^6*f*g*h^14*sqrt(-(50625*b^4*e^7*f^8 - 1266300*b^4*d*e^6*f^6*g^2 + 8469846 *b^4*d^2*e^5*f^4*g^4 - 6894300*b^4*d^3*e^4*f^2*g^6 + 1500625*b^4*d^4*e^...
Timed out. \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 1110, normalized size of antiderivative = 1.15 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=\text {Too large to display} \]
-1/21*b*e*f^2*p*(3*(sqrt(2)*e^(3/4)*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) - sqrt(2)*e^(3/4)*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 ) + sqrt(2)*e*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(d)*h) + sqrt(2)*e*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(d)*h))/d + 8/((h*x)^(3/2)*d))/h^3 - 2/3*b*g^2*x^3*log((e *x^2 + d)^p*c)/(h*x)^(9/2) + 2/5*b*e*f*g*p*(e*(sqrt(2)*log(sqrt(e)*h*x + s qrt(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) + sq rt(d)*h)/((d*h^2)^(1/4)*e^(3/4)) - sqrt(2)*log(-(sqrt(2)*sqrt(-sqrt(d)*sqr t(e)*h) + sqrt(2)*(d*h^2)^(1/4)*e^(1/4) - 2*sqrt(h*x)*sqrt(e))/(sqrt(2)*sq rt(-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))/(sqr t(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)))/d - 8/(sqrt(h*x)*d))/h^...
Time = 0.44 (sec) , antiderivative size = 688, normalized size of antiderivative = 0.71 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=-\frac {\frac {2 \, {\left (15 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b e^{2} f^{2} h p - 35 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e g^{2} h p + 42 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b f g 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^{2} e h} + \frac {2 \, {\left (15 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b e^{2} f^{2} h p - 35 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e g^{2} h p + 42 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b f g 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^{2} e h} + \frac {{\left (15 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b e^{2} f^{2} h p - 35 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e g^{2} h p - 42 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b f g 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^{2} e h} - \frac {{\left (15 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b e^{2} f^{2} h p - 35 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e g^{2} h p - 42 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b f g 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^{2} e h} + \frac {2 \, {\left (35 \, b g^{2} h^{4} p x^{2} + 42 \, b f g h^{4} p x + 15 \, b f^{2} h^{4} p\right )} \log \left (e h^{2} x^{2} + d h^{2}\right )}{\sqrt {h x} h^{3} x^{3}} + \frac {2 \, {\left (168 \, b e f g h^{4} p x^{3} - 35 \, b d g^{2} h^{4} p x^{2} \log \left (h^{2}\right ) + 20 \, b e f^{2} h^{4} p x^{2} - 42 \, b d f g h^{4} p x \log \left (h^{2}\right ) + 35 \, b d g^{2} h^{4} x^{2} \log \left (c\right ) + 35 \, a d g^{2} h^{4} x^{2} - 15 \, b d f^{2} h^{4} p \log \left (h^{2}\right ) + 42 \, b d f g h^{4} x \log \left (c\right ) + 42 \, a d f g h^{4} x + 15 \, b d f^{2} h^{4} \log \left (c\right ) + 15 \, a d f^{2} h^{4}\right )}}{\sqrt {h x} d h^{3} x^{3}}}{105 \, h^{5}} \]
-1/105*(2*(15*sqrt(2)*(d*e^3*h^2)^(1/4)*b*e^2*f^2*h*p - 35*sqrt(2)*(d*e^3* h^2)^(1/4)*b*d*e*g^2*h*p + 42*sqrt(2)*(d*e^3*h^2)^(3/4)*b*f*g*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^2*e* h) + 2*(15*sqrt(2)*(d*e^3*h^2)^(1/4)*b*e^2*f^2*h*p - 35*sqrt(2)*(d*e^3*h^2 )^(1/4)*b*d*e*g^2*h*p + 42*sqrt(2)*(d*e^3*h^2)^(3/4)*b*f*g*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^2*e*h) + (15*sqrt(2)*(d*e^3*h^2)^(1/4)*b*e^2*f^2*h*p - 35*sqrt(2)*(d*e^3*h^2)^(1 /4)*b*d*e*g^2*h*p - 42*sqrt(2)*(d*e^3*h^2)^(3/4)*b*f*g*p)*log(h*x + sqrt(2 )*(d*h^2/e)^(1/4)*sqrt(h*x) + sqrt(d*h^2/e))/(d^2*e*h) - (15*sqrt(2)*(d*e^ 3*h^2)^(1/4)*b*e^2*f^2*h*p - 35*sqrt(2)*(d*e^3*h^2)^(1/4)*b*d*e*g^2*h*p - 42*sqrt(2)*(d*e^3*h^2)^(3/4)*b*f*g*p)*log(h*x - sqrt(2)*(d*h^2/e)^(1/4)*sq rt(h*x) + sqrt(d*h^2/e))/(d^2*e*h) + 2*(35*b*g^2*h^4*p*x^2 + 42*b*f*g*h^4* p*x + 15*b*f^2*h^4*p)*log(e*h^2*x^2 + d*h^2)/(sqrt(h*x)*h^3*x^3) + 2*(168* b*e*f*g*h^4*p*x^3 - 35*b*d*g^2*h^4*p*x^2*log(h^2) + 20*b*e*f^2*h^4*p*x^2 - 42*b*d*f*g*h^4*p*x*log(h^2) + 35*b*d*g^2*h^4*x^2*log(c) + 35*a*d*g^2*h^4* x^2 - 15*b*d*f^2*h^4*p*log(h^2) + 42*b*d*f*g*h^4*x*log(c) + 42*a*d*f*g*h^4 *x + 15*b*d*f^2*h^4*log(c) + 15*a*d*f^2*h^4)/(sqrt(h*x)*d*h^3*x^3))/h^5
Timed out. \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^2\,\left (a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\right )}{{\left (h\,x\right )}^{9/2}} \,d x \]