Integrand size = 31, antiderivative size = 935 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx=-\frac {8 b e f^2 p}{5 d h^3 \sqrt {h x}}+\frac {2 \sqrt {2} b e^{5/4} f^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{7/2}}-\frac {4 \sqrt {2} b e^{3/4} f g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{7/2}}-\frac {2 \sqrt {2} b \sqrt [4]{e} g^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{7/2}}-\frac {2 \sqrt {2} b e^{5/4} f^2 p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{7/2}}+\frac {4 \sqrt {2} b e^{3/4} f g p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{7/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} g^2 p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{7/2}}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h (h x)^{5/2}}-\frac {4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2 (h x)^{3/2}}-\frac {2 g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h^3 \sqrt {h x}}-\frac {\sqrt {2} b e^{5/4} 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 )}{5 d^{5/4} h^{7/2}}-\frac {2 \sqrt {2} b e^{3/4} 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 )}{3 d^{3/4} h^{7/2}}+\frac {\sqrt {2} b \sqrt [4]{e} 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 )}{\sqrt [4]{d} h^{7/2}}+\frac {\sqrt {2} b e^{5/4} 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 )}{5 d^{5/4} h^{7/2}}+\frac {2 \sqrt {2} b e^{3/4} 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 )}{3 d^{3/4} h^{7/2}}-\frac {\sqrt {2} b \sqrt [4]{e} 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 )}{\sqrt [4]{d} h^{7/2}} \]
-2/5*f^2*(a+b*ln(c*(e*x^2+d)^p))/h/(h*x)^(5/2)-4/3*f*g*(a+b*ln(c*(e*x^2+d) ^p))/h^2/(h*x)^(3/2)+2/5*b*e^(5/4)*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^(5/4)/h^(7/2)-4/3*b*e^(3/4)*f*g*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^(7/2)-2*b*e ^(1/4)*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^(1/4)/h^(7/2)-2/5*b*e^(5/4)*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^(5/4)/h^(7/2)+4/3*b*e^(3/4)*f*g*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^(7/2)+2*b*e^(1/ 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^( 1/4)/h^(7/2)-1/5*b*e^(5/4)*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^(5/4)/h^(7/2)-2/3*b*e^(3/4)*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^(3/4)/h^(7/2)+b*e^(1/4)*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^(1/4)/h^(7/2)+1/5*b*e ^(5/4)*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^(5/4)/h^(7/2)+2/3*b*e^(3/4)*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^(3/4)/h ^(7/2)-b*e^(1/4)*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^(1/4)/h^(7/2)-8/5*b*e*f^2*p/d/h^3/(h*x)^( 1/2)-2*g^2*(a+b*ln(c*(e*x^2+d)^p))/h^3/(h*x)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.64 (sec) , antiderivative size = 340, normalized size of antiderivative = 0.36 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx=\frac {2 x^{7/2} \left (\frac {2 b \sqrt [4]{e} g^2 p \left (\arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )+\text {arctanh}\left (\frac {d \sqrt [4]{e} \sqrt {x}}{(-d)^{5/4}}\right )\right )}{\sqrt [4]{-d}}-\frac {4 b e f^2 p \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\frac {e x^2}{d}\right )}{5 d \sqrt {x}}-\frac {\sqrt {2} b e^{3/4} f g p \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )+\log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )-\log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )\right )}{3 d^{3/4}}-\frac {f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 x^{5/2}}-\frac {2 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 x^{3/2}}-\frac {g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {x}}\right )}{(h x)^{7/2}} \]
(2*x^(7/2)*((2*b*e^(1/4)*g^2*p*(ArcTan[(e^(1/4)*Sqrt[x])/(-d)^(1/4)] + Arc Tanh[(d*e^(1/4)*Sqrt[x])/(-d)^(5/4)]))/(-d)^(1/4) - (4*b*e*f^2*p*Hypergeom etric2F1[-1/4, 1, 3/4, -((e*x^2)/d)])/(5*d*Sqrt[x]) - (Sqrt[2]*b*e^(3/4)*f *g*p*(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] - Sqrt[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]))/(3*d^(3/4)) - (f^2*(a + b*Log[c*(d + e*x^2)^p]))/(5*x^(5/2)) - (2*f* g*(a + b*Log[c*(d + e*x^2)^p]))/(3*x^(3/2)) - (g^2*(a + b*Log[c*(d + e*x^2 )^p]))/Sqrt[x]))/(h*x)^(7/2)
Time = 1.13 (sec) , antiderivative size = 913, 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.
| \(\displaystyle \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 2917 |
| \(\displaystyle \frac {2 \int \frac {(f h+g x h)^2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{h^5 x^3}d\sqrt {h x}}{h}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \int \frac {(f h+g x h)^2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{h^3 x^3}d\sqrt {h x}}{h^3}\) |
| \(\Big \downarrow \) 2926 |
| \(\displaystyle \frac {2 \int \left (\frac {\left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f^2}{h x^3}+\frac {2 g \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f}{h x^2}+\frac {g^2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{h x}\right )d\sqrt {h x}}{h^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (\frac {\sqrt {2} b e^{5/4} p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right ) f^2}{5 d^{5/4} \sqrt {h}}-\frac {\sqrt {2} b e^{5/4} p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right ) f^2}{5 d^{5/4} \sqrt {h}}-\frac {h^2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f^2}{5 (h x)^{5/2}}-\frac {b e^{5/4} 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}{5 \sqrt {2} d^{5/4} \sqrt {h}}+\frac {b e^{5/4} 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}{5 \sqrt {2} d^{5/4} \sqrt {h}}-\frac {4 b e p f^2}{5 d \sqrt {h x}}-\frac {2 \sqrt {2} b e^{3/4} g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right ) f}{3 d^{3/4} \sqrt {h}}+\frac {2 \sqrt {2} b e^{3/4} g p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right ) f}{3 d^{3/4} \sqrt {h}}-\frac {2 g h \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f}{3 (h x)^{3/2}}-\frac {\sqrt {2} b e^{3/4} 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}{3 d^{3/4} \sqrt {h}}+\frac {\sqrt {2} b e^{3/4} 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}{3 d^{3/4} \sqrt {h}}-\frac {\sqrt {2} b \sqrt [4]{e} g^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} \sqrt {h}}+\frac {\sqrt {2} b \sqrt [4]{e} g^2 p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{\sqrt [4]{d} \sqrt {h}}-\frac {g^2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{\sqrt {h x}}+\frac {b \sqrt [4]{e} 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 )}{\sqrt {2} \sqrt [4]{d} \sqrt {h}}-\frac {b \sqrt [4]{e} 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 )}{\sqrt {2} \sqrt [4]{d} \sqrt {h}}\right )}{h^3}\) |
(2*((-4*b*e*f^2*p)/(5*d*Sqrt[h*x]) + (Sqrt[2]*b*e^(5/4)*f^2*p*ArcTan[1 - ( Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*d^(5/4)*Sqrt[h]) - (2*Sq rt[2]*b*e^(3/4)*f*g*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt [h])])/(3*d^(3/4)*Sqrt[h]) - (Sqrt[2]*b*e^(1/4)*g^2*p*ArcTan[1 - (Sqrt[2]* e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(d^(1/4)*Sqrt[h]) - (Sqrt[2]*b*e^(5 /4)*f^2*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*d^ (5/4)*Sqrt[h]) + (2*Sqrt[2]*b*e^(3/4)*f*g*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sq rt[h*x])/(d^(1/4)*Sqrt[h])])/(3*d^(3/4)*Sqrt[h]) + (Sqrt[2]*b*e^(1/4)*g^2* p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(d^(1/4)*Sqrt [h]) - (f^2*h^2*(a + b*Log[c*(d + e*x^2)^p]))/(5*(h*x)^(5/2)) - (2*f*g*h*( a + b*Log[c*(d + e*x^2)^p]))/(3*(h*x)^(3/2)) - (g^2*(a + b*Log[c*(d + e*x^ 2)^p]))/Sqrt[h*x] - (b*e^(5/4)*f^2*p*Log[Sqrt[d]*h + Sqrt[e]*h*x - Sqrt[2] *d^(1/4)*e^(1/4)*Sqrt[h]*Sqrt[h*x]])/(5*Sqrt[2]*d^(5/4)*Sqrt[h]) - (Sqrt[2 ]*b*e^(3/4)*f*g*p*Log[Sqrt[d]*h + Sqrt[e]*h*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sq rt[h]*Sqrt[h*x]])/(3*d^(3/4)*Sqrt[h]) + (b*e^(1/4)*g^2*p*Log[Sqrt[d]*h + S qrt[e]*h*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h]*Sqrt[h*x]])/(Sqrt[2]*d^(1/4)* Sqrt[h]) + (b*e^(5/4)*f^2*p*Log[Sqrt[d]*h + Sqrt[e]*h*x + Sqrt[2]*d^(1/4)* e^(1/4)*Sqrt[h]*Sqrt[h*x]])/(5*Sqrt[2]*d^(5/4)*Sqrt[h]) + (Sqrt[2]*b*e^(3/ 4)*f*g*p*Log[Sqrt[d]*h + Sqrt[e]*h*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h]*Sqr t[h*x]])/(3*d^(3/4)*Sqrt[h]) - (b*e^(1/4)*g^2*p*Log[Sqrt[d]*h + Sqrt[e]...
3.7.14.3.1 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]
\[\int \frac {\left (g x +f \right )^{2} \left (a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )\right )}{\left (h x \right )^{\frac {7}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 2205 vs. \(2 (653) = 1306\).
Time = 0.46 (sec) , antiderivative size = 2205, normalized size of antiderivative = 2.36 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx=\text {Too large to display} \]
-2/15*(d*h^4*x^3*sqrt((d^2*h^7*sqrt(-(81*b^4*e^5*f^8 - 3420*b^4*d*e^4*f^6* g^2 + 40150*b^4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e^2*f^2*g^6 + 50625*b^4*d^ 4*e*g^8)*p^4/(d^5*h^14)) + 60*(b^2*e^2*f^3*g - 5*b^2*d*e*f*g^3)*p^2)/(d^2* h^7))*log(32*(81*b^3*e^5*f^8 - 1620*b^3*d*e^4*f^6*g^2 + 2150*b^3*d^2*e^3*f ^4*g^4 - 40500*b^3*d^3*e^2*f^2*g^6 + 50625*b^3*d^4*e*g^8)*sqrt(h*x)*p^3 + 32*(3*(d^4*e*f^2 - 5*d^5*g^2)*h^11*sqrt(-(81*b^4*e^5*f^8 - 3420*b^4*d*e^4* f^6*g^2 + 40150*b^4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e^2*f^2*g^6 + 50625*b^ 4*d^4*e*g^8)*p^4/(d^5*h^14)) - 10*(9*b^2*d^2*e^3*f^5*g - 190*b^2*d^3*e^2*f ^3*g^3 + 225*b^2*d^4*e*f*g^5)*h^4*p^2)*sqrt((d^2*h^7*sqrt(-(81*b^4*e^5*f^8 - 3420*b^4*d*e^4*f^6*g^2 + 40150*b^4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e^2* f^2*g^6 + 50625*b^4*d^4*e*g^8)*p^4/(d^5*h^14)) + 60*(b^2*e^2*f^3*g - 5*b^2 *d*e*f*g^3)*p^2)/(d^2*h^7))) - d*h^4*x^3*sqrt((d^2*h^7*sqrt(-(81*b^4*e^5*f ^8 - 3420*b^4*d*e^4*f^6*g^2 + 40150*b^4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e^ 2*f^2*g^6 + 50625*b^4*d^4*e*g^8)*p^4/(d^5*h^14)) + 60*(b^2*e^2*f^3*g - 5*b ^2*d*e*f*g^3)*p^2)/(d^2*h^7))*log(32*(81*b^3*e^5*f^8 - 1620*b^3*d*e^4*f^6* g^2 + 2150*b^3*d^2*e^3*f^4*g^4 - 40500*b^3*d^3*e^2*f^2*g^6 + 50625*b^3*d^4 *e*g^8)*sqrt(h*x)*p^3 - 32*(3*(d^4*e*f^2 - 5*d^5*g^2)*h^11*sqrt(-(81*b^4*e ^5*f^8 - 3420*b^4*d*e^4*f^6*g^2 + 40150*b^4*d^2*e^3*f^4*g^4 - 85500*b^4*d^ 3*e^2*f^2*g^6 + 50625*b^4*d^4*e*g^8)*p^4/(d^5*h^14)) - 10*(9*b^2*d^2*e^3*f ^5*g - 190*b^2*d^3*e^2*f^3*g^3 + 225*b^2*d^4*e*f*g^5)*h^4*p^2)*sqrt((d^...
Timed out. \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 1088, normalized size of antiderivative = 1.16 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx=\text {Too large to display} \]
-2*b*g^2*x^3*log((e*x^2 + d)^p*c)/(h*x)^(7/2) + 1/5*b*e*f^2*p*(e*(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)*sqr t(h*x)*e^(1/4) + sqrt(d)*h)/((d*h^2)^(1/4)*e^(3/4)) - 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)*s qrt(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)))/d - 8/( sqrt(h*x)*d))/h^3 - b*e*g^2*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)) - 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)) - 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)))/(sqr t(-sqrt(d)*sqrt(e)*h)*sqrt(e)))/h^3 - 2*a*g^2*x^3/(h*x)^(7/2) - 4/3*b*f...
Time = 0.43 (sec) , antiderivative size = 675, normalized size of antiderivative = 0.72 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx=-\frac {\frac {2 \, {\left (15 \, b g^{2} h^{3} p x^{2} + 10 \, b f g h^{3} p x + 3 \, b f^{2} h^{3} p\right )} \log \left (e h^{2} x^{2} + d h^{2}\right )}{\sqrt {h x} h^{2} x^{2}} - \frac {2 \, {\left (10 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e^{2} f g h p - 3 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b e f^{2} p + 15 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b d g^{2} 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^{2} h} - \frac {2 \, {\left (10 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e^{2} f g h p - 3 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b e f^{2} p + 15 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b d g^{2} 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^{2} h} - \frac {{\left (10 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e^{2} f g h p + 3 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b e f^{2} p - 15 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b d g^{2} 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^{2} h} + \frac {{\left (10 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e^{2} f g h p + 3 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b e f^{2} p - 15 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b d g^{2} 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^{2} h} - \frac {2 \, {\left (15 \, b d g^{2} h^{3} p x^{2} \log \left (h^{2}\right ) - 12 \, b e f^{2} h^{3} p x^{2} + 10 \, b d f g h^{3} p x \log \left (h^{2}\right ) - 15 \, b d g^{2} h^{3} x^{2} \log \left (c\right ) - 15 \, a d g^{2} h^{3} x^{2} + 3 \, b d f^{2} h^{3} p \log \left (h^{2}\right ) - 10 \, b d f g h^{3} x \log \left (c\right ) - 10 \, a d f g h^{3} x - 3 \, b d f^{2} h^{3} \log \left (c\right ) - 3 \, a d f^{2} h^{3}\right )}}{\sqrt {h x} d h^{2} x^{2}}}{15 \, h^{4}} \]
-1/15*(2*(15*b*g^2*h^3*p*x^2 + 10*b*f*g*h^3*p*x + 3*b*f^2*h^3*p)*log(e*h^2 *x^2 + d*h^2)/(sqrt(h*x)*h^2*x^2) - 2*(10*sqrt(2)*(d*e^3*h^2)^(1/4)*b*d*e^ 2*f*g*h*p - 3*sqrt(2)*(d*e^3*h^2)^(3/4)*b*e*f^2*p + 15*sqrt(2)*(d*e^3*h^2) ^(3/4)*b*d*g^2*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^2*h) - 2*(10*sqrt(2)*(d*e^3*h^2)^(1/4)*b*d*e^2* f*g*h*p - 3*sqrt(2)*(d*e^3*h^2)^(3/4)*b*e*f^2*p + 15*sqrt(2)*(d*e^3*h^2)^( 3/4)*b*d*g^2*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^2*h) - (10*sqrt(2)*(d*e^3*h^2)^(1/4)*b*d*e^2*f*g *h*p + 3*sqrt(2)*(d*e^3*h^2)^(3/4)*b*e*f^2*p - 15*sqrt(2)*(d*e^3*h^2)^(3/4 )*b*d*g^2*p)*log(h*x + sqrt(2)*(d*h^2/e)^(1/4)*sqrt(h*x) + sqrt(d*h^2/e))/ (d^2*e^2*h) + (10*sqrt(2)*(d*e^3*h^2)^(1/4)*b*d*e^2*f*g*h*p + 3*sqrt(2)*(d *e^3*h^2)^(3/4)*b*e*f^2*p - 15*sqrt(2)*(d*e^3*h^2)^(3/4)*b*d*g^2*p)*log(h* x - sqrt(2)*(d*h^2/e)^(1/4)*sqrt(h*x) + sqrt(d*h^2/e))/(d^2*e^2*h) - 2*(15 *b*d*g^2*h^3*p*x^2*log(h^2) - 12*b*e*f^2*h^3*p*x^2 + 10*b*d*f*g*h^3*p*x*lo g(h^2) - 15*b*d*g^2*h^3*x^2*log(c) - 15*a*d*g^2*h^3*x^2 + 3*b*d*f^2*h^3*p* log(h^2) - 10*b*d*f*g*h^3*x*log(c) - 10*a*d*f*g*h^3*x - 3*b*d*f^2*h^3*log( c) - 3*a*d*f^2*h^3)/(sqrt(h*x)*d*h^2*x^2))/h^4
Timed out. \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^2\,\left (a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\right )}{{\left (h\,x\right )}^{7/2}} \,d x \]