Integrand size = 29, antiderivative size = 641 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=-\frac {8 b e f p}{21 d h^3 (h x)^{3/2}}-\frac {8 b e g p}{5 d h^4 \sqrt {h x}}+\frac {2 \sqrt {2} b e^{7/4} f p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{7 d^{7/4} h^{9/2}}+\frac {2 \sqrt {2} b e^{5/4} g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{9/2}}-\frac {2 \sqrt {2} b e^{7/4} f p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{7 d^{7/4} h^{9/2}}-\frac {2 \sqrt {2} b e^{5/4} g p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{9/2}}-\frac {2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac {2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}}+\frac {\sqrt {2} b e^{7/4} f p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{7 d^{7/4} h^{9/2}}-\frac {\sqrt {2} b e^{5/4} g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{5 d^{5/4} h^{9/2}}-\frac {\sqrt {2} b e^{7/4} f p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{7 d^{7/4} h^{9/2}}+\frac {\sqrt {2} b e^{5/4} g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{5 d^{5/4} h^{9/2}} \]
-8/21*b*e*f*p/d/h^3/(h*x)^(3/2)-2/7*f*(a+b*ln(c*(e*x^2+d)^p))/h/(h*x)^(7/2 )-2/5*g*(a+b*ln(c*(e*x^2+d)^p))/h^2/(h*x)^(5/2)+2/7*b*e^(7/4)*f*p*arctan(1 -e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/2)/d^(7/4)/h^(9/2)+2/5* b*e^(5/4)*g*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^(9/2)-2/7*b*e^(7/4)*f*p*arctan(1+e^(1/4)*2^(1/2)*(h*x)^(1/2)/d ^(1/4)/h^(1/2))*2^(1/2)/d^(7/4)/h^(9/2)-2/5*b*e^(5/4)*g*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^(9/2)+1/7*b*e^(7/4 )*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^(7/4)/h^(9/2)-1/5*b*e^(5/4)*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)/d^(5/4)/h^(9/2)-1/ 7*b*e^(7/4)*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^(7/4)/h^(9/2)+1/5*b*e^(5/4)*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)/d^(5/4)/ h^(9/2)-8/5*b*e*g*p/d/h^4/(h*x)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.16 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=-\frac {2 \sqrt {h x} \left (20 b e f p x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},-\frac {e x^2}{d}\right )+84 b e g p x^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\frac {e x^2}{d}\right )+3 d (5 f+7 g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )\right )}{105 d h^5 x^4} \]
(-2*Sqrt[h*x]*(20*b*e*f*p*x^2*Hypergeometric2F1[-3/4, 1, 1/4, -((e*x^2)/d) ] + 84*b*e*g*p*x^3*Hypergeometric2F1[-1/4, 1, 3/4, -((e*x^2)/d)] + 3*d*(5* f + 7*g*x)*(a + b*Log[c*(d + e*x^2)^p])))/(105*d*h^5*x^4)
Time = 0.86 (sec) , antiderivative size = 629, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, 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.
| \(\displaystyle \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2917 |
| \(\displaystyle \frac {2 \int \frac {(f h+g x h) \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{h^5 x^4}d\sqrt {h x}}{h}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \int \frac {(f h+g x h) \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{h^4 x^4}d\sqrt {h x}}{h^2}\) |
| \(\Big \downarrow \) 2926 |
| \(\displaystyle \frac {2 \int \left (\frac {g \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{h^3 x^3}+\frac {f \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{h^3 x^4}\right )d\sqrt {h x}}{h^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (-\frac {f h \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 (h x)^{7/2}}-\frac {g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 (h x)^{5/2}}+\frac {\sqrt {2} b e^{7/4} f p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{7 d^{7/4} h^{5/2}}-\frac {\sqrt {2} b e^{7/4} f p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{7 d^{7/4} h^{5/2}}+\frac {\sqrt {2} b e^{5/4} g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{5/2}}-\frac {\sqrt {2} b e^{5/4} g p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{5 d^{5/4} h^{5/2}}+\frac {b e^{7/4} f p \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h} \sqrt {h x}+\sqrt {d} h+\sqrt {e} h x\right )}{7 \sqrt {2} d^{7/4} h^{5/2}}-\frac {b e^{7/4} f p \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h} \sqrt {h x}+\sqrt {d} h+\sqrt {e} h x\right )}{7 \sqrt {2} d^{7/4} h^{5/2}}-\frac {b e^{5/4} g p \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h} \sqrt {h x}+\sqrt {d} h+\sqrt {e} h x\right )}{5 \sqrt {2} d^{5/4} h^{5/2}}+\frac {b e^{5/4} g p \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h} \sqrt {h x}+\sqrt {d} h+\sqrt {e} h x\right )}{5 \sqrt {2} d^{5/4} h^{5/2}}-\frac {4 b e f p}{21 d h (h x)^{3/2}}-\frac {4 b e g p}{5 d h^2 \sqrt {h x}}\right )}{h^2}\) |
(2*((-4*b*e*f*p)/(21*d*h*(h*x)^(3/2)) - (4*b*e*g*p)/(5*d*h^2*Sqrt[h*x]) + (Sqrt[2]*b*e^(7/4)*f*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqr t[h])])/(7*d^(7/4)*h^(5/2)) + (Sqrt[2]*b*e^(5/4)*g*p*ArcTan[1 - (Sqrt[2]*e ^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*d^(5/4)*h^(5/2)) - (Sqrt[2]*b*e^( 7/4)*f*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(7*d^( 7/4)*h^(5/2)) - (Sqrt[2]*b*e^(5/4)*g*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h* x])/(d^(1/4)*Sqrt[h])])/(5*d^(5/4)*h^(5/2)) - (f*h*(a + b*Log[c*(d + e*x^2 )^p]))/(7*(h*x)^(7/2)) - (g*(a + b*Log[c*(d + e*x^2)^p]))/(5*(h*x)^(5/2)) + (b*e^(7/4)*f*p*Log[Sqrt[d]*h + Sqrt[e]*h*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqr t[h]*Sqrt[h*x]])/(7*Sqrt[2]*d^(7/4)*h^(5/2)) - (b*e^(5/4)*g*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)*h^(5/2)) - (b*e^(7/4)*f*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^(5/2)) + (b*e^(5/4)* g*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)*h^(5/2))))/h^2
3.7.10.3.1 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]
\[\int \frac {\left (g x +f \right ) \left (a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )\right )}{\left (h x \right )^{\frac {9}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 1369 vs. \(2 (441) = 882\).
Time = 0.41 (sec) , antiderivative size = 1369, normalized size of antiderivative = 2.14 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=\text {Too large to display} \]
2/105*(3*d*h^5*x^4*sqrt(-(d^3*h^9*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6* f^2*g^2 + 2401*b^4*d^2*e^5*g^4)*p^4/(d^7*h^18)) + 70*b^2*e^3*f*g*p^2)/(d^3 *h^9))*log(-32*(625*b^3*e^6*f^4 - 2401*b^3*d^2*e^4*g^4)*sqrt(h*x)*p^3 + 32 *(7*d^6*g*h^14*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2*g^2 + 2401*b^4* d^2*e^5*g^4)*p^4/(d^7*h^18)) + 5*(25*b^2*d^2*e^4*f^3 - 49*b^2*d^3*e^3*f*g^ 2)*h^5*p^2)*sqrt(-(d^3*h^9*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2*g^2 + 2401*b^4*d^2*e^5*g^4)*p^4/(d^7*h^18)) + 70*b^2*e^3*f*g*p^2)/(d^3*h^9))) - 3*d*h^5*x^4*sqrt(-(d^3*h^9*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2* g^2 + 2401*b^4*d^2*e^5*g^4)*p^4/(d^7*h^18)) + 70*b^2*e^3*f*g*p^2)/(d^3*h^9 ))*log(-32*(625*b^3*e^6*f^4 - 2401*b^3*d^2*e^4*g^4)*sqrt(h*x)*p^3 - 32*(7* d^6*g*h^14*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2*g^2 + 2401*b^4*d^2* e^5*g^4)*p^4/(d^7*h^18)) + 5*(25*b^2*d^2*e^4*f^3 - 49*b^2*d^3*e^3*f*g^2)*h ^5*p^2)*sqrt(-(d^3*h^9*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2*g^2 + 2 401*b^4*d^2*e^5*g^4)*p^4/(d^7*h^18)) + 70*b^2*e^3*f*g*p^2)/(d^3*h^9))) - 3 *d*h^5*x^4*sqrt((d^3*h^9*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2*g^2 + 2401*b^4*d^2*e^5*g^4)*p^4/(d^7*h^18)) - 70*b^2*e^3*f*g*p^2)/(d^3*h^9))*lo g(-32*(625*b^3*e^6*f^4 - 2401*b^3*d^2*e^4*g^4)*sqrt(h*x)*p^3 + 32*(7*d^6*g *h^14*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2*g^2 + 2401*b^4*d^2*e^5*g ^4)*p^4/(d^7*h^18)) - 5*(25*b^2*d^2*e^4*f^3 - 49*b^2*d^3*e^3*f*g^2)*h^5*p^ 2)*sqrt((d^3*h^9*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2*g^2 + 2401...
Timed out. \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=\text {Timed out} \]
Time = 0.33 (sec) , antiderivative size = 739, normalized size of antiderivative = 1.15 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=-\frac {b e f p {\left (\frac {3 \, {\left (\frac {\sqrt {2} e^{\frac {3}{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}}} - \frac {\sqrt {2} e^{\frac {3}{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}}} + \frac {\sqrt {2} e \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} h} + \frac {\sqrt {2} e \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} h}\right )}}{d} + \frac {8}{\left (h x\right )^{\frac {3}{2}} d}\right )}}{21 \, h^{3}} + \frac {b e g p {\left (\frac {e {\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 )}}{d} - \frac {8}{\sqrt {h x} d}\right )}}{5 \, h^{4}} - \frac {2 \, b g x^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{5 \, \left (h x\right )^{\frac {9}{2}}} - \frac {2 \, a g x^{2}}{5 \, \left (h x\right )^{\frac {9}{2}}} - \frac {2 \, b f \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{7 \, \left (h x\right )^{\frac {7}{2}} h} - \frac {2 \, a f}{7 \, \left (h x\right )^{\frac {7}{2}} h} \]
-1/21*b*e*f*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 + 1/5*b*e*g*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)*s qrt(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*sq rt(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^4 - 2/5*b*g*x^2*log((e*x^2 + d)^p*c)/(h*x)^(9/2) - 2/...
Time = 0.42 (sec) , antiderivative size = 511, normalized size of antiderivative = 0.80 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=-\frac {\frac {6 \, {\left (5 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b e^{2} f h p + 7 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b g p\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} + 2 \, \sqrt {h x}\right )}}{2 \, \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}}}\right )}{d^{2} e h} + \frac {6 \, {\left (5 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b e^{2} f h p + 7 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b g p\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} - 2 \, \sqrt {h x}\right )}}{2 \, \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}}}\right )}{d^{2} e h} + \frac {3 \, {\left (5 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b e^{2} f h p - 7 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b g p\right )} \log \left (h x + \sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} \sqrt {h x} + \sqrt {\frac {d h^{2}}{e}}\right )}{d^{2} e h} - \frac {3 \, {\left (5 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b e^{2} f h p - 7 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b g p\right )} \log \left (h x - \sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} \sqrt {h x} + \sqrt {\frac {d h^{2}}{e}}\right )}{d^{2} e h} + \frac {6 \, {\left (7 \, b g h^{4} p x + 5 \, b f h^{4} p\right )} \log \left (e h^{2} x^{2} + d h^{2}\right )}{\sqrt {h x} h^{3} x^{3}} + \frac {2 \, {\left (84 \, b e g h^{4} p x^{3} + 20 \, b e f h^{4} p x^{2} - 21 \, b d g h^{4} p x \log \left (h^{2}\right ) - 15 \, b d f h^{4} p \log \left (h^{2}\right ) + 21 \, b d g h^{4} x \log \left (c\right ) + 21 \, a d g h^{4} x + 15 \, b d f h^{4} \log \left (c\right ) + 15 \, a d f h^{4}\right )}}{\sqrt {h x} d h^{3} x^{3}}}{105 \, h^{5}} \]
-1/105*(6*(5*sqrt(2)*(d*e^3*h^2)^(1/4)*b*e^2*f*h*p + 7*sqrt(2)*(d*e^3*h^2) ^(3/4)*b*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) + 6*(5*sqrt(2)*(d*e^3*h^2)^(1/4)*b*e^2*f*h*p + 7 *sqrt(2)*(d*e^3*h^2)^(3/4)*b*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) + 3*(5*sqrt(2)*(d*e^3*h^2)^ (1/4)*b*e^2*f*h*p - 7*sqrt(2)*(d*e^3*h^2)^(3/4)*b*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) - 3*(5*sqrt(2)*(d*e^3* h^2)^(1/4)*b*e^2*f*h*p - 7*sqrt(2)*(d*e^3*h^2)^(3/4)*b*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) + 6*(7*b*g*h^4*p* x + 5*b*f*h^4*p)*log(e*h^2*x^2 + d*h^2)/(sqrt(h*x)*h^3*x^3) + 2*(84*b*e*g* h^4*p*x^3 + 20*b*e*f*h^4*p*x^2 - 21*b*d*g*h^4*p*x*log(h^2) - 15*b*d*f*h^4* p*log(h^2) + 21*b*d*g*h^4*x*log(c) + 21*a*d*g*h^4*x + 15*b*d*f*h^4*log(c) + 15*a*d*f*h^4)/(sqrt(h*x)*d*h^3*x^3))/h^5
Timed out. \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=\int \frac {\left (f+g\,x\right )\,\left (a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\right )}{{\left (h\,x\right )}^{9/2}} \,d x \]