3.7.12 \(\int \frac {(f+g x)^2 (a+b \log (c (d+e x^2)^p))}{(h x)^{3/2}} \, dx\) [612]

3.7.12.1 Optimal result

Integrand size = 31, antiderivative size = 949 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=\frac {4 a f g \sqrt {h x}}{h^2}-\frac {16 b f g p \sqrt {h x}}{h^2}-\frac {8 b g^2 p (h x)^{3/2}}{9 h^3}-\frac {2 \sqrt {2} b \sqrt [4]{e} f^2 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 {4 \sqrt {2} b \sqrt [4]{d} f 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 d^{3/4} g^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} f^2 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 {4 \sqrt {2} b \sqrt [4]{d} f 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 d^{3/4} g^2 p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} h^{3/2}}+\frac {4 b f g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {2 g^2 (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3}+\frac {\sqrt {2} b \sqrt [4]{e} 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 )}{\sqrt [4]{d} h^{3/2}}-\frac {2 \sqrt {2} b \sqrt [4]{d} 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 )}{\sqrt [4]{e} h^{3/2}}+\frac {\sqrt {2} b d^{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 e^{3/4} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{e} 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 )}{\sqrt [4]{d} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{d} 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 )}{\sqrt [4]{e} h^{3/2}}-\frac {\sqrt {2} b d^{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 e^{3/4} h^{3/2}} \]

output

-8/9*b*g^2*p*(h*x)^(3/2)/h^3+2/3*g^2*(h*x)^(3/2)*(a+b*ln(c*(e*x^2+d)^p))/h 
^3-2*b*e^(1/4)*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^(1/4)/h^(3/2)-4*b*d^(1/4)*f*g*p*arctan(1-e^(1/4)*2^(1/2)*(h*x)^ 
(1/2)/d^(1/4)/h^(1/2))*2^(1/2)/e^(1/4)/h^(3/2)-2/3*b*d^(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)/e^(3/4)/h^(3/2)+2*b 
*e^(1/4)*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^(1/4)/h^(3/2)+4*b*d^(1/4)*f*g*p*arctan(1+e^(1/4)*2^(1/2)*(h*x)^(1/2)/ 
d^(1/4)/h^(1/2))*2^(1/2)/e^(1/4)/h^(3/2)+2/3*b*d^(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)/e^(3/4)/h^(3/2)+b*e^(1/4) 
*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^(1/4)/h^(3/2)-2*b*d^(1/4)*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)/e^(1/4)/h^(3/2)+1 
/3*b*d^(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)/e^(3/4)/h^(3/2)-b*e^(1/4)*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^(1/4) 
/h^(3/2)+2*b*d^(1/4)*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)/e^(1/4)/h^(3/2)-1/3*b*d^(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)/e^(3/4)/h^(3/2)-2*f^2*(a+b*ln(c*(e*x^2+d)^p))/h/(h*x)^(1/2)+4*a*f*g*( 
h*x)^(1/2)/h^2-16*b*f*g*p*(h*x)^(1/2)/h^2+4*b*f*g*ln(c*(e*x^2+d)^p)*(h*...
 

3.7.12.2 Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 478, normalized size of antiderivative = 0.50 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=\frac {2 x^{3/2} \left (2 a f g \sqrt {x}-8 b f g p \sqrt {x}+\frac {1}{3} a g^2 x^{3/2}-\frac {4}{9} b g^2 p x^{3/2}-\frac {2 \sqrt {2} b \sqrt [4]{d} f g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )}{\sqrt [4]{e}}+\frac {2 \sqrt {2} b \sqrt [4]{d} f g p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )}{\sqrt [4]{e}}-\frac {2 b (-d)^{3/4} g^2 p \arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )}{3 e^{3/4}}+\frac {2 b (-d)^{3/4} g^2 p \text {arctanh}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )}{3 e^{3/4}}+\frac {2 b \sqrt [4]{e} f^2 p \left (\arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )+\text {arctanh}\left (\frac {d \sqrt [4]{e} \sqrt {x}}{(-d)^{5/4}}\right )\right )}{\sqrt [4]{-d}}-\frac {\sqrt {2} b \sqrt [4]{d} f g p \log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )}{\sqrt [4]{e}}+\frac {\sqrt {2} b \sqrt [4]{d} f g p \log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )}{\sqrt [4]{e}}+2 b f g \sqrt {x} \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} b g^2 x^{3/2} \log \left (c \left (d+e x^2\right )^p\right )-\frac {f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {x}}\right )}{(h x)^{3/2}} \]

input

Integrate[((f + g*x)^2*(a + b*Log[c*(d + e*x^2)^p]))/(h*x)^(3/2),x]
 

output

(2*x^(3/2)*(2*a*f*g*Sqrt[x] - 8*b*f*g*p*Sqrt[x] + (a*g^2*x^(3/2))/3 - (4*b 
*g^2*p*x^(3/2))/9 - (2*Sqrt[2]*b*d^(1/4)*f*g*p*ArcTan[1 - (Sqrt[2]*e^(1/4) 
*Sqrt[x])/d^(1/4)])/e^(1/4) + (2*Sqrt[2]*b*d^(1/4)*f*g*p*ArcTan[1 + (Sqrt[ 
2]*e^(1/4)*Sqrt[x])/d^(1/4)])/e^(1/4) - (2*b*(-d)^(3/4)*g^2*p*ArcTan[(e^(1 
/4)*Sqrt[x])/(-d)^(1/4)])/(3*e^(3/4)) + (2*b*(-d)^(3/4)*g^2*p*ArcTanh[(e^( 
1/4)*Sqrt[x])/(-d)^(1/4)])/(3*e^(3/4)) + (2*b*e^(1/4)*f^2*p*(ArcTan[(e^(1/ 
4)*Sqrt[x])/(-d)^(1/4)] + ArcTanh[(d*e^(1/4)*Sqrt[x])/(-d)^(5/4)]))/(-d)^( 
1/4) - (Sqrt[2]*b*d^(1/4)*f*g*p*Log[Sqrt[d] - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt 
[x] + Sqrt[e]*x])/e^(1/4) + (Sqrt[2]*b*d^(1/4)*f*g*p*Log[Sqrt[d] + Sqrt[2] 
*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x])/e^(1/4) + 2*b*f*g*Sqrt[x]*Log[c*(d 
+ e*x^2)^p] + (b*g^2*x^(3/2)*Log[c*(d + e*x^2)^p])/3 - (f^2*(a + b*Log[c*( 
d + e*x^2)^p]))/Sqrt[x]))/(h*x)^(3/2)
 

3.7.12.3 Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 922, normalized size of antiderivative = 0.97, 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.

\(\displaystyle \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx\)

\(\Big \downarrow \) 2917

\(\displaystyle \frac {2 \int \frac {(f h+g x h)^2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{h^3 x}d\sqrt {h x}}{h}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \int \frac {(f h+g x h)^2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{h x}d\sqrt {h x}}{h^3}\)

\(\Big \downarrow \) 2926

\(\displaystyle \frac {2 \int \left (\frac {h \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f^2}{x}+2 g h \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f+g^2 h x \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )\right )d\sqrt {h x}}{h^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (-\frac {\sqrt {2} b \sqrt [4]{e} h^{3/2} p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right ) f^2}{\sqrt [4]{d}}+\frac {\sqrt {2} b \sqrt [4]{e} h^{3/2} p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right ) f^2}{\sqrt [4]{d}}-\frac {h^2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f^2}{\sqrt {h x}}+\frac {b \sqrt [4]{e} h^{3/2} p \log \left (\sqrt {e} x h+\sqrt {d} h-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x} \sqrt {h}\right ) f^2}{\sqrt {2} \sqrt [4]{d}}-\frac {b \sqrt [4]{e} h^{3/2} p \log \left (\sqrt {e} x h+\sqrt {d} h+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x} \sqrt {h}\right ) f^2}{\sqrt {2} \sqrt [4]{d}}-\frac {2 \sqrt {2} b \sqrt [4]{d} g h^{3/2} p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right ) f}{\sqrt [4]{e}}+\frac {2 \sqrt {2} b \sqrt [4]{d} g h^{3/2} p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right ) f}{\sqrt [4]{e}}+2 b g h \sqrt {h x} \log \left (c \left (e x^2+d\right )^p\right ) f-\frac {\sqrt {2} b \sqrt [4]{d} g h^{3/2} p \log \left (\sqrt {e} x h+\sqrt {d} h-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x} \sqrt {h}\right ) f}{\sqrt [4]{e}}+\frac {\sqrt {2} b \sqrt [4]{d} g h^{3/2} p \log \left (\sqrt {e} x h+\sqrt {d} h+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x} \sqrt {h}\right ) f}{\sqrt [4]{e}}+2 a g h \sqrt {h x} f-8 b g h p \sqrt {h x} f-\frac {4}{9} b g^2 p (h x)^{3/2}-\frac {\sqrt {2} b d^{3/4} g^2 h^{3/2} p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4}}+\frac {\sqrt {2} b d^{3/4} g^2 h^{3/2} p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{3 e^{3/4}}+\frac {1}{3} g^2 (h x)^{3/2} \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )+\frac {b d^{3/4} g^2 h^{3/2} 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} e^{3/4}}-\frac {b d^{3/4} g^2 h^{3/2} 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} e^{3/4}}\right )}{h^3}\)

input

Int[((f + g*x)^2*(a + b*Log[c*(d + e*x^2)^p]))/(h*x)^(3/2),x]
 

output

(2*(2*a*f*g*h*Sqrt[h*x] - 8*b*f*g*h*p*Sqrt[h*x] - (4*b*g^2*p*(h*x)^(3/2))/ 
9 - (Sqrt[2]*b*e^(1/4)*f^2*h^(3/2)*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x] 
)/(d^(1/4)*Sqrt[h])])/d^(1/4) - (2*Sqrt[2]*b*d^(1/4)*f*g*h^(3/2)*p*ArcTan[ 
1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/e^(1/4) - (Sqrt[2]*b*d 
^(3/4)*g^2*h^(3/2)*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[ 
h])])/(3*e^(3/4)) + (Sqrt[2]*b*e^(1/4)*f^2*h^(3/2)*p*ArcTan[1 + (Sqrt[2]*e 
^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/d^(1/4) + (2*Sqrt[2]*b*d^(1/4)*f*g*h 
^(3/2)*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/e^(1/4 
) + (Sqrt[2]*b*d^(3/4)*g^2*h^(3/2)*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x] 
)/(d^(1/4)*Sqrt[h])])/(3*e^(3/4)) + 2*b*f*g*h*Sqrt[h*x]*Log[c*(d + e*x^2)^ 
p] - (f^2*h^2*(a + b*Log[c*(d + e*x^2)^p]))/Sqrt[h*x] + (g^2*(h*x)^(3/2)*( 
a + b*Log[c*(d + e*x^2)^p]))/3 + (b*e^(1/4)*f^2*h^(3/2)*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) 
) - (Sqrt[2]*b*d^(1/4)*f*g*h^(3/2)*p*Log[Sqrt[d]*h + Sqrt[e]*h*x - Sqrt[2] 
*d^(1/4)*e^(1/4)*Sqrt[h]*Sqrt[h*x]])/e^(1/4) + (b*d^(3/4)*g^2*h^(3/2)*p*Lo 
g[Sqrt[d]*h + Sqrt[e]*h*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h]*Sqrt[h*x]])/(3 
*Sqrt[2]*e^(3/4)) - (b*e^(1/4)*f^2*h^(3/2)*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)) + (Sqrt[2]* 
b*d^(1/4)*f*g*h^(3/2)*p*Log[Sqrt[d]*h + Sqrt[e]*h*x + Sqrt[2]*d^(1/4)*e^(1 
/4)*Sqrt[h]*Sqrt[h*x]])/e^(1/4) - (b*d^(3/4)*g^2*h^(3/2)*p*Log[Sqrt[d]*...
 

3.7.12.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2917

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]
 

rule 2926

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]
 

3.7.12.4 Maple [F]

\[\int \frac {\left (g x +f \right )^{2} \left (a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )\right )}{\left (h x \right )^{\frac {3}{2}}}d x\]

input

int((g*x+f)^2*(a+b*ln(c*(e*x^2+d)^p))/(h*x)^(3/2),x)
 

output

int((g*x+f)^2*(a+b*ln(c*(e*x^2+d)^p))/(h*x)^(3/2),x)
 

3.7.12.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2118 vs. \(2 (673) = 1346\).

Time = 0.46 (sec) , antiderivative size = 2118, normalized size of antiderivative = 2.23 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=\text {Too large to display} \]

input

integrate((g*x+f)^2*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(3/2),x, algorithm="fri 
cas")
 

output

-2/9*(3*h^2*x*sqrt(-(e*h^3*sqrt(-(81*b^4*e^4*f^8 - 540*b^4*d*e^3*f^6*g^2 + 
 918*b^4*d^2*e^2*f^4*g^4 - 60*b^4*d^3*e*f^2*g^6 + b^4*d^4*g^8)*p^4/(d*e^3* 
h^6)) + 12*(3*b^2*e*f^3*g + b^2*d*f*g^3)*p^2)/(e*h^3))*log(32*(81*b^3*e^4* 
f^8 + 108*b^3*d*e^3*f^6*g^2 - 1242*b^3*d^2*e^2*f^4*g^4 + 12*b^3*d^3*e*f^2* 
g^6 + b^3*d^4*g^8)*sqrt(h*x)*p^3 + 32*((3*d*e^3*f^2 + d^2*e^2*g^2)*h^5*sqr 
t(-(81*b^4*e^4*f^8 - 540*b^4*d*e^3*f^6*g^2 + 918*b^4*d^2*e^2*f^4*g^4 - 60* 
b^4*d^3*e*f^2*g^6 + b^4*d^4*g^8)*p^4/(d*e^3*h^6)) - 6*(9*b^2*d*e^3*f^5*g - 
 30*b^2*d^2*e^2*f^3*g^3 + b^2*d^3*e*f*g^5)*h^2*p^2)*sqrt(-(e*h^3*sqrt(-(81 
*b^4*e^4*f^8 - 540*b^4*d*e^3*f^6*g^2 + 918*b^4*d^2*e^2*f^4*g^4 - 60*b^4*d^ 
3*e*f^2*g^6 + b^4*d^4*g^8)*p^4/(d*e^3*h^6)) + 12*(3*b^2*e*f^3*g + b^2*d*f* 
g^3)*p^2)/(e*h^3))) - 3*h^2*x*sqrt(-(e*h^3*sqrt(-(81*b^4*e^4*f^8 - 540*b^4 
*d*e^3*f^6*g^2 + 918*b^4*d^2*e^2*f^4*g^4 - 60*b^4*d^3*e*f^2*g^6 + b^4*d^4* 
g^8)*p^4/(d*e^3*h^6)) + 12*(3*b^2*e*f^3*g + b^2*d*f*g^3)*p^2)/(e*h^3))*log 
(32*(81*b^3*e^4*f^8 + 108*b^3*d*e^3*f^6*g^2 - 1242*b^3*d^2*e^2*f^4*g^4 + 1 
2*b^3*d^3*e*f^2*g^6 + b^3*d^4*g^8)*sqrt(h*x)*p^3 - 32*((3*d*e^3*f^2 + d^2* 
e^2*g^2)*h^5*sqrt(-(81*b^4*e^4*f^8 - 540*b^4*d*e^3*f^6*g^2 + 918*b^4*d^2*e 
^2*f^4*g^4 - 60*b^4*d^3*e*f^2*g^6 + b^4*d^4*g^8)*p^4/(d*e^3*h^6)) - 6*(9*b 
^2*d*e^3*f^5*g - 30*b^2*d^2*e^2*f^3*g^3 + b^2*d^3*e*f*g^5)*h^2*p^2)*sqrt(- 
(e*h^3*sqrt(-(81*b^4*e^4*f^8 - 540*b^4*d*e^3*f^6*g^2 + 918*b^4*d^2*e^2*f^4 
*g^4 - 60*b^4*d^3*e*f^2*g^6 + b^4*d^4*g^8)*p^4/(d*e^3*h^6)) + 12*(3*b^2...
 

3.7.12.6 Sympy [F(-2)]

Exception generated. \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]

input

integrate((g*x+f)**2*(a+b*ln(c*(e*x**2+d)**p))/(h*x)**(3/2),x)
 

output

Exception raised: TypeError >> Invalid comparison of non-real zoo
 

3.7.12.7 Maxima [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 1119, normalized size of antiderivative = 1.18 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=\text {Too large to display} \]

input

integrate((g*x+f)^2*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(3/2),x, algorithm="max 
ima")
 

output

2/3*b*g^2*x^3*log((e*x^2 + d)^p*c)/(h*x)^(3/2) - b*e*f^2*p*(sqrt(2)*log(sq 
rt(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*sq 
rt(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 + 2/3*a*g^2 
*x^3/(h*x)^(3/2) + 4*b*f*g*x^2*log((e*x^2 + d)^p*c)/(h*x)^(3/2) + 4*a*f*g* 
x^2/(h*x)^(3/2) - 2*b*f^2*log((e*x^2 + d)^p*c)/(sqrt(h*x)*h) - 2*(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) + sqrt(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)) + 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)*sq 
rt(e)*h)*sqrt(d)) + sqrt(2)*h^3*log(-(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) - s 
qrt(2)*(d*h^2)^(1/4)*e^(1/4) - 2*sqrt(h*x)*sqrt(e))/(sqrt(2)*sqrt(-sqrt...
 

3.7.12.8 Giac [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 577, normalized size of antiderivative = 0.61 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=\frac {6 \, {\left (\frac {\sqrt {h x} b g^{2} p x}{h} - \frac {3 \, b f^{2} p}{\sqrt {h x}} + \frac {6 \, \sqrt {h x} b f g p}{h}\right )} \log \left (e h^{2} x^{2} + d h^{2}\right ) - \frac {2 \, {\left (3 \, b g^{2} p \log \left (h^{2}\right ) + 4 \, b g^{2} p - 3 \, b g^{2} \log \left (c\right ) - 3 \, a g^{2}\right )} \sqrt {h x} x}{h} + \frac {6 \, {\left (6 \, \sqrt {2} b d e^{2} f g h p + 3 \, \sqrt {2} \sqrt {d e} b e^{2} f^{2} h p + \sqrt {2} \sqrt {d e} b d e g^{2} h 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 )}{\left (d e^{3} h^{2}\right )^{\frac {3}{4}}} + \frac {6 \, {\left (6 \, \sqrt {2} b d e^{2} f g h p + 3 \, \sqrt {2} \sqrt {d e} b e^{2} f^{2} h p + \sqrt {2} \sqrt {d e} b d e g^{2} h 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 )}{\left (d e^{3} h^{2}\right )^{\frac {3}{4}}} + \frac {3 \, {\left (6 \, \sqrt {2} b d e^{2} f g h p + 3 \, \sqrt {2} \sqrt {d e} b e^{2} f^{2} h p + \sqrt {2} \sqrt {d e} b d e g^{2} h 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 )}{\left (d e^{3} h^{2}\right )^{\frac {3}{4}}} - \frac {3 \, {\left (6 \, \sqrt {2} b d e^{2} f g h p - 3 \, \sqrt {2} \sqrt {d e} b e^{2} f^{2} h p - \sqrt {2} \sqrt {d e} b d e g^{2} h 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 )}{\left (d e^{3} h^{2}\right )^{\frac {3}{4}}} + \frac {18 \, {\left (b f^{2} p \log \left (h^{2}\right ) - b f^{2} \log \left (c\right ) - a f^{2}\right )}}{\sqrt {h x}} - \frac {36 \, {\left (b f g p \log \left (h^{2}\right ) + 4 \, b f g p - b f g \log \left (c\right ) - a f g\right )} \sqrt {h x}}{h}}{9 \, h} \]

input

integrate((g*x+f)^2*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(3/2),x, algorithm="gia 
c")
 

output

1/9*(6*(sqrt(h*x)*b*g^2*p*x/h - 3*b*f^2*p/sqrt(h*x) + 6*sqrt(h*x)*b*f*g*p/ 
h)*log(e*h^2*x^2 + d*h^2) - 2*(3*b*g^2*p*log(h^2) + 4*b*g^2*p - 3*b*g^2*lo 
g(c) - 3*a*g^2)*sqrt(h*x)*x/h + 6*(6*sqrt(2)*b*d*e^2*f*g*h*p + 3*sqrt(2)*s 
qrt(d*e)*b*e^2*f^2*h*p + sqrt(2)*sqrt(d*e)*b*d*e*g^2*h*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^3*h^2)^(3 
/4) + 6*(6*sqrt(2)*b*d*e^2*f*g*h*p + 3*sqrt(2)*sqrt(d*e)*b*e^2*f^2*h*p + s 
qrt(2)*sqrt(d*e)*b*d*e*g^2*h*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^3*h^2)^(3/4) + 3*(6*sqrt(2)*b*d*e^ 
2*f*g*h*p + 3*sqrt(2)*sqrt(d*e)*b*e^2*f^2*h*p + sqrt(2)*sqrt(d*e)*b*d*e*g^ 
2*h*p)*log(h*x + sqrt(2)*(d*h^2/e)^(1/4)*sqrt(h*x) + sqrt(d*h^2/e))/(d*e^3 
*h^2)^(3/4) - 3*(6*sqrt(2)*b*d*e^2*f*g*h*p - 3*sqrt(2)*sqrt(d*e)*b*e^2*f^2 
*h*p - sqrt(2)*sqrt(d*e)*b*d*e*g^2*h*p)*log(h*x - sqrt(2)*(d*h^2/e)^(1/4)* 
sqrt(h*x) + sqrt(d*h^2/e))/(d*e^3*h^2)^(3/4) + 18*(b*f^2*p*log(h^2) - b*f^ 
2*log(c) - a*f^2)/sqrt(h*x) - 36*(b*f*g*p*log(h^2) + 4*b*f*g*p - b*f*g*log 
(c) - a*f*g)*sqrt(h*x)/h)/h
 

3.7.12.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^2\,\left (a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\right )}{{\left (h\,x\right )}^{3/2}} \,d x \]

input

int(((f + g*x)^2*(a + b*log(c*(d + e*x^2)^p)))/(h*x)^(3/2),x)
 

output

int(((f + g*x)^2*(a + b*log(c*(d + e*x^2)^p)))/(h*x)^(3/2), x)