\(\int \frac {(f+g x)^2 (a+b \log (c (d+e x^2)^p))}{(h x)^{5/2}} \, dx\) [613]

Optimal result

Integrand size = 31, antiderivative size = 692 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{5/2}} \, dx=\frac {2 a g^2 \sqrt {h x}}{h^3}-\frac {8 b g^2 p \sqrt {h x}}{h^3}-\frac {2 \sqrt {2} b e^{3/4} f^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{5/2}}-\frac {4 \sqrt {2} b \sqrt [4]{e} f g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{5/2}}-\frac {2 \sqrt {2} b \sqrt [4]{d} g^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} h^{5/2}}+\frac {2 \sqrt {2} b e^{3/4} f^2 p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{5/2}}+\frac {4 \sqrt {2} b \sqrt [4]{e} f g p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{5/2}}+\frac {2 \sqrt {2} b \sqrt [4]{d} g^2 p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} h^{5/2}}+\frac {2 \sqrt {2} b e^{3/4} f^2 p \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}}{\sqrt {h} \left (\sqrt {d}+\sqrt {e} x\right )}\right )}{3 d^{3/4} h^{5/2}}-\frac {4 \sqrt {2} b \sqrt [4]{e} f g p \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}}{\sqrt {h} \left (\sqrt {d}+\sqrt {e} x\right )}\right )}{\sqrt [4]{d} h^{5/2}}+\frac {2 \sqrt {2} b \sqrt [4]{d} g^2 p \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}}{\sqrt {h} \left (\sqrt {d}+\sqrt {e} x\right )}\right )}{\sqrt [4]{e} h^{5/2}}+\frac {2 b g^2 \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^3}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h (h x)^{3/2}}-\frac {4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h^2 \sqrt {h x}} \] Output:

2*a*g^2*(h*x)^(1/2)/h^3-8*b*g^2*p*(h*x)^(1/2)/h^3-2/3*2^(1/2)*b*e^(3/4)*f^ 
2*p*arctan(1-2^(1/2)*e^(1/4)*(h*x)^(1/2)/d^(1/4)/h^(1/2))/d^(3/4)/h^(5/2)- 
4*2^(1/2)*b*e^(1/4)*f*g*p*arctan(1-2^(1/2)*e^(1/4)*(h*x)^(1/2)/d^(1/4)/h^( 
1/2))/d^(1/4)/h^(5/2)-2*2^(1/2)*b*d^(1/4)*g^2*p*arctan(1-2^(1/2)*e^(1/4)*( 
h*x)^(1/2)/d^(1/4)/h^(1/2))/e^(1/4)/h^(5/2)+2/3*2^(1/2)*b*e^(3/4)*f^2*p*ar 
ctan(1+2^(1/2)*e^(1/4)*(h*x)^(1/2)/d^(1/4)/h^(1/2))/d^(3/4)/h^(5/2)+4*2^(1 
/2)*b*e^(1/4)*f*g*p*arctan(1+2^(1/2)*e^(1/4)*(h*x)^(1/2)/d^(1/4)/h^(1/2))/ 
d^(1/4)/h^(5/2)+2*2^(1/2)*b*d^(1/4)*g^2*p*arctan(1+2^(1/2)*e^(1/4)*(h*x)^( 
1/2)/d^(1/4)/h^(1/2))/e^(1/4)/h^(5/2)+2/3*2^(1/2)*b*e^(3/4)*f^2*p*arctanh( 
2^(1/2)*d^(1/4)*e^(1/4)*(h*x)^(1/2)/h^(1/2)/(d^(1/2)+e^(1/2)*x))/d^(3/4)/h 
^(5/2)-4*2^(1/2)*b*e^(1/4)*f*g*p*arctanh(2^(1/2)*d^(1/4)*e^(1/4)*(h*x)^(1/ 
2)/h^(1/2)/(d^(1/2)+e^(1/2)*x))/d^(1/4)/h^(5/2)+2*2^(1/2)*b*d^(1/4)*g^2*p* 
arctanh(2^(1/2)*d^(1/4)*e^(1/4)*(h*x)^(1/2)/h^(1/2)/(d^(1/2)+e^(1/2)*x))/e 
^(1/4)/h^(5/2)+2*b*g^2*(h*x)^(1/2)*ln(c*(e*x^2+d)^p)/h^3-2/3*f^2*(a+b*ln(c 
*(e*x^2+d)^p))/h/(h*x)^(3/2)-4*f*g*(a+b*ln(c*(e*x^2+d)^p))/h^2/(h*x)^(1/2)
 

Mathematica [A] (verified)

Time = 1.21 (sec) , antiderivative size = 527, normalized size of antiderivative = 0.76 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{5/2}} \, dx=\frac {2 x^{5/2} \left (a g^2 \sqrt {x}-4 b g^2 p \sqrt {x}-\frac {\sqrt {2} b \sqrt [4]{d} g^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )}{\sqrt [4]{e}}+\frac {\sqrt {2} b \sqrt [4]{d} g^2 p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )}{\sqrt [4]{e}}+\frac {4 b \sqrt [4]{e} f g p \left (\arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )+\text {arctanh}\left (\frac {d \sqrt [4]{e} \sqrt {x}}{(-d)^{5/4}}\right )\right )}{\sqrt [4]{-d}}-\frac {b \sqrt [4]{d} g^2 p \log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )}{\sqrt {2} \sqrt [4]{e}}-\frac {b e^{3/4} f^2 p \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )+\log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )-\log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )\right )}{3 \sqrt {2} d^{3/4}}+\frac {b \sqrt [4]{d} g^2 p \log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )}{\sqrt {2} \sqrt [4]{e}}+b g^2 \sqrt {x} \log \left (c \left (d+e x^2\right )^p\right )-\frac {f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 x^{3/2}}-\frac {2 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {x}}\right )}{(h x)^{5/2}} \] Input:

Integrate[((f + g*x)^2*(a + b*Log[c*(d + e*x^2)^p]))/(h*x)^(5/2),x]
 

Output:

(2*x^(5/2)*(a*g^2*Sqrt[x] - 4*b*g^2*p*Sqrt[x] - (Sqrt[2]*b*d^(1/4)*g^2*p*A 
rcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)])/e^(1/4) + (Sqrt[2]*b*d^(1/4) 
*g^2*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)])/e^(1/4) + (4*b*e^(1/ 
4)*f*g*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^2*p*Log[Sqrt[d] - Sqrt[2]*d^(1/ 
4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x])/(Sqrt[2]*e^(1/4)) - (b*e^(3/4)*f^2*p*(2*A 
rcTan[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] + Sqr 
t[e]*x] - Log[Sqrt[d] + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x]))/(3* 
Sqrt[2]*d^(3/4)) + (b*d^(1/4)*g^2*p*Log[Sqrt[d] + Sqrt[2]*d^(1/4)*e^(1/4)* 
Sqrt[x] + Sqrt[e]*x])/(Sqrt[2]*e^(1/4)) + b*g^2*Sqrt[x]*Log[c*(d + e*x^2)^ 
p] - (f^2*(a + b*Log[c*(d + e*x^2)^p]))/(3*x^(3/2)) - (2*f*g*(a + b*Log[c* 
(d + e*x^2)^p]))/Sqrt[x]))/(h*x)^(5/2)
 

Rubi [A] (verified)

Time = 2.10 (sec) , antiderivative size = 904, normalized size of antiderivative = 1.31, 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.

\(\displaystyle \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{5/2}} \, dx\)

\(\Big \downarrow \) 2917

\(\displaystyle \frac {2 \int \frac {(f h+g x h)^2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{h^4 x^2}d\sqrt {h x}}{h}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2926

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

\(\Big \downarrow \) 2009

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

Input:

Int[((f + g*x)^2*(a + b*Log[c*(d + e*x^2)^p]))/(h*x)^(5/2),x]
 

Output:

(2*(a*g^2*Sqrt[h*x] - 4*b*g^2*p*Sqrt[h*x] - (Sqrt[2]*b*e^(3/4)*f^2*Sqrt[h] 
*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*d^(3/4)) 
- (2*Sqrt[2]*b*e^(1/4)*f*g*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^2*Sqrt[h]*p*ArcTan[1 
- (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/e^(1/4) + (Sqrt[2]*b*e^( 
3/4)*f^2*Sqrt[h]*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h] 
)])/(3*d^(3/4)) + (2*Sqrt[2]*b*e^(1/4)*f*g*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^2*Sqr 
t[h]*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/e^(1/4) 
+ b*g^2*Sqrt[h*x]*Log[c*(d + e*x^2)^p] - (f^2*h^2*(a + b*Log[c*(d + e*x^2) 
^p]))/(3*(h*x)^(3/2)) - (2*f*g*h*(a + b*Log[c*(d + e*x^2)^p]))/Sqrt[h*x] - 
 (b*e^(3/4)*f^2*Sqrt[h]*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)) + (Sqrt[2]*b*e^(1/4)*f*g*Sqr 
t[h]*p*Log[Sqrt[d]*h + Sqrt[e]*h*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h]*Sqrt[ 
h*x]])/d^(1/4) - (b*d^(1/4)*g^2*Sqrt[h]*p*Log[Sqrt[d]*h + Sqrt[e]*h*x - Sq 
rt[2]*d^(1/4)*e^(1/4)*Sqrt[h]*Sqrt[h*x]])/(Sqrt[2]*e^(1/4)) + (b*e^(3/4)*f 
^2*Sqrt[h]*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)) - (Sqrt[2]*b*e^(1/4)*f*g*Sqrt[h]*p*Log[Sq 
rt[d]*h + Sqrt[e]*h*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h]*Sqrt[h*x]])/d^(1/4 
) + (b*d^(1/4)*g^2*Sqrt[h]*p*Log[Sqrt[d]*h + Sqrt[e]*h*x + Sqrt[2]*d^(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]
 

Maple [F]

\[\int \frac {\left (g x +f \right )^{2} \left (a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )\right )}{\left (h x \right )^{\frac {5}{2}}}d x\]

Input:

int((g*x+f)^2*(a+b*ln(c*(e*x^2+d)^p))/(h*x)^(5/2),x)
 

Output:

int((g*x+f)^2*(a+b*ln(c*(e*x^2+d)^p))/(h*x)^(5/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2112 vs. \(2 (500) = 1000\).

Time = 0.19 (sec) , antiderivative size = 2112, normalized size of antiderivative = 3.05 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)^2*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(5/2),x, algorithm="fri 
cas")
 

Output:

2/3*(h^3*x^2*sqrt(-(d*h^5*sqrt(-(b^4*e^4*f^8 - 60*b^4*d*e^3*f^6*g^2 + 918* 
b^4*d^2*e^2*f^4*g^4 - 540*b^4*d^3*e*f^2*g^6 + 81*b^4*d^4*g^8)*p^4/(d^3*e*h 
^10)) + 12*(b^2*e*f^3*g + 3*b^2*d*f*g^3)*p^2)/(d*h^5))*log(16*(b^3*e^4*f^8 
 + 12*b^3*d*e^3*f^6*g^2 - 1242*b^3*d^2*e^2*f^4*g^4 + 108*b^3*d^3*e*f^2*g^6 
 + 81*b^3*d^4*g^8)*sqrt(h*x)*p^3 + 16*(6*d^3*e*f*g*h^8*sqrt(-(b^4*e^4*f^8 
- 60*b^4*d*e^3*f^6*g^2 + 918*b^4*d^2*e^2*f^4*g^4 - 540*b^4*d^3*e*f^2*g^6 + 
 81*b^4*d^4*g^8)*p^4/(d^3*e*h^10)) + (b^2*d*e^3*f^6 - 27*b^2*d^2*e^2*f^4*g 
^2 - 81*b^2*d^3*e*f^2*g^4 + 27*b^2*d^4*g^6)*h^3*p^2)*sqrt(-(d*h^5*sqrt(-(b 
^4*e^4*f^8 - 60*b^4*d*e^3*f^6*g^2 + 918*b^4*d^2*e^2*f^4*g^4 - 540*b^4*d^3* 
e*f^2*g^6 + 81*b^4*d^4*g^8)*p^4/(d^3*e*h^10)) + 12*(b^2*e*f^3*g + 3*b^2*d* 
f*g^3)*p^2)/(d*h^5))) - h^3*x^2*sqrt(-(d*h^5*sqrt(-(b^4*e^4*f^8 - 60*b^4*d 
*e^3*f^6*g^2 + 918*b^4*d^2*e^2*f^4*g^4 - 540*b^4*d^3*e*f^2*g^6 + 81*b^4*d^ 
4*g^8)*p^4/(d^3*e*h^10)) + 12*(b^2*e*f^3*g + 3*b^2*d*f*g^3)*p^2)/(d*h^5))* 
log(16*(b^3*e^4*f^8 + 12*b^3*d*e^3*f^6*g^2 - 1242*b^3*d^2*e^2*f^4*g^4 + 10 
8*b^3*d^3*e*f^2*g^6 + 81*b^3*d^4*g^8)*sqrt(h*x)*p^3 - 16*(6*d^3*e*f*g*h^8* 
sqrt(-(b^4*e^4*f^8 - 60*b^4*d*e^3*f^6*g^2 + 918*b^4*d^2*e^2*f^4*g^4 - 540* 
b^4*d^3*e*f^2*g^6 + 81*b^4*d^4*g^8)*p^4/(d^3*e*h^10)) + (b^2*d*e^3*f^6 - 2 
7*b^2*d^2*e^2*f^4*g^2 - 81*b^2*d^3*e*f^2*g^4 + 27*b^2*d^4*g^6)*h^3*p^2)*sq 
rt(-(d*h^5*sqrt(-(b^4*e^4*f^8 - 60*b^4*d*e^3*f^6*g^2 + 918*b^4*d^2*e^2*f^4 
*g^4 - 540*b^4*d^3*e*f^2*g^6 + 81*b^4*d^4*g^8)*p^4/(d^3*e*h^10)) + 12*(...
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((g*x+f)**2*(a+b*ln(c*(e*x**2+d)**p))/(h*x)**(5/2),x)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1102 vs. \(2 (500) = 1000\).

Time = 0.14 (sec) , antiderivative size = 1102, normalized size of antiderivative = 1.59 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)^2*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(5/2),x, algorithm="max 
ima")
 

Output:

2*b*g^2*x^3*log((e*x^2 + d)^p*c)/(h*x)^(5/2) - 2*b*e*f*g*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 + 2*a*g^2 
*x^3/(h*x)^(5/2) - 4*b*f*g*x^2*log((e*x^2 + d)^p*c)/(h*x)^(5/2) + 1/3*(sqr 
t(2)*h^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)^(3/4)*e^(1/4)) - sqrt(2)*h^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)^(3/4)*e^(1/4)) + sqrt(2) 
*h*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)) 
+ sqrt(2)*h*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)*...
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 631, normalized size of antiderivative = 0.91 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate((g*x+f)^2*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(5/2),x, algorithm="gia 
c")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{5/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^2\,\left (a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\right )}{{\left (h\,x\right )}^{5/2}} \,d x \] Input:

int(((f + g*x)^2*(a + b*log(c*(d + e*x^2)^p)))/(h*x)^(5/2),x)
 

Output:

int(((f + g*x)^2*(a + b*log(c*(d + e*x^2)^p)))/(h*x)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 595, normalized size of antiderivative = 0.86 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{5/2}} \, dx =\text {Too large to display} \] Input:

int((g*x+f)^2*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(5/2),x)
 

Output:

(sqrt(h)*( - 12*sqrt(x)*e**(1/4)*d**(3/4)*sqrt(2)*atan((e**(1/4)*d**(1/4)* 
sqrt(2) - 2*sqrt(x)*sqrt(e))/(e**(1/4)*d**(1/4)*sqrt(2)))*b*e*f*g*p*x - 6* 
sqrt(x)*e**(3/4)*d**(1/4)*sqrt(2)*atan((e**(1/4)*d**(1/4)*sqrt(2) - 2*sqrt 
(x)*sqrt(e))/(e**(1/4)*d**(1/4)*sqrt(2)))*b*d*g**2*p*x - 2*sqrt(x)*e**(3/4 
)*d**(1/4)*sqrt(2)*atan((e**(1/4)*d**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(e))/(e 
**(1/4)*d**(1/4)*sqrt(2)))*b*e*f**2*p*x + 12*sqrt(x)*e**(1/4)*d**(3/4)*sqr 
t(2)*atan((e**(1/4)*d**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(e))/(e**(1/4)*d**(1/ 
4)*sqrt(2)))*b*e*f*g*p*x + 6*sqrt(x)*e**(3/4)*d**(1/4)*sqrt(2)*atan((e**(1 
/4)*d**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(e))/(e**(1/4)*d**(1/4)*sqrt(2)))*b*d 
*g**2*p*x + 2*sqrt(x)*e**(3/4)*d**(1/4)*sqrt(2)*atan((e**(1/4)*d**(1/4)*sq 
rt(2) + 2*sqrt(x)*sqrt(e))/(e**(1/4)*d**(1/4)*sqrt(2)))*b*e*f**2*p*x + 12* 
sqrt(x)*e**(1/4)*d**(3/4)*sqrt(2)*log( - sqrt(x)*e**(1/4)*d**(1/4)*sqrt(2) 
 + sqrt(d) + sqrt(e)*x)*b*e*f*g*p*x - 6*sqrt(x)*e**(1/4)*d**(3/4)*sqrt(2)* 
log((d + e*x**2)**p*c)*b*e*f*g*x - 6*sqrt(x)*e**(3/4)*d**(1/4)*sqrt(2)*log 
( - sqrt(x)*e**(1/4)*d**(1/4)*sqrt(2) + sqrt(d) + sqrt(e)*x)*b*d*g**2*p*x 
- 2*sqrt(x)*e**(3/4)*d**(1/4)*sqrt(2)*log( - sqrt(x)*e**(1/4)*d**(1/4)*sqr 
t(2) + sqrt(d) + sqrt(e)*x)*b*e*f**2*p*x + 3*sqrt(x)*e**(3/4)*d**(1/4)*sqr 
t(2)*log((d + e*x**2)**p*c)*b*d*g**2*x + sqrt(x)*e**(3/4)*d**(1/4)*sqrt(2) 
*log((d + e*x**2)**p*c)*b*e*f**2*x - 2*log((d + e*x**2)**p*c)*b*d*e*f**2 - 
 12*log((d + e*x**2)**p*c)*b*d*e*f*g*x + 6*log((d + e*x**2)**p*c)*b*d*e...