\(\int \frac {(f+g x)^2 (a+b \log (c (d+e x^2)^p))}{\sqrt {h x}} \, dx\) [611]

Optimal result

Integrand size = 31, antiderivative size = 760 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\frac {2 a f^2 \sqrt {h x}}{h}-\frac {8 b f^2 p \sqrt {h x}}{h}+\frac {8 b d g^2 p \sqrt {h x}}{5 e h}-\frac {16 b f g p (h x)^{3/2}}{9 h^2}-\frac {8 b g^2 p (h x)^{5/2}}{25 h^3}-\frac {2 \sqrt {2} b \sqrt [4]{d} f^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} \sqrt {h}}-\frac {4 \sqrt {2} b d^{3/4} f g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} \sqrt {h}}+\frac {2 \sqrt {2} b d^{5/4} g^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 e^{5/4} \sqrt {h}}+\frac {2 \sqrt {2} b \sqrt [4]{d} f^2 p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} \sqrt {h}}+\frac {4 \sqrt {2} b d^{3/4} f g p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} \sqrt {h}}-\frac {2 \sqrt {2} b d^{5/4} g^2 p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 e^{5/4} \sqrt {h}}+\frac {2 \sqrt {2} b \sqrt [4]{d} 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 )}{\sqrt [4]{e} \sqrt {h}}-\frac {4 \sqrt {2} b d^{3/4} 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 )}{3 e^{3/4} \sqrt {h}}-\frac {2 \sqrt {2} b d^{5/4} 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 )}{5 e^{5/4} \sqrt {h}}+\frac {2 b f^2 \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac {4 f g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}+\frac {2 g^2 (h x)^{5/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^3} \] Output:

2*a*f^2*(h*x)^(1/2)/h-8*b*f^2*p*(h*x)^(1/2)/h+8/5*b*d*g^2*p*(h*x)^(1/2)/e/ 
h-16/9*b*f*g*p*(h*x)^(3/2)/h^2-8/25*b*g^2*p*(h*x)^(5/2)/h^3-2*2^(1/2)*b*d^ 
(1/4)*f^2*p*arctan(1-2^(1/2)*e^(1/4)*(h*x)^(1/2)/d^(1/4)/h^(1/2))/e^(1/4)/ 
h^(1/2)-4/3*2^(1/2)*b*d^(3/4)*f*g*p*arctan(1-2^(1/2)*e^(1/4)*(h*x)^(1/2)/d 
^(1/4)/h^(1/2))/e^(3/4)/h^(1/2)+2/5*2^(1/2)*b*d^(5/4)*g^2*p*arctan(1-2^(1/ 
2)*e^(1/4)*(h*x)^(1/2)/d^(1/4)/h^(1/2))/e^(5/4)/h^(1/2)+2*2^(1/2)*b*d^(1/4 
)*f^2*p*arctan(1+2^(1/2)*e^(1/4)*(h*x)^(1/2)/d^(1/4)/h^(1/2))/e^(1/4)/h^(1 
/2)+4/3*2^(1/2)*b*d^(3/4)*f*g*p*arctan(1+2^(1/2)*e^(1/4)*(h*x)^(1/2)/d^(1/ 
4)/h^(1/2))/e^(3/4)/h^(1/2)-2/5*2^(1/2)*b*d^(5/4)*g^2*p*arctan(1+2^(1/2)*e 
^(1/4)*(h*x)^(1/2)/d^(1/4)/h^(1/2))/e^(5/4)/h^(1/2)+2*2^(1/2)*b*d^(1/4)*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 
))/e^(1/4)/h^(1/2)-4/3*2^(1/2)*b*d^(3/4)*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))/e^(3/4)/h^(1/2)-2/5*2^(1/2)* 
b*d^(5/4)*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^(5/4)/h^(1/2)+2*b*f^2*(h*x)^(1/2)*ln(c*(e*x^2+d)^p)/h+4/3 
*f*g*(h*x)^(3/2)*(a+b*ln(c*(e*x^2+d)^p))/h^2+2/5*g^2*(h*x)^(5/2)*(a+b*ln(c 
*(e*x^2+d)^p))/h^3
 

Mathematica [A] (verified)

Time = 0.91 (sec) , antiderivative size = 628, normalized size of antiderivative = 0.83 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\frac {2 \sqrt {x} \left (a f^2 \sqrt {x}-4 b f^2 p \sqrt {x}+\frac {2}{3} a f g x^{3/2}-\frac {8}{9} b f g p x^{3/2}-\frac {\sqrt {2} b \sqrt [4]{d} f^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )}{\sqrt [4]{e}}+\frac {\sqrt {2} b \sqrt [4]{d} f^2 p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )}{\sqrt [4]{e}}-\frac {4 b (-d)^{3/4} f g p \arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )}{3 e^{3/4}}+\frac {4 b (-d)^{3/4} f g p \text {arctanh}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )}{3 e^{3/4}}-\frac {b \sqrt [4]{d} f^2 p \log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )}{\sqrt {2} \sqrt [4]{e}}+\frac {b \sqrt [4]{d} f^2 p \log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )}{\sqrt {2} \sqrt [4]{e}}-\frac {b g^2 p \left (-40 d \sqrt [4]{e} \sqrt {x}+8 e^{5/4} x^{5/2}-10 \sqrt {2} d^{5/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )+10 \sqrt {2} d^{5/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )-5 \sqrt {2} d^{5/4} \log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )+5 \sqrt {2} d^{5/4} \log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )\right )}{50 e^{5/4}}+b f^2 \sqrt {x} \log \left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} b f g x^{3/2} \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^{5/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )\right )}{\sqrt {h x}} \] Input:

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

Output:

(2*Sqrt[x]*(a*f^2*Sqrt[x] - 4*b*f^2*p*Sqrt[x] + (2*a*f*g*x^(3/2))/3 - (8*b 
*f*g*p*x^(3/2))/9 - (Sqrt[2]*b*d^(1/4)*f^2*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*S 
qrt[x])/d^(1/4)])/e^(1/4) + (Sqrt[2]*b*d^(1/4)*f^2*p*ArcTan[1 + (Sqrt[2]*e 
^(1/4)*Sqrt[x])/d^(1/4)])/e^(1/4) - (4*b*(-d)^(3/4)*f*g*p*ArcTan[(e^(1/4)* 
Sqrt[x])/(-d)^(1/4)])/(3*e^(3/4)) + (4*b*(-d)^(3/4)*f*g*p*ArcTanh[(e^(1/4) 
*Sqrt[x])/(-d)^(1/4)])/(3*e^(3/4)) - (b*d^(1/4)*f^2*p*Log[Sqrt[d] - Sqrt[2 
]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x])/(Sqrt[2]*e^(1/4)) + (b*d^(1/4)*f^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*p*(-40*d*e^(1/4)*Sqrt[x] + 8*e^(5/4)*x^(5/2) - 10*Sqrt[2]* 
d^(5/4)*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)] + 10*Sqrt[2]*d^(5/4) 
*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)] - 5*Sqrt[2]*d^(5/4)*Log[Sqr 
t[d] - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x] + 5*Sqrt[2]*d^(5/4)*Lo 
g[Sqrt[d] + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x]))/(50*e^(5/4)) + 
b*f^2*Sqrt[x]*Log[c*(d + e*x^2)^p] + (2*b*f*g*x^(3/2)*Log[c*(d + e*x^2)^p] 
)/3 + (g^2*x^(5/2)*(a + b*Log[c*(d + e*x^2)^p]))/5))/Sqrt[h*x]
 

Rubi [A] (verified)

Time = 2.12 (sec) , antiderivative size = 976, normalized size of antiderivative = 1.28, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2917, 27, 2921, 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.

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

\(\Big \downarrow \) 2917

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2921

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (-\frac {\sqrt {2} b \sqrt [4]{d} f^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right ) h^{5/2}}{\sqrt [4]{e}}+\frac {\sqrt {2} b d^{5/4} g^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right ) h^{5/2}}{5 e^{5/4}}-\frac {2 \sqrt {2} b d^{3/4} f g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right ) h^{5/2}}{3 e^{3/4}}+\frac {\sqrt {2} b \sqrt [4]{d} f^2 p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right ) h^{5/2}}{\sqrt [4]{e}}-\frac {\sqrt {2} b d^{5/4} g^2 p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right ) h^{5/2}}{5 e^{5/4}}+\frac {2 \sqrt {2} b d^{3/4} f g p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right ) h^{5/2}}{3 e^{3/4}}-\frac {b \sqrt [4]{d} f^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 ) h^{5/2}}{\sqrt {2} \sqrt [4]{e}}+\frac {b d^{5/4} 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 ) h^{5/2}}{5 \sqrt {2} e^{5/4}}+\frac {\sqrt {2} b d^{3/4} f 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 ) h^{5/2}}{3 e^{3/4}}+\frac {b \sqrt [4]{d} f^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 ) h^{5/2}}{\sqrt {2} \sqrt [4]{e}}-\frac {b d^{5/4} 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 ) h^{5/2}}{5 \sqrt {2} e^{5/4}}-\frac {\sqrt {2} b d^{3/4} f 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 ) h^{5/2}}{3 e^{3/4}}+b f^2 \sqrt {h x} \log \left (c \left (e x^2+d\right )^p\right ) h^2+a f^2 \sqrt {h x} h^2-4 b f^2 p \sqrt {h x} h^2+\frac {4 b d g^2 p \sqrt {h x} h^2}{5 e}-\frac {8}{9} b f g p (h x)^{3/2} h+\frac {2}{3} f g (h x)^{3/2} \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) h-\frac {4}{25} b g^2 p (h x)^{5/2}+\frac {1}{5} g^2 (h x)^{5/2} \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )\right )}{h^3}\)

Input:

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

Output:

(2*(a*f^2*h^2*Sqrt[h*x] - 4*b*f^2*h^2*p*Sqrt[h*x] + (4*b*d*g^2*h^2*p*Sqrt[ 
h*x])/(5*e) - (8*b*f*g*h*p*(h*x)^(3/2))/9 - (4*b*g^2*p*(h*x)^(5/2))/25 - ( 
Sqrt[2]*b*d^(1/4)*f^2*h^(5/2)*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^ 
(1/4)*Sqrt[h])])/e^(1/4) - (2*Sqrt[2]*b*d^(3/4)*f*g*h^(5/2)*p*ArcTan[1 - ( 
Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*e^(3/4)) + (Sqrt[2]*b*d^ 
(5/4)*g^2*h^(5/2)*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h 
])])/(5*e^(5/4)) + (Sqrt[2]*b*d^(1/4)*f^2*h^(5/2)*p*ArcTan[1 + (Sqrt[2]*e^ 
(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/e^(1/4) + (2*Sqrt[2]*b*d^(3/4)*f*g*h^ 
(5/2)*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*e^(3 
/4)) - (Sqrt[2]*b*d^(5/4)*g^2*h^(5/2)*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h 
*x])/(d^(1/4)*Sqrt[h])])/(5*e^(5/4)) + b*f^2*h^2*Sqrt[h*x]*Log[c*(d + e*x^ 
2)^p] + (2*f*g*h*(h*x)^(3/2)*(a + b*Log[c*(d + e*x^2)^p]))/3 + (g^2*(h*x)^ 
(5/2)*(a + b*Log[c*(d + e*x^2)^p]))/5 - (b*d^(1/4)*f^2*h^(5/2)*p*Log[Sqrt[ 
d]*h + Sqrt[e]*h*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h]*Sqrt[h*x]])/(Sqrt[2]* 
e^(1/4)) + (Sqrt[2]*b*d^(3/4)*f*g*h^(5/2)*p*Log[Sqrt[d]*h + Sqrt[e]*h*x - 
Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h]*Sqrt[h*x]])/(3*e^(3/4)) + (b*d^(5/4)*g^2*h 
^(5/2)*p*Log[Sqrt[d]*h + Sqrt[e]*h*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h]*Sqr 
t[h*x]])/(5*Sqrt[2]*e^(5/4)) + (b*d^(1/4)*f^2*h^(5/2)*p*Log[Sqrt[d]*h + Sq 
rt[e]*h*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h]*Sqrt[h*x]])/(Sqrt[2]*e^(1/4)) 
- (Sqrt[2]*b*d^(3/4)*f*g*h^(5/2)*p*Log[Sqrt[d]*h + Sqrt[e]*h*x + Sqrt[2...
 

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 2921

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + 
 (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> With[{t = ExpandIntegrand[(a + b*Log[ 
c*(d + e*x^n)^p])^q, (f + g*x^s)^r, x]}, Int[t, x] /; SumQ[t]] /; FreeQ[{a, 
 b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && Integ 
erQ[r] && IntegerQ[s] && (EqQ[q, 1] || (GtQ[r, 0] && GtQ[s, 1]) || (LtQ[s, 
0] && LtQ[r, 0]))
 

Maple [F]

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

Input:

int((g*x+f)^2*(a+b*ln(c*(e*x^2+d)^p))/(h*x)^(1/2),x)
 

Output:

int((g*x+f)^2*(a+b*ln(c*(e*x^2+d)^p))/(h*x)^(1/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2178 vs. \(2 (548) = 1096\).

Time = 0.14 (sec) , antiderivative size = 2178, normalized size of antiderivative = 2.87 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)^2*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(1/2),x, algorithm="fri 
cas")
 

Output:

2/225*(15*e*h*sqrt(-(e^2*h*sqrt(-(50625*b^4*d*e^4*f^8 - 85500*b^4*d^2*e^3* 
f^6*g^2 + 40150*b^4*d^3*e^2*f^4*g^4 - 3420*b^4*d^4*e*f^2*g^6 + 81*b^4*d^5* 
g^8)*p^4/(e^5*h^2)) + 60*(5*b^2*d*e*f^3*g - b^2*d^2*f*g^3)*p^2)/(e^2*h))*l 
og(16*(50625*b^3*e^4*f^8 - 40500*b^3*d*e^3*f^6*g^2 + 2150*b^3*d^2*e^2*f^4* 
g^4 - 1620*b^3*d^3*e*f^2*g^6 + 81*b^3*d^4*g^8)*sqrt(h*x)*p^3 + 16*(10*e^4* 
f*g*h^2*sqrt(-(50625*b^4*d*e^4*f^8 - 85500*b^4*d^2*e^3*f^6*g^2 + 40150*b^4 
*d^3*e^2*f^4*g^4 - 3420*b^4*d^4*e*f^2*g^6 + 81*b^4*d^5*g^8)*p^4/(e^5*h^2)) 
 + 3*(1125*b^2*e^4*f^6 - 1175*b^2*d*e^3*f^4*g^2 + 235*b^2*d^2*e^2*f^2*g^4 
- 9*b^2*d^3*e*g^6)*h*p^2)*sqrt(-(e^2*h*sqrt(-(50625*b^4*d*e^4*f^8 - 85500* 
b^4*d^2*e^3*f^6*g^2 + 40150*b^4*d^3*e^2*f^4*g^4 - 3420*b^4*d^4*e*f^2*g^6 + 
 81*b^4*d^5*g^8)*p^4/(e^5*h^2)) + 60*(5*b^2*d*e*f^3*g - b^2*d^2*f*g^3)*p^2 
)/(e^2*h))) - 15*e*h*sqrt(-(e^2*h*sqrt(-(50625*b^4*d*e^4*f^8 - 85500*b^4*d 
^2*e^3*f^6*g^2 + 40150*b^4*d^3*e^2*f^4*g^4 - 3420*b^4*d^4*e*f^2*g^6 + 81*b 
^4*d^5*g^8)*p^4/(e^5*h^2)) + 60*(5*b^2*d*e*f^3*g - b^2*d^2*f*g^3)*p^2)/(e^ 
2*h))*log(16*(50625*b^3*e^4*f^8 - 40500*b^3*d*e^3*f^6*g^2 + 2150*b^3*d^2*e 
^2*f^4*g^4 - 1620*b^3*d^3*e*f^2*g^6 + 81*b^3*d^4*g^8)*sqrt(h*x)*p^3 - 16*( 
10*e^4*f*g*h^2*sqrt(-(50625*b^4*d*e^4*f^8 - 85500*b^4*d^2*e^3*f^6*g^2 + 40 
150*b^4*d^3*e^2*f^4*g^4 - 3420*b^4*d^4*e*f^2*g^6 + 81*b^4*d^5*g^8)*p^4/(e^ 
5*h^2)) + 3*(1125*b^2*e^4*f^6 - 1175*b^2*d*e^3*f^4*g^2 + 235*b^2*d^2*e^2*f 
^2*g^4 - 9*b^2*d^3*e*g^6)*h*p^2)*sqrt(-(e^2*h*sqrt(-(50625*b^4*d*e^4*f^...
 

Sympy [F(-2)]

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

integrate((g*x+f)**2*(a+b*ln(c*(e*x**2+d)**p))/(h*x)**(1/2),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. 1165 vs. \(2 (548) = 1096\).

Time = 0.14 (sec) , antiderivative size = 1165, normalized size of antiderivative = 1.53 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)^2*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(1/2),x, algorithm="max 
ima")
 

Output:

2/5*b*g^2*x^3*log((e*x^2 + d)^p*c)/sqrt(h*x) + 2/5*a*g^2*x^3/sqrt(h*x) + 4 
/3*b*f*g*x^2*log((e*x^2 + d)^p*c)/sqrt(h*x) + 4/3*a*f*g*x^2/sqrt(h*x) + 2* 
sqrt(h*x)*b*f^2*log((e*x^2 + d)^p*c)/h - (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*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(d)) + 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)*sqrt(e)*h)*sqrt 
(d)))*d/e)*b*e*f^2*p/h^3 + 2*sqrt(h*x)*a*f^2/h - 2/9*(3*d*h^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(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)*s 
qrt(-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(-(sq 
rt(2)*sqrt(-sqrt(d)*sqrt(e)*h) - sqrt(2)*(d*h^2)^(1/4)*e^(1/4) - 2*sqrt...
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 900, normalized size of antiderivative = 1.18 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)^2*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(1/2),x, algorithm="gia 
c")
 

Output:

1/225*(90*sqrt(h*x)*b*g^2*x^2*log(c) + 90*sqrt(h*x)*a*g^2*x^2 + 300*sqrt(h 
*x)*b*f*g*x*log(c) + 225*(e*(2*sqrt(2)*(d*e^3*h^2)^(1/4)*arctan(1/2*sqrt(2 
)*(sqrt(2)*(d*h^2/e)^(1/4) + 2*sqrt(h*x))/(d*h^2/e)^(1/4))/e^2 + 2*sqrt(2) 
*(d*e^3*h^2)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(d*h^2/e)^(1/4) - 2*sqrt(h 
*x))/(d*h^2/e)^(1/4))/e^2 + sqrt(2)*(d*e^3*h^2)^(1/4)*log(h*x + sqrt(2)*(d 
*h^2/e)^(1/4)*sqrt(h*x) + sqrt(d*h^2/e))/e^2 - sqrt(2)*(d*e^3*h^2)^(1/4)*l 
og(h*x - sqrt(2)*(d*h^2/e)^(1/4)*sqrt(h*x) + sqrt(d*h^2/e))/e^2 - 8*sqrt(h 
*x)/e) + 2*sqrt(h*x)*log(e*x^2 + d))*b*f^2*p + 300*sqrt(h*x)*a*f*g*x + 450 
*sqrt(h*x)*b*f^2*log(c) + 50*(6*sqrt(h*x)*h*x*log(e*x^2 + d) - (8*sqrt(h*x 
)*h*x/e - 6*sqrt(2)*(d*e^3*h^2)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(d*h^2/e 
)^(1/4) + 2*sqrt(h*x))/(d*h^2/e)^(1/4))/e^4 - 6*sqrt(2)*(d*e^3*h^2)^(3/4)* 
arctan(-1/2*sqrt(2)*(sqrt(2)*(d*h^2/e)^(1/4) - 2*sqrt(h*x))/(d*h^2/e)^(1/4 
))/e^4 + 3*sqrt(2)*(d*e^3*h^2)^(3/4)*log(h*x + sqrt(2)*(d*h^2/e)^(1/4)*sqr 
t(h*x) + sqrt(d*h^2/e))/e^4 - 3*sqrt(2)*(d*e^3*h^2)^(3/4)*log(h*x - sqrt(2 
)*(d*h^2/e)^(1/4)*sqrt(h*x) + sqrt(d*h^2/e))/e^4)*e)*b*f*g*p/h + 450*sqrt( 
h*x)*a*f^2 + 9*(10*sqrt(h*x)*h^2*x^2*log(e*x^2 + d) - (10*sqrt(2)*(d*e^3*h 
^2)^(1/4)*d*h^2*arctan(1/2*sqrt(2)*(sqrt(2)*(d*h^2/e)^(1/4) + 2*sqrt(h*x)) 
/(d*h^2/e)^(1/4))/e^3 + 10*sqrt(2)*(d*e^3*h^2)^(1/4)*d*h^2*arctan(-1/2*sqr 
t(2)*(sqrt(2)*(d*h^2/e)^(1/4) - 2*sqrt(h*x))/(d*h^2/e)^(1/4))/e^3 + 5*sqrt 
(2)*(d*e^3*h^2)^(1/4)*d*h^2*log(h*x + sqrt(2)*(d*h^2/e)^(1/4)*sqrt(h*x)...
 

Mupad [F(-1)]

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

int(((f + g*x)^2*(a + b*log(c*(d + e*x^2)^p)))/(h*x)^(1/2),x)
 

Output:

int(((f + g*x)^2*(a + b*log(c*(d + e*x^2)^p)))/(h*x)^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 606, normalized size of antiderivative = 0.80 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx =\text {Too large to display} \] Input:

int((g*x+f)^2*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(1/2),x)
 

Output:

(sqrt(h)*( - 300*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 + 90*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 - 450*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 + 300*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 - 90*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 + 450*e**(3/4)*d**(1/4)*sqrt(2)*at 
an((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 + 300*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 - 150*e**(1/4)*d**(3/4)*s 
qrt(2)*log((d + e*x**2)**p*c)*b*e*f*g + 90*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 - 45 
0*e**(3/4)*d**(1/4)*sqrt(2)*log( - sqrt(x)*e**(1/4)*d**(1/4)*sqrt(2) + sqr 
t(d) + sqrt(e)*x)*b*e*f**2*p - 45*e**(3/4)*d**(1/4)*sqrt(2)*log((d + e*x** 
2)**p*c)*b*d*g**2 + 225*e**(3/4)*d**(1/4)*sqrt(2)*log((d + e*x**2)**p*c)*b 
*e*f**2 + 450*sqrt(x)*log((d + e*x**2)**p*c)*b*e**2*f**2 + 300*sqrt(x)*log 
((d + e*x**2)**p*c)*b*e**2*f*g*x + 90*sqrt(x)*log((d + e*x**2)**p*c)*b*e** 
2*g**2*x**2 + 450*sqrt(x)*a*e**2*f**2 + 300*sqrt(x)*a*e**2*f*g*x + 90*s...