\(\int \frac {(f+g x)^2 (a+b \log (c (d+e x^2)^p))}{(h x)^{7/2}} \, dx\) [614]

Optimal result

Integrand size = 31, antiderivative size = 693 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx=-\frac {8 b e f^2 p}{5 d h^3 \sqrt {h x}}+\frac {2 \sqrt {2} b e^{5/4} f^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{7/2}}-\frac {4 \sqrt {2} b e^{3/4} f g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{7/2}}-\frac {2 \sqrt {2} b \sqrt [4]{e} g^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{7/2}}-\frac {2 \sqrt {2} b e^{5/4} f^2 p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{7/2}}+\frac {4 \sqrt {2} b e^{3/4} f g p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{7/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} g^2 p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{7/2}}+\frac {2 \sqrt {2} b e^{5/4} f^2 p \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}}{\sqrt {h} \left (\sqrt {d}+\sqrt {e} x\right )}\right )}{5 d^{5/4} h^{7/2}}+\frac {4 \sqrt {2} b e^{3/4} f g p \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}}{\sqrt {h} \left (\sqrt {d}+\sqrt {e} x\right )}\right )}{3 d^{3/4} h^{7/2}}-\frac {2 \sqrt {2} b \sqrt [4]{e} g^2 p \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}}{\sqrt {h} \left (\sqrt {d}+\sqrt {e} x\right )}\right )}{\sqrt [4]{d} h^{7/2}}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h (h x)^{5/2}}-\frac {4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2 (h x)^{3/2}}-\frac {2 g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h^3 \sqrt {h x}} \] Output:

-8/5*b*e*f^2*p/d/h^3/(h*x)^(1/2)+2/5*2^(1/2)*b*e^(5/4)*f^2*p*arctan(1-2^(1 
/2)*e^(1/4)*(h*x)^(1/2)/d^(1/4)/h^(1/2))/d^(5/4)/h^(7/2)-4/3*2^(1/2)*b*e^( 
3/4)*f*g*p*arctan(1-2^(1/2)*e^(1/4)*(h*x)^(1/2)/d^(1/4)/h^(1/2))/d^(3/4)/h 
^(7/2)-2*2^(1/2)*b*e^(1/4)*g^2*p*arctan(1-2^(1/2)*e^(1/4)*(h*x)^(1/2)/d^(1 
/4)/h^(1/2))/d^(1/4)/h^(7/2)-2/5*2^(1/2)*b*e^(5/4)*f^2*p*arctan(1+2^(1/2)* 
e^(1/4)*(h*x)^(1/2)/d^(1/4)/h^(1/2))/d^(5/4)/h^(7/2)+4/3*2^(1/2)*b*e^(3/4) 
*f*g*p*arctan(1+2^(1/2)*e^(1/4)*(h*x)^(1/2)/d^(1/4)/h^(1/2))/d^(3/4)/h^(7/ 
2)+2*2^(1/2)*b*e^(1/4)*g^2*p*arctan(1+2^(1/2)*e^(1/4)*(h*x)^(1/2)/d^(1/4)/ 
h^(1/2))/d^(1/4)/h^(7/2)+2/5*2^(1/2)*b*e^(5/4)*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^(5/4)/h^(7/2)+4/3*2^ 
(1/2)*b*e^(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))/d^(3/4)/h^(7/2)-2*2^(1/2)*b*e^(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))/d^(1/4)/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*g^2*(a+b*ln(c*(e*x^2+d)^p))/h^3/(h*x)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 1.08 (sec) , antiderivative size = 340, normalized size of antiderivative = 0.49 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx=\frac {2 x^{7/2} \left (\frac {2 b \sqrt [4]{e} g^2 p \left (\arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )+\text {arctanh}\left (\frac {d \sqrt [4]{e} \sqrt {x}}{(-d)^{5/4}}\right )\right )}{\sqrt [4]{-d}}-\frac {4 b e f^2 p \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\frac {e x^2}{d}\right )}{5 d \sqrt {x}}-\frac {\sqrt {2} b e^{3/4} f g p \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )+\log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )-\log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )\right )}{3 d^{3/4}}-\frac {f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 x^{5/2}}-\frac {2 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 x^{3/2}}-\frac {g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {x}}\right )}{(h x)^{7/2}} \] Input:

Integrate[((f + g*x)^2*(a + b*Log[c*(d + e*x^2)^p]))/(h*x)^(7/2),x]
 

Output:

(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)
 

Rubi [A] (verified)

Time = 2.03 (sec) , antiderivative size = 913, normalized size of antiderivative = 1.32, 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}\)

Input:

Int[((f + g*x)^2*(a + b*Log[c*(d + e*x^2)^p]))/(h*x)^(7/2),x]
 

Output:

(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]...
 

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]

\[\int \frac {\left (g x +f \right )^{2} \left (a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )\right )}{\left (h x \right )^{\frac {7}{2}}}d x\]

Input:

int((g*x+f)^2*(a+b*ln(c*(e*x^2+d)^p))/(h*x)^(7/2),x)
 

Output:

int((g*x+f)^2*(a+b*ln(c*(e*x^2+d)^p))/(h*x)^(7/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2205 vs. \(2 (493) = 986\).

Time = 0.26 (sec) , antiderivative size = 2205, normalized size of antiderivative = 3.18 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)^2*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(7/2),x, algorithm="fri 
cas")
 

Output:

-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^...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx=\text {Timed out} \] Input:

integrate((g*x+f)**2*(a+b*ln(c*(e*x**2+d)**p))/(h*x)**(7/2),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1088 vs. \(2 (493) = 986\).

Time = 0.14 (sec) , antiderivative size = 1088, normalized size of antiderivative = 1.57 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)^2*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(7/2),x, algorithm="max 
ima")
 

Output:

-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...
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 675, normalized size of antiderivative = 0.97 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx =\text {Too large to display} \] Input:

integrate((g*x+f)^2*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(7/2),x, algorithm="gia 
c")
 

Output:

-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
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^2\,\left (a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\right )}{{\left (h\,x\right )}^{7/2}} \,d x \] Input:

int(((f + g*x)^2*(a + b*log(c*(d + e*x^2)^p)))/(h*x)^(7/2),x)
 

Output:

int(((f + g*x)^2*(a + b*log(c*(d + e*x^2)^p)))/(h*x)^(7/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 623, normalized size of antiderivative = 0.90 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx =\text {Too large to display} \] Input:

int((g*x+f)^2*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(7/2),x)
 

Output:

(sqrt(h)*( - 30*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*d*g**2*p*x**2 
+ 6*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**2*p*x**2 - 20*sqrt(x) 
*e**(3/4)*d**(1/4)*sqrt(2)*atan((e**(1/4)*d**(1/4)*sqrt(2) - 2*sqrt(x)*sqr 
t(e))/(e**(1/4)*d**(1/4)*sqrt(2)))*b*d*f*g*p*x**2 + 30*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*d*g**2*p*x**2 - 6*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**2*p*x**2 + 20*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*f*g*p*x**2 + 30*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*d*g**2*p*x**2 - 6*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**2*p*x**2 - 15*sqrt(x)*e**(1/4)*d**(3/4)*sqrt(2)*log(( 
d + e*x**2)**p*c)*b*d*g**2*x**2 + 3*sqrt(x)*e**(1/4)*d**(3/4)*sqrt(2)*log( 
(d + e*x**2)**p*c)*b*e*f**2*x**2 - 20*sqrt(x)*e**(3/4)*d**(1/4)*sqrt(2)*lo 
g( - sqrt(x)*e**(1/4)*d**(1/4)*sqrt(2) + sqrt(d) + sqrt(e)*x)*b*d*f*g*p*x* 
*2 + 10*sqrt(x)*e**(3/4)*d**(1/4)*sqrt(2)*log((d + e*x**2)**p*c)*b*d*f*g*x 
**2 - 6*log((d + e*x**2)**p*c)*b*d**2*f**2 - 20*log((d + e*x**2)**p*c)*...