\(\int \frac {(f+g x) (a+b \log (c (d+e x^2)^p))}{(h x)^{5/2}} \, dx\) [608]

Optimal result

Integrand size = 29, antiderivative size = 431 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{5/2}} \, dx=-\frac {2 \sqrt {2} b e^{3/4} f p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{5/2}}-\frac {2 \sqrt {2} b \sqrt [4]{e} g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{5/2}}+\frac {2 \sqrt {2} b e^{3/4} f p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{5/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} g p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{5/2}}+\frac {2 \sqrt {2} b e^{3/4} f p \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}}{\sqrt {h} \left (\sqrt {d}+\sqrt {e} x\right )}\right )}{3 d^{3/4} h^{5/2}}-\frac {2 \sqrt {2} b \sqrt [4]{e} g p \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}}{\sqrt {h} \left (\sqrt {d}+\sqrt {e} x\right )}\right )}{\sqrt [4]{d} h^{5/2}}-\frac {2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h (h x)^{3/2}}-\frac {2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h^2 \sqrt {h x}} \] Output:

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

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.63 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{5/2}} \, dx=\frac {2 x^{5/2} \left (\frac {2 b \sqrt [4]{e} g p \left (\arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )+\text {arctanh}\left (\frac {d \sqrt [4]{e} \sqrt {x}}{(-d)^{5/4}}\right )\right )}{\sqrt [4]{-d}}-\frac {b e^{3/4} f p \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )+\log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )-\log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )\right )}{3 \sqrt {2} d^{3/4}}-\frac {f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 x^{3/2}}-\frac {g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {x}}\right )}{(h x)^{5/2}} \] Input:

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

Output:

(2*x^(5/2)*((2*b*e^(1/4)*g*p*(ArcTan[(e^(1/4)*Sqrt[x])/(-d)^(1/4)] + ArcTa 
nh[(d*e^(1/4)*Sqrt[x])/(-d)^(5/4)]))/(-d)^(1/4) - (b*e^(3/4)*f*p*(2*ArcTan 
[1 - (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sq 
rt[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*Sqrt[ 
2]*d^(3/4)) - (f*(a + b*Log[c*(d + e*x^2)^p]))/(3*x^(3/2)) - (g*(a + b*Log 
[c*(d + e*x^2)^p]))/Sqrt[x]))/(h*x)^(5/2)
 

Rubi [A] (verified)

Time = 1.41 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.33, 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.

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

\(\Big \downarrow \) 2917

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2926

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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

Input:

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

Output:

int((g*x+f)*(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. 1236 vs. \(2 (303) = 606\).

Time = 0.13 (sec) , antiderivative size = 1236, normalized size of antiderivative = 2.87 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-2/3*(h^3*x^2*sqrt(-(6*b^2*e*f*g*p^2 + d*h^5*sqrt(-(b^4*e^3*f^4 - 18*b^4*d 
*e^2*f^2*g^2 + 81*b^4*d^2*e*g^4)*p^4/(d^3*h^10)))/(d*h^5))*log(-32*(b^3*e^ 
3*f^4 - 81*b^3*d^2*e*g^4)*sqrt(h*x)*p^3 + 32*(3*d^3*g*h^8*sqrt(-(b^4*e^3*f 
^4 - 18*b^4*d*e^2*f^2*g^2 + 81*b^4*d^2*e*g^4)*p^4/(d^3*h^10)) + (b^2*d*e^2 
*f^3 - 9*b^2*d^2*e*f*g^2)*h^3*p^2)*sqrt(-(6*b^2*e*f*g*p^2 + d*h^5*sqrt(-(b 
^4*e^3*f^4 - 18*b^4*d*e^2*f^2*g^2 + 81*b^4*d^2*e*g^4)*p^4/(d^3*h^10)))/(d* 
h^5))) - h^3*x^2*sqrt(-(6*b^2*e*f*g*p^2 + d*h^5*sqrt(-(b^4*e^3*f^4 - 18*b^ 
4*d*e^2*f^2*g^2 + 81*b^4*d^2*e*g^4)*p^4/(d^3*h^10)))/(d*h^5))*log(-32*(b^3 
*e^3*f^4 - 81*b^3*d^2*e*g^4)*sqrt(h*x)*p^3 - 32*(3*d^3*g*h^8*sqrt(-(b^4*e^ 
3*f^4 - 18*b^4*d*e^2*f^2*g^2 + 81*b^4*d^2*e*g^4)*p^4/(d^3*h^10)) + (b^2*d* 
e^2*f^3 - 9*b^2*d^2*e*f*g^2)*h^3*p^2)*sqrt(-(6*b^2*e*f*g*p^2 + d*h^5*sqrt( 
-(b^4*e^3*f^4 - 18*b^4*d*e^2*f^2*g^2 + 81*b^4*d^2*e*g^4)*p^4/(d^3*h^10)))/ 
(d*h^5))) - h^3*x^2*sqrt(-(6*b^2*e*f*g*p^2 - d*h^5*sqrt(-(b^4*e^3*f^4 - 18 
*b^4*d*e^2*f^2*g^2 + 81*b^4*d^2*e*g^4)*p^4/(d^3*h^10)))/(d*h^5))*log(-32*( 
b^3*e^3*f^4 - 81*b^3*d^2*e*g^4)*sqrt(h*x)*p^3 + 32*(3*d^3*g*h^8*sqrt(-(b^4 
*e^3*f^4 - 18*b^4*d*e^2*f^2*g^2 + 81*b^4*d^2*e*g^4)*p^4/(d^3*h^10)) - (b^2 
*d*e^2*f^3 - 9*b^2*d^2*e*f*g^2)*h^3*p^2)*sqrt(-(6*b^2*e*f*g*p^2 - d*h^5*sq 
rt(-(b^4*e^3*f^4 - 18*b^4*d*e^2*f^2*g^2 + 81*b^4*d^2*e*g^4)*p^4/(d^3*h^10) 
))/(d*h^5))) + h^3*x^2*sqrt(-(6*b^2*e*f*g*p^2 - d*h^5*sqrt(-(b^4*e^3*f^4 - 
 18*b^4*d*e^2*f^2*g^2 + 81*b^4*d^2*e*g^4)*p^4/(d^3*h^10)))/(d*h^5))*log...
 

Sympy [F(-2)]

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

integrate((g*x+f)*(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. 707 vs. \(2 (303) = 606\).

Time = 0.13 (sec) , antiderivative size = 707, normalized size of antiderivative = 1.64 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.06 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{5/2}} \, dx=-\frac {\frac {2 \, {\left (3 \, b g h p x + b f h p\right )} \log \left (e h^{2} x^{2} + d h^{2}\right )}{\sqrt {h x} h^{2} x} - \frac {2 \, {\left (3 \, b g h p x \log \left (h^{2}\right ) + b f h p \log \left (h^{2}\right ) - 3 \, b g h x \log \left (c\right ) - 3 \, a g h x - b f h \log \left (c\right ) - a f h\right )}}{\sqrt {h x} h^{2} x} - \frac {2 \, {\left (\sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b e^{2} f h p + 3 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b g p\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} + 2 \, \sqrt {h x}\right )}}{2 \, \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}}}\right )}{d e^{2} h^{3}} - \frac {2 \, {\left (\sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b e^{2} f h p + 3 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b g p\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} - 2 \, \sqrt {h x}\right )}}{2 \, \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}}}\right )}{d e^{2} h^{3}} - \frac {{\left (\sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b e^{2} f h p - 3 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b g p\right )} \log \left (h x + \sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} \sqrt {h x} + \sqrt {\frac {d h^{2}}{e}}\right )}{d e^{2} h^{3}} + \frac {{\left (\sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b e^{2} f h p - 3 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b g p\right )} \log \left (h x - \sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} \sqrt {h x} + \sqrt {\frac {d h^{2}}{e}}\right )}{d e^{2} h^{3}}}{3 \, h} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 365, normalized size of antiderivative = 0.85 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{5/2}} \, dx=\frac {\sqrt {h}\, \left (-6 \sqrt {x}\, e^{\frac {1}{4}} d^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {e}}{e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}}\right ) b g p x -2 \sqrt {x}\, e^{\frac {3}{4}} d^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {e}}{e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}}\right ) b f p x +6 \sqrt {x}\, e^{\frac {1}{4}} d^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {e}}{e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}}\right ) b g p x +2 \sqrt {x}\, e^{\frac {3}{4}} d^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {e}}{e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}}\right ) b f p x +6 \sqrt {x}\, e^{\frac {1}{4}} d^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}+\sqrt {d}+\sqrt {e}\, x \right ) b g p x -3 \sqrt {x}\, e^{\frac {1}{4}} d^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) b g x -2 \sqrt {x}\, e^{\frac {3}{4}} d^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}+\sqrt {d}+\sqrt {e}\, x \right ) b f p x +\sqrt {x}\, e^{\frac {3}{4}} d^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) b f x -2 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) b d f -6 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) b d g x -2 a d f -6 a d g x \right )}{3 \sqrt {x}\, d \,h^{3} x} \] Input:

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

Output:

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