3.7.16 \(\int \frac {\sqrt {h x} (a+b \log (c (d+e x^2)^p))}{f+g x} \, dx\) [616]

3.7.16.1 Optimal result
3.7.16.2 Mathematica [A] (verified)
3.7.16.3 Rubi [A] (verified)
3.7.16.4 Maple [F]
3.7.16.5 Fricas [F]
3.7.16.6 Sympy [F(-1)]
3.7.16.7 Maxima [F]
3.7.16.8 Giac [F]
3.7.16.9 Mupad [F(-1)]

3.7.16.1 Optimal result

Integrand size = 31, antiderivative size = 1680 \[ \int \frac {\sqrt {h x} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f+g x} \, dx =\text {Too large to display} \]

output
-2*b*d^(1/4)*p*arctan(1-e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/ 
2)*h^(1/2)/e^(1/4)/g+2*b*d^(1/4)*p*arctan(1+e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^ 
(1/4)/h^(1/2))*2^(1/2)*h^(1/2)/e^(1/4)/g-b*d^(1/4)*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)*h^(1/2)/e^(1/ 
4)/g+b*d^(1/4)*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)*h^(1/2)/e^(1/4)/g-2*arctan(g^(1/2)*(h*x)^(1/2)/f^ 
(1/2)/h^(1/2))*(a+b*ln(c*(e*x^2+d)^p))*f^(1/2)*h^(1/2)/g^(3/2)-8*b*p*arcta 
n(g^(1/2)*(h*x)^(1/2)/f^(1/2)/h^(1/2))*ln(2*f^(1/2)*h^(1/2)/(f^(1/2)*h^(1/ 
2)-I*g^(1/2)*(h*x)^(1/2)))*f^(1/2)*h^(1/2)/g^(3/2)+2*b*p*arctan(g^(1/2)*(h 
*x)^(1/2)/f^(1/2)/h^(1/2))*ln(2*f^(1/2)*g^(1/2)*h^(1/2)*((-d)^(1/4)*(-h)^( 
1/2)-e^(1/4)*(h*x)^(1/2))/((-d)^(1/4)*g^(1/2)*(-h)^(1/2)-I*e^(1/4)*f^(1/2) 
*h^(1/2))/(f^(1/2)*h^(1/2)-I*g^(1/2)*(h*x)^(1/2)))*f^(1/2)*h^(1/2)/g^(3/2) 
+2*b*p*arctan(g^(1/2)*(h*x)^(1/2)/f^(1/2)/h^(1/2))*ln(-2*f^(1/2)*g^(1/2)*( 
(-d)^(1/4)*h^(1/2)-e^(1/4)*(h*x)^(1/2))/(I*e^(1/4)*f^(1/2)-(-d)^(1/4)*g^(1 
/2))/(f^(1/2)*h^(1/2)-I*g^(1/2)*(h*x)^(1/2)))*f^(1/2)*h^(1/2)/g^(3/2)+2*b* 
p*arctan(g^(1/2)*(h*x)^(1/2)/f^(1/2)/h^(1/2))*ln(2*f^(1/2)*g^(1/2)*h^(1/2) 
*((-d)^(1/4)*(-h)^(1/2)+e^(1/4)*(h*x)^(1/2))/((-d)^(1/4)*g^(1/2)*(-h)^(1/2 
)+I*e^(1/4)*f^(1/2)*h^(1/2))/(f^(1/2)*h^(1/2)-I*g^(1/2)*(h*x)^(1/2)))*f^(1 
/2)*h^(1/2)/g^(3/2)+2*b*p*arctan(g^(1/2)*(h*x)^(1/2)/f^(1/2)/h^(1/2))*ln(2 
*f^(1/2)*g^(1/2)*((-d)^(1/4)*h^(1/2)+e^(1/4)*(h*x)^(1/2))/(I*e^(1/4)*f^...
 
3.7.16.2 Mathematica [A] (verified)

Time = 0.91 (sec) , antiderivative size = 1506, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {h x} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f+g x} \, dx =\text {Too large to display} \]

input
Integrate[(Sqrt[h*x]*(a + b*Log[c*(d + e*x^2)^p]))/(f + g*x),x]
 
output
(Sqrt[h*x]*(2*a*Sqrt[g]*Sqrt[x] - 8*b*Sqrt[g]*p*Sqrt[x] - (2*Sqrt[2]*b*d^( 
1/4)*Sqrt[g]*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)])/e^(1/4) + (2 
*Sqrt[2]*b*d^(1/4)*Sqrt[g]*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)] 
)/e^(1/4) - (Sqrt[2]*b*d^(1/4)*Sqrt[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)*Sqrt[g]*p*Log[Sqrt 
[d] + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x])/e^(1/4) + 2*b*Sqrt[g]* 
Sqrt[x]*Log[c*(d + e*x^2)^p] + Sqrt[-f]*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]]*(a 
 + b*Log[c*(d + e*x^2)^p]) - Sqrt[-f]*Log[Sqrt[-f] + Sqrt[g]*Sqrt[x]]*(a + 
 b*Log[c*(d + e*x^2)^p]) - b*Sqrt[-f]*p*(Log[(Sqrt[g]*((-d)^(1/4) - e^(1/4 
)*Sqrt[x]))/(-(e^(1/4)*Sqrt[-f]) + (-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] - Sqr 
t[g]*Sqrt[x]] + Log[(Sqrt[g]*((-d)^(1/4) + I*e^(1/4)*Sqrt[x]))/(I*e^(1/4)* 
Sqrt[-f] + (-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]] + Log[(Sqr 
t[g]*(I*(-d)^(1/4) + e^(1/4)*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + I*(-d)^(1/4)*Sq 
rt[g])]*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]] + Log[(Sqrt[g]*((-d)^(1/4) + e^(1/ 
4)*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + (-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] - Sqrt[ 
g]*Sqrt[x]] + PolyLog[2, (e^(1/4)*(Sqrt[-f] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*S 
qrt[-f] - (-d)^(1/4)*Sqrt[g])] + PolyLog[2, (e^(1/4)*(Sqrt[-f] - Sqrt[g]*S 
qrt[x]))/(e^(1/4)*Sqrt[-f] - I*(-d)^(1/4)*Sqrt[g])] + PolyLog[2, (e^(1/4)* 
(Sqrt[-f] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + I*(-d)^(1/4)*Sqrt[g])] + 
 PolyLog[2, (e^(1/4)*(Sqrt[-f] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + ...
 
3.7.16.3 Rubi [A] (verified)

Time = 2.52 (sec) , antiderivative size = 1677, normalized size of antiderivative = 1.00, 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.

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

\(\Big \downarrow \) 2917

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2926

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

\(\Big \downarrow \) 2009

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

input
Int[(Sqrt[h*x]*(a + b*Log[c*(d + e*x^2)^p]))/(f + g*x),x]
 
output
2*((a*Sqrt[h*x])/g - (4*b*p*Sqrt[h*x])/g - (Sqrt[2]*b*d^(1/4)*Sqrt[h]*p*Ar 
cTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(e^(1/4)*g) + (Sq 
rt[2]*b*d^(1/4)*Sqrt[h]*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)* 
Sqrt[h])])/(e^(1/4)*g) + (b*Sqrt[h*x]*Log[c*(d + e*x^2)^p])/g - (Sqrt[f]*S 
qrt[h]*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*(a + b*Log[c*(d + e*x 
^2)^p]))/g^(3/2) - (4*b*Sqrt[f]*Sqrt[h]*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt 
[f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqrt[h])/(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h* 
x])])/g^(3/2) + (b*Sqrt[f]*Sqrt[h]*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*S 
qrt[h])]*Log[(2*Sqrt[f]*Sqrt[g]*Sqrt[h]*((-d)^(1/4)*Sqrt[-h] - e^(1/4)*Sqr 
t[h*x]))/(((-d)^(1/4)*Sqrt[g]*Sqrt[-h] - I*e^(1/4)*Sqrt[f]*Sqrt[h])*(Sqrt[ 
f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/g^(3/2) + (b*Sqrt[f]*Sqrt[h]*p*ArcTan 
[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(-2*Sqrt[f]*Sqrt[g]*((-d)^(1/4 
)*Sqrt[h] - e^(1/4)*Sqrt[h*x]))/((I*e^(1/4)*Sqrt[f] - (-d)^(1/4)*Sqrt[g])* 
(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/g^(3/2) + (b*Sqrt[f]*Sqrt[h]*p* 
ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqrt[g]*Sqrt[ 
h]*((-d)^(1/4)*Sqrt[-h] + e^(1/4)*Sqrt[h*x]))/(((-d)^(1/4)*Sqrt[g]*Sqrt[-h 
] + I*e^(1/4)*Sqrt[f]*Sqrt[h])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/ 
g^(3/2) + (b*Sqrt[f]*Sqrt[h]*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h] 
)]*Log[(2*Sqrt[f]*Sqrt[g]*((-d)^(1/4)*Sqrt[h] + e^(1/4)*Sqrt[h*x]))/((I*e^ 
(1/4)*Sqrt[f] + (-d)^(1/4)*Sqrt[g])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h...
 

3.7.16.3.1 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]
 
3.7.16.4 Maple [F]

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

input
int((h*x)^(1/2)*(a+b*ln(c*(e*x^2+d)^p))/(g*x+f),x)
 
output
int((h*x)^(1/2)*(a+b*ln(c*(e*x^2+d)^p))/(g*x+f),x)
 
3.7.16.5 Fricas [F]

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

input
integrate((h*x)^(1/2)*(a+b*log(c*(e*x^2+d)^p))/(g*x+f),x, algorithm="frica 
s")
 
output
integral((sqrt(h*x)*b*log((e*x^2 + d)^p*c) + sqrt(h*x)*a)/(g*x + f), x)
 
3.7.16.6 Sympy [F(-1)]

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

input
integrate((h*x)**(1/2)*(a+b*ln(c*(e*x**2+d)**p))/(g*x+f),x)
 
output
Timed out
 
3.7.16.7 Maxima [F]

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

input
integrate((h*x)^(1/2)*(a+b*log(c*(e*x^2+d)^p))/(g*x+f),x, algorithm="maxim 
a")
 
output
b*integrate((sqrt(h)*sqrt(x)*log((e*x^2 + d)^p) + sqrt(h)*sqrt(x)*log(c))/ 
(g*x + f), x) - 2*(f*h^2*arctan(sqrt(h*x)*g/sqrt(f*g*h))/(sqrt(f*g*h)*g) - 
 sqrt(h*x)*h/g)*a/h
 
3.7.16.8 Giac [F]

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

input
integrate((h*x)^(1/2)*(a+b*log(c*(e*x^2+d)^p))/(g*x+f),x, algorithm="giac" 
)
 
output
integrate(sqrt(h*x)*(b*log((e*x^2 + d)^p*c) + a)/(g*x + f), x)
 
3.7.16.9 Mupad [F(-1)]

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

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