\(\int \frac {a+b \log (c (d+e x^2)^p)}{(h x)^{3/2} (f+g x)} \, dx\) [618]

Optimal result

Integrand size = 31, antiderivative size = 1580 \[ \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{(h x)^{3/2} (f+g x)} \, dx =\text {Too large to display} \] Output:

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

Mathematica [A] (verified)

Time = 1.73 (sec) , antiderivative size = 1336, normalized size of antiderivative = 0.85 \[ \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{(h x)^{3/2} (f+g x)} \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*Log[c*(d + e*x^2)^p])/((h*x)^(3/2)*(f + g*x)),x]
 

Output:

(x^(3/2)*((4*b*e^(1/4)*p*(ArcTan[(e^(1/4)*Sqrt[x])/(-d)^(1/4)] + ArcTanh[( 
d*e^(1/4)*Sqrt[x])/(-d)^(5/4)]))/(-d)^(1/4) - (2*(a + b*Log[c*(d + e*x^2)^ 
p]))/Sqrt[x] + (f*Sqrt[g]*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]]*(a + b*Log[c*(d 
+ e*x^2)^p]))/(-f)^(3/2) + (Sqrt[g]*Log[Sqrt[-f] + Sqrt[g]*Sqrt[x]]*(a + b 
*Log[c*(d + e*x^2)^p]))/Sqrt[-f] + (b*Sqrt[g]*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] - Sqrt[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[(Sqrt[g]*(I*(-d)^(1/4) + e^(1/4)*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + I*(-d)^ 
(1/4)*Sqrt[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)*Sqrt[-f] - (-d)^(1/4)*Sqrt[g])] + PolyLog[2, (e^(1/4)*(Sqrt[-f] - S 
qrt[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] + I*(-d)^(1/4)*Sqr 
t[g])] + PolyLog[2, (e^(1/4)*(Sqrt[-f] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[- 
f] + (-d)^(1/4)*Sqrt[g])]))/Sqrt[-f] + (b*f*Sqrt[g]*p*(Log[(Sqrt[g]*((-d)^ 
(1/4) - e^(1/4)*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + (-d)^(1/4)*Sqrt[g])]*Log[Sqr 
t[-f] + Sqrt[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[...
 

Rubi [A] (verified)

Time = 4.18 (sec) , antiderivative size = 1658, normalized size of antiderivative = 1.05, 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.

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

\(\Big \downarrow \) 2917

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2926

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*Log[c*(d + e*x^2)^p])/((h*x)^(3/2)*(f + g*x)),x]
 

Output:

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

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]

\[\int \frac {a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{\left (h x \right )^{\frac {3}{2}} \left (g x +f \right )}d x\]

Input:

int((a+b*ln(c*(e*x^2+d)^p))/(h*x)^(3/2)/(g*x+f),x)
 

Output:

int((a+b*ln(c*(e*x^2+d)^p))/(h*x)^(3/2)/(g*x+f),x)
 

Fricas [F]

\[ \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{(h x)^{3/2} (f+g x)} \, dx=\int { \frac {b \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) + a}{{\left (g x + f\right )} \left (h x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((a+b*log(c*(e*x^2+d)^p))/(h*x)^(3/2)/(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*h^2*x^3 + f*h 
^2*x^2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{(h x)^{3/2} (f+g x)} \, dx=\text {Timed out} \] Input:

integrate((a+b*ln(c*(e*x**2+d)**p))/(h*x)**(3/2)/(g*x+f),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{(h x)^{3/2} (f+g x)} \, dx=\int { \frac {b \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) + a}{{\left (g x + f\right )} \left (h x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((a+b*log(c*(e*x^2+d)^p))/(h*x)^(3/2)/(g*x+f),x, algorithm="maxim 
a")
 

Output:

b*integrate((sqrt(h)*log((e*x^2 + d)^p) + sqrt(h)*log(c))/(g*h^2*x^(5/2) + 
 f*h^2*x^(3/2)), x) - 2*a*(g*arctan(sqrt(h*x)*g/sqrt(f*g*h))/(sqrt(f*g*h)* 
f) + 1/(sqrt(h*x)*f))/h
 

Giac [F]

\[ \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{(h x)^{3/2} (f+g x)} \, dx=\int { \frac {b \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) + a}{{\left (g x + f\right )} \left (h x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((a+b*log(c*(e*x^2+d)^p))/(h*x)^(3/2)/(g*x+f),x, algorithm="giac" 
)
 

Output:

integrate((b*log((e*x^2 + d)^p*c) + a)/((g*x + f)*(h*x)^(3/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{(h x)^{3/2} (f+g x)} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{\left (f+g\,x\right )\,{\left (h\,x\right )}^{3/2}} \,d x \] Input:

int((a + b*log(c*(d + e*x^2)^p))/((f + g*x)*(h*x)^(3/2)),x)
 

Output:

int((a + b*log(c*(d + e*x^2)^p))/((f + g*x)*(h*x)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{(h x)^{3/2} (f+g x)} \, dx=\frac {\sqrt {h}\, \left (-2 \sqrt {x}\, e^{\frac {1}{4}} d^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {e}}{e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}}\right ) b f p +2 \sqrt {x}\, e^{\frac {1}{4}} d^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {e}}{e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}}\right ) b f p -2 \sqrt {x}\, \sqrt {g}\, \sqrt {f}\, \mathit {atan} \left (\frac {\sqrt {x}\, g}{\sqrt {g}\, \sqrt {f}}\right ) a d +\sqrt {x}\, e^{\frac {1}{4}} d^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}+\sqrt {d}+\sqrt {e}\, x \right ) b f p -\sqrt {x}\, e^{\frac {1}{4}} d^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}+\sqrt {d}+\sqrt {e}\, x \right ) b f p -4 \sqrt {x}\, \left (\int \frac {\sqrt {x}}{e g \,x^{4}+e f \,x^{3}+d g \,x^{2}+d f x}d x \right ) b \,d^{2} f g p -4 \sqrt {x}\, \left (\int \frac {\sqrt {x}}{e g \,x^{3}+e f \,x^{2}+d g x +d f}d x \right ) b \,d^{2} g^{2} p +\sqrt {x}\, \left (\int \frac {\sqrt {x}\, \mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) x^{2}}{e g \,x^{3}+e f \,x^{2}+d g x +d f}d x \right ) b d e \,g^{2}+\sqrt {x}\, \left (\int \frac {\sqrt {x}\, \mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right )}{e g \,x^{3}+e f \,x^{2}+d g x +d f}d x \right ) b \,d^{2} g^{2}-2 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) b d f -2 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) b d g x -2 a d f +8 b d g p x \right )}{\sqrt {x}\, d \,f^{2} h^{2}} \] Input:

int((a+b*log(c*(e*x^2+d)^p))/(h*x)^(3/2)/(g*x+f),x)
 

Output:

(sqrt(h)*( - 2*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*f*p + 2*sqrt(x) 
*e**(1/4)*d**(3/4)*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*f*p - 2*sqrt(x)*sqrt(g)*sqrt(f)*atan( 
(sqrt(x)*g)/(sqrt(g)*sqrt(f)))*a*d + 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*f*p - 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*f*p - 4*sqrt(x)*int(sqrt(x)/(d*f*x + d*g*x**2 + e*f*x**3 
 + e*g*x**4),x)*b*d**2*f*g*p - 4*sqrt(x)*int(sqrt(x)/(d*f + d*g*x + e*f*x* 
*2 + e*g*x**3),x)*b*d**2*g**2*p + sqrt(x)*int((sqrt(x)*log((d + e*x**2)**p 
*c)*x**2)/(d*f + d*g*x + e*f*x**2 + e*g*x**3),x)*b*d*e*g**2 + sqrt(x)*int( 
(sqrt(x)*log((d + e*x**2)**p*c))/(d*f + d*g*x + e*f*x**2 + e*g*x**3),x)*b* 
d**2*g**2 - 2*log((d + e*x**2)**p*c)*b*d*f - 2*log((d + e*x**2)**p*c)*b*d* 
g*x - 2*a*d*f + 8*b*d*g*p*x))/(sqrt(x)*d*f**2*h**2)