\(\int \frac {(f+g x) (a+b \log (c (d+e x^2)^p))}{(h x)^{3/2}} \, dx\) [607]

Optimal result

Integrand size = 29, antiderivative size = 449 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=\frac {2 a g \sqrt {h x}}{h^2}-\frac {8 b g p \sqrt {h x}}{h^2}-\frac {2 \sqrt {2} b \sqrt [4]{e} f p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}-\frac {2 \sqrt {2} b \sqrt [4]{d} g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} f p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{d} g p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} h^{3/2}}-\frac {2 \sqrt {2} b \sqrt [4]{e} f p \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}}{\sqrt {h} \left (\sqrt {d}+\sqrt {e} x\right )}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{d} g p \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}}{\sqrt {h} \left (\sqrt {d}+\sqrt {e} x\right )}\right )}{\sqrt [4]{e} h^{3/2}}+\frac {2 b g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}-\frac {2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}} \] Output:

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

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.74 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=\frac {2 x^{3/2} \left (a g \sqrt {x}-4 b g p \sqrt {x}-\frac {\sqrt {2} b \sqrt [4]{d} g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )}{\sqrt [4]{e}}+\frac {\sqrt {2} b \sqrt [4]{d} g p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )}{\sqrt [4]{e}}+\frac {2 b \sqrt [4]{e} f p \left (\arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )+\text {arctanh}\left (\frac {d \sqrt [4]{e} \sqrt {x}}{(-d)^{5/4}}\right )\right )}{\sqrt [4]{-d}}-\frac {b \sqrt [4]{d} g p \log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )}{\sqrt {2} \sqrt [4]{e}}+\frac {b \sqrt [4]{d} g p \log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )}{\sqrt {2} \sqrt [4]{e}}+b g \sqrt {x} \log \left (c \left (d+e x^2\right )^p\right )-\frac {f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {x}}\right )}{(h x)^{3/2}} \] Input:

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

Output:

(2*x^(3/2)*(a*g*Sqrt[x] - 4*b*g*p*Sqrt[x] - (Sqrt[2]*b*d^(1/4)*g*p*ArcTan[ 
1 - (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)])/e^(1/4) + (Sqrt[2]*b*d^(1/4)*g*p*A 
rcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)])/e^(1/4) + (2*b*e^(1/4)*f*p*( 
ArcTan[(e^(1/4)*Sqrt[x])/(-d)^(1/4)] + ArcTanh[(d*e^(1/4)*Sqrt[x])/(-d)^(5 
/4)]))/(-d)^(1/4) - (b*d^(1/4)*g*p*Log[Sqrt[d] - Sqrt[2]*d^(1/4)*e^(1/4)*S 
qrt[x] + Sqrt[e]*x])/(Sqrt[2]*e^(1/4)) + (b*d^(1/4)*g*p*Log[Sqrt[d] + Sqrt 
[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x])/(Sqrt[2]*e^(1/4)) + b*g*Sqrt[x]* 
Log[c*(d + e*x^2)^p] - (f*(a + b*Log[c*(d + e*x^2)^p]))/Sqrt[x]))/(h*x)^(3 
/2)
 

Rubi [A] (verified)

Time = 1.47 (sec) , antiderivative size = 581, normalized size of antiderivative = 1.29, 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.

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

\(\Big \downarrow \) 2917

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2926

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1162 vs. \(2 (325) = 650\).

Time = 0.12 (sec) , antiderivative size = 1162, normalized size of antiderivative = 2.59 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

2*(h^2*x*sqrt(-(2*b^2*f*g*p^2 + h^3*sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 
 + b^4*d^2*g^4)*p^4/(d*e*h^6)))/h^3)*log(-32*(b^3*e^2*f^4 - b^3*d^2*g^4)*s 
qrt(h*x)*p^3 + 32*(d*e*f*h^5*sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^4* 
d^2*g^4)*p^4/(d*e*h^6)) - (b^2*d*e*f^2*g - b^2*d^2*g^3)*h^2*p^2)*sqrt(-(2* 
b^2*f*g*p^2 + h^3*sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p^ 
4/(d*e*h^6)))/h^3)) - h^2*x*sqrt(-(2*b^2*f*g*p^2 + h^3*sqrt(-(b^4*e^2*f^4 
- 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p^4/(d*e*h^6)))/h^3)*log(-32*(b^3*e^2*f 
^4 - b^3*d^2*g^4)*sqrt(h*x)*p^3 - 32*(d*e*f*h^5*sqrt(-(b^4*e^2*f^4 - 2*b^4 
*d*e*f^2*g^2 + b^4*d^2*g^4)*p^4/(d*e*h^6)) - (b^2*d*e*f^2*g - b^2*d^2*g^3) 
*h^2*p^2)*sqrt(-(2*b^2*f*g*p^2 + h^3*sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^ 
2 + b^4*d^2*g^4)*p^4/(d*e*h^6)))/h^3)) - h^2*x*sqrt(-(2*b^2*f*g*p^2 - h^3* 
sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p^4/(d*e*h^6)))/h^3) 
*log(-32*(b^3*e^2*f^4 - b^3*d^2*g^4)*sqrt(h*x)*p^3 + 32*(d*e*f*h^5*sqrt(-( 
b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p^4/(d*e*h^6)) + (b^2*d*e*f 
^2*g - b^2*d^2*g^3)*h^2*p^2)*sqrt(-(2*b^2*f*g*p^2 - h^3*sqrt(-(b^4*e^2*f^4 
 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p^4/(d*e*h^6)))/h^3)) + h^2*x*sqrt(-(2 
*b^2*f*g*p^2 - h^3*sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p 
^4/(d*e*h^6)))/h^3)*log(-32*(b^3*e^2*f^4 - b^3*d^2*g^4)*sqrt(h*x)*p^3 - 32 
*(d*e*f*h^5*sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p^4/(d*e 
*h^6)) + (b^2*d*e*f^2*g - b^2*d^2*g^3)*h^2*p^2)*sqrt(-(2*b^2*f*g*p^2 - ...
 

Sympy [F(-2)]

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

integrate((g*x+f)*(a+b*ln(c*(e*x**2+d)**p))/(h*x)**(3/2),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. 731 vs. \(2 (325) = 650\).

Time = 0.13 (sec) , antiderivative size = 731, normalized size of antiderivative = 1.63 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

-b*e*f*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*b*g*x^2*log((e*x^2 + d)^p*c)/(h*x)^(3/2) + 2*a*g*x^2/(h*x 
)^(3/2) - 2*b*f*log((e*x^2 + d)^p*c)/(sqrt(h*x)*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) + s 
qrt(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)) + sqr 
t(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)*sqr 
t(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)*...
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.00 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=-\frac {2 \, {\left (\frac {b f p}{\sqrt {h x}} - \frac {\sqrt {h x} b g p}{h}\right )} \log \left (e h^{2} x^{2} + d h^{2}\right ) - \frac {2 \, {\left (b f p \log \left (h^{2}\right ) - b f \log \left (c\right ) - a f\right )}}{\sqrt {h x}} + \frac {2 \, {\left (b g p \log \left (h^{2}\right ) + 4 \, b g p - b g \log \left (c\right ) - a g\right )} \sqrt {h x}}{h} - \frac {2 \, {\left (\sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e g h p + \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b f p\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} + 2 \, \sqrt {h x}\right )}}{2 \, \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}}}\right )}{d e^{2} h^{2}} - \frac {2 \, {\left (\sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e g h p + \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b f p\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} - 2 \, \sqrt {h x}\right )}}{2 \, \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}}}\right )}{d e^{2} h^{2}} - \frac {{\left (\sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e g h p - \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b f p\right )} \log \left (h x + \sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} \sqrt {h x} + \sqrt {\frac {d h^{2}}{e}}\right )}{d e^{2} h^{2}} + \frac {{\left (\sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e g h p - \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b f p\right )} \log \left (h x - \sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} \sqrt {h x} + \sqrt {\frac {d h^{2}}{e}}\right )}{d e^{2} h^{2}}}{h} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 368, normalized size of antiderivative = 0.82 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=\frac {\sqrt {h}\, \left (-2 \sqrt {x}\, e^{\frac {5}{4}} d^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {e}}{e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}}\right ) b f p -2 \sqrt {x}\, e^{\frac {3}{4}} d^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {e}}{e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}}\right ) b g p +2 \sqrt {x}\, e^{\frac {5}{4}} d^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {e}}{e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}}\right ) b f p +2 \sqrt {x}\, e^{\frac {3}{4}} d^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {e}}{e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}}\right ) b g p +2 \sqrt {x}\, e^{\frac {5}{4}} d^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}+\sqrt {d}+\sqrt {e}\, x \right ) b f p -\sqrt {x}\, e^{\frac {5}{4}} d^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) b f -2 \sqrt {x}\, e^{\frac {3}{4}} d^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}+\sqrt {d}+\sqrt {e}\, x \right ) b g p +\sqrt {x}\, e^{\frac {3}{4}} d^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) b g -2 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) b d e f +2 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) b d e g x -2 a d e f +2 a d e g x -8 b d e g p x \right )}{\sqrt {x}\, d e \,h^{2}} \] Input:

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