\(\int \frac {(f+g x) (a+b \log (c (d+e x^2)^p))}{\sqrt {h x}} \, dx\) [606]

Optimal result

Integrand size = 29, antiderivative size = 474 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\frac {2 a f \sqrt {h x}}{h}-\frac {8 b f p \sqrt {h x}}{h}-\frac {8 b g p (h x)^{3/2}}{9 h^2}-\frac {2 \sqrt {2} b \sqrt [4]{d} f p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} \sqrt {h}}-\frac {2 \sqrt {2} b d^{3/4} g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} \sqrt {h}}+\frac {2 \sqrt {2} b \sqrt [4]{d} f p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} \sqrt {h}}+\frac {2 \sqrt {2} b d^{3/4} g p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} \sqrt {h}}+\frac {2 \sqrt {2} b \sqrt [4]{d} f p \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}}{\sqrt {h} \left (\sqrt {d}+\sqrt {e} x\right )}\right )}{\sqrt [4]{e} \sqrt {h}}-\frac {2 \sqrt {2} b d^{3/4} g p \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}}{\sqrt {h} \left (\sqrt {d}+\sqrt {e} x\right )}\right )}{3 e^{3/4} \sqrt {h}}+\frac {2 b f \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac {2 g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2} \] Output:

2*a*f*(h*x)^(1/2)/h-8*b*f*p*(h*x)^(1/2)/h-8/9*b*g*p*(h*x)^(3/2)/h^2-2*2^(1 
/2)*b*d^(1/4)*f*p*arctan(1-2^(1/2)*e^(1/4)*(h*x)^(1/2)/d^(1/4)/h^(1/2))/e^ 
(1/4)/h^(1/2)-2/3*2^(1/2)*b*d^(3/4)*g*p*arctan(1-2^(1/2)*e^(1/4)*(h*x)^(1/ 
2)/d^(1/4)/h^(1/2))/e^(3/4)/h^(1/2)+2*2^(1/2)*b*d^(1/4)*f*p*arctan(1+2^(1/ 
2)*e^(1/4)*(h*x)^(1/2)/d^(1/4)/h^(1/2))/e^(1/4)/h^(1/2)+2/3*2^(1/2)*b*d^(3 
/4)*g*p*arctan(1+2^(1/2)*e^(1/4)*(h*x)^(1/2)/d^(1/4)/h^(1/2))/e^(3/4)/h^(1 
/2)+2*2^(1/2)*b*d^(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))/e^(1/4)/h^(1/2)-2/3*2^(1/2)*b*d^(3/4)*g*p*arcta 
nh(2^(1/2)*d^(1/4)*e^(1/4)*(h*x)^(1/2)/h^(1/2)/(d^(1/2)+e^(1/2)*x))/e^(3/4 
)/h^(1/2)+2*b*f*(h*x)^(1/2)*ln(c*(e*x^2+d)^p)/h+2/3*g*(h*x)^(3/2)*(a+b*ln( 
c*(e*x^2+d)^p))/h^2
 

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 373, normalized size of antiderivative = 0.79 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\frac {2 \sqrt {x} \left (a f \sqrt {x}-4 b f p \sqrt {x}+\frac {1}{3} a g x^{3/2}-\frac {4}{9} b g p x^{3/2}-\frac {\sqrt {2} b \sqrt [4]{d} f p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )}{\sqrt [4]{e}}+\frac {\sqrt {2} b \sqrt [4]{d} f p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )}{\sqrt [4]{e}}-\frac {2 b (-d)^{3/4} g p \arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )}{3 e^{3/4}}+\frac {2 b (-d)^{3/4} g p \text {arctanh}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )}{3 e^{3/4}}-\frac {b \sqrt [4]{d} f p \log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )}{\sqrt {2} \sqrt [4]{e}}+\frac {b \sqrt [4]{d} f p \log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )}{\sqrt {2} \sqrt [4]{e}}+b f \sqrt {x} \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} b g x^{3/2} \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \] Input:

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

Output:

(2*Sqrt[x]*(a*f*Sqrt[x] - 4*b*f*p*Sqrt[x] + (a*g*x^(3/2))/3 - (4*b*g*p*x^( 
3/2))/9 - (Sqrt[2]*b*d^(1/4)*f*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1 
/4)])/e^(1/4) + (Sqrt[2]*b*d^(1/4)*f*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[x] 
)/d^(1/4)])/e^(1/4) - (2*b*(-d)^(3/4)*g*p*ArcTan[(e^(1/4)*Sqrt[x])/(-d)^(1 
/4)])/(3*e^(3/4)) + (2*b*(-d)^(3/4)*g*p*ArcTanh[(e^(1/4)*Sqrt[x])/(-d)^(1/ 
4)])/(3*e^(3/4)) - (b*d^(1/4)*f*p*Log[Sqrt[d] - Sqrt[2]*d^(1/4)*e^(1/4)*Sq 
rt[x] + Sqrt[e]*x])/(Sqrt[2]*e^(1/4)) + (b*d^(1/4)*f*p*Log[Sqrt[d] + Sqrt[ 
2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x])/(Sqrt[2]*e^(1/4)) + b*f*Sqrt[x]*L 
og[c*(d + e*x^2)^p] + (b*g*x^(3/2)*Log[c*(d + e*x^2)^p])/3))/Sqrt[h*x]
 

Rubi [A] (verified)

Time = 1.57 (sec) , antiderivative size = 609, normalized size of antiderivative = 1.28, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2917, 27, 2921, 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.

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

\(\Big \downarrow \) 2917

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2921

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (\frac {1}{3} g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )+a f h \sqrt {h x}-\frac {\sqrt {2} b d^{3/4} g h^{3/2} p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4}}+\frac {\sqrt {2} b d^{3/4} g h^{3/2} p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{3 e^{3/4}}-\frac {\sqrt {2} b \sqrt [4]{d} f h^{3/2} p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e}}+\frac {\sqrt {2} b \sqrt [4]{d} f h^{3/2} p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{\sqrt [4]{e}}+b f h \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )+\frac {b d^{3/4} g h^{3/2} p \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h} \sqrt {h x}+\sqrt {d} h+\sqrt {e} h x\right )}{3 \sqrt {2} e^{3/4}}-\frac {b d^{3/4} g h^{3/2} p \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h} \sqrt {h x}+\sqrt {d} h+\sqrt {e} h x\right )}{3 \sqrt {2} e^{3/4}}-\frac {b \sqrt [4]{d} f h^{3/2} p \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h} \sqrt {h x}+\sqrt {d} h+\sqrt {e} h x\right )}{\sqrt {2} \sqrt [4]{e}}+\frac {b \sqrt [4]{d} f h^{3/2} p \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h} \sqrt {h x}+\sqrt {d} h+\sqrt {e} h x\right )}{\sqrt {2} \sqrt [4]{e}}-4 b f h p \sqrt {h x}-\frac {4}{9} b g p (h x)^{3/2}\right )}{h^2}\)

Input:

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

Output:

(2*(a*f*h*Sqrt[h*x] - 4*b*f*h*p*Sqrt[h*x] - (4*b*g*p*(h*x)^(3/2))/9 - (Sqr 
t[2]*b*d^(1/4)*f*h^(3/2)*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4) 
*Sqrt[h])])/e^(1/4) - (Sqrt[2]*b*d^(3/4)*g*h^(3/2)*p*ArcTan[1 - (Sqrt[2]*e 
^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*e^(3/4)) + (Sqrt[2]*b*d^(1/4)*f*h 
^(3/2)*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/e^(1/4 
) + (Sqrt[2]*b*d^(3/4)*g*h^(3/2)*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/ 
(d^(1/4)*Sqrt[h])])/(3*e^(3/4)) + b*f*h*Sqrt[h*x]*Log[c*(d + e*x^2)^p] + ( 
g*(h*x)^(3/2)*(a + b*Log[c*(d + e*x^2)^p]))/3 - (b*d^(1/4)*f*h^(3/2)*p*Log 
[Sqrt[d]*h + Sqrt[e]*h*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h]*Sqrt[h*x]])/(Sq 
rt[2]*e^(1/4)) + (b*d^(3/4)*g*h^(3/2)*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]*e^(3/4)) + (b*d^(1/4)*f 
*h^(3/2)*p*Log[Sqrt[d]*h + Sqrt[e]*h*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h]*S 
qrt[h*x]])/(Sqrt[2]*e^(1/4)) - (b*d^(3/4)*g*h^(3/2)*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]*e^(3/4))) 
)/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 2921

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + 
 (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> With[{t = ExpandIntegrand[(a + b*Log[ 
c*(d + e*x^n)^p])^q, (f + g*x^s)^r, x]}, Int[t, x] /; SumQ[t]] /; FreeQ[{a, 
 b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && Integ 
erQ[r] && IntegerQ[s] && (EqQ[q, 1] || (GtQ[r, 0] && GtQ[s, 1]) || (LtQ[s, 
0] && LtQ[r, 0]))
 

Maple [F]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1196 vs. \(2 (338) = 676\).

Time = 0.12 (sec) , antiderivative size = 1196, normalized size of antiderivative = 2.52 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(1/2),x, algorithm="frica 
s")
 

Output:

-2/9*(3*h*sqrt(-(6*b^2*d*f*g*p^2 + e*h*sqrt(-(81*b^4*d*e^2*f^4 - 18*b^4*d^ 
2*e*f^2*g^2 + b^4*d^3*g^4)*p^4/(e^3*h^2)))/(e*h))*log(-32*(81*b^3*e^2*f^4 
- b^3*d^2*g^4)*sqrt(h*x)*p^3 + 32*(e^2*g*h^2*sqrt(-(81*b^4*d*e^2*f^4 - 18* 
b^4*d^2*e*f^2*g^2 + b^4*d^3*g^4)*p^4/(e^3*h^2)) + 3*(9*b^2*e^2*f^3 - b^2*d 
*e*f*g^2)*h*p^2)*sqrt(-(6*b^2*d*f*g*p^2 + e*h*sqrt(-(81*b^4*d*e^2*f^4 - 18 
*b^4*d^2*e*f^2*g^2 + b^4*d^3*g^4)*p^4/(e^3*h^2)))/(e*h))) - 3*h*sqrt(-(6*b 
^2*d*f*g*p^2 + e*h*sqrt(-(81*b^4*d*e^2*f^4 - 18*b^4*d^2*e*f^2*g^2 + b^4*d^ 
3*g^4)*p^4/(e^3*h^2)))/(e*h))*log(-32*(81*b^3*e^2*f^4 - b^3*d^2*g^4)*sqrt( 
h*x)*p^3 - 32*(e^2*g*h^2*sqrt(-(81*b^4*d*e^2*f^4 - 18*b^4*d^2*e*f^2*g^2 + 
b^4*d^3*g^4)*p^4/(e^3*h^2)) + 3*(9*b^2*e^2*f^3 - b^2*d*e*f*g^2)*h*p^2)*sqr 
t(-(6*b^2*d*f*g*p^2 + e*h*sqrt(-(81*b^4*d*e^2*f^4 - 18*b^4*d^2*e*f^2*g^2 + 
 b^4*d^3*g^4)*p^4/(e^3*h^2)))/(e*h))) - 3*h*sqrt(-(6*b^2*d*f*g*p^2 - e*h*s 
qrt(-(81*b^4*d*e^2*f^4 - 18*b^4*d^2*e*f^2*g^2 + b^4*d^3*g^4)*p^4/(e^3*h^2) 
))/(e*h))*log(-32*(81*b^3*e^2*f^4 - b^3*d^2*g^4)*sqrt(h*x)*p^3 + 32*(e^2*g 
*h^2*sqrt(-(81*b^4*d*e^2*f^4 - 18*b^4*d^2*e*f^2*g^2 + b^4*d^3*g^4)*p^4/(e^ 
3*h^2)) - 3*(9*b^2*e^2*f^3 - b^2*d*e*f*g^2)*h*p^2)*sqrt(-(6*b^2*d*f*g*p^2 
- e*h*sqrt(-(81*b^4*d*e^2*f^4 - 18*b^4*d^2*e*f^2*g^2 + b^4*d^3*g^4)*p^4/(e 
^3*h^2)))/(e*h))) + 3*h*sqrt(-(6*b^2*d*f*g*p^2 - e*h*sqrt(-(81*b^4*d*e^2*f 
^4 - 18*b^4*d^2*e*f^2*g^2 + b^4*d^3*g^4)*p^4/(e^3*h^2)))/(e*h))*log(-32*(8 
1*b^3*e^2*f^4 - b^3*d^2*g^4)*sqrt(h*x)*p^3 - 32*(e^2*g*h^2*sqrt(-(81*b^...
 

Sympy [F(-2)]

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

integrate((g*x+f)*(a+b*ln(c*(e*x**2+d)**p))/(h*x)**(1/2),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. 754 vs. \(2 (338) = 676\).

Time = 0.13 (sec) , antiderivative size = 754, normalized size of antiderivative = 1.59 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx =\text {Too large to display} \] Input:

integrate((g*x+f)*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(1/2),x, algorithm="maxim 
a")
 

Output:

2/3*b*g*x^2*log((e*x^2 + d)^p*c)/sqrt(h*x) + 2/3*a*g*x^2/sqrt(h*x) + 2*sqr 
t(h*x)*b*f*log((e*x^2 + d)^p*c)/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) + sqrt(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)*sqr 
t(h*x)*e^(1/4) + sqrt(d)*h)/((d*h^2)^(3/4)*e^(1/4)) + sqrt(2)*h^3*log(-(sq 
rt(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)) + 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)*sqrt(e)*h)*sqrt(d))) 
*d/e)*b*e*f*p/h^3 + 2*sqrt(h*x)*a*f/h - 1/9*(3*d*h^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(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))/(sqr 
t(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)*sqr 
t(-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) + ...
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 556, normalized size of antiderivative = 1.17 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx =\text {Too large to display} \] Input:

integrate((g*x+f)*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(1/2),x, algorithm="giac" 
)
 

Output:

1/9*(6*sqrt(h*x)*b*g*x*log(c) + 9*(e*(2*sqrt(2)*(d*e^3*h^2)^(1/4)*arctan(1 
/2*sqrt(2)*(sqrt(2)*(d*h^2/e)^(1/4) + 2*sqrt(h*x))/(d*h^2/e)^(1/4))/e^2 + 
2*sqrt(2)*(d*e^3*h^2)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(d*h^2/e)^(1/4) - 
 2*sqrt(h*x))/(d*h^2/e)^(1/4))/e^2 + sqrt(2)*(d*e^3*h^2)^(1/4)*log(h*x + s 
qrt(2)*(d*h^2/e)^(1/4)*sqrt(h*x) + sqrt(d*h^2/e))/e^2 - sqrt(2)*(d*e^3*h^2 
)^(1/4)*log(h*x - sqrt(2)*(d*h^2/e)^(1/4)*sqrt(h*x) + sqrt(d*h^2/e))/e^2 - 
 8*sqrt(h*x)/e) + 2*sqrt(h*x)*log(e*x^2 + d))*b*f*p + 6*sqrt(h*x)*a*g*x + 
18*sqrt(h*x)*b*f*log(c) + (6*sqrt(h*x)*h*x*log(e*x^2 + d) - (8*sqrt(h*x)*h 
*x/e - 6*sqrt(2)*(d*e^3*h^2)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(d*h^2/e)^( 
1/4) + 2*sqrt(h*x))/(d*h^2/e)^(1/4))/e^4 - 6*sqrt(2)*(d*e^3*h^2)^(3/4)*arc 
tan(-1/2*sqrt(2)*(sqrt(2)*(d*h^2/e)^(1/4) - 2*sqrt(h*x))/(d*h^2/e)^(1/4))/ 
e^4 + 3*sqrt(2)*(d*e^3*h^2)^(3/4)*log(h*x + sqrt(2)*(d*h^2/e)^(1/4)*sqrt(h 
*x) + sqrt(d*h^2/e))/e^4 - 3*sqrt(2)*(d*e^3*h^2)^(3/4)*log(h*x - sqrt(2)*( 
d*h^2/e)^(1/4)*sqrt(h*x) + sqrt(d*h^2/e))/e^4)*e)*b*g*p/h + 18*sqrt(h*x)*a 
*f)/h
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.76 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\frac {\sqrt {h}\, \left (-6 e^{\frac {1}{4}} d^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {e}}{e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}}\right ) b g p -18 e^{\frac {3}{4}} d^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {e}}{e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}}\right ) b f p +6 e^{\frac {1}{4}} d^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {e}}{e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}}\right ) b g p +18 e^{\frac {3}{4}} d^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {e}}{e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}}\right ) b f p +6 e^{\frac {1}{4}} d^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}+\sqrt {d}+\sqrt {e}\, x \right ) b g p -3 e^{\frac {1}{4}} d^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) b g -18 e^{\frac {3}{4}} d^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}+\sqrt {d}+\sqrt {e}\, x \right ) b f p +9 e^{\frac {3}{4}} d^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) b f +18 \sqrt {x}\, \mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) b e f +6 \sqrt {x}\, \mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) b e g x +18 \sqrt {x}\, a e f +6 \sqrt {x}\, a e g x -72 \sqrt {x}\, b e f p -8 \sqrt {x}\, b e g p x \right )}{9 e h} \] Input:

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

Output:

(sqrt(h)*( - 6*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 - 18*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 + 6*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 + 18 
*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*f*p + 6*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 - 3*e** 
(1/4)*d**(3/4)*sqrt(2)*log((d + e*x**2)**p*c)*b*g - 18*e**(3/4)*d**(1/4)*s 
qrt(2)*log( - sqrt(x)*e**(1/4)*d**(1/4)*sqrt(2) + sqrt(d) + sqrt(e)*x)*b*f 
*p + 9*e**(3/4)*d**(1/4)*sqrt(2)*log((d + e*x**2)**p*c)*b*f + 18*sqrt(x)*l 
og((d + e*x**2)**p*c)*b*e*f + 6*sqrt(x)*log((d + e*x**2)**p*c)*b*e*g*x + 1 
8*sqrt(x)*a*e*f + 6*sqrt(x)*a*e*g*x - 72*sqrt(x)*b*e*f*p - 8*sqrt(x)*b*e*g 
*p*x))/(9*e*h)