\(\int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{3/2}} \, dx\) [172]

Optimal result

Integrand size = 29, antiderivative size = 170 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{3/2}} \, dx=-\frac {2 (f g-e h)}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {2 \sqrt {b} (b g-a h) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{(b c-a d) (b e-a f)^{3/2}}+\frac {2 \sqrt {d} (d g-c h) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(b c-a d) (d e-c f)^{3/2}} \] Output:

(2*e*h-2*f*g)/(-a*f+b*e)/(-c*f+d*e)/(f*x+e)^(1/2)-2*b^(1/2)*(-a*h+b*g)*arc 
tanh(b^(1/2)*(f*x+e)^(1/2)/(-a*f+b*e)^(1/2))/(-a*d+b*c)/(-a*f+b*e)^(3/2)+2 
*d^(1/2)*(-c*h+d*g)*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/(-a*d+ 
b*c)/(-c*f+d*e)^(3/2)
 

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.99 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{3/2}} \, dx=2 \left (\frac {-f g+e h}{(b e-a f) (d e-c f) \sqrt {e+f x}}+\frac {\sqrt {b} (b g-a h) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{(-b c+a d) (-b e+a f)^{3/2}}+\frac {\sqrt {d} (d g-c h) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{(b c-a d) (-d e+c f)^{3/2}}\right ) \] Input:

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

Output:

2*((-(f*g) + e*h)/((b*e - a*f)*(d*e - c*f)*Sqrt[e + f*x]) + (Sqrt[b]*(b*g 
- a*h)*ArcTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[-(b*e) + a*f]])/((-(b*c) + a*d) 
*(-(b*e) + a*f)^(3/2)) + (Sqrt[d]*(d*g - c*h)*ArcTan[(Sqrt[d]*Sqrt[e + f*x 
])/Sqrt[-(d*e) + c*f]])/((b*c - a*d)*(-(d*e) + c*f)^(3/2)))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.22, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {169, 27, 174, 73, 221}

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.

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

\(\Big \downarrow \) 169

\(\displaystyle -\frac {2 \int -\frac {b d e g-b c f g-a d f g+a c f h-b d (f g-e h) x}{2 (a+b x) (c+d x) \sqrt {e+f x}}dx}{(b e-a f) (d e-c f)}-\frac {2 (f g-e h)}{\sqrt {e+f x} (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {b (d e-c f) g-a f (d g-c h)-b d (f g-e h) x}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{(b e-a f) (d e-c f)}-\frac {2 (f g-e h)}{\sqrt {e+f x} (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {b (b g-a h) (d e-c f) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b c-a d}-\frac {d (b e-a f) (d g-c h) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{b c-a d}}{(b e-a f) (d e-c f)}-\frac {2 (f g-e h)}{\sqrt {e+f x} (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {2 b (b g-a h) (d e-c f) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{f (b c-a d)}-\frac {2 d (b e-a f) (d g-c h) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{f (b c-a d)}}{(b e-a f) (d e-c f)}-\frac {2 (f g-e h)}{\sqrt {e+f x} (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {2 \sqrt {d} (b e-a f) (d g-c h) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(b c-a d) \sqrt {d e-c f}}-\frac {2 \sqrt {b} (b g-a h) (d e-c f) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{(b c-a d) \sqrt {b e-a f}}}{(b e-a f) (d e-c f)}-\frac {2 (f g-e h)}{\sqrt {e+f x} (b e-a f) (d e-c f)}\)

Input:

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

Output:

(-2*(f*g - e*h))/((b*e - a*f)*(d*e - c*f)*Sqrt[e + f*x]) + ((-2*Sqrt[b]*(d 
*e - c*f)*(b*g - a*h)*ArcTanh[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/(( 
b*c - a*d)*Sqrt[b*e - a*f]) + (2*Sqrt[d]*(b*e - a*f)*(d*g - c*h)*ArcTanh[( 
Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/((b*c - a*d)*Sqrt[d*e - c*f]))/(( 
b*e - a*f)*(d*e - c*f))
 

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.99

method	result	size
derivativedivides	\(-\frac {2 b \left (a h -b g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{\left (a d -b c \right ) \left (a f -b e \right ) \sqrt {\left (a f -b e \right ) b}}-\frac {2 \left (-e h +f g \right )}{\left (a f -b e \right ) \left (c f -d e \right ) \sqrt {f x +e}}+\frac {2 d \left (c h -d g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (a d -b c \right ) \left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}\)	\(169\)
default	\(-\frac {2 b \left (a h -b g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{\left (a d -b c \right ) \left (a f -b e \right ) \sqrt {\left (a f -b e \right ) b}}-\frac {2 \left (-e h +f g \right )}{\left (a f -b e \right ) \left (c f -d e \right ) \sqrt {f x +e}}+\frac {2 d \left (c h -d g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (a d -b c \right ) \left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}\)	\(169\)
pseudoelliptic	\(-\frac {2 b \left (a h -b g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{\left (a d -b c \right ) \left (a f -b e \right ) \sqrt {\left (a f -b e \right ) b}}+\frac {2 e h -2 f g}{\left (a f -b e \right ) \left (c f -d e \right ) \sqrt {f x +e}}+\frac {2 d \left (c h -d g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (a d -b c \right ) \left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}\)	\(169\)

Input:

int((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-2*b*(a*h-b*g)/(a*d-b*c)/(a*f-b*e)/((a*f-b*e)*b)^(1/2)*arctan(b*(f*x+e)^(1 
/2)/((a*f-b*e)*b)^(1/2))-2*(-e*h+f*g)/(a*f-b*e)/(c*f-d*e)/(f*x+e)^(1/2)+2* 
d*(c*h-d*g)/(a*d-b*c)/(c*f-d*e)/((c*f-d*e)*d)^(1/2)*arctan(d*(f*x+e)^(1/2) 
/((c*f-d*e)*d)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (148) = 296\).

Time = 20.21 (sec) , antiderivative size = 1604, normalized size of antiderivative = 9.44 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[-(((b*d*e^2 - b*c*e*f)*g - (a*d*e^2 - a*c*e*f)*h + ((b*d*e*f - b*c*f^2)*g 
 - (a*d*e*f - a*c*f^2)*h)*x)*sqrt(b/(b*e - a*f))*log((b*f*x + 2*b*e - a*f 
+ 2*(b*e - a*f)*sqrt(f*x + e)*sqrt(b/(b*e - a*f)))/(b*x + a)) + ((b*d*e^2 
- a*d*e*f)*g - (b*c*e^2 - a*c*e*f)*h + ((b*d*e*f - a*d*f^2)*g - (b*c*e*f - 
 a*c*f^2)*h)*x)*sqrt(d/(d*e - c*f))*log((d*f*x + 2*d*e - c*f - 2*(d*e - c* 
f)*sqrt(f*x + e)*sqrt(d/(d*e - c*f)))/(d*x + c)) + 2*((b*c - a*d)*f*g - (b 
*c - a*d)*e*h)*sqrt(f*x + e))/((b^2*c*d - a*b*d^2)*e^3 - (b^2*c^2 - a^2*d^ 
2)*e^2*f + (a*b*c^2 - a^2*c*d)*e*f^2 + ((b^2*c*d - a*b*d^2)*e^2*f - (b^2*c 
^2 - a^2*d^2)*e*f^2 + (a*b*c^2 - a^2*c*d)*f^3)*x), -(2*((b*d*e^2 - a*d*e*f 
)*g - (b*c*e^2 - a*c*e*f)*h + ((b*d*e*f - a*d*f^2)*g - (b*c*e*f - a*c*f^2) 
*h)*x)*sqrt(-d/(d*e - c*f))*arctan(sqrt(f*x + e)*sqrt(-d/(d*e - c*f))) + ( 
(b*d*e^2 - b*c*e*f)*g - (a*d*e^2 - a*c*e*f)*h + ((b*d*e*f - b*c*f^2)*g - ( 
a*d*e*f - a*c*f^2)*h)*x)*sqrt(b/(b*e - a*f))*log((b*f*x + 2*b*e - a*f + 2* 
(b*e - a*f)*sqrt(f*x + e)*sqrt(b/(b*e - a*f)))/(b*x + a)) + 2*((b*c - a*d) 
*f*g - (b*c - a*d)*e*h)*sqrt(f*x + e))/((b^2*c*d - a*b*d^2)*e^3 - (b^2*c^2 
 - a^2*d^2)*e^2*f + (a*b*c^2 - a^2*c*d)*e*f^2 + ((b^2*c*d - a*b*d^2)*e^2*f 
 - (b^2*c^2 - a^2*d^2)*e*f^2 + (a*b*c^2 - a^2*c*d)*f^3)*x), (2*((b*d*e^2 - 
 b*c*e*f)*g - (a*d*e^2 - a*c*e*f)*h + ((b*d*e*f - b*c*f^2)*g - (a*d*e*f - 
a*c*f^2)*h)*x)*sqrt(-b/(b*e - a*f))*arctan(sqrt(f*x + e)*sqrt(-b/(b*e - a* 
f))) - ((b*d*e^2 - a*d*e*f)*g - (b*c*e^2 - a*c*e*f)*h + ((b*d*e*f - a*d...
 

Sympy [A] (verification not implemented)

Time = 18.18 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.20 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {f \left (e h - f g\right )}{\sqrt {e + f x} \left (a f - b e\right ) \left (c f - d e\right )} + \frac {f \left (c h - d g\right ) \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{\sqrt {\frac {c f - d e}{d}} \left (a d - b c\right ) \left (c f - d e\right )} - \frac {f \left (a h - b g\right ) \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {a f - b e}{b}}} \right )}}{\sqrt {\frac {a f - b e}{b}} \left (a d - b c\right ) \left (a f - b e\right )}\right )}{f} & \text {for}\: f \neq 0 \\\frac {\frac {\left (a h - b g\right ) \left (\begin {cases} \frac {x}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b x \right )}}{b} & \text {otherwise} \end {cases}\right )}{a d - b c} - \frac {\left (c h - d g\right ) \left (\begin {cases} \frac {x}{c} & \text {for}\: d = 0 \\\frac {\log {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right )}{a d - b c}}{e^{\frac {3}{2}}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((2*(f*(e*h - f*g)/(sqrt(e + f*x)*(a*f - b*e)*(c*f - d*e)) + f*(c 
*h - d*g)*atan(sqrt(e + f*x)/sqrt((c*f - d*e)/d))/(sqrt((c*f - d*e)/d)*(a* 
d - b*c)*(c*f - d*e)) - f*(a*h - b*g)*atan(sqrt(e + f*x)/sqrt((a*f - b*e)/ 
b))/(sqrt((a*f - b*e)/b)*(a*d - b*c)*(a*f - b*e)))/f, Ne(f, 0)), (((a*h - 
b*g)*Piecewise((x/a, Eq(b, 0)), (log(a + b*x)/b, True))/(a*d - b*c) - (c*h 
 - d*g)*Piecewise((x/c, Eq(d, 0)), (log(c + d*x)/d, True))/(a*d - b*c))/e* 
*(3/2), True))
 

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(c*f-d*e>0)', see `assume?` for m 
ore detail
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.16 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{3/2}} \, dx=\frac {2 \, {\left (b^{2} g - a b h\right )} \arctan \left (\frac {\sqrt {f x + e} b}{\sqrt {-b^{2} e + a b f}}\right )}{{\left (b^{2} c e - a b d e - a b c f + a^{2} d f\right )} \sqrt {-b^{2} e + a b f}} - \frac {2 \, {\left (d^{2} g - c d h\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{{\left (b c d e - a d^{2} e - b c^{2} f + a c d f\right )} \sqrt {-d^{2} e + c d f}} - \frac {2 \, {\left (f g - e h\right )}}{{\left (b d e^{2} - b c e f - a d e f + a c f^{2}\right )} \sqrt {f x + e}} \] Input:

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

Output:

2*(b^2*g - a*b*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^2*c*e - 
 a*b*d*e - a*b*c*f + a^2*d*f)*sqrt(-b^2*e + a*b*f)) - 2*(d^2*g - c*d*h)*ar 
ctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/((b*c*d*e - a*d^2*e - b*c^2*f + 
 a*c*d*f)*sqrt(-d^2*e + c*d*f)) - 2*(f*g - e*h)/((b*d*e^2 - b*c*e*f - a*d* 
e*f + a*c*f^2)*sqrt(f*x + e))
 

Mupad [B] (verification not implemented)

Time = 6.50 (sec) , antiderivative size = 30821, normalized size of antiderivative = 181.30 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

atan((a^2*b^5*d^5*e^7*(e + f*x)^(1/2)*((b^3*g^2 + a^2*b*h^2 - 2*a*b^2*g*h) 
/(b^5*c^2*e^3 - a^5*d^2*f^3 + a^2*b^3*d^2*e^3 - a^3*b^2*c^2*f^3 - 2*a*b^4* 
c*d*e^3 + 2*a^4*b*c*d*f^3 - 3*a*b^4*c^2*e^2*f + 3*a^4*b*d^2*e*f^2 + 3*a^2* 
b^3*c^2*e*f^2 - 3*a^3*b^2*d^2*e^2*f + 6*a^2*b^3*c*d*e^2*f - 6*a^3*b^2*c*d* 
e*f^2))^(3/2)*2i - a^4*b^3*c^5*f^7*(e + f*x)^(1/2)*((b^3*g^2 + a^2*b*h^2 - 
 2*a*b^2*g*h)/(b^5*c^2*e^3 - a^5*d^2*f^3 + a^2*b^3*d^2*e^3 - a^3*b^2*c^2*f 
^3 - 2*a*b^4*c*d*e^3 + 2*a^4*b*c*d*f^3 - 3*a*b^4*c^2*e^2*f + 3*a^4*b*d^2*e 
*f^2 + 3*a^2*b^3*c^2*e*f^2 - 3*a^3*b^2*d^2*e^2*f + 6*a^2*b^3*c*d*e^2*f - 6 
*a^3*b^2*c*d*e*f^2))^(3/2)*1i - a^7*c^2*d^3*f^7*(e + f*x)^(1/2)*((b^3*g^2 
+ a^2*b*h^2 - 2*a*b^2*g*h)/(b^5*c^2*e^3 - a^5*d^2*f^3 + a^2*b^3*d^2*e^3 - 
a^3*b^2*c^2*f^3 - 2*a*b^4*c*d*e^3 + 2*a^4*b*c*d*f^3 - 3*a*b^4*c^2*e^2*f + 
3*a^4*b*d^2*e*f^2 + 3*a^2*b^3*c^2*e*f^2 - 3*a^3*b^2*d^2*e^2*f + 6*a^2*b^3* 
c*d*e^2*f - 6*a^3*b^2*c*d*e*f^2))^(3/2)*1i + b^7*c^2*d^3*e^7*(e + f*x)^(1/ 
2)*((b^3*g^2 + a^2*b*h^2 - 2*a*b^2*g*h)/(b^5*c^2*e^3 - a^5*d^2*f^3 + a^2*b 
^3*d^2*e^3 - a^3*b^2*c^2*f^3 - 2*a*b^4*c*d*e^3 + 2*a^4*b*c*d*f^3 - 3*a*b^4 
*c^2*e^2*f + 3*a^4*b*d^2*e*f^2 + 3*a^2*b^3*c^2*e*f^2 - 3*a^3*b^2*d^2*e^2*f 
 + 6*a^2*b^3*c*d*e^2*f - 6*a^3*b^2*c*d*e*f^2))^(3/2)*2i - a^7*d^5*e^2*f^5* 
(e + f*x)^(1/2)*((b^3*g^2 + a^2*b*h^2 - 2*a*b^2*g*h)/(b^5*c^2*e^3 - a^5*d^ 
2*f^3 + a^2*b^3*d^2*e^3 - a^3*b^2*c^2*f^3 - 2*a*b^4*c*d*e^3 + 2*a^4*b*c*d* 
f^3 - 3*a*b^4*c^2*e^2*f + 3*a^4*b*d^2*e*f^2 + 3*a^2*b^3*c^2*e*f^2 - 3*a...
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 909, normalized size of antiderivative = 5.35 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

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