\(\int \frac {(d+e x)^{3/2}}{(f+g x) (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [46]

Optimal result

Integrand size = 46, antiderivative size = 133 \[ \int \frac {(d+e x)^{3/2}}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 \sqrt {g} \arctan \left (\frac {\sqrt {c d f-a e g} \sqrt {d+e x}}{\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{(c d f-a e g)^{3/2}} \] Output:

-2*(e*x+d)^(1/2)/(-a*e*g+c*d*f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+2* 
g^(1/2)*arctan(1/g^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*(-a*e*g+c 
*d*f)^(1/2)*(e*x+d)^(1/2))/(-a*e*g+c*d*f)^(3/2)
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.82 \[ \int \frac {(d+e x)^{3/2}}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} \left (\sqrt {c d f-a e g}+\sqrt {g} \sqrt {a e+c d x} \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )\right )}{(c d f-a e g)^{3/2} \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1252, 1255, 218}

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.

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

\(\Big \downarrow \) 1252

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

\(\Big \downarrow \) 1255

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

\(\Big \downarrow \) 218

\(\displaystyle -\frac {2 \sqrt {g} \arctan \left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{(c d f-a e g)^{3/2}}-\frac {2 \sqrt {d+e x}}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 218

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

rule 1252

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
+ (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e^2*(d + e*x)^(m - 1)*(f + g*x)^(n 
+ 1)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(c*e*f + c*d*g - b*e*g))), x] + Si 
mp[e^2*g*((m - n - 2)/((p + 1)*(c*e*f + c*d*g - b*e*g)))   Int[(d + e*x)^(m 
 - 1)*(f + g*x)^n*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e 
, f, g, n}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p, 0] && LtQ[p, 
-1] && RationalQ[n]
 

rule 1255

Int[Sqrt[(d_) + (e_.)*(x_)]/(((f_.) + (g_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + 
 (c_.)*(x_)^2]), x_Symbol] :> Simp[2*e^2   Subst[Int[1/(c*(e*f + d*g) - b*e 
*g + e^2*g*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{ 
a, b, c, d, e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
 

Maple [A] (verified)

Time = 2.73 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.89

method	result	size
default	\(-\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (g \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) \sqrt {c d x +a e}-\sqrt {\left (a e g -d f c \right ) g}\right )}{\sqrt {e x +d}\, \left (c d x +a e \right ) \left (a e g -d f c \right ) \sqrt {\left (a e g -d f c \right ) g}}\)	\(118\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (117) = 234\).

Time = 0.10 (sec) , antiderivative size = 553, normalized size of antiderivative = 4.16 \[ \int \frac {(d+e x)^{3/2}}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\left [-\frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-\frac {g}{c d f - a e g}} \log \left (-\frac {c d e g x^{2} - c d^{2} f + 2 \, a d e g + 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d f - a e g\right )} \sqrt {e x + d} \sqrt {-\frac {g}{c d f - a e g}} - {\left (c d e f - {\left (c d^{2} + 2 \, a e^{2}\right )} g\right )} x}{e g x^{2} + d f + {\left (e f + d g\right )} x}\right ) + 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{a c d^{2} e f - a^{2} d e^{2} g + {\left (c^{2} d^{2} e f - a c d e^{2} g\right )} x^{2} + {\left ({\left (c^{2} d^{3} + a c d e^{2}\right )} f - {\left (a c d^{2} e + a^{2} e^{3}\right )} g\right )} x}, -\frac {2 \, {\left ({\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {\frac {g}{c d f - a e g}} \arctan \left (-\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d f - a e g\right )} \sqrt {e x + d} \sqrt {\frac {g}{c d f - a e g}}}{c d e g x^{2} + a d e g + {\left (c d^{2} + a e^{2}\right )} g x}\right ) + \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}\right )}}{a c d^{2} e f - a^{2} d e^{2} g + {\left (c^{2} d^{2} e f - a c d e^{2} g\right )} x^{2} + {\left ({\left (c^{2} d^{3} + a c d e^{2}\right )} f - {\left (a c d^{2} e + a^{2} e^{3}\right )} g\right )} x}\right ] \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.62 \[ \int \frac {(d+e x)^{3/2}}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \, g \arctan \left (\frac {\sqrt {c d x + a e} g}{\sqrt {c d f g - a e g^{2}}}\right )}{\sqrt {c d f g - a e g^{2}} {\left (c d f - a e g\right )}} - \frac {2}{{\left (c d f - a e g\right )} \sqrt {c d x + a e}} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.80 \[ \int \frac {(d+e x)^{3/2}}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {-2 \sqrt {g}\, \sqrt {c d x +a e}\, \sqrt {-a e g +c d f}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {g}\, \sqrt {-a e g +c d f}}\right )+2 a e g -2 c d f}{\sqrt {c d x +a e}\, \left (a^{2} e^{2} g^{2}-2 a c d e f g +c^{2} d^{2} f^{2}\right )} \] Input:

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

Output:

(2*( - sqrt(g)*sqrt(a*e + c*d*x)*sqrt( - a*e*g + c*d*f)*atan((sqrt(a*e + c 
*d*x)*g)/(sqrt(g)*sqrt( - a*e*g + c*d*f))) + a*e*g - c*d*f))/(sqrt(a*e + c 
*d*x)*(a**2*e**2*g**2 - 2*a*c*d*e*f*g + c**2*d**2*f**2))