3.8.24 \(\int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [724]

3.8.24.1 Optimal result

Integrand size = 48, antiderivative size = 124 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {4 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(c d f-a e g)^2 \sqrt {d+e x} \sqrt {f+g x}} \]

output

-2*(e*x+d)^(1/2)/(-a*e*g+c*d*f)/(g*x+f)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e 
*x^2)^(1/2)-4*g*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^2/( 
e*x+d)^(1/2)/(g*x+f)^(1/2)
 

3.8.24.2 Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.52 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} (a e g+c d (f+2 g x))}{(c d f-a e g)^2 \sqrt {(a e+c d x) (d+e x)} \sqrt {f+g x}} \]

input

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

output

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

3.8.24.3 Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {1252, 1248}

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.

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

\(\Big \downarrow \) 1252

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

\(\Big \downarrow \) 1248

\(\displaystyle -\frac {4 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^2}-\frac {2 \sqrt {d+e x}}{\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}\)

input

Int[(d + e*x)^(3/2)/((f + g*x)^(3/2)*(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[f + g*x]*Sqrt[a*d*e + (c*d^2 + a* 
e^2)*x + c*d*e*x^2]) - (4*g*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/( 
(c*d*f - a*e*g)^2*Sqrt[d + e*x]*Sqrt[f + g*x])
 

3.8.24.3.1 Defintions of rubi rules used

rule 1248

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)/((n + 1)*(c*e*f + c*d*g - b*e*g))), x] / 
; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] 
 && EqQ[m + p, 0] && EqQ[m - n - 2, 0]
 

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]
 

3.8.24.4 Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.56

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (2 c d g x +a e g +c d f \right )}{\sqrt {e x +d}\, \sqrt {g x +f}\, \left (c d x +a e \right ) \left (a e g -c d f \right )^{2}}\)	\(70\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (2 c d g x +a e g +c d f \right ) \left (e x +d \right )^{\frac {3}{2}}}{\sqrt {g x +f}\, \left (a^{2} e^{2} g^{2}-2 a c d e f g +c^{2} d^{2} f^{2}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\)	\(97\)

input

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

output

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

3.8.24.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (112) = 224\).

Time = 0.35 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.62 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d g x + c d f + a e g\right )} \sqrt {e x + d} \sqrt {g x + f}}{a c^{2} d^{3} e f^{3} - 2 \, a^{2} c d^{2} e^{2} f^{2} g + a^{3} d e^{3} f g^{2} + {\left (c^{3} d^{3} e f^{2} g - 2 \, a c^{2} d^{2} e^{2} f g^{2} + a^{2} c d e^{3} g^{3}\right )} x^{3} + {\left (c^{3} d^{3} e f^{3} + {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} f^{2} g - {\left (2 \, a c^{2} d^{3} e + a^{2} c d e^{3}\right )} f g^{2} + {\left (a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} g^{3}\right )} x^{2} + {\left (a^{3} d e^{3} g^{3} + {\left (c^{3} d^{4} + a c^{2} d^{2} e^{2}\right )} f^{3} - {\left (a c^{2} d^{3} e + 2 \, a^{2} c d e^{3}\right )} f^{2} g - {\left (a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f g^{2}\right )} x} \]

input

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

output

-2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*g*x + c*d*f + a*e*g) 
*sqrt(e*x + d)*sqrt(g*x + f)/(a*c^2*d^3*e*f^3 - 2*a^2*c*d^2*e^2*f^2*g + a^ 
3*d*e^3*f*g^2 + (c^3*d^3*e*f^2*g - 2*a*c^2*d^2*e^2*f*g^2 + a^2*c*d*e^3*g^3 
)*x^3 + (c^3*d^3*e*f^3 + (c^3*d^4 - a*c^2*d^2*e^2)*f^2*g - (2*a*c^2*d^3*e 
+ a^2*c*d*e^3)*f*g^2 + (a^2*c*d^2*e^2 + a^3*e^4)*g^3)*x^2 + (a^3*d*e^3*g^3 
 + (c^3*d^4 + a*c^2*d^2*e^2)*f^3 - (a*c^2*d^3*e + 2*a^2*c*d*e^3)*f^2*g - ( 
a^2*c*d^2*e^2 - a^3*e^4)*f*g^2)*x)
 

3.8.24.6 Sympy [F]

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

input

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

output

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

3.8.24.7 Maxima [F]

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

input

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

3.8.24.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 793 vs. \(2 (112) = 224\).

Time = 0.38 (sec) , antiderivative size = 793, normalized size of antiderivative = 6.40 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-2 \, {\left (\frac {\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} c d g^{2}}{{\left (c^{2} d^{2} e^{2} f^{2} {\left | g \right |} - 2 \, a c d e^{3} f g {\left | g \right |} + a^{2} e^{4} g^{2} {\left | g \right |}\right )} \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g}} + \frac {2 \, \sqrt {c d g} g^{2}}{{\left (c d e^{2} f g - a e^{3} g^{2} + {\left (\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \sqrt {c d g} - \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g}\right )}^{2}\right )} {\left (c d f {\left | g \right |} - a e g {\left | g \right |}\right )}}\right )} e^{2} + \frac {2 \, {\left (\sqrt {e^{2} f - d e g} c^{2} d^{2} e f g^{2} - \sqrt {e^{2} f - d e g} c^{2} d^{3} g^{3} + \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \sqrt {c d g} c d^{2} g^{2} - \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \sqrt {c d g} a e^{2} g^{2}\right )}}{\sqrt {e^{2} f - d e g} \sqrt {c d g} c^{3} d^{4} f^{2} g {\left | g \right |} - \sqrt {e^{2} f - d e g} \sqrt {c d g} a c^{2} d^{2} e^{2} f^{2} g {\left | g \right |} - 2 \, \sqrt {e^{2} f - d e g} \sqrt {c d g} a c^{2} d^{3} e f g^{2} {\left | g \right |} + 2 \, \sqrt {e^{2} f - d e g} \sqrt {c d g} a^{2} c d e^{3} f g^{2} {\left | g \right |} + \sqrt {e^{2} f - d e g} \sqrt {c d g} a^{2} c d^{2} e^{2} g^{3} {\left | g \right |} - \sqrt {e^{2} f - d e g} \sqrt {c d g} a^{3} e^{4} g^{3} {\left | g \right |} + \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} c^{3} d^{3} e f^{3} {\left | g \right |} - \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} c^{3} d^{4} f^{2} g {\left | g \right |} - 2 \, \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} a c^{2} d^{2} e^{2} f^{2} g {\left | g \right |} + 2 \, \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} a c^{2} d^{3} e f g^{2} {\left | g \right |} + \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} a^{2} c d e^{3} f g^{2} {\left | g \right |} - \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} a^{2} c d^{2} e^{2} g^{3} {\left | g \right |}} \]

input

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

output

-2*(sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g^2/((c^2*d^2*e^2*f^2*abs(g) - 
 2*a*c*d*e^3*f*g*abs(g) + a^2*e^4*g^2*abs(g))*sqrt(-c*d*e^2*f*g + a*e^3*g^ 
2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)) + 2*sqrt(c*d*g)*g^2/((c*d*e^2* 
f*g - a*e^3*g^2 + (sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) - sqrt( 
-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g))^2)*(c*d 
*f*abs(g) - a*e*g*abs(g))))*e^2 + 2*(sqrt(e^2*f - d*e*g)*c^2*d^2*e*f*g^2 - 
 sqrt(e^2*f - d*e*g)*c^2*d^3*g^3 + sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d 
*g)*c*d^2*g^2 - sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*a*e^2*g^2)/(sqr 
t(e^2*f - d*e*g)*sqrt(c*d*g)*c^3*d^4*f^2*g*abs(g) - sqrt(e^2*f - d*e*g)*sq 
rt(c*d*g)*a*c^2*d^2*e^2*f^2*g*abs(g) - 2*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a 
*c^2*d^3*e*f*g^2*abs(g) + 2*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c*d*e^3*f* 
g^2*abs(g) + sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c*d^2*e^2*g^3*abs(g) - sq 
rt(e^2*f - d*e*g)*sqrt(c*d*g)*a^3*e^4*g^3*abs(g) + sqrt(-c*d^2*e*g^2 + a*e 
^3*g^2)*c^3*d^3*e*f^3*abs(g) - sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*c^3*d^4*f^2* 
g*abs(g) - 2*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a*c^2*d^2*e^2*f^2*g*abs(g) + 2 
*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a*c^2*d^3*e*f*g^2*abs(g) + sqrt(-c*d^2*e*g 
^2 + a*e^3*g^2)*a^2*c*d*e^3*f*g^2*abs(g) - sqrt(-c*d^2*e*g^2 + a*e^3*g^2)* 
a^2*c*d^2*e^2*g^3*abs(g))
 

3.8.24.9 Mupad [B] (verification not implemented)

Time = 13.59 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.22 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {\left (\frac {4\,g\,x\,\sqrt {d+e\,x}}{e\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {\left (2\,a\,e\,g+2\,c\,d\,f\right )\,\sqrt {d+e\,x}}{c\,d\,e\,{\left (a\,e\,g-c\,d\,f\right )}^2}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^2\,\sqrt {f+g\,x}+\frac {a\,\sqrt {f+g\,x}}{c}+\frac {x\,\sqrt {f+g\,x}\,\left (c\,d^2+a\,e^2\right )}{c\,d\,e}} \]

input

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

output

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