\(\int \frac {\sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [88]

Optimal result

Integrand size = 48, antiderivative size = 129 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g) \sqrt {d+e x} (f+g x)^{3/2}}+\frac {4 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g)^2 \sqrt {d+e x} \sqrt {f+g x}} \] Output:

2/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)/(e*x+d)^(1/2)/( 
g*x+f)^(3/2)+4/3*c*d*(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)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)} (-a e g+c d (3 f+2 g x))}{3 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{3/2}} \] Input:

Integrate[Sqrt[d + e*x]/((f + g*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + 
c*d*e*x^2]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 129, 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 = {1254, 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.

\(\displaystyle \int \frac {\sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \, dx\)

\(\Big \downarrow \) 1254

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

\(\Big \downarrow \) 1248

\(\displaystyle \frac {4 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^2}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)}\)

Input:

Int[Sqrt[d + e*x]/((f + g*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e* 
x^2]),x]
 

Output:

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

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 1254

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

Maple [A] (verified)

Time = 2.98 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.47

method	result	size
default	\(-\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (-2 c d g x +a e g -3 d f c \right )}{3 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {3}{2}} \left (a e g -d f c \right )^{2}}\)	\(61\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-2 c d g x +a e g -3 d f c \right ) \sqrt {e x +d}}{3 \left (g x +f \right )^{\frac {3}{2}} \left (a^{2} e^{2} g^{2}-2 a c d e f g +f^{2} c^{2} d^{2}\right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}\)	\(98\)
orering	\(-\frac {2 \left (-2 c d g x +a e g -3 d f c \right ) \left (c d x +a e \right ) \sqrt {e x +d}}{3 \left (a^{2} e^{2} g^{2}-2 a c d e f g +f^{2} c^{2} d^{2}\right ) \left (g x +f \right )^{\frac {3}{2}} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\)	\(99\)

Input:

int((e*x+d)^(1/2)/(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2),x, 
method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (113) = 226\).

Time = 0.11 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.23 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d g x + 3 \, c d f - a e g\right )} \sqrt {e x + d} \sqrt {g x + f}}{3 \, {\left (c^{2} d^{3} f^{4} - 2 \, a c d^{2} e f^{3} g + a^{2} d e^{2} f^{2} g^{2} + {\left (c^{2} d^{2} e f^{2} g^{2} - 2 \, a c d e^{2} f g^{3} + a^{2} e^{3} g^{4}\right )} x^{3} + {\left (2 \, c^{2} d^{2} e f^{3} g + a^{2} d e^{2} g^{4} + {\left (c^{2} d^{3} - 4 \, a c d e^{2}\right )} f^{2} g^{2} - 2 \, {\left (a c d^{2} e - a^{2} e^{3}\right )} f g^{3}\right )} x^{2} + {\left (c^{2} d^{2} e f^{4} + 2 \, a^{2} d e^{2} f g^{3} + 2 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} f^{3} g - {\left (4 \, a c d^{2} e - a^{2} e^{3}\right )} f^{2} g^{2}\right )} x\right )}} \] Input:

integrate((e*x+d)^(1/2)/(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1 
/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {\sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {\sqrt {d + e x}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (f + g x\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((e*x+d)**(1/2)/(g*x+f)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x** 
2)**(1/2),x)
                                                                                    
                                                                                    
 

Output:

Integral(sqrt(d + e*x)/(sqrt((d + e*x)*(a*e + c*d*x))*(f + g*x)**(5/2)), x 
)
 

Maxima [F]

\[ \int \frac {\sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((e*x+d)^(1/2)/(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1 
/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x 
+ f)^(5/2)), x)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.64 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {8 \, {\left (c d e^{2} f g - a e^{3} g^{2} + 3 \, {\left (\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \sqrt {c d g} - \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g}\right )}^{2}\right )} \sqrt {c d g} c d e^{4} g^{2}}{3 \, {\left (c d e^{2} f g - a e^{3} g^{2} + {\left (\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \sqrt {c d g} - \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g}\right )}^{2}\right )}^{3} {\left | g \right |}} \] Input:

integrate((e*x+d)^(1/2)/(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1 
/2),x, algorithm="giac")
 

Output:

8/3*(c*d*e^2*f*g - a*e^3*g^2 + 3*(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)*sqrt(c*d*g)*c*d*e^4*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)^3*abs(g))
 

Mupad [B] (verification not implemented)

Time = 7.49 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.14 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {\left (\frac {\left (2\,a\,e\,g-6\,c\,d\,f\right )\,\sqrt {d+e\,x}}{3\,e\,g\,{\left (a\,e\,g-c\,d\,f\right )}^2}-\frac {4\,c\,d\,x\,\sqrt {d+e\,x}}{3\,e\,{\left (a\,e\,g-c\,d\,f\right )}^2}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^2\,\sqrt {f+g\,x}+\frac {d\,f\,\sqrt {f+g\,x}}{e\,g}+\frac {x\,\sqrt {f+g\,x}\,\left (d\,g+e\,f\right )}{e\,g}} \] Input:

int((d + e*x)^(1/2)/((f + g*x)^(5/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^ 
2)^(1/2)),x)
 

Output:

-((((2*a*e*g - 6*c*d*f)*(d + e*x)^(1/2))/(3*e*g*(a*e*g - c*d*f)^2) - (4*c* 
d*x*(d + e*x)^(1/2))/(3*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) + (d*f*(f + g*x)^(1/2))/(e*g) + (x 
*(f + g*x)^(1/2)*(d*g + e*f))/(e*g))
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.79 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {-\frac {2 \sqrt {g x +f}\, \sqrt {c d x +a e}\, a e \,g^{2}}{3}+2 \sqrt {g x +f}\, \sqrt {c d x +a e}\, c d f g +\frac {4 \sqrt {g x +f}\, \sqrt {c d x +a e}\, c d \,g^{2} x}{3}-\frac {4 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, c d \,f^{2}}{3}-\frac {8 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, c d f g x}{3}-\frac {4 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, c d \,g^{2} x^{2}}{3}}{g \left (a^{2} e^{2} g^{4} x^{2}-2 a c d e f \,g^{3} x^{2}+c^{2} d^{2} f^{2} g^{2} x^{2}+2 a^{2} e^{2} f \,g^{3} x -4 a c d e \,f^{2} g^{2} x +2 c^{2} d^{2} f^{3} g x +a^{2} e^{2} f^{2} g^{2}-2 a c d e \,f^{3} g +c^{2} d^{2} f^{4}\right )} \] Input:

int((e*x+d)^(1/2)/(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)
 

Output:

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