\(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{5/2}} \, dx\) [738]

Optimal result

Integrand size = 48, antiderivative size = 63 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{5/2}} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {874} \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{5/2}} \, dx=\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)} \]

[In]

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

[Out]

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

Rule 874

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] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d
*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0] && EqQ[m - n - 2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{5/2}} \, dx=\frac {2 ((a e+c d x) (d+e x))^{3/2}}{3 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.84

method	result	size
default	\(-\frac {2 \left (c d x +a e \right ) \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}}{3 \left (g x +f \right )^{\frac {3}{2}} \left (a e g -c d f \right ) \sqrt {e x +d}}\)	\(53\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{3 \left (g x +f \right )^{\frac {3}{2}} \left (a e g -c d f \right ) \sqrt {e x +d}}\)	\(63\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (55) = 110\).

Time = 0.31 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.68 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{5/2}} \, dx=\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d x + a e\right )} \sqrt {e x + d} \sqrt {g x + f}}{3 \, {\left (c d^{2} f^{3} - a d e f^{2} g + {\left (c d e f g^{2} - a e^{2} g^{3}\right )} x^{3} + {\left (2 \, c d e f^{2} g - a d e g^{3} + {\left (c d^{2} - 2 \, a e^{2}\right )} f g^{2}\right )} x^{2} + {\left (c d e f^{3} - 2 \, a d e f g^{2} + {\left (2 \, c d^{2} - a e^{2}\right )} f^{2} g\right )} x\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (55) = 110\).

Time = 0.51 (sec) , antiderivative size = 311, normalized size of antiderivative = 4.94 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {{\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{2} d^{2} e^{4} g {\left | c \right |} {\left | d \right |}}{{\left (c^{2} d^{2} e^{2} f - a c d e^{3} g + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} c d g\right )}^{\frac {3}{2}} {\left (c d e^{2} f g {\left | e \right |} - a e^{3} g^{2} {\left | e \right |}\right )}} + \frac {\sqrt {-c d^{2} e + a e^{3}} c d^{2} e^{2} {\left | c \right |} {\left | d \right |} - \sqrt {-c d^{2} e + a e^{3}} a e^{4} {\left | c \right |} {\left | d \right |}}{\sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} c d e f^{2} {\left | e \right |} - \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} c d^{2} f g {\left | e \right |} - \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a e^{2} f g {\left | e \right |} + \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a d e g^{2} {\left | e \right |}}\right )} {\left | e \right |}}{3 \, e^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 12.99 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.16 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{5/2}} \, dx=-\frac {\left (\frac {2\,a\,e}{3\,a\,e\,g^2-3\,c\,d\,f\,g}+\frac {2\,c\,d\,x}{3\,a\,e\,g^2-3\,c\,d\,f\,g}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}-\frac {\sqrt {f+g\,x}\,\left (3\,c\,d\,f^2-3\,a\,e\,f\,g\right )\,\sqrt {d+e\,x}}{3\,a\,e\,g^2-3\,c\,d\,f\,g}} \]

[In]

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

[Out]

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