\(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{7/2}} \, dx\) [739]

Optimal result

Integrand size = 48, antiderivative size = 129 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{7/2}} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{5/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{15 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{3/2}} \]

[Out]

2/5*(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)^(5/2)+4/15*c*d*(a*d*e+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(3/2)/(-a*e*g+c*d*f)^2/(e*x+d)^(3/2)/(g*x+f)^(3/2)

Rubi [A] (verified)

Time = 0.08 (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 = {886, 874} \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{7/2}} \, dx=\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{15 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)} \]

[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)^(7/2)),x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(5*(c*d*f - a*e*g)*(d + e*x)^(3/2)*(f + g*x)^(5/2)) + (4*c*d
*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(15*(c*d*f - a*e*g)^2*(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]

Rule 886

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] - Dist[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] && 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] && LtQ[n, -1] && IntegerQ[2*p]

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}}{5 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{5/2}}+\frac {(2 c d) \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}{5 (c d f-a e g)} \\ & = \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{5/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{15 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{7/2}} \, dx=\frac {2 ((a e+c d x) (d+e x))^{3/2} (-3 a e g+c d (5 f+2 g x))}{15 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{5/2}} \]

[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)^(7/2)),x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(-3*a*e*g + c*d*(5*f + 2*g*x)))/(15*(c*d*f - a*e*g)^2*(d + e*x)^(3/2)*(f +
g*x)^(5/2))

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.54

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c d x +a e \right ) \left (-2 c d g x +3 a e g -5 c d f \right )}{15 \left (g x +f \right )^{\frac {5}{2}} \sqrt {e x +d}\, \left (a e g -c d f \right )^{2}}\)	\(70\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-2 c d g x +3 a e g -5 c d f \right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{15 \left (g x +f \right )^{\frac {5}{2}} \left (a^{2} e^{2} g^{2}-2 a c d e f g +c^{2} d^{2} f^{2}\right ) \sqrt {e x +d}}\)	\(99\)

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^(7/2)/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/15*((c*d*x+a*e)*(e*x+d))^(1/2)/(g*x+f)^(5/2)/(e*x+d)^(1/2)*(c*d*x+a*e)*(-2*c*d*g*x+3*a*e*g-5*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. 402 vs. \(2 (113) = 226\).

Time = 0.35 (sec) , antiderivative size = 402, normalized size of antiderivative = 3.12 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{7/2}} \, dx=\frac {2 \, {\left (2 \, c^{2} d^{2} g x^{2} + 5 \, a c d e f - 3 \, a^{2} e^{2} g + {\left (5 \, c^{2} d^{2} f - a c d e g\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f}}{15 \, {\left (c^{2} d^{3} f^{5} - 2 \, a c d^{2} e f^{4} g + a^{2} d e^{2} f^{3} g^{2} + {\left (c^{2} d^{2} e f^{2} g^{3} - 2 \, a c d e^{2} f g^{4} + a^{2} e^{3} g^{5}\right )} x^{4} + {\left (3 \, c^{2} d^{2} e f^{3} g^{2} + a^{2} d e^{2} g^{5} + {\left (c^{2} d^{3} - 6 \, a c d e^{2}\right )} f^{2} g^{3} - {\left (2 \, a c d^{2} e - 3 \, a^{2} e^{3}\right )} f g^{4}\right )} x^{3} + 3 \, {\left (c^{2} d^{2} e f^{4} g + a^{2} d e^{2} f g^{4} + {\left (c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{3} g^{2} - {\left (2 \, a c d^{2} e - a^{2} e^{3}\right )} f^{2} g^{3}\right )} x^{2} + {\left (c^{2} d^{2} e f^{5} + 3 \, a^{2} d e^{2} f^{2} g^{3} + {\left (3 \, c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{4} g - {\left (6 \, a c d^{2} e - a^{2} e^{3}\right )} f^{3} g^{2}\right )} x\right )}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^(7/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/15*(2*c^2*d^2*g*x^2 + 5*a*c*d*e*f - 3*a^2*e^2*g + (5*c^2*d^2*f - a*c*d*e*g)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d
^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f)/(c^2*d^3*f^5 - 2*a*c*d^2*e*f^4*g + a^2*d*e^2*f^3*g^2 + (c^2*d^2*e*f
^2*g^3 - 2*a*c*d*e^2*f*g^4 + a^2*e^3*g^5)*x^4 + (3*c^2*d^2*e*f^3*g^2 + a^2*d*e^2*g^5 + (c^2*d^3 - 6*a*c*d*e^2)
*f^2*g^3 - (2*a*c*d^2*e - 3*a^2*e^3)*f*g^4)*x^3 + 3*(c^2*d^2*e*f^4*g + a^2*d*e^2*f*g^4 + (c^2*d^3 - 2*a*c*d*e^
2)*f^3*g^2 - (2*a*c*d^2*e - a^2*e^3)*f^2*g^3)*x^2 + (c^2*d^2*e*f^5 + 3*a^2*d*e^2*f^2*g^3 + (3*c^2*d^3 - 2*a*c*
d*e^2)*f^4*g - (6*a*c*d^2*e - a^2*e^3)*f^3*g^2)*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{7/2}} \, dx=\text {Timed out} \]

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(g*x+f)**(7/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{7/2}} \, dx=\int { \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{\sqrt {e x + d} {\left (g x + f\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^(7/2)/(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)^(7/2)), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 762 vs. \(2 (113) = 226\).

Time = 0.84 (sec) , antiderivative size = 762, normalized size of antiderivative = 5.91 \[ \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)^{7/2}} \, dx=\frac {2 \, {\left (\frac {5 \, \sqrt {-c d^{2} e + a e^{3}} c^{2} d^{3} e^{3} f {\left | c \right |} {\left | d \right |} - 5 \, \sqrt {-c d^{2} e + a e^{3}} a c d e^{5} f {\left | c \right |} {\left | d \right |} - 2 \, \sqrt {-c d^{2} e + a e^{3}} c^{2} d^{4} e^{2} g {\left | c \right |} {\left | d \right |} - \sqrt {-c d^{2} e + a e^{3}} a c d^{2} e^{4} g {\left | c \right |} {\left | d \right |} + 3 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} e^{6} g {\left | c \right |} {\left | d \right |}}{\sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} c^{2} d^{2} e^{2} f^{4} {\left | e \right |} - 2 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} c^{2} d^{3} e f^{3} g {\left | e \right |} - 2 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a c d e^{3} f^{3} g {\left | e \right |} + \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} c^{2} d^{4} f^{2} g^{2} {\left | e \right |} + 4 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a c d^{2} e^{2} f^{2} g^{2} {\left | e \right |} + \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a^{2} e^{4} f^{2} g^{2} {\left | e \right |} - 2 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a c d^{3} e f g^{3} {\left | e \right |} - 2 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a^{2} d e^{3} f g^{3} {\left | e \right |} + \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a^{2} d^{2} e^{2} g^{4} {\left | e \right |}} + \frac {{\left (\frac {2 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} c^{4} d^{4} e^{6} g^{3} {\left | c \right |} {\left | d \right |}}{c^{2} d^{2} e^{4} f^{2} g^{2} {\left | e \right |} - 2 \, a c d e^{5} f g^{3} {\left | e \right |} + a^{2} e^{6} g^{4} {\left | e \right |}} + \frac {5 \, {\left (c^{5} d^{5} e^{8} f g^{2} {\left | c \right |} {\left | d \right |} - a c^{4} d^{4} e^{9} g^{3} {\left | c \right |} {\left | d \right |}\right )}}{c^{2} d^{2} e^{4} f^{2} g^{2} {\left | e \right |} - 2 \, a c d e^{5} f g^{3} {\left | e \right |} + a^{2} e^{6} g^{4} {\left | e \right |}}\right )} {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}}}{{\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 {5}{2}}}\right )} {\left | e \right |}}{15 \, e^{2}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^(7/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/15*((5*sqrt(-c*d^2*e + a*e^3)*c^2*d^3*e^3*f*abs(c)*abs(d) - 5*sqrt(-c*d^2*e + a*e^3)*a*c*d*e^5*f*abs(c)*abs(
d) - 2*sqrt(-c*d^2*e + a*e^3)*c^2*d^4*e^2*g*abs(c)*abs(d) - sqrt(-c*d^2*e + a*e^3)*a*c*d^2*e^4*g*abs(c)*abs(d)
+ 3*sqrt(-c*d^2*e + a*e^3)*a^2*e^6*g*abs(c)*abs(d))/(sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^2*d^2*e^2*f^4*abs(e)
- 2*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^2*d^3*e*f^3*g*abs(e) - 2*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c*d*e^3*
f^3*g*abs(e) + sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^2*d^4*f^2*g^2*abs(e) + 4*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*
a*c*d^2*e^2*f^2*g^2*abs(e) + sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^2*e^4*f^2*g^2*abs(e) - 2*sqrt(c^2*d^2*e^2*f -
c^2*d^3*e*g)*a*c*d^3*e*f*g^3*abs(e) - 2*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^2*d*e^3*f*g^3*abs(e) + sqrt(c^2*d
^2*e^2*f - c^2*d^3*e*g)*a^2*d^2*e^2*g^4*abs(e)) + (2*((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^4*d^4*e^6*g^3*abs(c
)*abs(d)/(c^2*d^2*e^4*f^2*g^2*abs(e) - 2*a*c*d*e^5*f*g^3*abs(e) + a^2*e^6*g^4*abs(e)) + 5*(c^5*d^5*e^8*f*g^2*a
bs(c)*abs(d) - a*c^4*d^4*e^9*g^3*abs(c)*abs(d))/(c^2*d^2*e^4*f^2*g^2*abs(e) - 2*a*c*d*e^5*f*g^3*abs(e) + a^2*e
^6*g^4*abs(e)))*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)/(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)^(5/2))*abs(e)/e^2

Mupad [B] (verification not implemented)

Time = 13.35 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.45 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{7/2}} \, dx=\frac {\left (\frac {x\,\left (10\,c^2\,d^2\,f-2\,a\,c\,d\,e\,g\right )}{15\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^2}-\frac {6\,a^2\,e^2\,g-10\,a\,c\,d\,e\,f}{15\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {4\,c^2\,d^2\,x^2}{15\,g\,{\left (a\,e\,g-c\,d\,f\right )}^2}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}+\frac {f^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^2}+\frac {2\,f\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g}} \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/((f + g*x)^(7/2)*(d + e*x)^(1/2)),x)

[Out]

(((x*(10*c^2*d^2*f - 2*a*c*d*e*g))/(15*g^2*(a*e*g - c*d*f)^2) - (6*a^2*e^2*g - 10*a*c*d*e*f)/(15*g^2*(a*e*g -
c*d*f)^2) + (4*c^2*d^2*x^2)/(15*g*(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 + e*x)^(1/2) + (f^2*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^2 + (2*f*x*(f + g*x)^(1/2)*(d + e*x)^(1/
2))/g)