\(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{9/2}} \, dx\) [740]

Optimal result

Integrand size = 48, antiderivative size = 198 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{9/2}} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{7/2}}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{5/2}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 (c d f-a e g)^3 (d+e x)^{3/2} (f+g x)^{3/2}} \]

[Out]

2/7*(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)^(7/2)+8/35*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)^(5/2)+16/105*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(3/2)/(-a*e*g+c*d*f)^3/(e*x+d)^(3/2)/(g*x+f)^(3/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {886, 874} \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{9/2}} \, dx=\frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)^3}+\frac {8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{35 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 (d+e x)^{3/2} (f+g x)^{7/2} (c d f-a e g)} \]

[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)^(9/2)),x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(7*(c*d*f - a*e*g)*(d + e*x)^(3/2)*(f + g*x)^(7/2)) + (8*c*d
*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(35*(c*d*f - a*e*g)^2*(d + e*x)^(3/2)*(f + g*x)^(5/2)) + (16*c
^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(105*(c*d*f - a*e*g)^3*(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}}{7 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{7/2}}+\frac {(4 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)^{7/2}} \, dx}{7 (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}}{7 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{7/2}}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{5/2}}+\frac {\left (8 c^2 d^2\right ) \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}{35 (c d f-a e g)^2} \\ & = \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{7/2}}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{5/2}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 (c d f-a e g)^3 (d+e x)^{3/2} (f+g x)^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{9/2}} \, dx=\frac {2 ((a e+c d x) (d+e x))^{3/2} \left (15 a^2 e^2 g^2-6 a c d e g (7 f+2 g x)+c^2 d^2 \left (35 f^2+28 f g x+8 g^2 x^2\right )\right )}{105 (c d f-a e g)^3 (d+e x)^{3/2} (f+g x)^{7/2}} \]

[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)^(9/2)),x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(15*a^2*e^2*g^2 - 6*a*c*d*e*g*(7*f + 2*g*x) + c^2*d^2*(35*f^2 + 28*f*g*x +
8*g^2*x^2)))/(105*(c*d*f - a*e*g)^3*(d + e*x)^(3/2)*(f + g*x)^(7/2))

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.60

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c d x +a e \right ) \left (8 g^{2} x^{2} c^{2} d^{2}-12 a c d e \,g^{2} x +28 c^{2} d^{2} f g x +15 a^{2} e^{2} g^{2}-42 a c d e f g +35 c^{2} d^{2} f^{2}\right )}{105 \left (g x +f \right )^{\frac {7}{2}} \sqrt {e x +d}\, \left (a e g -c d f \right )^{3}}\)	\(119\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (8 g^{2} x^{2} c^{2} d^{2}-12 a c d e \,g^{2} x +28 c^{2} d^{2} f g x +15 a^{2} e^{2} g^{2}-42 a c d e f g +35 c^{2} d^{2} f^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{105 \left (g x +f \right )^{\frac {7}{2}} \left (a^{3} e^{3} g^{3}-3 a^{2} c d \,e^{2} f \,g^{2}+3 a \,c^{2} d^{2} e \,f^{2} g -f^{3} c^{3} d^{3}\right ) \sqrt {e x +d}}\)	\(169\)

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^(9/2)/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/105*((c*d*x+a*e)*(e*x+d))^(1/2)/(g*x+f)^(7/2)/(e*x+d)^(1/2)*(c*d*x+a*e)*(8*c^2*d^2*g^2*x^2-12*a*c*d*e*g^2*x
+28*c^2*d^2*f*g*x+15*a^2*e^2*g^2-42*a*c*d*e*f*g+35*c^2*d^2*f^2)/(a*e*g-c*d*f)^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 748 vs. \(2 (174) = 348\).

Time = 0.55 (sec) , antiderivative size = 748, normalized size of antiderivative = 3.78 \[ \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)^{9/2}} \, dx=\frac {2 \, {\left (8 \, c^{3} d^{3} g^{2} x^{3} + 35 \, a c^{2} d^{2} e f^{2} - 42 \, a^{2} c d e^{2} f g + 15 \, a^{3} e^{3} g^{2} + 4 \, {\left (7 \, c^{3} d^{3} f g - a c^{2} d^{2} e g^{2}\right )} x^{2} + {\left (35 \, c^{3} d^{3} f^{2} - 14 \, a c^{2} d^{2} e f g + 3 \, a^{2} c d e^{2} g^{2}\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}}{105 \, {\left (c^{3} d^{4} f^{7} - 3 \, a c^{2} d^{3} e f^{6} g + 3 \, a^{2} c d^{2} e^{2} f^{5} g^{2} - a^{3} d e^{3} f^{4} g^{3} + {\left (c^{3} d^{3} e f^{3} g^{4} - 3 \, a c^{2} d^{2} e^{2} f^{2} g^{5} + 3 \, a^{2} c d e^{3} f g^{6} - a^{3} e^{4} g^{7}\right )} x^{5} + {\left (4 \, c^{3} d^{3} e f^{4} g^{3} - a^{3} d e^{3} g^{7} + {\left (c^{3} d^{4} - 12 \, a c^{2} d^{2} e^{2}\right )} f^{3} g^{4} - 3 \, {\left (a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} f^{2} g^{5} + {\left (3 \, a^{2} c d^{2} e^{2} - 4 \, a^{3} e^{4}\right )} f g^{6}\right )} x^{4} + 2 \, {\left (3 \, c^{3} d^{3} e f^{5} g^{2} - 2 \, a^{3} d e^{3} f g^{6} + {\left (2 \, c^{3} d^{4} - 9 \, a c^{2} d^{2} e^{2}\right )} f^{4} g^{3} - 3 \, {\left (2 \, a c^{2} d^{3} e - 3 \, a^{2} c d e^{3}\right )} f^{3} g^{4} + 3 \, {\left (2 \, a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f^{2} g^{5}\right )} x^{3} + 2 \, {\left (2 \, c^{3} d^{3} e f^{6} g - 3 \, a^{3} d e^{3} f^{2} g^{5} + 3 \, {\left (c^{3} d^{4} - 2 \, a c^{2} d^{2} e^{2}\right )} f^{5} g^{2} - 3 \, {\left (3 \, a c^{2} d^{3} e - 2 \, a^{2} c d e^{3}\right )} f^{4} g^{3} + {\left (9 \, a^{2} c d^{2} e^{2} - 2 \, a^{3} e^{4}\right )} f^{3} g^{4}\right )} x^{2} + {\left (c^{3} d^{3} e f^{7} - 4 \, a^{3} d e^{3} f^{3} g^{4} + {\left (4 \, c^{3} d^{4} - 3 \, a c^{2} d^{2} e^{2}\right )} f^{6} g - 3 \, {\left (4 \, a c^{2} d^{3} e - a^{2} c d e^{3}\right )} f^{5} g^{2} + {\left (12 \, a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f^{4} g^{3}\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)^(9/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/105*(8*c^3*d^3*g^2*x^3 + 35*a*c^2*d^2*e*f^2 - 42*a^2*c*d*e^2*f*g + 15*a^3*e^3*g^2 + 4*(7*c^3*d^3*f*g - a*c^2
*d^2*e*g^2)*x^2 + (35*c^3*d^3*f^2 - 14*a*c^2*d^2*e*f*g + 3*a^2*c*d*e^2*g^2)*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^3*d^4*f^7 - 3*a*c^2*d^3*e*f^6*g + 3*a^2*c*d^2*e^2*f^5*g^2 - a^3*d*
e^3*f^4*g^3 + (c^3*d^3*e*f^3*g^4 - 3*a*c^2*d^2*e^2*f^2*g^5 + 3*a^2*c*d*e^3*f*g^6 - a^3*e^4*g^7)*x^5 + (4*c^3*d
^3*e*f^4*g^3 - a^3*d*e^3*g^7 + (c^3*d^4 - 12*a*c^2*d^2*e^2)*f^3*g^4 - 3*(a*c^2*d^3*e - 4*a^2*c*d*e^3)*f^2*g^5
+ (3*a^2*c*d^2*e^2 - 4*a^3*e^4)*f*g^6)*x^4 + 2*(3*c^3*d^3*e*f^5*g^2 - 2*a^3*d*e^3*f*g^6 + (2*c^3*d^4 - 9*a*c^2
*d^2*e^2)*f^4*g^3 - 3*(2*a*c^2*d^3*e - 3*a^2*c*d*e^3)*f^3*g^4 + 3*(2*a^2*c*d^2*e^2 - a^3*e^4)*f^2*g^5)*x^3 + 2
*(2*c^3*d^3*e*f^6*g - 3*a^3*d*e^3*f^2*g^5 + 3*(c^3*d^4 - 2*a*c^2*d^2*e^2)*f^5*g^2 - 3*(3*a*c^2*d^3*e - 2*a^2*c
*d*e^3)*f^4*g^3 + (9*a^2*c*d^2*e^2 - 2*a^3*e^4)*f^3*g^4)*x^2 + (c^3*d^3*e*f^7 - 4*a^3*d*e^3*f^3*g^4 + (4*c^3*d
^4 - 3*a*c^2*d^2*e^2)*f^6*g - 3*(4*a*c^2*d^3*e - a^2*c*d*e^3)*f^5*g^2 + (12*a^2*c*d^2*e^2 - a^3*e^4)*f^4*g^3)*
x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{9/2}} \, dx=\text {Timed out} \]

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(g*x+f)**(9/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{9/2}} \, dx=\int { \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{\sqrt {e x + d} {\left (g x + f\right )}^{\frac {9}{2}}} \,d x } \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^(9/2)/(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)^(9/2)), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1426 vs. \(2 (174) = 348\).

Time = 0.79 (sec) , antiderivative size = 1426, normalized size of antiderivative = 7.20 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{9/2}} \, dx=\text {Too large to display} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^(9/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/105*((35*sqrt(-c*d^2*e + a*e^3)*c^3*d^4*e^4*f^2*abs(c)*abs(d) - 35*sqrt(-c*d^2*e + a*e^3)*a*c^2*d^2*e^6*f^2*
abs(c)*abs(d) - 28*sqrt(-c*d^2*e + a*e^3)*c^3*d^5*e^3*f*g*abs(c)*abs(d) - 14*sqrt(-c*d^2*e + a*e^3)*a*c^2*d^3*
e^5*f*g*abs(c)*abs(d) + 42*sqrt(-c*d^2*e + a*e^3)*a^2*c*d*e^7*f*g*abs(c)*abs(d) + 8*sqrt(-c*d^2*e + a*e^3)*c^3
*d^6*e^2*g^2*abs(c)*abs(d) + 4*sqrt(-c*d^2*e + a*e^3)*a*c^2*d^4*e^4*g^2*abs(c)*abs(d) + 3*sqrt(-c*d^2*e + a*e^
3)*a^2*c*d^2*e^6*g^2*abs(c)*abs(d) - 15*sqrt(-c*d^2*e + a*e^3)*a^3*e^8*g^2*abs(c)*abs(d))/(sqrt(c^2*d^2*e^2*f
- c^2*d^3*e*g)*c^3*d^3*e^3*f^6*abs(e) - 3*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^3*d^4*e^2*f^5*g*abs(e) - 3*sqrt(
c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c^2*d^2*e^4*f^5*g*abs(e) + 3*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^3*d^5*e*f^4*g^
2*abs(e) + 9*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c^2*d^3*e^3*f^4*g^2*abs(e) + 3*sqrt(c^2*d^2*e^2*f - c^2*d^3*e
*g)*a^2*c*d*e^5*f^4*g^2*abs(e) - sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^3*d^6*f^3*g^3*abs(e) - 9*sqrt(c^2*d^2*e^2
*f - c^2*d^3*e*g)*a*c^2*d^4*e^2*f^3*g^3*abs(e) - 9*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^2*c*d^2*e^4*f^3*g^3*abs
(e) - sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^3*e^6*f^3*g^3*abs(e) + 3*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c^2*d^5
*e*f^2*g^4*abs(e) + 9*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^2*c*d^3*e^3*f^2*g^4*abs(e) + 3*sqrt(c^2*d^2*e^2*f -
c^2*d^3*e*g)*a^3*d*e^5*f^2*g^4*abs(e) - 3*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^2*c*d^4*e^2*f*g^5*abs(e) - 3*sqr
t(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^3*d^2*e^4*f*g^5*abs(e) + sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^3*d^3*e^3*g^6*ab
s(e)) + ((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*(4*(2*((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^6*d^6*e^8*g^5*ab
s(c)*abs(d)/(c^3*d^3*e^6*f^3*g^3*abs(e) - 3*a*c^2*d^2*e^7*f^2*g^4*abs(e) + 3*a^2*c*d*e^8*f*g^5*abs(e) - a^3*e^
9*g^6*abs(e)) + 7*(c^7*d^7*e^10*f*g^4*abs(c)*abs(d) - a*c^6*d^6*e^11*g^5*abs(c)*abs(d))/(c^3*d^3*e^6*f^3*g^3*a
bs(e) - 3*a*c^2*d^2*e^7*f^2*g^4*abs(e) + 3*a^2*c*d*e^8*f*g^5*abs(e) - a^3*e^9*g^6*abs(e)))*((e*x + d)*c*d*e -
c*d^2*e + a*e^3) + 35*(c^8*d^8*e^12*f^2*g^3*abs(c)*abs(d) - 2*a*c^7*d^7*e^13*f*g^4*abs(c)*abs(d) + a^2*c^6*d^6
*e^14*g^5*abs(c)*abs(d))/(c^3*d^3*e^6*f^3*g^3*abs(e) - 3*a*c^2*d^2*e^7*f^2*g^4*abs(e) + 3*a^2*c*d*e^8*f*g^5*ab
s(e) - a^3*e^9*g^6*abs(e)))/(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)^(7/2))*a
bs(e)/e^2

Mupad [B] (verification not implemented)

Time = 13.57 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.46 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{9/2}} \, dx=-\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {30\,a^3\,e^3\,g^2-84\,a^2\,c\,d\,e^2\,f\,g+70\,a\,c^2\,d^2\,e\,f^2}{105\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {x\,\left (6\,a^2\,c\,d\,e^2\,g^2-28\,a\,c^2\,d^2\,e\,f\,g+70\,c^3\,d^3\,f^2\right )}{105\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {16\,c^3\,d^3\,x^3}{105\,g\,{\left (a\,e\,g-c\,d\,f\right )}^3}-\frac {8\,c^2\,d^2\,x^2\,\left (a\,e\,g-7\,c\,d\,f\right )}{105\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^3}\right )}{x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}+\frac {f^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^3}+\frac {3\,f\,x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g}+\frac {3\,f^2\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^2}} \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/((f + g*x)^(9/2)*(d + e*x)^(1/2)),x)

[Out]

-((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((30*a^3*e^3*g^2 + 70*a*c^2*d^2*e*f^2 - 84*a^2*c*d*e^2*f*g)/(1
05*g^3*(a*e*g - c*d*f)^3) + (x*(70*c^3*d^3*f^2 + 6*a^2*c*d*e^2*g^2 - 28*a*c^2*d^2*e*f*g))/(105*g^3*(a*e*g - c*
d*f)^3) + (16*c^3*d^3*x^3)/(105*g*(a*e*g - c*d*f)^3) - (8*c^2*d^2*x^2*(a*e*g - 7*c*d*f))/(105*g^2*(a*e*g - c*d
*f)^3)))/(x^3*(f + g*x)^(1/2)*(d + e*x)^(1/2) + (f^3*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^3 + (3*f*x^2*(f + g*x)
^(1/2)*(d + e*x)^(1/2))/g + (3*f^2*x*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^2)