\(\int \frac {\sqrt {d+e x}}{(f+g x)^{9/2} \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [719]

Optimal result

Integrand size = 48, antiderivative size = 267 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{9/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 (c d f-a e g) \sqrt {d+e x} (f+g x)^{7/2}}+\frac {12 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{5/2}}+\frac {16 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{3/2}}+\frac {32 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^4 \sqrt {d+e x} \sqrt {f+g x}} \]

[Out]

2/7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)/(g*x+f)^(7/2)/(e*x+d)^(1/2)+12/35*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/(g*x+f)^(5/2)/(e*x+d)^(1/2)+16/35*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^3/(g*x+f)^(3/2)/(e*x+d)^(1/2)+32/35*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2)/(-a*e*g+c*d*f)^4/(e*x+d)^(1/2)/(g*x+f)^(1/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {886, 874} \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{9/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {32 c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 \sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^4}+\frac {16 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)^3}+\frac {12 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 \sqrt {d+e x} (f+g x)^{5/2} (c d f-a e g)^2}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 \sqrt {d+e x} (f+g x)^{7/2} (c d f-a e g)} \]

[In]

Int[Sqrt[d + e*x]/((f + g*x)^(9/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

(2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(7*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^(7/2)) + (12*c*d*Sq
rt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^(5/2)) + (16*c^2*d^2*
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)^(3/2)) + (32*c^3*d^
3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*(c*d*f - a*e*g)^4*Sqrt[d + e*x]*Sqrt[f + g*x])

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 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 (c d f-a e g) \sqrt {d+e x} (f+g x)^{7/2}}+\frac {(6 c d) \int \frac {\sqrt {d+e x}}{(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{7 (c d f-a e g)} \\ & = \frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 (c d f-a e g) \sqrt {d+e x} (f+g x)^{7/2}}+\frac {12 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{5/2}}+\frac {\left (24 c^2 d^2\right ) \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}{35 (c d f-a e g)^2} \\ & = \frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 (c d f-a e g) \sqrt {d+e x} (f+g x)^{7/2}}+\frac {12 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{5/2}}+\frac {16 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{3/2}}+\frac {\left (16 c^3 d^3\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{35 (c d f-a e g)^3} \\ & = \frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 (c d f-a e g) \sqrt {d+e x} (f+g x)^{7/2}}+\frac {12 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{5/2}}+\frac {16 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{3/2}}+\frac {32 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^4 \sqrt {d+e x} \sqrt {f+g x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.57 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{9/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)} \left (-5 a^3 e^3 g^3+3 a^2 c d e^2 g^2 (7 f+2 g x)-a c^2 d^2 e g \left (35 f^2+28 f g x+8 g^2 x^2\right )+c^3 d^3 \left (35 f^3+70 f^2 g x+56 f g^2 x^2+16 g^3 x^3\right )\right )}{35 (c d f-a e g)^4 \sqrt {d+e x} (f+g x)^{7/2}} \]

[In]

Integrate[Sqrt[d + e*x]/((f + g*x)^(9/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-5*a^3*e^3*g^3 + 3*a^2*c*d*e^2*g^2*(7*f + 2*g*x) - a*c^2*d^2*e*g*(35*f^2 + 2
8*f*g*x + 8*g^2*x^2) + c^3*d^3*(35*f^3 + 70*f^2*g*x + 56*f*g^2*x^2 + 16*g^3*x^3)))/(35*(c*d*f - a*e*g)^4*Sqrt[
d + e*x]*(f + g*x)^(7/2))

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.69

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-16 g^{3} x^{3} c^{3} d^{3}+8 a \,c^{2} d^{2} e \,g^{3} x^{2}-56 c^{3} d^{3} f \,g^{2} x^{2}-6 a^{2} c d \,e^{2} g^{3} x +28 a \,c^{2} d^{2} e f \,g^{2} x -70 c^{3} d^{3} f^{2} g x +5 a^{3} e^{3} g^{3}-21 a^{2} c d \,e^{2} f \,g^{2}+35 a \,c^{2} d^{2} e \,f^{2} g -35 f^{3} c^{3} d^{3}\right )}{35 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {7}{2}} \left (a e g -c d f \right )^{4}}\)	\(183\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-16 g^{3} x^{3} c^{3} d^{3}+8 a \,c^{2} d^{2} e \,g^{3} x^{2}-56 c^{3} d^{3} f \,g^{2} x^{2}-6 a^{2} c d \,e^{2} g^{3} x +28 a \,c^{2} d^{2} e f \,g^{2} x -70 c^{3} d^{3} f^{2} g x +5 a^{3} e^{3} g^{3}-21 a^{2} c d \,e^{2} f \,g^{2}+35 a \,c^{2} d^{2} e \,f^{2} g -35 f^{3} c^{3} d^{3}\right ) \sqrt {e x +d}}{35 \left (g x +f \right )^{\frac {7}{2}} \left (a^{4} e^{4} g^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} c^{4} d^{4}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}\)	\(260\)

[In]

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

[Out]

-2/35/(e*x+d)^(1/2)/(g*x+f)^(7/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(-16*c^3*d^3*g^3*x^3+8*a*c^2*d^2*e*g^3*x^2-56*c^
3*d^3*f*g^2*x^2-6*a^2*c*d*e^2*g^3*x+28*a*c^2*d^2*e*f*g^2*x-70*c^3*d^3*f^2*g*x+5*a^3*e^3*g^3-21*a^2*c*d*e^2*f*g
^2+35*a*c^2*d^2*e*f^2*g-35*c^3*d^3*f^3)/(a*e*g-c*d*f)^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 953 vs. \(2 (235) = 470\).

Time = 2.75 (sec) , antiderivative size = 953, normalized size of antiderivative = 3.57 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{9/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, {\left (16 \, c^{3} d^{3} g^{3} x^{3} + 35 \, c^{3} d^{3} f^{3} - 35 \, a c^{2} d^{2} e f^{2} g + 21 \, a^{2} c d e^{2} f g^{2} - 5 \, a^{3} e^{3} g^{3} + 8 \, {\left (7 \, c^{3} d^{3} f g^{2} - a c^{2} d^{2} e g^{3}\right )} x^{2} + 2 \, {\left (35 \, c^{3} d^{3} f^{2} g - 14 \, a c^{2} d^{2} e f g^{2} + 3 \, a^{2} c d e^{2} g^{3}\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}}{35 \, {\left (c^{4} d^{5} f^{8} - 4 \, a c^{3} d^{4} e f^{7} g + 6 \, a^{2} c^{2} d^{3} e^{2} f^{6} g^{2} - 4 \, a^{3} c d^{2} e^{3} f^{5} g^{3} + a^{4} d e^{4} f^{4} g^{4} + {\left (c^{4} d^{4} e f^{4} g^{4} - 4 \, a c^{3} d^{3} e^{2} f^{3} g^{5} + 6 \, a^{2} c^{2} d^{2} e^{3} f^{2} g^{6} - 4 \, a^{3} c d e^{4} f g^{7} + a^{4} e^{5} g^{8}\right )} x^{5} + {\left (4 \, c^{4} d^{4} e f^{5} g^{3} + a^{4} d e^{4} g^{8} + {\left (c^{4} d^{5} - 16 \, a c^{3} d^{3} e^{2}\right )} f^{4} g^{4} - 4 \, {\left (a c^{3} d^{4} e - 6 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{3} g^{5} + 2 \, {\left (3 \, a^{2} c^{2} d^{3} e^{2} - 8 \, a^{3} c d e^{4}\right )} f^{2} g^{6} - 4 \, {\left (a^{3} c d^{2} e^{3} - a^{4} e^{5}\right )} f g^{7}\right )} x^{4} + 2 \, {\left (3 \, c^{4} d^{4} e f^{6} g^{2} + 2 \, a^{4} d e^{4} f g^{7} + 2 \, {\left (c^{4} d^{5} - 6 \, a c^{3} d^{3} e^{2}\right )} f^{5} g^{3} - 2 \, {\left (4 \, a c^{3} d^{4} e - 9 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{4} g^{4} + 12 \, {\left (a^{2} c^{2} d^{3} e^{2} - a^{3} c d e^{4}\right )} f^{3} g^{5} - {\left (8 \, a^{3} c d^{2} e^{3} - 3 \, a^{4} e^{5}\right )} f^{2} g^{6}\right )} x^{3} + 2 \, {\left (2 \, c^{4} d^{4} e f^{7} g + 3 \, a^{4} d e^{4} f^{2} g^{6} + {\left (3 \, c^{4} d^{5} - 8 \, a c^{3} d^{3} e^{2}\right )} f^{6} g^{2} - 12 \, {\left (a c^{3} d^{4} e - a^{2} c^{2} d^{2} e^{3}\right )} f^{5} g^{3} + 2 \, {\left (9 \, a^{2} c^{2} d^{3} e^{2} - 4 \, a^{3} c d e^{4}\right )} f^{4} g^{4} - 2 \, {\left (6 \, a^{3} c d^{2} e^{3} - a^{4} e^{5}\right )} f^{3} g^{5}\right )} x^{2} + {\left (c^{4} d^{4} e f^{8} + 4 \, a^{4} d e^{4} f^{3} g^{5} + 4 \, {\left (c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} f^{7} g - 2 \, {\left (8 \, a c^{3} d^{4} e - 3 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{6} g^{2} + 4 \, {\left (6 \, a^{2} c^{2} d^{3} e^{2} - a^{3} c d e^{4}\right )} f^{5} g^{3} - {\left (16 \, a^{3} c d^{2} e^{3} - a^{4} e^{5}\right )} f^{4} g^{4}\right )} x\right )}} \]

[In]

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

[Out]

2/35*(16*c^3*d^3*g^3*x^3 + 35*c^3*d^3*f^3 - 35*a*c^2*d^2*e*f^2*g + 21*a^2*c*d*e^2*f*g^2 - 5*a^3*e^3*g^3 + 8*(7
*c^3*d^3*f*g^2 - a*c^2*d^2*e*g^3)*x^2 + 2*(35*c^3*d^3*f^2*g - 14*a*c^2*d^2*e*f*g^2 + 3*a^2*c*d*e^2*g^3)*x)*sqr
t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f)/(c^4*d^5*f^8 - 4*a*c^3*d^4*e*f^7*g + 6*a^
2*c^2*d^3*e^2*f^6*g^2 - 4*a^3*c*d^2*e^3*f^5*g^3 + a^4*d*e^4*f^4*g^4 + (c^4*d^4*e*f^4*g^4 - 4*a*c^3*d^3*e^2*f^3
*g^5 + 6*a^2*c^2*d^2*e^3*f^2*g^6 - 4*a^3*c*d*e^4*f*g^7 + a^4*e^5*g^8)*x^5 + (4*c^4*d^4*e*f^5*g^3 + a^4*d*e^4*g
^8 + (c^4*d^5 - 16*a*c^3*d^3*e^2)*f^4*g^4 - 4*(a*c^3*d^4*e - 6*a^2*c^2*d^2*e^3)*f^3*g^5 + 2*(3*a^2*c^2*d^3*e^2
- 8*a^3*c*d*e^4)*f^2*g^6 - 4*(a^3*c*d^2*e^3 - a^4*e^5)*f*g^7)*x^4 + 2*(3*c^4*d^4*e*f^6*g^2 + 2*a^4*d*e^4*f*g^
7 + 2*(c^4*d^5 - 6*a*c^3*d^3*e^2)*f^5*g^3 - 2*(4*a*c^3*d^4*e - 9*a^2*c^2*d^2*e^3)*f^4*g^4 + 12*(a^2*c^2*d^3*e^
2 - a^3*c*d*e^4)*f^3*g^5 - (8*a^3*c*d^2*e^3 - 3*a^4*e^5)*f^2*g^6)*x^3 + 2*(2*c^4*d^4*e*f^7*g + 3*a^4*d*e^4*f^2
*g^6 + (3*c^4*d^5 - 8*a*c^3*d^3*e^2)*f^6*g^2 - 12*(a*c^3*d^4*e - a^2*c^2*d^2*e^3)*f^5*g^3 + 2*(9*a^2*c^2*d^3*e
^2 - 4*a^3*c*d*e^4)*f^4*g^4 - 2*(6*a^3*c*d^2*e^3 - a^4*e^5)*f^3*g^5)*x^2 + (c^4*d^4*e*f^8 + 4*a^4*d*e^4*f^3*g^
5 + 4*(c^4*d^5 - a*c^3*d^3*e^2)*f^7*g - 2*(8*a*c^3*d^4*e - 3*a^2*c^2*d^2*e^3)*f^6*g^2 + 4*(6*a^2*c^2*d^3*e^2 -
a^3*c*d*e^4)*f^5*g^3 - (16*a^3*c*d^2*e^3 - a^4*e^5)*f^4*g^4)*x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2971 vs. \(2 (235) = 470\).

Time = 0.55 (sec) , antiderivative size = 2971, normalized size of antiderivative = 11.13 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{9/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

64/35*(c^3*d^3*e^6*f^3*g^3 - 3*a*c^2*d^2*e^7*f^2*g^4 + 3*a^2*c*d*e^8*f*g^5 - a^3*e^9*g^6 + 7*(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^
2*d^2*e^4*f^2*g^2 - 14*(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*a*c*d*e^5*f*g^3 + 7*(sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) - s
qrt(-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*a^2*e^6*g^4 + 21*(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))^4*c*d*e
^2*f*g - 21*(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))^4*a*e^3*g^2 + 35*(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))^6)*sqrt(c*d*g)*c^3*d^3*e^8*g^4/((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)^7*abs(g)) - 2/35*(16*sqrt(c*d*g)*c^3*d^3*e^8*f^3*g - 203*sqrt(c*d*g)*c^3*d^4*e^7*f^2*g^2 +
155*sqrt(c*d*g)*a*c^2*d^2*e^9*f^2*g^2 + 462*sqrt(c*d*g)*c^3*d^5*e^6*f*g^3 - 518*sqrt(c*d*g)*a*c^2*d^3*e^8*f*g
^3 + 104*sqrt(c*d*g)*a^2*c*d*e^10*f*g^3 - 280*sqrt(c*d*g)*c^3*d^6*e^5*g^4 + 378*sqrt(c*d*g)*a*c^2*d^4*e^7*g^4
- 119*sqrt(c*d*g)*a^2*c*d^2*e^9*g^4 + 5*sqrt(c*d*g)*a^3*e^11*g^4 - 77*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*
f - d*e*g)*c^3*d^3*e^6*f^2*g + 322*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*c^3*d^4*e^5*f*g^2 - 168*
sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*a*c^2*d^2*e^7*f*g^2 - 280*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sq
rt(e^2*f - d*e*g)*c^3*d^5*e^4*g^3 + 238*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*a*c^2*d^3*e^6*g^3 -
35*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*a^2*c*d*e^8*g^3)/(c^4*d^4*e^8*f^7*g*abs(g) - 28*c^4*d^5
*e^7*f^6*g^2*abs(g) + 21*a*c^3*d^3*e^9*f^6*g^2*abs(g) + 182*c^4*d^6*e^6*f^5*g^3*abs(g) - 196*a*c^3*d^4*e^8*f^5
*g^3*abs(g) + 35*a^2*c^2*d^2*e^10*f^5*g^3*abs(g) - 532*c^4*d^7*e^5*f^4*g^4*abs(g) + 686*a*c^3*d^5*e^7*f^4*g^4*
abs(g) - 196*a^2*c^2*d^3*e^9*f^4*g^4*abs(g) + 7*a^3*c*d*e^11*f^4*g^4*abs(g) + 833*c^4*d^8*e^4*f^3*g^5*abs(g) -
1204*a*c^3*d^6*e^6*f^3*g^5*abs(g) + 434*a^2*c^2*d^4*e^8*f^3*g^5*abs(g) - 28*a^3*c*d^2*e^10*f^3*g^5*abs(g) - 7
28*c^4*d^9*e^3*f^2*g^6*abs(g) + 1141*a*c^3*d^7*e^5*f^2*g^6*abs(g) - 476*a^2*c^2*d^5*e^7*f^2*g^6*abs(g) + 42*a^
3*c*d^3*e^9*f^2*g^6*abs(g) + 336*c^4*d^10*e^2*f*g^7*abs(g) - 560*a*c^3*d^8*e^4*f*g^7*abs(g) + 259*a^2*c^2*d^6*
e^6*f*g^7*abs(g) - 28*a^3*c*d^4*e^8*f*g^7*abs(g) - 64*c^4*d^11*e*g^8*abs(g) + 112*a*c^3*d^9*e^3*g^8*abs(g) - 5
6*a^2*c^2*d^7*e^5*g^8*abs(g) + 7*a^3*c*d^5*e^7*g^8*abs(g) - 7*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*
g)*sqrt(c*d*g)*c^3*d^3*e^6*f^6*abs(g) + 77*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^3*
d^4*e^5*f^5*g*abs(g) - 35*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c^2*d^2*e^7*f^5*g*a
bs(g) - 301*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^3*d^5*e^4*f^4*g^2*abs(g) + 217*sq
rt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c^2*d^3*e^6*f^4*g^2*abs(g) - 21*sqrt(-c*d^2*e*g
^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c*d*e^8*f^4*g^2*abs(g) + 575*sqrt(-c*d^2*e*g^2 + a*e^3*g^2
)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^3*d^6*e^3*f^3*g^3*abs(g) - 521*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f -
d*e*g)*sqrt(c*d*g)*a*c^2*d^4*e^5*f^3*g^3*abs(g) + 87*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(
c*d*g)*a^2*c*d^2*e^7*f^3*g^3*abs(g) - sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^3*e^9*f
^3*g^3*abs(g) - 584*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^3*d^7*e^2*f^2*g^4*abs(g)
+ 611*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c^2*d^5*e^4*f^2*g^4*abs(g) - 135*sqrt(-
c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c*d^3*e^6*f^2*g^4*abs(g) + 3*sqrt(-c*d^2*e*g^2 +
a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^3*d*e^8*f^2*g^4*abs(g) + 304*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(
e^2*f - d*e*g)*sqrt(c*d*g)*c^3*d^8*e*f*g^5*abs(g) - 352*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqr
t(c*d*g)*a*c^2*d^6*e^3*f*g^5*abs(g) + 93*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c*
d^4*e^5*f*g^5*abs(g) - 3*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^3*d^2*e^7*f*g^5*abs(
g) - 64*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^3*d^9*g^6*abs(g) + 80*sqrt(-c*d^2*e*g
^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c^2*d^7*e^2*g^6*abs(g) - 24*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*s
qrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c*d^5*e^4*g^6*abs(g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*s
qrt(c*d*g)*a^3*d^3*e^6*g^6*abs(g))

Mupad [B] (verification not implemented)

Time = 14.09 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{9/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {\sqrt {d+e\,x}\,\left (10\,a^3\,e^3\,g^3-42\,a^2\,c\,d\,e^2\,f\,g^2+70\,a\,c^2\,d^2\,e\,f^2\,g-70\,c^3\,d^3\,f^3\right )}{35\,e\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {32\,c^3\,d^3\,x^3\,\sqrt {d+e\,x}}{35\,e\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {4\,c\,d\,x\,\sqrt {d+e\,x}\,\left (3\,a^2\,e^2\,g^2-14\,a\,c\,d\,e\,f\,g+35\,c^2\,d^2\,f^2\right )}{35\,e\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {16\,c^2\,d^2\,x^2\,\left (a\,e\,g-7\,c\,d\,f\right )\,\sqrt {d+e\,x}}{35\,e\,g\,{\left (a\,e\,g-c\,d\,f\right )}^4}\right )}{x^4\,\sqrt {f+g\,x}+\frac {d\,f^3\,\sqrt {f+g\,x}}{e\,g^3}+\frac {x^3\,\sqrt {f+g\,x}\,\left (d\,g+3\,e\,f\right )}{e\,g}+\frac {3\,f\,x^2\,\sqrt {f+g\,x}\,\left (d\,g+e\,f\right )}{e\,g^2}+\frac {f^2\,x\,\sqrt {f+g\,x}\,\left (3\,d\,g+e\,f\right )}{e\,g^3}} \]

[In]

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

[Out]

-((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*(((d + e*x)^(1/2)*(10*a^3*e^3*g^3 - 70*c^3*d^3*f^3 + 70*a*c^2*
d^2*e*f^2*g - 42*a^2*c*d*e^2*f*g^2))/(35*e*g^3*(a*e*g - c*d*f)^4) - (32*c^3*d^3*x^3*(d + e*x)^(1/2))/(35*e*(a*
e*g - c*d*f)^4) - (4*c*d*x*(d + e*x)^(1/2)*(3*a^2*e^2*g^2 + 35*c^2*d^2*f^2 - 14*a*c*d*e*f*g))/(35*e*g^2*(a*e*g
- c*d*f)^4) + (16*c^2*d^2*x^2*(a*e*g - 7*c*d*f)*(d + e*x)^(1/2))/(35*e*g*(a*e*g - c*d*f)^4)))/(x^4*(f + g*x)^
(1/2) + (d*f^3*(f + g*x)^(1/2))/(e*g^3) + (x^3*(f + g*x)^(1/2)*(d*g + 3*e*f))/(e*g) + (3*f*x^2*(f + g*x)^(1/2)
*(d*g + e*f))/(e*g^2) + (f^2*x*(f + g*x)^(1/2)*(3*d*g + e*f))/(e*g^3))