\(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{11/2}} \, dx\) [741]

Optimal result

Integrand size = 48, antiderivative size = 267 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{11/2}} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{9 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{9/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{21 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{7/2}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 (c d f-a e g)^3 (d+e x)^{3/2} (f+g x)^{5/2}}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{315 (c d f-a e g)^4 (d+e x)^{3/2} (f+g x)^{3/2}} \]

[Out]

2/9*(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)^(9/2)+4/21*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)^(7/2)+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)^(5/2)+32/315*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(3/2)/(-a*e*g+c*d*f)^4/(e*x+d)^(3/2)/(g*x+f)^(3/2)

Rubi [A] (verified)

Time = 0.20 (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} \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{11/2}} \, dx=\frac {32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{315 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)^4}+\frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)^3}+\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{21 (d+e x)^{3/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 (d+e x)^{3/2} (f+g x)^{9/2} (c d f-a e g)} \]

[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)^(11/2)),x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(9*(c*d*f - a*e*g)*(d + e*x)^(3/2)*(f + g*x)^(9/2)) + (4*c*d
*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(21*(c*d*f - a*e*g)^2*(d + e*x)^(3/2)*(f + g*x)^(7/2)) + (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)^(5/2))
+ (32*c^3*d^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(315*(c*d*f - a*e*g)^4*(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}}{9 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{9/2}}+\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)^{9/2}} \, dx}{3 (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}}{9 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{9/2}}+\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}}{21 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{7/2}}+\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)^{7/2}} \, dx}{21 (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}}{9 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{9/2}}+\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}}{21 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{7/2}}+\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)^{5/2}}+\frac {\left (16 c^3 d^3\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}{105 (c d f-a e g)^3} \\ & = \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{9 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{9/2}}+\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}}{21 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{7/2}}+\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)^{5/2}}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{315 (c d f-a e g)^4 (d+e x)^{3/2} (f+g x)^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.57 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{11/2}} \, dx=\frac {2 ((a e+c d x) (d+e x))^{3/2} \left (-35 a^3 e^3 g^3+15 a^2 c d e^2 g^2 (9 f+2 g x)-3 a c^2 d^2 e g \left (63 f^2+36 f g x+8 g^2 x^2\right )+c^3 d^3 \left (105 f^3+126 f^2 g x+72 f g^2 x^2+16 g^3 x^3\right )\right )}{315 (c d f-a e g)^4 (d+e x)^{3/2} (f+g x)^{9/2}} \]

[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)^(11/2)),x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(-35*a^3*e^3*g^3 + 15*a^2*c*d*e^2*g^2*(9*f + 2*g*x) - 3*a*c^2*d^2*e*g*(63*f
^2 + 36*f*g*x + 8*g^2*x^2) + c^3*d^3*(105*f^3 + 126*f^2*g*x + 72*f*g^2*x^2 + 16*g^3*x^3)))/(315*(c*d*f - a*e*g
)^4*(d + e*x)^(3/2)*(f + g*x)^(9/2))

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.72

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c d x +a e \right ) \left (-16 g^{3} x^{3} c^{3} d^{3}+24 a \,c^{2} d^{2} e \,g^{3} x^{2}-72 c^{3} d^{3} f \,g^{2} x^{2}-30 a^{2} c d \,e^{2} g^{3} x +108 a \,c^{2} d^{2} e f \,g^{2} x -126 c^{3} d^{3} f^{2} g x +35 a^{3} e^{3} g^{3}-135 a^{2} c d \,e^{2} f \,g^{2}+189 a \,c^{2} d^{2} e \,f^{2} g -105 f^{3} c^{3} d^{3}\right )}{315 \left (g x +f \right )^{\frac {9}{2}} \sqrt {e x +d}\, \left (a e g -c d f \right )^{4}}\)	\(191\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-16 g^{3} x^{3} c^{3} d^{3}+24 a \,c^{2} d^{2} e \,g^{3} x^{2}-72 c^{3} d^{3} f \,g^{2} x^{2}-30 a^{2} c d \,e^{2} g^{3} x +108 a \,c^{2} d^{2} e f \,g^{2} x -126 c^{3} d^{3} f^{2} g x +35 a^{3} e^{3} g^{3}-135 a^{2} c d \,e^{2} f \,g^{2}+189 a \,c^{2} d^{2} e \,f^{2} g -105 f^{3} c^{3} d^{3}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{315 \left (g x +f \right )^{\frac {9}{2}} \left (a^{4} e^{4} g^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} c^{4} d^{4}\right ) \sqrt {e x +d}}\)	\(260\)

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^(11/2)/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/315*((c*d*x+a*e)*(e*x+d))^(1/2)/(g*x+f)^(9/2)/(e*x+d)^(1/2)*(c*d*x+a*e)*(-16*c^3*d^3*g^3*x^3+24*a*c^2*d^2*e
*g^3*x^2-72*c^3*d^3*f*g^2*x^2-30*a^2*c*d*e^2*g^3*x+108*a*c^2*d^2*e*f*g^2*x-126*c^3*d^3*f^2*g*x+35*a^3*e^3*g^3-
135*a^2*c*d*e^2*f*g^2+189*a*c^2*d^2*e*f^2*g-105*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. 1179 vs. \(2 (235) = 470\).

Time = 1.09 (sec) , antiderivative size = 1179, normalized size of antiderivative = 4.42 \[ \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)^{11/2}} \, dx=\frac {2 \, {\left (16 \, c^{4} d^{4} g^{3} x^{4} + 105 \, a c^{3} d^{3} e f^{3} - 189 \, a^{2} c^{2} d^{2} e^{2} f^{2} g + 135 \, a^{3} c d e^{3} f g^{2} - 35 \, a^{4} e^{4} g^{3} + 8 \, {\left (9 \, c^{4} d^{4} f g^{2} - a c^{3} d^{3} e g^{3}\right )} x^{3} + 6 \, {\left (21 \, c^{4} d^{4} f^{2} g - 6 \, a c^{3} d^{3} e f g^{2} + a^{2} c^{2} d^{2} e^{2} g^{3}\right )} x^{2} + {\left (105 \, c^{4} d^{4} f^{3} - 63 \, a c^{3} d^{3} e f^{2} g + 27 \, a^{2} c^{2} d^{2} e^{2} f g^{2} - 5 \, a^{3} c d e^{3} 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}}{315 \, {\left (c^{4} d^{5} f^{9} - 4 \, a c^{3} d^{4} e f^{8} g + 6 \, a^{2} c^{2} d^{3} e^{2} f^{7} g^{2} - 4 \, a^{3} c d^{2} e^{3} f^{6} g^{3} + a^{4} d e^{4} f^{5} g^{4} + {\left (c^{4} d^{4} e f^{4} g^{5} - 4 \, a c^{3} d^{3} e^{2} f^{3} g^{6} + 6 \, a^{2} c^{2} d^{2} e^{3} f^{2} g^{7} - 4 \, a^{3} c d e^{4} f g^{8} + a^{4} e^{5} g^{9}\right )} x^{6} + {\left (5 \, c^{4} d^{4} e f^{5} g^{4} + a^{4} d e^{4} g^{9} + {\left (c^{4} d^{5} - 20 \, a c^{3} d^{3} e^{2}\right )} f^{4} g^{5} - 2 \, {\left (2 \, a c^{3} d^{4} e - 15 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{3} g^{6} + 2 \, {\left (3 \, a^{2} c^{2} d^{3} e^{2} - 10 \, a^{3} c d e^{4}\right )} f^{2} g^{7} - {\left (4 \, a^{3} c d^{2} e^{3} - 5 \, a^{4} e^{5}\right )} f g^{8}\right )} x^{5} + 5 \, {\left (2 \, c^{4} d^{4} e f^{6} g^{3} + a^{4} d e^{4} f g^{8} + {\left (c^{4} d^{5} - 8 \, a c^{3} d^{3} e^{2}\right )} f^{5} g^{4} - 4 \, {\left (a c^{3} d^{4} e - 3 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{4} g^{5} + 2 \, {\left (3 \, a^{2} c^{2} d^{3} e^{2} - 4 \, a^{3} c d e^{4}\right )} f^{3} g^{6} - 2 \, {\left (2 \, a^{3} c d^{2} e^{3} - a^{4} e^{5}\right )} f^{2} g^{7}\right )} x^{4} + 10 \, {\left (c^{4} d^{4} e f^{7} g^{2} + a^{4} d e^{4} f^{2} g^{7} + {\left (c^{4} d^{5} - 4 \, a c^{3} d^{3} e^{2}\right )} f^{6} g^{3} - 2 \, {\left (2 \, a c^{3} d^{4} e - 3 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{5} g^{4} + 2 \, {\left (3 \, a^{2} c^{2} d^{3} e^{2} - 2 \, a^{3} c d e^{4}\right )} f^{4} g^{5} - {\left (4 \, a^{3} c d^{2} e^{3} - a^{4} e^{5}\right )} f^{3} g^{6}\right )} x^{3} + 5 \, {\left (c^{4} d^{4} e f^{8} g + 2 \, a^{4} d e^{4} f^{3} g^{6} + 2 \, {\left (c^{4} d^{5} - 2 \, a c^{3} d^{3} e^{2}\right )} f^{7} g^{2} - 2 \, {\left (4 \, a c^{3} d^{4} e - 3 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{6} g^{3} + 4 \, {\left (3 \, a^{2} c^{2} d^{3} e^{2} - a^{3} c d e^{4}\right )} f^{5} g^{4} - {\left (8 \, a^{3} c d^{2} e^{3} - a^{4} e^{5}\right )} f^{4} g^{5}\right )} x^{2} + {\left (c^{4} d^{4} e f^{9} + 5 \, a^{4} d e^{4} f^{4} g^{5} + {\left (5 \, c^{4} d^{5} - 4 \, a c^{3} d^{3} e^{2}\right )} f^{8} g - 2 \, {\left (10 \, a c^{3} d^{4} e - 3 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{7} g^{2} + 2 \, {\left (15 \, a^{2} c^{2} d^{3} e^{2} - 2 \, a^{3} c d e^{4}\right )} f^{6} g^{3} - {\left (20 \, a^{3} c d^{2} e^{3} - a^{4} e^{5}\right )} f^{5} g^{4}\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)^(11/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/315*(16*c^4*d^4*g^3*x^4 + 105*a*c^3*d^3*e*f^3 - 189*a^2*c^2*d^2*e^2*f^2*g + 135*a^3*c*d*e^3*f*g^2 - 35*a^4*e
^4*g^3 + 8*(9*c^4*d^4*f*g^2 - a*c^3*d^3*e*g^3)*x^3 + 6*(21*c^4*d^4*f^2*g - 6*a*c^3*d^3*e*f*g^2 + a^2*c^2*d^2*e
^2*g^3)*x^2 + (105*c^4*d^4*f^3 - 63*a*c^3*d^3*e*f^2*g + 27*a^2*c^2*d^2*e^2*f*g^2 - 5*a^3*c*d*e^3*g^3)*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^4*d^5*f^9 - 4*a*c^3*d^4*e*f^8*g + 6*a^2*
c^2*d^3*e^2*f^7*g^2 - 4*a^3*c*d^2*e^3*f^6*g^3 + a^4*d*e^4*f^5*g^4 + (c^4*d^4*e*f^4*g^5 - 4*a*c^3*d^3*e^2*f^3*g
^6 + 6*a^2*c^2*d^2*e^3*f^2*g^7 - 4*a^3*c*d*e^4*f*g^8 + a^4*e^5*g^9)*x^6 + (5*c^4*d^4*e*f^5*g^4 + a^4*d*e^4*g^9
+ (c^4*d^5 - 20*a*c^3*d^3*e^2)*f^4*g^5 - 2*(2*a*c^3*d^4*e - 15*a^2*c^2*d^2*e^3)*f^3*g^6 + 2*(3*a^2*c^2*d^3*e^
2 - 10*a^3*c*d*e^4)*f^2*g^7 - (4*a^3*c*d^2*e^3 - 5*a^4*e^5)*f*g^8)*x^5 + 5*(2*c^4*d^4*e*f^6*g^3 + a^4*d*e^4*f*
g^8 + (c^4*d^5 - 8*a*c^3*d^3*e^2)*f^5*g^4 - 4*(a*c^3*d^4*e - 3*a^2*c^2*d^2*e^3)*f^4*g^5 + 2*(3*a^2*c^2*d^3*e^2
- 4*a^3*c*d*e^4)*f^3*g^6 - 2*(2*a^3*c*d^2*e^3 - a^4*e^5)*f^2*g^7)*x^4 + 10*(c^4*d^4*e*f^7*g^2 + a^4*d*e^4*f^2
*g^7 + (c^4*d^5 - 4*a*c^3*d^3*e^2)*f^6*g^3 - 2*(2*a*c^3*d^4*e - 3*a^2*c^2*d^2*e^3)*f^5*g^4 + 2*(3*a^2*c^2*d^3*
e^2 - 2*a^3*c*d*e^4)*f^4*g^5 - (4*a^3*c*d^2*e^3 - a^4*e^5)*f^3*g^6)*x^3 + 5*(c^4*d^4*e*f^8*g + 2*a^4*d*e^4*f^3
*g^6 + 2*(c^4*d^5 - 2*a*c^3*d^3*e^2)*f^7*g^2 - 2*(4*a*c^3*d^4*e - 3*a^2*c^2*d^2*e^3)*f^6*g^3 + 4*(3*a^2*c^2*d^
3*e^2 - a^3*c*d*e^4)*f^5*g^4 - (8*a^3*c*d^2*e^3 - a^4*e^5)*f^4*g^5)*x^2 + (c^4*d^4*e*f^9 + 5*a^4*d*e^4*f^4*g^5
+ (5*c^4*d^5 - 4*a*c^3*d^3*e^2)*f^8*g - 2*(10*a*c^3*d^4*e - 3*a^2*c^2*d^2*e^3)*f^7*g^2 + 2*(15*a^2*c^2*d^3*e^
2 - 2*a^3*c*d*e^4)*f^6*g^3 - (20*a^3*c*d^2*e^3 - a^4*e^5)*f^5*g^4)*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{11/2}} \, dx=\text {Timed out} \]

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(g*x+f)**(11/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{11/2}} \, dx=\int { \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{\sqrt {e x + d} {\left (g x + f\right )}^{\frac {11}{2}}} \,d x } \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^(11/2)/(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)^(11/2)), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2280 vs. \(2 (235) = 470\).

Time = 1.18 (sec) , antiderivative size = 2280, normalized size of antiderivative = 8.54 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{11/2}} \, dx=\text {Too large to display} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^(11/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/315*((105*sqrt(-c*d^2*e + a*e^3)*c^4*d^5*e^5*f^3*abs(c)*abs(d) - 105*sqrt(-c*d^2*e + a*e^3)*a*c^3*d^3*e^7*f^
3*abs(c)*abs(d) - 126*sqrt(-c*d^2*e + a*e^3)*c^4*d^6*e^4*f^2*g*abs(c)*abs(d) - 63*sqrt(-c*d^2*e + a*e^3)*a*c^3
*d^4*e^6*f^2*g*abs(c)*abs(d) + 189*sqrt(-c*d^2*e + a*e^3)*a^2*c^2*d^2*e^8*f^2*g*abs(c)*abs(d) + 72*sqrt(-c*d^2
*e + a*e^3)*c^4*d^7*e^3*f*g^2*abs(c)*abs(d) + 36*sqrt(-c*d^2*e + a*e^3)*a*c^3*d^5*e^5*f*g^2*abs(c)*abs(d) + 27
*sqrt(-c*d^2*e + a*e^3)*a^2*c^2*d^3*e^7*f*g^2*abs(c)*abs(d) - 135*sqrt(-c*d^2*e + a*e^3)*a^3*c*d*e^9*f*g^2*abs
(c)*abs(d) - 16*sqrt(-c*d^2*e + a*e^3)*c^4*d^8*e^2*g^3*abs(c)*abs(d) - 8*sqrt(-c*d^2*e + a*e^3)*a*c^3*d^6*e^4*
g^3*abs(c)*abs(d) - 6*sqrt(-c*d^2*e + a*e^3)*a^2*c^2*d^4*e^6*g^3*abs(c)*abs(d) - 5*sqrt(-c*d^2*e + a*e^3)*a^3*
c*d^2*e^8*g^3*abs(c)*abs(d) + 35*sqrt(-c*d^2*e + a*e^3)*a^4*e^10*g^3*abs(c)*abs(d))/(sqrt(c^2*d^2*e^2*f - c^2*
d^3*e*g)*c^4*d^4*e^4*f^8*abs(e) - 4*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^4*d^5*e^3*f^7*g*abs(e) - 4*sqrt(c^2*d^
2*e^2*f - c^2*d^3*e*g)*a*c^3*d^3*e^5*f^7*g*abs(e) + 6*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^4*d^6*e^2*f^6*g^2*ab
s(e) + 16*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c^3*d^4*e^4*f^6*g^2*abs(e) + 6*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)
*a^2*c^2*d^2*e^6*f^6*g^2*abs(e) - 4*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^4*d^7*e*f^5*g^3*abs(e) - 24*sqrt(c^2*d
^2*e^2*f - c^2*d^3*e*g)*a*c^3*d^5*e^3*f^5*g^3*abs(e) - 24*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^2*c^2*d^3*e^5*f^
5*g^3*abs(e) - 4*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^3*c*d*e^7*f^5*g^3*abs(e) + sqrt(c^2*d^2*e^2*f - c^2*d^3*e
*g)*c^4*d^8*f^4*g^4*abs(e) + 16*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c^3*d^6*e^2*f^4*g^4*abs(e) + 36*sqrt(c^2*d
^2*e^2*f - c^2*d^3*e*g)*a^2*c^2*d^4*e^4*f^4*g^4*abs(e) + 16*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^3*c*d^2*e^6*f^
4*g^4*abs(e) + sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^4*e^8*f^4*g^4*abs(e) - 4*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*
a*c^3*d^7*e*f^3*g^5*abs(e) - 24*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^2*c^2*d^5*e^3*f^3*g^5*abs(e) - 24*sqrt(c^2
*d^2*e^2*f - c^2*d^3*e*g)*a^3*c*d^3*e^5*f^3*g^5*abs(e) - 4*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^4*d*e^7*f^3*g^5
*abs(e) + 6*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^2*c^2*d^6*e^2*f^2*g^6*abs(e) + 16*sqrt(c^2*d^2*e^2*f - c^2*d^3
*e*g)*a^3*c*d^4*e^4*f^2*g^6*abs(e) + 6*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^4*d^2*e^6*f^2*g^6*abs(e) - 4*sqrt(c
^2*d^2*e^2*f - c^2*d^3*e*g)*a^3*c*d^5*e^3*f*g^7*abs(e) - 4*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^4*d^3*e^5*f*g^7
*abs(e) + sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^4*d^4*e^4*g^8*abs(e)) + ((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2
)*(2*((e*x + d)*c*d*e - c*d^2*e + a*e^3)*(4*(2*((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^8*d^8*e^10*g^7*abs(c)*abs
(d)/(c^4*d^4*e^8*f^4*g^4*abs(e) - 4*a*c^3*d^3*e^9*f^3*g^5*abs(e) + 6*a^2*c^2*d^2*e^10*f^2*g^6*abs(e) - 4*a^3*c
*d*e^11*f*g^7*abs(e) + a^4*e^12*g^8*abs(e)) + 9*(c^9*d^9*e^12*f*g^6*abs(c)*abs(d) - a*c^8*d^8*e^13*g^7*abs(c)*
abs(d))/(c^4*d^4*e^8*f^4*g^4*abs(e) - 4*a*c^3*d^3*e^9*f^3*g^5*abs(e) + 6*a^2*c^2*d^2*e^10*f^2*g^6*abs(e) - 4*a
^3*c*d*e^11*f*g^7*abs(e) + a^4*e^12*g^8*abs(e)))*((e*x + d)*c*d*e - c*d^2*e + a*e^3) + 63*(c^10*d^10*e^14*f^2*
g^5*abs(c)*abs(d) - 2*a*c^9*d^9*e^15*f*g^6*abs(c)*abs(d) + a^2*c^8*d^8*e^16*g^7*abs(c)*abs(d))/(c^4*d^4*e^8*f^
4*g^4*abs(e) - 4*a*c^3*d^3*e^9*f^3*g^5*abs(e) + 6*a^2*c^2*d^2*e^10*f^2*g^6*abs(e) - 4*a^3*c*d*e^11*f*g^7*abs(e
) + a^4*e^12*g^8*abs(e))) + 105*(c^11*d^11*e^16*f^3*g^4*abs(c)*abs(d) - 3*a*c^10*d^10*e^17*f^2*g^5*abs(c)*abs(
d) + 3*a^2*c^9*d^9*e^18*f*g^6*abs(c)*abs(d) - a^3*c^8*d^8*e^19*g^7*abs(c)*abs(d))/(c^4*d^4*e^8*f^4*g^4*abs(e)
- 4*a*c^3*d^3*e^9*f^3*g^5*abs(e) + 6*a^2*c^2*d^2*e^10*f^2*g^6*abs(e) - 4*a^3*c*d*e^11*f*g^7*abs(e) + a^4*e^12*
g^8*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)^(9/2))*abs(e)/e^2

Mupad [B] (verification not implemented)

Time = 13.46 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.53 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{11/2}} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {x\,\left (-10\,a^3\,c\,d\,e^3\,g^3+54\,a^2\,c^2\,d^2\,e^2\,f\,g^2-126\,a\,c^3\,d^3\,e\,f^2\,g+210\,c^4\,d^4\,f^3\right )}{315\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {70\,a^4\,e^4\,g^3-270\,a^3\,c\,d\,e^3\,f\,g^2+378\,a^2\,c^2\,d^2\,e^2\,f^2\,g-210\,a\,c^3\,d^3\,e\,f^3}{315\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {32\,c^4\,d^4\,x^4}{315\,g\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {4\,c^2\,d^2\,x^2\,\left (a^2\,e^2\,g^2-6\,a\,c\,d\,e\,f\,g+21\,c^2\,d^2\,f^2\right )}{105\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {16\,c^3\,d^3\,x^3\,\left (a\,e\,g-9\,c\,d\,f\right )}{315\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^4}\right )}{x^4\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}+\frac {f^4\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^4}+\frac {4\,f\,x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g}+\frac {4\,f^3\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^3}+\frac {6\,f^2\,x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^2}} \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/((f + g*x)^(11/2)*(d + e*x)^(1/2)),x)

[Out]

((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((x*(210*c^4*d^4*f^3 - 10*a^3*c*d*e^3*g^3 + 54*a^2*c^2*d^2*e^2*
f*g^2 - 126*a*c^3*d^3*e*f^2*g))/(315*g^4*(a*e*g - c*d*f)^4) - (70*a^4*e^4*g^3 - 210*a*c^3*d^3*e*f^3 + 378*a^2*
c^2*d^2*e^2*f^2*g - 270*a^3*c*d*e^3*f*g^2)/(315*g^4*(a*e*g - c*d*f)^4) + (32*c^4*d^4*x^4)/(315*g*(a*e*g - c*d*
f)^4) + (4*c^2*d^2*x^2*(a^2*e^2*g^2 + 21*c^2*d^2*f^2 - 6*a*c*d*e*f*g))/(105*g^3*(a*e*g - c*d*f)^4) - (16*c^3*d
^3*x^3*(a*e*g - 9*c*d*f))/(315*g^2*(a*e*g - c*d*f)^4)))/(x^4*(f + g*x)^(1/2)*(d + e*x)^(1/2) + (f^4*(f + g*x)^
(1/2)*(d + e*x)^(1/2))/g^4 + (4*f*x^3*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g + (4*f^3*x*(f + g*x)^(1/2)*(d + e*x)^
(1/2))/g^3 + (6*f^2*x^2*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^2)