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

Optimal. Leaf size=267 \[ \frac {32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{315 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)^4}+\frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)^3}+\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{21 (d+e x)^{3/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 (d+e x)^{3/2} (f+g x)^{9/2} (c d f-a e g)} \]

[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]  time = 0.31, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {872, 860} \[ \frac {32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{315 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)^4}+\frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)^3}+\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{21 (d+e x)^{3/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 (d+e x)^{3/2} (f+g x)^{9/2} (c d f-a e g)} \]

Antiderivative was successfully verified.

[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 860

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 872

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*} \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 {(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]  time = 0.14, size = 152, normalized size = 0.57 \[ \frac {2 ((d+e x) (a e+c d x))^{3/2} \left (-35 a^3 e^3 g^3+15 a^2 c d e^2 g^2 (9 f+2 g x)-3 a c^2 d^2 e g \left (63 f^2+36 f g x+8 g^2 x^2\right )+c^3 d^3 \left (105 f^3+126 f^2 g x+72 f g^2 x^2+16 g^3 x^3\right )\right )}{315 (d+e x)^{3/2} (f+g x)^{9/2} (c d f-a e g)^4} \]

Antiderivative was successfully verified.

[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))

________________________________________________________________________________________

fricas [B]  time = 0.91, size = 1179, normalized size = 4.42 \[ \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 )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[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]

Timed out

________________________________________________________________________________________

maple [A]  time = 0.01, size = 260, normalized size = 0.97 \[ -\frac {2 \left (c d x +a e \right ) \left (-16 g^{3} x^{3} c^{3} d^{3}+24 a \,c^{2} d^{2} e \,g^{3} x^{2}-72 c^{3} d^{3} f \,g^{2} x^{2}-30 a^{2} c d \,e^{2} g^{3} x +108 a \,c^{2} d^{2} e f \,g^{2} x -126 c^{3} d^{3} f^{2} g x +35 a^{3} e^{3} g^{3}-135 a^{2} c d \,e^{2} f \,g^{2}+189 a \,c^{2} d^{2} e \,f^{2} g -105 f^{3} c^{3} d^{3}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{315 \left (g x +f \right )^{\frac {9}{2}} \left (g^{4} e^{4} a^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} c^{4} d^{4}\right ) \sqrt {e x +d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(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)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)/(g*x+f)^(9/2)/(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+c^4*d^4*f^4)/(e*x+d)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{\sqrt {e x + d} {\left (g x + f\right )}^{\frac {11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 4.50, size = 409, normalized size = 1.53 \[ \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {x\,\left (-10\,a^3\,c\,d\,e^3\,g^3+54\,a^2\,c^2\,d^2\,e^2\,f\,g^2-126\,a\,c^3\,d^3\,e\,f^2\,g+210\,c^4\,d^4\,f^3\right )}{315\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {70\,a^4\,e^4\,g^3-270\,a^3\,c\,d\,e^3\,f\,g^2+378\,a^2\,c^2\,d^2\,e^2\,f^2\,g-210\,a\,c^3\,d^3\,e\,f^3}{315\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {32\,c^4\,d^4\,x^4}{315\,g\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {4\,c^2\,d^2\,x^2\,\left (a^2\,e^2\,g^2-6\,a\,c\,d\,e\,f\,g+21\,c^2\,d^2\,f^2\right )}{105\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {16\,c^3\,d^3\,x^3\,\left (a\,e\,g-9\,c\,d\,f\right )}{315\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^4}\right )}{x^4\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}+\frac {f^4\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^4}+\frac {4\,f\,x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g}+\frac {4\,f^3\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^3}+\frac {6\,f^2\,x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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