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

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

[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]
time = 0.21, 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 = {886, 874} \begin {gather*} \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)} \end {gather*}

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 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*} \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 = 0.17, size = 152, normalized size = 0.57 \begin {gather*} \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}} \end {gather*}

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))

________________________________________________________________________________________

Maple [A]
time = 0.14, size = 191, normalized size = 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	\(-\frac {2 \left (c d x +a e \right ) \left (-16 g^{3} x^{3} c^{3} d^{3}+24 a \,c^{2} d^{2} e \,g^{3} x^{2}-72 c^{3} d^{3} f \,g^{2} x^{2}-30 a^{2} c d \,e^{2} g^{3} x +108 a \,c^{2} d^{2} e f \,g^{2} x -126 c^{3} d^{3} f^{2} g x +35 a^{3} e^{3} g^{3}-135 a^{2} c d \,e^{2} f \,g^{2}+189 a \,c^{2} d^{2} e \,f^{2} g -105 f^{3} c^{3} d^{3}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{315 \left (g x +f \right )^{\frac {9}{2}} \left (g^{4} e^{4} a^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} c^{4} d^{4}\right ) \sqrt {e x +d}}\)	\(260\)

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1243 vs. \(2 (247) = 494\).
time = 5.17, size = 1243, normalized size = 4.66 \begin {gather*} \frac {2 \, {\left (16 \, c^{4} d^{4} g^{3} x^{4} + 72 \, c^{4} d^{4} f g^{2} x^{3} + 126 \, c^{4} d^{4} f^{2} g x^{2} + 105 \, c^{4} d^{4} f^{3} x - 35 \, a^{4} g^{3} e^{4} - 5 \, {\left (a^{3} c d g^{3} x - 27 \, a^{3} c d f g^{2}\right )} e^{3} + 3 \, {\left (2 \, a^{2} c^{2} d^{2} g^{3} x^{2} + 9 \, a^{2} c^{2} d^{2} f g^{2} x - 63 \, a^{2} c^{2} d^{2} f^{2} g\right )} e^{2} - {\left (8 \, a c^{3} d^{3} g^{3} x^{3} + 36 \, a c^{3} d^{3} f g^{2} x^{2} + 63 \, a c^{3} d^{3} f^{2} g x - 105 \, a c^{3} d^{3} f^{3}\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {g x + f} \sqrt {x e + d}}{315 \, {\left (c^{4} d^{5} f^{4} g^{5} x^{5} + 5 \, c^{4} d^{5} f^{5} g^{4} x^{4} + 10 \, c^{4} d^{5} f^{6} g^{3} x^{3} + 10 \, c^{4} d^{5} f^{7} g^{2} x^{2} + 5 \, c^{4} d^{5} f^{8} g x + c^{4} d^{5} f^{9} + {\left (a^{4} g^{9} x^{6} + 5 \, a^{4} f g^{8} x^{5} + 10 \, a^{4} f^{2} g^{7} x^{4} + 10 \, a^{4} f^{3} g^{6} x^{3} + 5 \, a^{4} f^{4} g^{5} x^{2} + a^{4} f^{5} g^{4} x\right )} e^{5} - {\left (4 \, a^{3} c d f g^{8} x^{6} - a^{4} d f^{5} g^{4} + {\left (20 \, a^{3} c d f^{2} g^{7} - a^{4} d g^{9}\right )} x^{5} + 5 \, {\left (8 \, a^{3} c d f^{3} g^{6} - a^{4} d f g^{8}\right )} x^{4} + 10 \, {\left (4 \, a^{3} c d f^{4} g^{5} - a^{4} d f^{2} g^{7}\right )} x^{3} + 10 \, {\left (2 \, a^{3} c d f^{5} g^{4} - a^{4} d f^{3} g^{6}\right )} x^{2} + {\left (4 \, a^{3} c d f^{6} g^{3} - 5 \, a^{4} d f^{4} g^{5}\right )} x\right )} e^{4} + 2 \, {\left (3 \, a^{2} c^{2} d^{2} f^{2} g^{7} x^{6} - 2 \, a^{3} c d^{2} f^{6} g^{3} + {\left (15 \, a^{2} c^{2} d^{2} f^{3} g^{6} - 2 \, a^{3} c d^{2} f g^{8}\right )} x^{5} + 10 \, {\left (3 \, a^{2} c^{2} d^{2} f^{4} g^{5} - a^{3} c d^{2} f^{2} g^{7}\right )} x^{4} + 10 \, {\left (3 \, a^{2} c^{2} d^{2} f^{5} g^{4} - 2 \, a^{3} c d^{2} f^{3} g^{6}\right )} x^{3} + 5 \, {\left (3 \, a^{2} c^{2} d^{2} f^{6} g^{3} - 4 \, a^{3} c d^{2} f^{4} g^{5}\right )} x^{2} + {\left (3 \, a^{2} c^{2} d^{2} f^{7} g^{2} - 10 \, a^{3} c d^{2} f^{5} g^{4}\right )} x\right )} e^{3} - 2 \, {\left (2 \, a c^{3} d^{3} f^{3} g^{6} x^{6} - 3 \, a^{2} c^{2} d^{3} f^{7} g^{2} + {\left (10 \, a c^{3} d^{3} f^{4} g^{5} - 3 \, a^{2} c^{2} d^{3} f^{2} g^{7}\right )} x^{5} + 5 \, {\left (4 \, a c^{3} d^{3} f^{5} g^{4} - 3 \, a^{2} c^{2} d^{3} f^{3} g^{6}\right )} x^{4} + 10 \, {\left (2 \, a c^{3} d^{3} f^{6} g^{3} - 3 \, a^{2} c^{2} d^{3} f^{4} g^{5}\right )} x^{3} + 10 \, {\left (a c^{3} d^{3} f^{7} g^{2} - 3 \, a^{2} c^{2} d^{3} f^{5} g^{4}\right )} x^{2} + {\left (2 \, a c^{3} d^{3} f^{8} g - 15 \, a^{2} c^{2} d^{3} f^{6} g^{3}\right )} x\right )} e^{2} + {\left (c^{4} d^{4} f^{4} g^{5} x^{6} - 4 \, a c^{3} d^{4} f^{8} g + {\left (5 \, c^{4} d^{4} f^{5} g^{4} - 4 \, a c^{3} d^{4} f^{3} g^{6}\right )} x^{5} + 10 \, {\left (c^{4} d^{4} f^{6} g^{3} - 2 \, a c^{3} d^{4} f^{4} g^{5}\right )} x^{4} + 10 \, {\left (c^{4} d^{4} f^{7} g^{2} - 4 \, a c^{3} d^{4} f^{5} g^{4}\right )} x^{3} + 5 \, {\left (c^{4} d^{4} f^{8} g - 8 \, a c^{3} d^{4} f^{6} g^{3}\right )} x^{2} + {\left (c^{4} d^{4} f^{9} - 20 \, a c^{3} d^{4} f^{7} g^{2}\right )} x\right )} e\right )}} \end {gather*}

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 + 72*c^4*d^4*f*g^2*x^3 + 126*c^4*d^4*f^2*g*x^2 + 105*c^4*d^4*f^3*x - 35*a^4*g^3*e^4
- 5*(a^3*c*d*g^3*x - 27*a^3*c*d*f*g^2)*e^3 + 3*(2*a^2*c^2*d^2*g^3*x^2 + 9*a^2*c^2*d^2*f*g^2*x - 63*a^2*c^2*d^2
*f^2*g)*e^2 - (8*a*c^3*d^3*g^3*x^3 + 36*a*c^3*d^3*f*g^2*x^2 + 63*a*c^3*d^3*f^2*g*x - 105*a*c^3*d^3*f^3)*e)*sqr
t(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(g*x + f)*sqrt(x*e + d)/(c^4*d^5*f^4*g^5*x^5 + 5*c^4*d^5*f^5*g^4*
x^4 + 10*c^4*d^5*f^6*g^3*x^3 + 10*c^4*d^5*f^7*g^2*x^2 + 5*c^4*d^5*f^8*g*x + c^4*d^5*f^9 + (a^4*g^9*x^6 + 5*a^4
*f*g^8*x^5 + 10*a^4*f^2*g^7*x^4 + 10*a^4*f^3*g^6*x^3 + 5*a^4*f^4*g^5*x^2 + a^4*f^5*g^4*x)*e^5 - (4*a^3*c*d*f*g
^8*x^6 - a^4*d*f^5*g^4 + (20*a^3*c*d*f^2*g^7 - a^4*d*g^9)*x^5 + 5*(8*a^3*c*d*f^3*g^6 - a^4*d*f*g^8)*x^4 + 10*(
4*a^3*c*d*f^4*g^5 - a^4*d*f^2*g^7)*x^3 + 10*(2*a^3*c*d*f^5*g^4 - a^4*d*f^3*g^6)*x^2 + (4*a^3*c*d*f^6*g^3 - 5*a
^4*d*f^4*g^5)*x)*e^4 + 2*(3*a^2*c^2*d^2*f^2*g^7*x^6 - 2*a^3*c*d^2*f^6*g^3 + (15*a^2*c^2*d^2*f^3*g^6 - 2*a^3*c*
d^2*f*g^8)*x^5 + 10*(3*a^2*c^2*d^2*f^4*g^5 - a^3*c*d^2*f^2*g^7)*x^4 + 10*(3*a^2*c^2*d^2*f^5*g^4 - 2*a^3*c*d^2*
f^3*g^6)*x^3 + 5*(3*a^2*c^2*d^2*f^6*g^3 - 4*a^3*c*d^2*f^4*g^5)*x^2 + (3*a^2*c^2*d^2*f^7*g^2 - 10*a^3*c*d^2*f^5
*g^4)*x)*e^3 - 2*(2*a*c^3*d^3*f^3*g^6*x^6 - 3*a^2*c^2*d^3*f^7*g^2 + (10*a*c^3*d^3*f^4*g^5 - 3*a^2*c^2*d^3*f^2*
g^7)*x^5 + 5*(4*a*c^3*d^3*f^5*g^4 - 3*a^2*c^2*d^3*f^3*g^6)*x^4 + 10*(2*a*c^3*d^3*f^6*g^3 - 3*a^2*c^2*d^3*f^4*g
^5)*x^3 + 10*(a*c^3*d^3*f^7*g^2 - 3*a^2*c^2*d^3*f^5*g^4)*x^2 + (2*a*c^3*d^3*f^8*g - 15*a^2*c^2*d^3*f^6*g^3)*x)
*e^2 + (c^4*d^4*f^4*g^5*x^6 - 4*a*c^3*d^4*f^8*g + (5*c^4*d^4*f^5*g^4 - 4*a*c^3*d^4*f^3*g^6)*x^5 + 10*(c^4*d^4*
f^6*g^3 - 2*a*c^3*d^4*f^4*g^5)*x^4 + 10*(c^4*d^4*f^7*g^2 - 4*a*c^3*d^4*f^5*g^4)*x^3 + 5*(c^4*d^4*f^8*g - 8*a*c
^3*d^4*f^6*g^3)*x^2 + (c^4*d^4*f^9 - 20*a*c^3*d^4*f^7*g^2)*x)*e)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

________________________________________________________________________________________

Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 4.50, size = 409, normalized size = 1.53 \begin {gather*} \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}} \end {gather*}

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)

________________________________________________________________________________________