3.2252 \(\int \sqrt{d+e x} (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=343 \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{195 c^2 e^2 \sqrt{d+e x}}-\frac{4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{715 c^3 e^2 (d+e x)^{3/2}}-\frac{16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{6435 c^4 e^2 (d+e x)^{5/2}}-\frac{32 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{45045 c^5 e^2 (d+e x)^{7/2}}-\frac{2 g \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e^2} \]

[Out]

(-32*(2*c*d - b*e)^3*(15*c*e*f + c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(45045*c^5*e^2*
(d + e*x)^(7/2)) - (16*(2*c*d - b*e)^2*(15*c*e*f + c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2
))/(6435*c^4*e^2*(d + e*x)^(5/2)) - (4*(2*c*d - b*e)*(15*c*e*f + c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c
*e^2*x^2)^(7/2))/(715*c^3*e^2*(d + e*x)^(3/2)) - (2*(15*c*e*f + c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*
e^2*x^2)^(7/2))/(195*c^2*e^2*Sqrt[d + e*x]) - (2*g*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/
(15*c*e^2)

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Rubi [A]  time = 0.574929, antiderivative size = 343, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {794, 656, 648} \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{195 c^2 e^2 \sqrt{d+e x}}-\frac{4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{715 c^3 e^2 (d+e x)^{3/2}}-\frac{16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{6435 c^4 e^2 (d+e x)^{5/2}}-\frac{32 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{45045 c^5 e^2 (d+e x)^{7/2}}-\frac{2 g \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(-32*(2*c*d - b*e)^3*(15*c*e*f + c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(45045*c^5*e^2*
(d + e*x)^(7/2)) - (16*(2*c*d - b*e)^2*(15*c*e*f + c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2
))/(6435*c^4*e^2*(d + e*x)^(5/2)) - (4*(2*c*d - b*e)*(15*c*e*f + c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c
*e^2*x^2)^(7/2))/(715*c^3*e^2*(d + e*x)^(3/2)) - (2*(15*c*e*f + c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*
e^2*x^2)^(7/2))/(195*c^2*e^2*Sqrt[d + e*x]) - (2*g*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/
(15*c*e^2)

Rule 794

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)
*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g
, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 0])

Rule 656

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[(Simplify[m + p]*(2*c*d - b*e))/(c*(m + 2*p + 1)), In
t[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rule 648

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rubi steps

\begin{align*} \int \sqrt{d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx &=-\frac{2 g \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e^2}-\frac{\left (2 \left (\frac{7}{2} e \left (-2 c e^2 f+b e^2 g\right )+\frac{1}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int \sqrt{d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx}{15 c e^3}\\ &=-\frac{2 (15 c e f+c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{195 c^2 e^2 \sqrt{d+e x}}-\frac{2 g \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e^2}+\frac{(2 (2 c d-b e) (15 c e f+c d g-8 b e g)) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{\sqrt{d+e x}} \, dx}{65 c^2 e}\\ &=-\frac{4 (2 c d-b e) (15 c e f+c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{715 c^3 e^2 (d+e x)^{3/2}}-\frac{2 (15 c e f+c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{195 c^2 e^2 \sqrt{d+e x}}-\frac{2 g \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e^2}+\frac{\left (8 (2 c d-b e)^2 (15 c e f+c d g-8 b e g)\right ) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx}{715 c^3 e}\\ &=-\frac{16 (2 c d-b e)^2 (15 c e f+c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{6435 c^4 e^2 (d+e x)^{5/2}}-\frac{4 (2 c d-b e) (15 c e f+c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{715 c^3 e^2 (d+e x)^{3/2}}-\frac{2 (15 c e f+c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{195 c^2 e^2 \sqrt{d+e x}}-\frac{2 g \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e^2}+\frac{\left (16 (2 c d-b e)^3 (15 c e f+c d g-8 b e g)\right ) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx}{6435 c^4 e}\\ &=-\frac{32 (2 c d-b e)^3 (15 c e f+c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{45045 c^5 e^2 (d+e x)^{7/2}}-\frac{16 (2 c d-b e)^2 (15 c e f+c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{6435 c^4 e^2 (d+e x)^{5/2}}-\frac{4 (2 c d-b e) (15 c e f+c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{715 c^3 e^2 (d+e x)^{3/2}}-\frac{2 (15 c e f+c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{195 c^2 e^2 \sqrt{d+e x}}-\frac{2 g \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e^2}\\ \end{align*}

Mathematica [A]  time = 0.314809, size = 264, normalized size = 0.77 \[ \frac{2 (b e-c d+c e x)^3 \sqrt{(d+e x) (c (d-e x)-b e)} \left (24 b^2 c^2 e^2 \left (187 d^2 g+d e (95 f+161 g x)+7 e^2 x (5 f+6 g x)\right )-16 b^3 c e^3 (77 d g+15 e f+28 e g x)+128 b^4 e^4 g-2 b c^3 e \left (d^2 e (4065 f+5922 g x)+3611 d^3 g+21 d e^2 x (170 f+183 g x)+21 e^3 x^2 (45 f+44 g x)\right )+c^4 \left (147 d^2 e^2 x (145 f+129 g x)+d^3 e (12525 f+13433 g x)+3838 d^4 g+21 d e^3 x^2 (675 f+583 g x)+231 e^4 x^3 (15 f+13 g x)\right )\right )}{45045 c^5 e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + e*x]*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(128*b^4*e^4*g - 16*b^3*c*e^3*(15*e*f + 77*
d*g + 28*e*g*x) + 24*b^2*c^2*e^2*(187*d^2*g + 7*e^2*x*(5*f + 6*g*x) + d*e*(95*f + 161*g*x)) - 2*b*c^3*e*(3611*
d^3*g + 21*e^3*x^2*(45*f + 44*g*x) + 21*d*e^2*x*(170*f + 183*g*x) + d^2*e*(4065*f + 5922*g*x)) + c^4*(3838*d^4
*g + 231*e^4*x^3*(15*f + 13*g*x) + 147*d^2*e^2*x*(145*f + 129*g*x) + 21*d*e^3*x^2*(675*f + 583*g*x) + d^3*e*(1
2525*f + 13433*g*x))))/(45045*c^5*e^2*Sqrt[d + e*x])

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Maple [A]  time = 0.008, size = 367, normalized size = 1.1 \begin{align*}{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 3003\,g{e}^{4}{x}^{4}{c}^{4}-1848\,b{c}^{3}{e}^{4}g{x}^{3}+12243\,{c}^{4}d{e}^{3}g{x}^{3}+3465\,{c}^{4}{e}^{4}f{x}^{3}+1008\,{b}^{2}{c}^{2}{e}^{4}g{x}^{2}-7686\,b{c}^{3}d{e}^{3}g{x}^{2}-1890\,b{c}^{3}{e}^{4}f{x}^{2}+18963\,{c}^{4}{d}^{2}{e}^{2}g{x}^{2}+14175\,{c}^{4}d{e}^{3}f{x}^{2}-448\,{b}^{3}c{e}^{4}gx+3864\,{b}^{2}{c}^{2}d{e}^{3}gx+840\,{b}^{2}{c}^{2}{e}^{4}fx-11844\,b{c}^{3}{d}^{2}{e}^{2}gx-7140\,b{c}^{3}d{e}^{3}fx+13433\,{c}^{4}{d}^{3}egx+21315\,{c}^{4}{d}^{2}{e}^{2}fx+128\,{b}^{4}{e}^{4}g-1232\,{b}^{3}cd{e}^{3}g-240\,{b}^{3}c{e}^{4}f+4488\,{b}^{2}{c}^{2}{d}^{2}{e}^{2}g+2280\,{b}^{2}{c}^{2}d{e}^{3}f-7222\,b{c}^{3}{d}^{3}eg-8130\,b{c}^{3}{d}^{2}{e}^{2}f+3838\,{c}^{4}{d}^{4}g+12525\,f{d}^{3}{c}^{4}e \right ) }{45045\,{c}^{5}{e}^{2}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

[Out]

2/45045*(c*e*x+b*e-c*d)*(3003*c^4*e^4*g*x^4-1848*b*c^3*e^4*g*x^3+12243*c^4*d*e^3*g*x^3+3465*c^4*e^4*f*x^3+1008
*b^2*c^2*e^4*g*x^2-7686*b*c^3*d*e^3*g*x^2-1890*b*c^3*e^4*f*x^2+18963*c^4*d^2*e^2*g*x^2+14175*c^4*d*e^3*f*x^2-4
48*b^3*c*e^4*g*x+3864*b^2*c^2*d*e^3*g*x+840*b^2*c^2*e^4*f*x-11844*b*c^3*d^2*e^2*g*x-7140*b*c^3*d*e^3*f*x+13433
*c^4*d^3*e*g*x+21315*c^4*d^2*e^2*f*x+128*b^4*e^4*g-1232*b^3*c*d*e^3*g-240*b^3*c*e^4*f+4488*b^2*c^2*d^2*e^2*g+2
280*b^2*c^2*d*e^3*f-7222*b*c^3*d^3*e*g-8130*b*c^3*d^2*e^2*f+3838*c^4*d^4*g+12525*c^4*d^3*e*f)*(-c*e^2*x^2-b*e^
2*x-b*d*e+c*d^2)^(5/2)/c^5/e^2/(e*x+d)^(5/2)

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Maxima [B]  time = 1.35112, size = 1185, normalized size = 3.45 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="maxima")

[Out]

2/3003*(231*c^6*e^6*x^6 - 835*c^6*d^6 + 3047*b*c^5*d^5*e - 4283*b^2*c^4*d^4*e^2 + 2933*b^3*c^3*d^3*e^3 - 1046*
b^4*c^2*d^2*e^4 + 200*b^5*c*d*e^5 - 16*b^6*e^6 + 63*(4*c^6*d*e^5 + 9*b*c^5*e^6)*x^5 - 7*(103*c^6*d^2*e^4 - 193
*b*c^5*d*e^5 - 53*b^2*c^4*e^6)*x^4 - (824*c^6*d^3*e^3 + 206*b*c^5*d^2*e^4 - 1454*b^2*c^4*d*e^5 - 5*b^3*c^3*e^6
)*x^3 + 3*(271*c^6*d^4*e^2 - 954*b*c^5*d^3*e^3 + 664*b^2*c^4*d^2*e^4 + 21*b^3*c^3*d*e^5 - 2*b^4*c^2*e^6)*x^2 +
(1084*c^6*d^5*e - 1897*b*c^5*d^4*e^2 + 466*b^2*c^4*d^3*e^3 + 431*b^3*c^3*d^2*e^4 - 92*b^4*c^2*d*e^5 + 8*b^5*c
*e^6)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*f/(c^4*e^2*x + c^4*d*e) + 2/45045*(3003*c^7*e^7*x^7 - 3838*c^7*d^7
+ 18736*b*c^6*d^6*e - 37668*b^2*c^5*d^5*e^2 + 40200*b^3*c^4*d^4*e^3 - 24510*b^4*c^3*d^3*e^4 + 8568*b^5*c^2*d^
2*e^5 - 1616*b^6*c*d*e^6 + 128*b^7*e^7 + 231*(14*c^7*d*e^6 + 31*b*c^6*e^7)*x^6 - 63*(139*c^7*d^2*e^5 - 263*b*c
^6*d*e^6 - 71*b^2*c^5*e^7)*x^5 - 35*(278*c^7*d^3*e^4 + 54*b*c^6*d^2*e^5 - 474*b^2*c^5*d*e^6 - b^3*c^4*e^7)*x^4
+ 5*(1637*c^7*d^4*e^3 - 5930*b*c^6*d^3*e^4 + 4224*b^2*c^5*d^2*e^5 + 77*b^3*c^4*d*e^6 - 8*b^4*c^3*e^7)*x^3 + 3
*(3274*c^7*d^5*e^2 - 6125*b*c^6*d^4*e^3 + 2290*b^2*c^5*d^3*e^4 + 715*b^3*c^4*d^2*e^5 - 170*b^4*c^3*d*e^6 + 16*
b^5*c^2*e^7)*x^2 - (1919*c^7*d^6*e - 7449*b*c^6*d^5*e^2 + 11385*b^2*c^5*d^4*e^3 - 8715*b^3*c^4*d^3*e^4 + 3540*
b^4*c^3*d^2*e^5 - 744*b^5*c^2*d*e^6 + 64*b^6*c*e^7)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*g/(c^5*e^3*x + c^5*d
*e^2)

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Fricas [B]  time = 1.56204, size = 1933, normalized size = 5.64 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")

[Out]

2/45045*(3003*c^7*e^7*g*x^7 + 231*(15*c^7*e^7*f + (14*c^7*d*e^6 + 31*b*c^6*e^7)*g)*x^6 + 63*(15*(4*c^7*d*e^6 +
9*b*c^6*e^7)*f - (139*c^7*d^2*e^5 - 263*b*c^6*d*e^6 - 71*b^2*c^5*e^7)*g)*x^5 - 35*(3*(103*c^7*d^2*e^5 - 193*b
*c^6*d*e^6 - 53*b^2*c^5*e^7)*f + (278*c^7*d^3*e^4 + 54*b*c^6*d^2*e^5 - 474*b^2*c^5*d*e^6 - b^3*c^4*e^7)*g)*x^4
- 5*(3*(824*c^7*d^3*e^4 + 206*b*c^6*d^2*e^5 - 1454*b^2*c^5*d*e^6 - 5*b^3*c^4*e^7)*f - (1637*c^7*d^4*e^3 - 593
0*b*c^6*d^3*e^4 + 4224*b^2*c^5*d^2*e^5 + 77*b^3*c^4*d*e^6 - 8*b^4*c^3*e^7)*g)*x^3 + 3*(15*(271*c^7*d^4*e^3 - 9
54*b*c^6*d^3*e^4 + 664*b^2*c^5*d^2*e^5 + 21*b^3*c^4*d*e^6 - 2*b^4*c^3*e^7)*f + (3274*c^7*d^5*e^2 - 6125*b*c^6*
d^4*e^3 + 2290*b^2*c^5*d^3*e^4 + 715*b^3*c^4*d^2*e^5 - 170*b^4*c^3*d*e^6 + 16*b^5*c^2*e^7)*g)*x^2 - 15*(835*c^
7*d^6*e - 3047*b*c^6*d^5*e^2 + 4283*b^2*c^5*d^4*e^3 - 2933*b^3*c^4*d^3*e^4 + 1046*b^4*c^3*d^2*e^5 - 200*b^5*c^
2*d*e^6 + 16*b^6*c*e^7)*f - 2*(1919*c^7*d^7 - 9368*b*c^6*d^6*e + 18834*b^2*c^5*d^5*e^2 - 20100*b^3*c^4*d^4*e^3
+ 12255*b^4*c^3*d^3*e^4 - 4284*b^5*c^2*d^2*e^5 + 808*b^6*c*d*e^6 - 64*b^7*e^7)*g + (15*(1084*c^7*d^5*e^2 - 18
97*b*c^6*d^4*e^3 + 466*b^2*c^5*d^3*e^4 + 431*b^3*c^4*d^2*e^5 - 92*b^4*c^3*d*e^6 + 8*b^5*c^2*e^7)*f - (1919*c^7
*d^6*e - 7449*b*c^6*d^5*e^2 + 11385*b^2*c^5*d^4*e^3 - 8715*b^3*c^4*d^3*e^4 + 3540*b^4*c^3*d^2*e^5 - 744*b^5*c^
2*d*e^6 + 64*b^6*c*e^7)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)/(c^5*e^3*x + c^5*d*e^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(1/2)*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")

[Out]

Timed out