3.2240 \(\int (d+e x)^{5/2} (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=419 \[ -\frac{256 (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{45045 c^6 e^2 (d+e x)^{5/2}}-\frac{128 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{9009 c^5 e^2 (d+e x)^{3/2}}-\frac{32 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{1287 c^4 e^2 \sqrt{d+e x}}-\frac{16 \sqrt{d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{429 c^3 e^2}-\frac{2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{39 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2} \]

[Out]

(-256*(2*c*d - b*e)^4*(3*c*e*f + c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(45045*c^6*e^2*
(d + e*x)^(5/2)) - (128*(2*c*d - b*e)^3*(3*c*e*f + c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2
))/(9009*c^5*e^2*(d + e*x)^(3/2)) - (32*(2*c*d - b*e)^2*(3*c*e*f + c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x -
c*e^2*x^2)^(5/2))/(1287*c^4*e^2*Sqrt[d + e*x]) - (16*(2*c*d - b*e)*(3*c*e*f + c*d*g - 2*b*e*g)*Sqrt[d + e*x]*
(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(429*c^3*e^2) - (2*(3*c*e*f + c*d*g - 2*b*e*g)*(d + e*x)^(3/2)*(d
*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(39*c^2*e^2) - (2*g*(d + e*x)^(5/2)*(d*(c*d - b*e) - b*e^2*x - c*e^
2*x^2)^(5/2))/(15*c*e^2)

________________________________________________________________________________________

Rubi [A]  time = 0.72607, antiderivative size = 419, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {794, 656, 648} \[ -\frac{256 (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{45045 c^6 e^2 (d+e x)^{5/2}}-\frac{128 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{9009 c^5 e^2 (d+e x)^{3/2}}-\frac{32 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{1287 c^4 e^2 \sqrt{d+e x}}-\frac{16 \sqrt{d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{429 c^3 e^2}-\frac{2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{39 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]

[Out]

(-256*(2*c*d - b*e)^4*(3*c*e*f + c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(45045*c^6*e^2*
(d + e*x)^(5/2)) - (128*(2*c*d - b*e)^3*(3*c*e*f + c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2
))/(9009*c^5*e^2*(d + e*x)^(3/2)) - (32*(2*c*d - b*e)^2*(3*c*e*f + c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x -
c*e^2*x^2)^(5/2))/(1287*c^4*e^2*Sqrt[d + e*x]) - (16*(2*c*d - b*e)*(3*c*e*f + c*d*g - 2*b*e*g)*Sqrt[d + e*x]*
(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(429*c^3*e^2) - (2*(3*c*e*f + c*d*g - 2*b*e*g)*(d + e*x)^(3/2)*(d
*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(39*c^2*e^2) - (2*g*(d + e*x)^(5/2)*(d*(c*d - b*e) - b*e^2*x - c*e^
2*x^2)^(5/2))/(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 (d+e x)^{5/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx &=-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2}-\frac{\left (2 \left (\frac{5}{2} e \left (-2 c e^2 f+b e^2 g\right )+\frac{5}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int (d+e x)^{5/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{15 c e^3}\\ &=-\frac{2 (3 c e f+c d g-2 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{39 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2}+\frac{(8 (2 c d-b e) (3 c e f+c d g-2 b e g)) \int (d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{39 c^2 e}\\ &=-\frac{16 (2 c d-b e) (3 c e f+c d g-2 b e g) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{429 c^3 e^2}-\frac{2 (3 c e f+c d g-2 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{39 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2}+\frac{\left (16 (2 c d-b e)^2 (3 c e f+c d g-2 b e g)\right ) \int \sqrt{d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{143 c^3 e}\\ &=-\frac{32 (2 c d-b e)^2 (3 c e f+c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{1287 c^4 e^2 \sqrt{d+e x}}-\frac{16 (2 c d-b e) (3 c e f+c d g-2 b e g) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{429 c^3 e^2}-\frac{2 (3 c e f+c d g-2 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{39 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2}+\frac{\left (64 (2 c d-b e)^3 (3 c e f+c d g-2 b e g)\right ) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx}{1287 c^4 e}\\ &=-\frac{128 (2 c d-b e)^3 (3 c e f+c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9009 c^5 e^2 (d+e x)^{3/2}}-\frac{32 (2 c d-b e)^2 (3 c e f+c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{1287 c^4 e^2 \sqrt{d+e x}}-\frac{16 (2 c d-b e) (3 c e f+c d g-2 b e g) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{429 c^3 e^2}-\frac{2 (3 c e f+c d g-2 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{39 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2}+\frac{\left (128 (2 c d-b e)^4 (3 c e f+c d g-2 b e g)\right ) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{9009 c^5 e}\\ &=-\frac{256 (2 c d-b e)^4 (3 c e f+c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{45045 c^6 e^2 (d+e x)^{5/2}}-\frac{128 (2 c d-b e)^3 (3 c e f+c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9009 c^5 e^2 (d+e x)^{3/2}}-\frac{32 (2 c d-b e)^2 (3 c e f+c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{1287 c^4 e^2 \sqrt{d+e x}}-\frac{16 (2 c d-b e) (3 c e f+c d g-2 b e g) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{429 c^3 e^2}-\frac{2 (3 c e f+c d g-2 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{39 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2}\\ \end{align*}

Mathematica [A]  time = 0.390578, size = 364, normalized size = 0.87 \[ -\frac{2 (b e-c d+c e x)^2 \sqrt{(d+e x) (c (d-e x)-b e)} \left (16 b^2 c^3 e^2 \left (3 d^2 e (347 f+515 g x)+1724 d^3 g+30 d e^2 x (19 f+21 g x)+105 e^3 x^2 (f+g x)\right )-32 b^3 c^2 e^3 \left (389 d^2 g+2 d e (63 f+100 g x)+5 e^2 x (6 f+7 g x)\right )+128 b^4 c e^4 (22 d g+3 e f+5 e g x)-256 b^5 e^5 g-2 b c^4 e \left (30 d^2 e^2 x (542 f+553 g x)+4 d^3 e (4131 f+5530 g x)+15191 d^4 g+420 d e^3 x^2 (17 f+16 g x)+105 e^4 x^3 (12 f+11 g x)\right )+c^5 \left (210 d^2 e^3 x^2 (203 f+173 g x)+20 d^3 e^2 x (2505 f+2212 g x)+d^4 e (29049 f+31715 g x)+12686 d^5 g+210 d e^4 x^3 (90 f+77 g x)+231 e^5 x^4 (15 f+13 g x)\right )\right )}{45045 c^6 e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(5/2)*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]

[Out]

(-2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-256*b^5*e^5*g + 128*b^4*c*e^4*(3*e*f + 2
2*d*g + 5*e*g*x) - 32*b^3*c^2*e^3*(389*d^2*g + 5*e^2*x*(6*f + 7*g*x) + 2*d*e*(63*f + 100*g*x)) + 16*b^2*c^3*e^
2*(1724*d^3*g + 105*e^3*x^2*(f + g*x) + 30*d*e^2*x*(19*f + 21*g*x) + 3*d^2*e*(347*f + 515*g*x)) - 2*b*c^4*e*(1
5191*d^4*g + 105*e^4*x^3*(12*f + 11*g*x) + 420*d*e^3*x^2*(17*f + 16*g*x) + 30*d^2*e^2*x*(542*f + 553*g*x) + 4*
d^3*e*(4131*f + 5530*g*x)) + c^5*(12686*d^5*g + 231*e^5*x^4*(15*f + 13*g*x) + 210*d*e^4*x^3*(90*f + 77*g*x) +
210*d^2*e^3*x^2*(203*f + 173*g*x) + 20*d^3*e^2*x*(2505*f + 2212*g*x) + d^4*e*(29049*f + 31715*g*x))))/(45045*c
^6*e^2*Sqrt[d + e*x])

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Maple [A]  time = 0.009, size = 535, normalized size = 1.3 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -3003\,g{e}^{5}{x}^{5}{c}^{5}+2310\,b{c}^{4}{e}^{5}g{x}^{4}-16170\,{c}^{5}d{e}^{4}g{x}^{4}-3465\,{c}^{5}{e}^{5}f{x}^{4}-1680\,{b}^{2}{c}^{3}{e}^{5}g{x}^{3}+13440\,b{c}^{4}d{e}^{4}g{x}^{3}+2520\,b{c}^{4}{e}^{5}f{x}^{3}-36330\,{c}^{5}{d}^{2}{e}^{3}g{x}^{3}-18900\,{c}^{5}d{e}^{4}f{x}^{3}+1120\,{b}^{3}{c}^{2}{e}^{5}g{x}^{2}-10080\,{b}^{2}{c}^{3}d{e}^{4}g{x}^{2}-1680\,{b}^{2}{c}^{3}{e}^{5}f{x}^{2}+33180\,b{c}^{4}{d}^{2}{e}^{3}g{x}^{2}+14280\,b{c}^{4}d{e}^{4}f{x}^{2}-44240\,{c}^{5}{d}^{3}{e}^{2}g{x}^{2}-42630\,{c}^{5}{d}^{2}{e}^{3}f{x}^{2}-640\,{b}^{4}c{e}^{5}gx+6400\,{b}^{3}{c}^{2}d{e}^{4}gx+960\,{b}^{3}{c}^{2}{e}^{5}fx-24720\,{b}^{2}{c}^{3}{d}^{2}{e}^{3}gx-9120\,{b}^{2}{c}^{3}d{e}^{4}fx+44240\,b{c}^{4}{d}^{3}{e}^{2}gx+32520\,b{c}^{4}{d}^{2}{e}^{3}fx-31715\,{c}^{5}{d}^{4}egx-50100\,{c}^{5}{d}^{3}{e}^{2}fx+256\,{b}^{5}{e}^{5}g-2816\,{b}^{4}cd{e}^{4}g-384\,{b}^{4}c{e}^{5}f+12448\,{b}^{3}{c}^{2}{d}^{2}{e}^{3}g+4032\,{b}^{3}{c}^{2}d{e}^{4}f-27584\,{b}^{2}{c}^{3}{d}^{3}{e}^{2}g-16656\,{b}^{2}{c}^{3}{d}^{2}{e}^{3}f+30382\,b{c}^{4}{d}^{4}eg+33048\,b{c}^{4}{d}^{3}{e}^{2}f-12686\,{c}^{5}{d}^{5}g-29049\,f{d}^{4}{c}^{5}e \right ) }{45045\,{c}^{6}{e}^{2}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)

[Out]

-2/45045*(c*e*x+b*e-c*d)*(-3003*c^5*e^5*g*x^5+2310*b*c^4*e^5*g*x^4-16170*c^5*d*e^4*g*x^4-3465*c^5*e^5*f*x^4-16
80*b^2*c^3*e^5*g*x^3+13440*b*c^4*d*e^4*g*x^3+2520*b*c^4*e^5*f*x^3-36330*c^5*d^2*e^3*g*x^3-18900*c^5*d*e^4*f*x^
3+1120*b^3*c^2*e^5*g*x^2-10080*b^2*c^3*d*e^4*g*x^2-1680*b^2*c^3*e^5*f*x^2+33180*b*c^4*d^2*e^3*g*x^2+14280*b*c^
4*d*e^4*f*x^2-44240*c^5*d^3*e^2*g*x^2-42630*c^5*d^2*e^3*f*x^2-640*b^4*c*e^5*g*x+6400*b^3*c^2*d*e^4*g*x+960*b^3
*c^2*e^5*f*x-24720*b^2*c^3*d^2*e^3*g*x-9120*b^2*c^3*d*e^4*f*x+44240*b*c^4*d^3*e^2*g*x+32520*b*c^4*d^2*e^3*f*x-
31715*c^5*d^4*e*g*x-50100*c^5*d^3*e^2*f*x+256*b^5*e^5*g-2816*b^4*c*d*e^4*g-384*b^4*c*e^5*f+12448*b^3*c^2*d^2*e
^3*g+4032*b^3*c^2*d*e^4*f-27584*b^2*c^3*d^3*e^2*g-16656*b^2*c^3*d^2*e^3*f+30382*b*c^4*d^4*e*g+33048*b*c^4*d^3*
e^2*f-12686*c^5*d^5*g-29049*c^5*d^4*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c^6/e^2/(e*x+d)^(3/2)

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Maxima [B]  time = 1.812, size = 1181, normalized size = 2.82 \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)^(5/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="maxima")

[Out]

-2/15015*(1155*c^6*e^6*x^6 + 9683*c^6*d^6 - 30382*b*c^5*d^5*e + 37267*b^2*c^4*d^4*e^2 - 23464*b^3*c^3*d^3*e^3
+ 8368*b^4*c^2*d^2*e^4 - 1600*b^5*c*d*e^5 + 128*b^6*e^6 + 210*(19*c^6*d*e^5 + 7*b*c^5*e^6)*x^5 + 35*(79*c^6*d^
2*e^4 + 206*b*c^5*d*e^5 + b^2*c^4*e^6)*x^4 - 20*(271*c^6*d^3*e^3 - 683*b*c^5*d^2*e^4 - 19*b^2*c^4*d*e^5 + 2*b^
3*c^3*e^6)*x^3 - 3*(3169*c^6*d^4*e^2 - 3628*b*c^5*d^3*e^3 - 694*b^2*c^4*d^2*e^4 + 168*b^3*c^3*d*e^5 - 16*b^4*c
^2*e^6)*x^2 - 2*(1333*c^6*d^5*e + 1421*b*c^5*d^4*e^2 - 4142*b^2*c^4*d^3*e^3 + 1724*b^3*c^3*d^2*e^4 - 368*b^4*c
^2*d*e^5 + 32*b^5*c*e^6)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*f/(c^5*e^2*x + c^5*d*e) - 2/45045*(3003*c^7*e^7
*x^7 + 12686*c^7*d^7 - 55754*b*c^6*d^6*e + 101034*b^2*c^5*d^5*e^2 - 97998*b^3*c^4*d^4*e^3 + 55296*b^4*c^3*d^3*
e^4 - 18336*b^5*c^2*d^2*e^5 + 3328*b^6*c*d*e^6 - 256*b^7*e^7 + 924*(11*c^7*d*e^6 + 4*b*c^6*e^7)*x^6 + 63*(111*
c^7*d^2*e^5 + 278*b*c^6*d*e^6 + b^2*c^5*e^7)*x^5 - 70*(175*c^7*d^3*e^4 - 453*b*c^6*d^2*e^5 - 9*b^2*c^5*d*e^6 +
b^3*c^4*e^7)*x^4 - 5*(4087*c^7*d^4*e^3 - 4900*b*c^6*d^3*e^4 - 618*b^2*c^5*d^2*e^5 + 160*b^3*c^4*d*e^6 - 16*b^
4*c^3*e^7)*x^3 - 12*(542*c^7*d^5*e^2 + 11*b*c^6*d^4*e^3 - 862*b^2*c^5*d^3*e^4 + 389*b^3*c^4*d^2*e^5 - 88*b^4*c
^3*d*e^6 + 8*b^5*c^2*e^7)*x^2 + (6343*c^7*d^6*e - 21534*b*c^6*d^5*e^2 + 28983*b^2*c^5*d^4*e^3 - 20016*b^3*c^4*
d^3*e^4 + 7632*b^4*c^3*d^2*e^5 - 1536*b^5*c^2*d*e^6 + 128*b^6*c*e^7)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*g/(
c^6*e^3*x + c^6*d*e^2)

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Fricas [B]  time = 1.75167, size = 1948, normalized size = 4.65 \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)^(5/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

-2/45045*(3003*c^7*e^7*g*x^7 + 231*(15*c^7*e^7*f + 4*(11*c^7*d*e^6 + 4*b*c^6*e^7)*g)*x^6 + 63*(10*(19*c^7*d*e^
6 + 7*b*c^6*e^7)*f + (111*c^7*d^2*e^5 + 278*b*c^6*d*e^6 + b^2*c^5*e^7)*g)*x^5 + 35*(3*(79*c^7*d^2*e^5 + 206*b*
c^6*d*e^6 + b^2*c^5*e^7)*f - 2*(175*c^7*d^3*e^4 - 453*b*c^6*d^2*e^5 - 9*b^2*c^5*d*e^6 + b^3*c^4*e^7)*g)*x^4 -
5*(12*(271*c^7*d^3*e^4 - 683*b*c^6*d^2*e^5 - 19*b^2*c^5*d*e^6 + 2*b^3*c^4*e^7)*f + (4087*c^7*d^4*e^3 - 4900*b*
c^6*d^3*e^4 - 618*b^2*c^5*d^2*e^5 + 160*b^3*c^4*d*e^6 - 16*b^4*c^3*e^7)*g)*x^3 - 3*(3*(3169*c^7*d^4*e^3 - 3628
*b*c^6*d^3*e^4 - 694*b^2*c^5*d^2*e^5 + 168*b^3*c^4*d*e^6 - 16*b^4*c^3*e^7)*f + 4*(542*c^7*d^5*e^2 + 11*b*c^6*d
^4*e^3 - 862*b^2*c^5*d^3*e^4 + 389*b^3*c^4*d^2*e^5 - 88*b^4*c^3*d*e^6 + 8*b^5*c^2*e^7)*g)*x^2 + 3*(9683*c^7*d^
6*e - 30382*b*c^6*d^5*e^2 + 37267*b^2*c^5*d^4*e^3 - 23464*b^3*c^4*d^3*e^4 + 8368*b^4*c^3*d^2*e^5 - 1600*b^5*c^
2*d*e^6 + 128*b^6*c*e^7)*f + 2*(6343*c^7*d^7 - 27877*b*c^6*d^6*e + 50517*b^2*c^5*d^5*e^2 - 48999*b^3*c^4*d^4*e
^3 + 27648*b^4*c^3*d^3*e^4 - 9168*b^5*c^2*d^2*e^5 + 1664*b^6*c*d*e^6 - 128*b^7*e^7)*g - (6*(1333*c^7*d^5*e^2 +
1421*b*c^6*d^4*e^3 - 4142*b^2*c^5*d^3*e^4 + 1724*b^3*c^4*d^2*e^5 - 368*b^4*c^3*d*e^6 + 32*b^5*c^2*e^7)*f - (6
343*c^7*d^6*e - 21534*b*c^6*d^5*e^2 + 28983*b^2*c^5*d^4*e^3 - 20016*b^3*c^4*d^3*e^4 + 7632*b^4*c^3*d^2*e^5 - 1
536*b^5*c^2*d*e^6 + 128*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^6*e^3*x +
c^6*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)**(5/2)*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),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)^(5/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="giac")

[Out]

Timed out