3.2241 \(\int (d+e x)^{3/2} (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=347 \[ -\frac{32 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{15015 c^5 e^2 (d+e x)^{5/2}}-\frac{16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{3003 c^4 e^2 (d+e x)^{3/2}}-\frac{4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{429 c^3 e^2 \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{143 c^2 e^2}-\frac{2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2} \]

[Out]

(-32*(2*c*d - b*e)^3*(13*c*e*f + 3*c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(15015*c^5*e^
2*(d + e*x)^(5/2)) - (16*(2*c*d - b*e)^2*(13*c*e*f + 3*c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^
(5/2))/(3003*c^4*e^2*(d + e*x)^(3/2)) - (4*(2*c*d - b*e)*(13*c*e*f + 3*c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2
*x - c*e^2*x^2)^(5/2))/(429*c^3*e^2*Sqrt[d + e*x]) - (2*(13*c*e*f + 3*c*d*g - 8*b*e*g)*Sqrt[d + e*x]*(d*(c*d -
b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(143*c^2*e^2) - (2*g*(d + e*x)^(3/2)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(5/2))/(13*c*e^2)

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Rubi [A]  time = 0.600538, antiderivative size = 347, 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} \[ -\frac{32 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{15015 c^5 e^2 (d+e x)^{5/2}}-\frac{16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{3003 c^4 e^2 (d+e x)^{3/2}}-\frac{4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{429 c^3 e^2 \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{143 c^2 e^2}-\frac{2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-32*(2*c*d - b*e)^3*(13*c*e*f + 3*c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(15015*c^5*e^
2*(d + e*x)^(5/2)) - (16*(2*c*d - b*e)^2*(13*c*e*f + 3*c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^
(5/2))/(3003*c^4*e^2*(d + e*x)^(3/2)) - (4*(2*c*d - b*e)*(13*c*e*f + 3*c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2
*x - c*e^2*x^2)^(5/2))/(429*c^3*e^2*Sqrt[d + e*x]) - (2*(13*c*e*f + 3*c*d*g - 8*b*e*g)*Sqrt[d + e*x]*(d*(c*d -
b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(143*c^2*e^2) - (2*g*(d + e*x)^(3/2)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(5/2))/(13*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)^{3/2} (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)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2}-\frac{\left (2 \left (\frac{5}{2} e \left (-2 c e^2 f+b e^2 g\right )+\frac{3}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \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}{13 c e^3}\\ &=-\frac{2 (13 c e f+3 c d g-8 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}}{143 c^2 e^2}-\frac{2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2}+\frac{(6 (2 c d-b e) (13 c e f+3 c d g-8 b e g)) \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^2 e}\\ &=-\frac{4 (2 c d-b e) (13 c e f+3 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{429 c^3 e^2 \sqrt{d+e x}}-\frac{2 (13 c e f+3 c d g-8 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}}{143 c^2 e^2}-\frac{2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2}+\frac{\left (8 (2 c d-b e)^2 (13 c e f+3 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 )^{3/2}}{\sqrt{d+e x}} \, dx}{429 c^3 e}\\ &=-\frac{16 (2 c d-b e)^2 (13 c e f+3 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3003 c^4 e^2 (d+e x)^{3/2}}-\frac{4 (2 c d-b e) (13 c e f+3 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{429 c^3 e^2 \sqrt{d+e x}}-\frac{2 (13 c e f+3 c d g-8 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}}{143 c^2 e^2}-\frac{2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2}+\frac{\left (16 (2 c d-b e)^3 (13 c e f+3 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 )^{3/2}}{(d+e x)^{3/2}} \, dx}{3003 c^4 e}\\ &=-\frac{32 (2 c d-b e)^3 (13 c e f+3 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15015 c^5 e^2 (d+e x)^{5/2}}-\frac{16 (2 c d-b e)^2 (13 c e f+3 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3003 c^4 e^2 (d+e x)^{3/2}}-\frac{4 (2 c d-b e) (13 c e f+3 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{429 c^3 e^2 \sqrt{d+e x}}-\frac{2 (13 c e f+3 c d g-8 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}}{143 c^2 e^2}-\frac{2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2}\\ \end{align*}

Mathematica [A]  time = 0.287462, size = 264, normalized size = 0.76 \[ -\frac{2 (b e-c d+c e x)^2 \sqrt{(d+e x) (c (d-e x)-b e)} \left (8 b^2 c^2 e^2 \left (473 d^2 g+d e (221 f+315 g x)+5 e^2 x (13 f+14 g x)\right )-16 b^3 c e^3 (71 d g+13 e f+20 e g x)+128 b^4 e^4 g-2 b c^3 e \left (d^2 e (2743 f+3470 g x)+2765 d^3 g+25 d e^2 x (78 f+77 g x)+35 e^3 x^2 (13 f+12 g x)\right )+c^4 \left (5 d^2 e^2 x (1963 f+1659 g x)+d^3 e (6929 f+6885 g x)+2754 d^4 g+35 d e^3 x^2 (169 f+141 g x)+105 e^4 x^3 (13 f+11 g x)\right )\right )}{15015 c^5 e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(3/2)*(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))]*(128*b^4*e^4*g - 16*b^3*c*e^3*(13*e*f + 71
*d*g + 20*e*g*x) + 8*b^2*c^2*e^2*(473*d^2*g + 5*e^2*x*(13*f + 14*g*x) + d*e*(221*f + 315*g*x)) - 2*b*c^3*e*(27
65*d^3*g + 35*e^3*x^2*(13*f + 12*g*x) + 25*d*e^2*x*(78*f + 77*g*x) + d^2*e*(2743*f + 3470*g*x)) + c^4*(2754*d^
4*g + 105*e^4*x^3*(13*f + 11*g*x) + 35*d*e^3*x^2*(169*f + 141*g*x) + 5*d^2*e^2*x*(1963*f + 1659*g*x) + d^3*e*(
6929*f + 6885*g*x))))/(15015*c^5*e^2*Sqrt[d + e*x])

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Maple [A]  time = 0.007, size = 367, normalized size = 1.1 \begin{align*}{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 1155\,g{e}^{4}{x}^{4}{c}^{4}-840\,b{c}^{3}{e}^{4}g{x}^{3}+4935\,{c}^{4}d{e}^{3}g{x}^{3}+1365\,{c}^{4}{e}^{4}f{x}^{3}+560\,{b}^{2}{c}^{2}{e}^{4}g{x}^{2}-3850\,b{c}^{3}d{e}^{3}g{x}^{2}-910\,b{c}^{3}{e}^{4}f{x}^{2}+8295\,{c}^{4}{d}^{2}{e}^{2}g{x}^{2}+5915\,{c}^{4}d{e}^{3}f{x}^{2}-320\,{b}^{3}c{e}^{4}gx+2520\,{b}^{2}{c}^{2}d{e}^{3}gx+520\,{b}^{2}{c}^{2}{e}^{4}fx-6940\,b{c}^{3}{d}^{2}{e}^{2}gx-3900\,b{c}^{3}d{e}^{3}fx+6885\,{c}^{4}{d}^{3}egx+9815\,{c}^{4}{d}^{2}{e}^{2}fx+128\,{b}^{4}{e}^{4}g-1136\,{b}^{3}cd{e}^{3}g-208\,{b}^{3}c{e}^{4}f+3784\,{b}^{2}{c}^{2}{d}^{2}{e}^{2}g+1768\,{b}^{2}{c}^{2}d{e}^{3}f-5530\,b{c}^{3}{d}^{3}eg-5486\,b{c}^{3}{d}^{2}{e}^{2}f+2754\,{c}^{4}{d}^{4}g+6929\,f{d}^{3}{c}^{4}e \right ) }{15015\,{c}^{5}{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)^(3/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)

[Out]

2/15015*(c*e*x+b*e-c*d)*(1155*c^4*e^4*g*x^4-840*b*c^3*e^4*g*x^3+4935*c^4*d*e^3*g*x^3+1365*c^4*e^4*f*x^3+560*b^
2*c^2*e^4*g*x^2-3850*b*c^3*d*e^3*g*x^2-910*b*c^3*e^4*f*x^2+8295*c^4*d^2*e^2*g*x^2+5915*c^4*d*e^3*f*x^2-320*b^3
*c*e^4*g*x+2520*b^2*c^2*d*e^3*g*x+520*b^2*c^2*e^4*f*x-6940*b*c^3*d^2*e^2*g*x-3900*b*c^3*d*e^3*f*x+6885*c^4*d^3
*e*g*x+9815*c^4*d^2*e^2*f*x+128*b^4*e^4*g-1136*b^3*c*d*e^3*g-208*b^3*c*e^4*f+3784*b^2*c^2*d^2*e^2*g+1768*b^2*c
^2*d*e^3*f-5530*b*c^3*d^3*e*g-5486*b*c^3*d^2*e^2*f+2754*c^4*d^4*g+6929*c^4*d^3*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+
c*d^2)^(3/2)/c^5/e^2/(e*x+d)^(3/2)

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Maxima [B]  time = 1.78618, size = 913, normalized size = 2.63 \begin{align*} -\frac{2 \,{\left (105 \, c^{5} e^{5} x^{5} + 533 \, c^{5} d^{5} - 1488 \, b c^{4} d^{4} e + 1513 \, b^{2} c^{3} d^{3} e^{2} - 710 \, b^{3} c^{2} d^{2} e^{3} + 168 \, b^{4} c d e^{4} - 16 \, b^{5} e^{5} + 35 \,{\left (7 \, c^{5} d e^{4} + 4 \, b c^{4} e^{5}\right )} x^{4} - 5 \,{\left (10 \, c^{5} d^{2} e^{3} - 108 \, b c^{4} d e^{4} - b^{2} c^{3} e^{5}\right )} x^{3} - 3 \,{\left (174 \, c^{5} d^{3} e^{2} - 236 \, b c^{4} d^{2} e^{3} - 17 \, b^{2} c^{3} d e^{4} + 2 \, b^{3} c^{2} e^{5}\right )} x^{2} -{\left (311 \, c^{5} d^{4} e - 100 \, b c^{4} d^{3} e^{2} - 279 \, b^{2} c^{3} d^{2} e^{3} + 76 \, b^{3} c^{2} d e^{4} - 8 \, b^{4} c e^{5}\right )} x\right )} \sqrt{-c e x + c d - b e}{\left (e x + d\right )} f}{1155 \,{\left (c^{4} e^{2} x + c^{4} d e\right )}} - \frac{2 \,{\left (1155 \, c^{6} e^{6} x^{6} + 2754 \, c^{6} d^{6} - 11038 \, b c^{5} d^{5} e + 17598 \, b^{2} c^{4} d^{4} e^{2} - 14234 \, b^{3} c^{3} d^{3} e^{3} + 6184 \, b^{4} c^{2} d^{2} e^{4} - 1392 \, b^{5} c d e^{5} + 128 \, b^{6} e^{6} + 105 \,{\left (25 \, c^{6} d e^{5} + 14 \, b c^{5} e^{6}\right )} x^{5} - 35 \,{\left (12 \, c^{6} d^{2} e^{4} - 154 \, b c^{5} d e^{5} - b^{2} c^{4} e^{6}\right )} x^{4} - 5 \,{\left (954 \, c^{6} d^{3} e^{3} - 1328 \, b c^{5} d^{2} e^{4} - 63 \, b^{2} c^{4} d e^{5} + 8 \, b^{3} c^{3} e^{6}\right )} x^{3} - 3 \,{\left (907 \, c^{6} d^{4} e^{2} - 560 \, b c^{5} d^{3} e^{3} - 473 \, b^{2} c^{4} d^{2} e^{4} + 142 \, b^{3} c^{3} d e^{5} - 16 \, b^{4} c^{2} e^{6}\right )} x^{2} +{\left (1377 \, c^{6} d^{5} e - 4142 \, b c^{5} d^{4} e^{2} + 4657 \, b^{2} c^{4} d^{3} e^{3} - 2460 \, b^{3} c^{3} d^{2} e^{4} + 632 \, b^{4} c^{2} d e^{5} - 64 \, b^{5} c e^{6}\right )} x\right )} \sqrt{-c e x + c d - b e}{\left (e x + d\right )} g}{15015 \,{\left (c^{5} e^{3} x + c^{5} d e^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="maxima")

[Out]

-2/1155*(105*c^5*e^5*x^5 + 533*c^5*d^5 - 1488*b*c^4*d^4*e + 1513*b^2*c^3*d^3*e^2 - 710*b^3*c^2*d^2*e^3 + 168*b
^4*c*d*e^4 - 16*b^5*e^5 + 35*(7*c^5*d*e^4 + 4*b*c^4*e^5)*x^4 - 5*(10*c^5*d^2*e^3 - 108*b*c^4*d*e^4 - b^2*c^3*e
^5)*x^3 - 3*(174*c^5*d^3*e^2 - 236*b*c^4*d^2*e^3 - 17*b^2*c^3*d*e^4 + 2*b^3*c^2*e^5)*x^2 - (311*c^5*d^4*e - 10
0*b*c^4*d^3*e^2 - 279*b^2*c^3*d^2*e^3 + 76*b^3*c^2*d*e^4 - 8*b^4*c*e^5)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*
f/(c^4*e^2*x + c^4*d*e) - 2/15015*(1155*c^6*e^6*x^6 + 2754*c^6*d^6 - 11038*b*c^5*d^5*e + 17598*b^2*c^4*d^4*e^2
- 14234*b^3*c^3*d^3*e^3 + 6184*b^4*c^2*d^2*e^4 - 1392*b^5*c*d*e^5 + 128*b^6*e^6 + 105*(25*c^6*d*e^5 + 14*b*c^
5*e^6)*x^5 - 35*(12*c^6*d^2*e^4 - 154*b*c^5*d*e^5 - b^2*c^4*e^6)*x^4 - 5*(954*c^6*d^3*e^3 - 1328*b*c^5*d^2*e^4
- 63*b^2*c^4*d*e^5 + 8*b^3*c^3*e^6)*x^3 - 3*(907*c^6*d^4*e^2 - 560*b*c^5*d^3*e^3 - 473*b^2*c^4*d^2*e^4 + 142*
b^3*c^3*d*e^5 - 16*b^4*c^2*e^6)*x^2 + (1377*c^6*d^5*e - 4142*b*c^5*d^4*e^2 + 4657*b^2*c^4*d^3*e^3 - 2460*b^3*c
^3*d^2*e^4 + 632*b^4*c^2*d*e^5 - 64*b^5*c*e^6)*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 = 1.44968, size = 1474, normalized size = 4.25 \begin{align*} -\frac{2 \,{\left (1155 \, c^{6} e^{6} g x^{6} + 105 \,{\left (13 \, c^{6} e^{6} f +{\left (25 \, c^{6} d e^{5} + 14 \, b c^{5} e^{6}\right )} g\right )} x^{5} + 35 \,{\left (13 \,{\left (7 \, c^{6} d e^{5} + 4 \, b c^{5} e^{6}\right )} f -{\left (12 \, c^{6} d^{2} e^{4} - 154 \, b c^{5} d e^{5} - b^{2} c^{4} e^{6}\right )} g\right )} x^{4} - 5 \,{\left (13 \,{\left (10 \, c^{6} d^{2} e^{4} - 108 \, b c^{5} d e^{5} - b^{2} c^{4} e^{6}\right )} f +{\left (954 \, c^{6} d^{3} e^{3} - 1328 \, b c^{5} d^{2} e^{4} - 63 \, b^{2} c^{4} d e^{5} + 8 \, b^{3} c^{3} e^{6}\right )} g\right )} x^{3} - 3 \,{\left (13 \,{\left (174 \, c^{6} d^{3} e^{3} - 236 \, b c^{5} d^{2} e^{4} - 17 \, b^{2} c^{4} d e^{5} + 2 \, b^{3} c^{3} e^{6}\right )} f +{\left (907 \, c^{6} d^{4} e^{2} - 560 \, b c^{5} d^{3} e^{3} - 473 \, b^{2} c^{4} d^{2} e^{4} + 142 \, b^{3} c^{3} d e^{5} - 16 \, b^{4} c^{2} e^{6}\right )} g\right )} x^{2} + 13 \,{\left (533 \, c^{6} d^{5} e - 1488 \, b c^{5} d^{4} e^{2} + 1513 \, b^{2} c^{4} d^{3} e^{3} - 710 \, b^{3} c^{3} d^{2} e^{4} + 168 \, b^{4} c^{2} d e^{5} - 16 \, b^{5} c e^{6}\right )} f + 2 \,{\left (1377 \, c^{6} d^{6} - 5519 \, b c^{5} d^{5} e + 8799 \, b^{2} c^{4} d^{4} e^{2} - 7117 \, b^{3} c^{3} d^{3} e^{3} + 3092 \, b^{4} c^{2} d^{2} e^{4} - 696 \, b^{5} c d e^{5} + 64 \, b^{6} e^{6}\right )} g -{\left (13 \,{\left (311 \, c^{6} d^{4} e^{2} - 100 \, b c^{5} d^{3} e^{3} - 279 \, b^{2} c^{4} d^{2} e^{4} + 76 \, b^{3} c^{3} d e^{5} - 8 \, b^{4} c^{2} e^{6}\right )} f -{\left (1377 \, c^{6} d^{5} e - 4142 \, b c^{5} d^{4} e^{2} + 4657 \, b^{2} c^{4} d^{3} e^{3} - 2460 \, b^{3} c^{3} d^{2} e^{4} + 632 \, b^{4} c^{2} d e^{5} - 64 \, b^{5} c e^{6}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d}}{15015 \,{\left (c^{5} e^{3} x + c^{5} d e^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

-2/15015*(1155*c^6*e^6*g*x^6 + 105*(13*c^6*e^6*f + (25*c^6*d*e^5 + 14*b*c^5*e^6)*g)*x^5 + 35*(13*(7*c^6*d*e^5
+ 4*b*c^5*e^6)*f - (12*c^6*d^2*e^4 - 154*b*c^5*d*e^5 - b^2*c^4*e^6)*g)*x^4 - 5*(13*(10*c^6*d^2*e^4 - 108*b*c^5
*d*e^5 - b^2*c^4*e^6)*f + (954*c^6*d^3*e^3 - 1328*b*c^5*d^2*e^4 - 63*b^2*c^4*d*e^5 + 8*b^3*c^3*e^6)*g)*x^3 - 3
*(13*(174*c^6*d^3*e^3 - 236*b*c^5*d^2*e^4 - 17*b^2*c^4*d*e^5 + 2*b^3*c^3*e^6)*f + (907*c^6*d^4*e^2 - 560*b*c^5
*d^3*e^3 - 473*b^2*c^4*d^2*e^4 + 142*b^3*c^3*d*e^5 - 16*b^4*c^2*e^6)*g)*x^2 + 13*(533*c^6*d^5*e - 1488*b*c^5*d
^4*e^2 + 1513*b^2*c^4*d^3*e^3 - 710*b^3*c^3*d^2*e^4 + 168*b^4*c^2*d*e^5 - 16*b^5*c*e^6)*f + 2*(1377*c^6*d^6 -
5519*b*c^5*d^5*e + 8799*b^2*c^4*d^4*e^2 - 7117*b^3*c^3*d^3*e^3 + 3092*b^4*c^2*d^2*e^4 - 696*b^5*c*d*e^5 + 64*b
^6*e^6)*g - (13*(311*c^6*d^4*e^2 - 100*b*c^5*d^3*e^3 - 279*b^2*c^4*d^2*e^4 + 76*b^3*c^3*d*e^5 - 8*b^4*c^2*e^6)
*f - (1377*c^6*d^5*e - 4142*b*c^5*d^4*e^2 + 4657*b^2*c^4*d^3*e^3 - 2460*b^3*c^3*d^2*e^4 + 632*b^4*c^2*d*e^5 -
64*b^5*c*e^6)*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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{3}{2}} \left (f + g x\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)^(3/2)*(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