3.2251 \(\int (d+e x)^{3/2} (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=424 \[ -\frac{256 (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-10 b e g+3 c d g+17 c e f)}{765765 c^6 e^2 (d+e x)^{7/2}}-\frac{128 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-10 b e g+3 c d g+17 c e f)}{109395 c^5 e^2 (d+e x)^{5/2}}-\frac{32 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-10 b e g+3 c d g+17 c e f)}{12155 c^4 e^2 (d+e x)^{3/2}}-\frac{16 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-10 b e g+3 c d g+17 c e f)}{3315 c^3 e^2 \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-10 b e g+3 c d g+17 c e f)}{255 c^2 e^2}-\frac{2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e^2} \]

[Out]

(-256*(2*c*d - b*e)^4*(17*c*e*f + 3*c*d*g - 10*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(765765*c^6
*e^2*(d + e*x)^(7/2)) - (128*(2*c*d - b*e)^3*(17*c*e*f + 3*c*d*g - 10*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*
x^2)^(7/2))/(109395*c^5*e^2*(d + e*x)^(5/2)) - (32*(2*c*d - b*e)^2*(17*c*e*f + 3*c*d*g - 10*b*e*g)*(d*(c*d - b
*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(12155*c^4*e^2*(d + e*x)^(3/2)) - (16*(2*c*d - b*e)*(17*c*e*f + 3*c*d*g - 10
*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(3315*c^3*e^2*Sqrt[d + e*x]) - (2*(17*c*e*f + 3*c*d*g - 1
0*b*e*g)*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(255*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)^(7/2))/(17*c*e^2)

________________________________________________________________________________________

Rubi [A]  time = 0.76409, antiderivative size = 424, 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} \[ -\frac{256 (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-10 b e g+3 c d g+17 c e f)}{765765 c^6 e^2 (d+e x)^{7/2}}-\frac{128 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-10 b e g+3 c d g+17 c e f)}{109395 c^5 e^2 (d+e x)^{5/2}}-\frac{32 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-10 b e g+3 c d g+17 c e f)}{12155 c^4 e^2 (d+e x)^{3/2}}-\frac{16 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-10 b e g+3 c d g+17 c e f)}{3315 c^3 e^2 \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-10 b e g+3 c d g+17 c e f)}{255 c^2 e^2}-\frac{2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e^2} \]

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)^(5/2),x]

[Out]

(-256*(2*c*d - b*e)^4*(17*c*e*f + 3*c*d*g - 10*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(765765*c^6
*e^2*(d + e*x)^(7/2)) - (128*(2*c*d - b*e)^3*(17*c*e*f + 3*c*d*g - 10*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*
x^2)^(7/2))/(109395*c^5*e^2*(d + e*x)^(5/2)) - (32*(2*c*d - b*e)^2*(17*c*e*f + 3*c*d*g - 10*b*e*g)*(d*(c*d - b
*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(12155*c^4*e^2*(d + e*x)^(3/2)) - (16*(2*c*d - b*e)*(17*c*e*f + 3*c*d*g - 10
*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(3315*c^3*e^2*Sqrt[d + e*x]) - (2*(17*c*e*f + 3*c*d*g - 1
0*b*e*g)*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(255*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)^(7/2))/(17*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 )^{5/2} \, 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 )^{7/2}}{17 c e^2}-\frac{\left (2 \left (\frac{7}{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 )^{5/2} \, dx}{17 c e^3}\\ &=-\frac{2 (17 c e f+3 c d g-10 b e g) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{255 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 )^{7/2}}{17 c e^2}+\frac{(8 (2 c d-b e) (17 c e f+3 c d g-10 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 )^{5/2} \, dx}{255 c^2 e}\\ &=-\frac{16 (2 c d-b e) (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3315 c^3 e^2 \sqrt{d+e x}}-\frac{2 (17 c e f+3 c d g-10 b e g) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{255 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 )^{7/2}}{17 c e^2}+\frac{\left (16 (2 c d-b e)^2 (17 c e f+3 c d g-10 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}}{\sqrt{d+e x}} \, dx}{1105 c^3 e}\\ &=-\frac{32 (2 c d-b e)^2 (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{12155 c^4 e^2 (d+e x)^{3/2}}-\frac{16 (2 c d-b e) (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3315 c^3 e^2 \sqrt{d+e x}}-\frac{2 (17 c e f+3 c d g-10 b e g) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{255 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 )^{7/2}}{17 c e^2}+\frac{\left (64 (2 c d-b e)^3 (17 c e f+3 c d g-10 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}{12155 c^4 e}\\ &=-\frac{128 (2 c d-b e)^3 (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{109395 c^5 e^2 (d+e x)^{5/2}}-\frac{32 (2 c d-b e)^2 (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{12155 c^4 e^2 (d+e x)^{3/2}}-\frac{16 (2 c d-b e) (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3315 c^3 e^2 \sqrt{d+e x}}-\frac{2 (17 c e f+3 c d g-10 b e g) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{255 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 )^{7/2}}{17 c e^2}+\frac{\left (128 (2 c d-b e)^4 (17 c e f+3 c d g-10 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}{109395 c^5 e}\\ &=-\frac{256 (2 c d-b e)^4 (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{765765 c^6 e^2 (d+e x)^{7/2}}-\frac{128 (2 c d-b e)^3 (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{109395 c^5 e^2 (d+e x)^{5/2}}-\frac{32 (2 c d-b e)^2 (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{12155 c^4 e^2 (d+e x)^{3/2}}-\frac{16 (2 c d-b e) (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3315 c^3 e^2 \sqrt{d+e x}}-\frac{2 (17 c e f+3 c d g-10 b e g) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{255 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 )^{7/2}}{17 c e^2}\\ \end{align*}

Mathematica [A]  time = 0.424684, size = 367, normalized size = 0.87 \[ \frac{2 (b e-c d+c e x)^3 \sqrt{(d+e x) (c (d-e x)-b e)} \left (16 b^2 c^3 e^2 \left (3 d^2 e (2397 f+4249 g x)+10864 d^3 g+294 d e^2 x (17 f+21 g x)+21 e^3 x^2 (51 f+55 g x)\right )-32 b^3 c^2 e^3 \left (2253 d^2 g+2 d e (391 f+756 g x)+7 e^2 x (34 f+45 g x)\right )+128 b^4 c e^4 (118 d g+17 e f+35 e g x)-1280 b^5 e^5 g-2 b c^4 e \left (42 d^2 e^2 x (3842 f+4287 g x)+4 d^3 e (32623 f+50554 g x)+104843 d^4 g+84 d e^3 x^2 (969 f+968 g x)+231 e^4 x^3 (68 f+65 g x)\right )+c^5 \left (126 d^2 e^3 x^2 (4471 f+3949 g x)+28 d^3 e^2 x (21097 f+19638 g x)+d^4 e (278171 f+329469 g x)+94134 d^5 g+462 d e^4 x^3 (578 f+507 g x)+3003 e^5 x^4 (17 f+15 g x)\right )\right )}{765765 c^6 e^2 \sqrt{d+e x}} \]

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)^(5/2),x]

[Out]

(2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-1280*b^5*e^5*g + 128*b^4*c*e^4*(17*e*f +
118*d*g + 35*e*g*x) - 32*b^3*c^2*e^3*(2253*d^2*g + 7*e^2*x*(34*f + 45*g*x) + 2*d*e*(391*f + 756*g*x)) + 16*b^2
*c^3*e^2*(10864*d^3*g + 294*d*e^2*x*(17*f + 21*g*x) + 21*e^3*x^2*(51*f + 55*g*x) + 3*d^2*e*(2397*f + 4249*g*x)
) - 2*b*c^4*e*(104843*d^4*g + 231*e^4*x^3*(68*f + 65*g*x) + 84*d*e^3*x^2*(969*f + 968*g*x) + 42*d^2*e^2*x*(384
2*f + 4287*g*x) + 4*d^3*e*(32623*f + 50554*g*x)) + c^5*(94134*d^5*g + 3003*e^5*x^4*(17*f + 15*g*x) + 462*d*e^4
*x^3*(578*f + 507*g*x) + 126*d^2*e^3*x^2*(4471*f + 3949*g*x) + 28*d^3*e^2*x*(21097*f + 19638*g*x) + d^4*e*(278
171*f + 329469*g*x))))/(765765*c^6*e^2*Sqrt[d + e*x])

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Maple [A]  time = 0.007, size = 535, normalized size = 1.3 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -45045\,g{e}^{5}{x}^{5}{c}^{5}+30030\,b{c}^{4}{e}^{5}g{x}^{4}-234234\,{c}^{5}d{e}^{4}g{x}^{4}-51051\,{c}^{5}{e}^{5}f{x}^{4}-18480\,{b}^{2}{c}^{3}{e}^{5}g{x}^{3}+162624\,b{c}^{4}d{e}^{4}g{x}^{3}+31416\,b{c}^{4}{e}^{5}f{x}^{3}-497574\,{c}^{5}{d}^{2}{e}^{3}g{x}^{3}-267036\,{c}^{5}d{e}^{4}f{x}^{3}+10080\,{b}^{3}{c}^{2}{e}^{5}g{x}^{2}-98784\,{b}^{2}{c}^{3}d{e}^{4}g{x}^{2}-17136\,{b}^{2}{c}^{3}{e}^{5}f{x}^{2}+360108\,b{c}^{4}{d}^{2}{e}^{3}g{x}^{2}+162792\,b{c}^{4}d{e}^{4}f{x}^{2}-549864\,{c}^{5}{d}^{3}{e}^{2}g{x}^{2}-563346\,{c}^{5}{d}^{2}{e}^{3}f{x}^{2}-4480\,{b}^{4}c{e}^{5}gx+48384\,{b}^{3}{c}^{2}d{e}^{4}gx+7616\,{b}^{3}{c}^{2}{e}^{5}fx-203952\,{b}^{2}{c}^{3}{d}^{2}{e}^{3}gx-79968\,{b}^{2}{c}^{3}d{e}^{4}fx+404432\,b{c}^{4}{d}^{3}{e}^{2}gx+322728\,b{c}^{4}{d}^{2}{e}^{3}fx-329469\,{c}^{5}{d}^{4}egx-590716\,{c}^{5}{d}^{3}{e}^{2}fx+1280\,{b}^{5}{e}^{5}g-15104\,{b}^{4}cd{e}^{4}g-2176\,{b}^{4}c{e}^{5}f+72096\,{b}^{3}{c}^{2}{d}^{2}{e}^{3}g+25024\,{b}^{3}{c}^{2}d{e}^{4}f-173824\,{b}^{2}{c}^{3}{d}^{3}{e}^{2}g-115056\,{b}^{2}{c}^{3}{d}^{2}{e}^{3}f+209686\,b{c}^{4}{d}^{4}eg+260984\,b{c}^{4}{d}^{3}{e}^{2}f-94134\,{c}^{5}{d}^{5}g-278171\,f{d}^{4}{c}^{5}e \right ) }{765765\,{c}^{6}{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)^(3/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

[Out]

-2/765765*(c*e*x+b*e-c*d)*(-45045*c^5*e^5*g*x^5+30030*b*c^4*e^5*g*x^4-234234*c^5*d*e^4*g*x^4-51051*c^5*e^5*f*x
^4-18480*b^2*c^3*e^5*g*x^3+162624*b*c^4*d*e^4*g*x^3+31416*b*c^4*e^5*f*x^3-497574*c^5*d^2*e^3*g*x^3-267036*c^5*
d*e^4*f*x^3+10080*b^3*c^2*e^5*g*x^2-98784*b^2*c^3*d*e^4*g*x^2-17136*b^2*c^3*e^5*f*x^2+360108*b*c^4*d^2*e^3*g*x
^2+162792*b*c^4*d*e^4*f*x^2-549864*c^5*d^3*e^2*g*x^2-563346*c^5*d^2*e^3*f*x^2-4480*b^4*c*e^5*g*x+48384*b^3*c^2
*d*e^4*g*x+7616*b^3*c^2*e^5*f*x-203952*b^2*c^3*d^2*e^3*g*x-79968*b^2*c^3*d*e^4*f*x+404432*b*c^4*d^3*e^2*g*x+32
2728*b*c^4*d^2*e^3*f*x-329469*c^5*d^4*e*g*x-590716*c^5*d^3*e^2*f*x+1280*b^5*e^5*g-15104*b^4*c*d*e^4*g-2176*b^4
*c*e^5*f+72096*b^3*c^2*d^2*e^3*g+25024*b^3*c^2*d*e^4*f-173824*b^2*c^3*d^3*e^2*g-115056*b^2*c^3*d^2*e^3*f+20968
6*b*c^4*d^4*e*g+260984*b*c^4*d^3*e^2*f-94134*c^5*d^5*g-278171*c^5*d^4*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5
/2)/c^6/e^2/(e*x+d)^(5/2)

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Maxima [B]  time = 1.47327, size = 1496, normalized size = 3.53 \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)^(3/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="maxima")

[Out]

2/45045*(3003*c^7*e^7*x^7 - 16363*c^7*d^7 + 64441*b*c^6*d^6*e - 101913*b^2*c^5*d^5*e^2 + 84195*b^3*c^4*d^4*e^3
- 40200*b^4*c^3*d^3*e^4 + 11568*b^5*c^2*d^2*e^5 - 1856*b^6*c*d*e^6 + 128*b^7*e^7 + 231*(29*c^7*d*e^6 + 31*b*c
^6*e^7)*x^6 - 63*(79*c^7*d^2*e^5 - 398*b*c^6*d*e^6 - 71*b^2*c^5*e^7)*x^5 - 35*(587*c^7*d^3*e^4 - 525*b*c^6*d^2
*e^5 - 633*b^2*c^5*d*e^6 - b^3*c^4*e^7)*x^4 - 5*(835*c^7*d^4*e^3 + 6548*b*c^6*d^3*e^4 - 8586*b^2*c^5*d^2*e^5 -
92*b^3*c^4*d*e^6 + 8*b^4*c^3*e^7)*x^3 + 3*(7339*c^7*d^5*e^2 - 20435*b*c^6*d^4*e^3 + 12250*b^2*c^5*d^3*e^4 + 1
030*b^3*c^4*d^2*e^5 - 200*b^4*c^3*d*e^6 + 16*b^5*c^2*e^7)*x^2 + (14341*c^7*d^6*e - 21006*b*c^6*d^5*e^2 - 4395*
b^2*c^5*d^4*e^3 + 15180*b^3*c^4*d^3*e^4 - 4920*b^4*c^3*d^2*e^5 + 864*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)*f/(c^5*e^2*x + c^5*d*e) + 2/765765*(45045*c^8*e^8*x^8 - 94134*c^8*d^8 + 492088*b*c^
7*d^7*e - 1085284*b^2*c^6*d^6*e^2 + 1316760*b^3*c^5*d^5*e^3 - 962550*b^4*c^4*d^4*e^4 + 436704*b^5*c^3*d^3*e^5
- 121248*b^6*c^2*d^2*e^6 + 18944*b^7*c*d*e^7 - 1280*b^8*e^8 + 3003*(33*c^8*d*e^7 + 35*b*c^7*e^8)*x^7 - 231*(30
3*c^8*d^2*e^6 - 1558*b*c^7*d*e^7 - 275*b^2*c^6*e^8)*x^6 - 63*(4527*c^8*d^3*e^5 - 4129*b*c^7*d^2*e^6 - 4813*b^2
*c^6*d*e^7 - 5*b^3*c^5*e^8)*x^5 - 35*(1761*c^8*d^4*e^4 + 11860*b*c^7*d^3*e^5 - 15954*b^2*c^6*d^2*e^6 - 108*b^3
*c^5*d*e^7 + 10*b^4*c^4*e^8)*x^4 + 5*(51549*c^8*d^5*e^3 - 146429*b*c^7*d^4*e^4 + 91238*b^2*c^6*d^3*e^5 + 4506*
b^3*c^5*d^2*e^6 - 944*b^4*c^4*d*e^7 + 80*b^5*c^3*e^8)*x^3 + 3*(52047*c^8*d^6*e^2 - 89650*b*c^7*d^5*e^3 + 15875
*b^2*c^6*d^4*e^4 + 30740*b^3*c^5*d^3*e^5 - 10900*b^4*c^4*d^2*e^6 + 2048*b^5*c^3*d*e^7 - 160*b^6*c^2*e^8)*x^2 -
(47067*c^8*d^7*e - 198977*b*c^7*d^6*e^2 + 343665*b^2*c^6*d^5*e^3 - 314715*b^3*c^5*d^4*e^4 + 166560*b^4*c^4*d^
3*e^5 - 51792*b^5*c^3*d^2*e^6 + 8832*b^6*c^2*d*e^7 - 640*b^7*c*e^8)*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 = 1.55421, size = 2543, normalized size = 6. \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)^(3/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")

[Out]

2/765765*(45045*c^8*e^8*g*x^8 + 3003*(17*c^8*e^8*f + (33*c^8*d*e^7 + 35*b*c^7*e^8)*g)*x^7 + 231*(17*(29*c^8*d*
e^7 + 31*b*c^7*e^8)*f - (303*c^8*d^2*e^6 - 1558*b*c^7*d*e^7 - 275*b^2*c^6*e^8)*g)*x^6 - 63*(17*(79*c^8*d^2*e^6
- 398*b*c^7*d*e^7 - 71*b^2*c^6*e^8)*f + (4527*c^8*d^3*e^5 - 4129*b*c^7*d^2*e^6 - 4813*b^2*c^6*d*e^7 - 5*b^3*c
^5*e^8)*g)*x^5 - 35*(17*(587*c^8*d^3*e^5 - 525*b*c^7*d^2*e^6 - 633*b^2*c^6*d*e^7 - b^3*c^5*e^8)*f + (1761*c^8*
d^4*e^4 + 11860*b*c^7*d^3*e^5 - 15954*b^2*c^6*d^2*e^6 - 108*b^3*c^5*d*e^7 + 10*b^4*c^4*e^8)*g)*x^4 - 5*(17*(83
5*c^8*d^4*e^4 + 6548*b*c^7*d^3*e^5 - 8586*b^2*c^6*d^2*e^6 - 92*b^3*c^5*d*e^7 + 8*b^4*c^4*e^8)*f - (51549*c^8*d
^5*e^3 - 146429*b*c^7*d^4*e^4 + 91238*b^2*c^6*d^3*e^5 + 4506*b^3*c^5*d^2*e^6 - 944*b^4*c^4*d*e^7 + 80*b^5*c^3*
e^8)*g)*x^3 + 3*(17*(7339*c^8*d^5*e^3 - 20435*b*c^7*d^4*e^4 + 12250*b^2*c^6*d^3*e^5 + 1030*b^3*c^5*d^2*e^6 - 2
00*b^4*c^4*d*e^7 + 16*b^5*c^3*e^8)*f + (52047*c^8*d^6*e^2 - 89650*b*c^7*d^5*e^3 + 15875*b^2*c^6*d^4*e^4 + 3074
0*b^3*c^5*d^3*e^5 - 10900*b^4*c^4*d^2*e^6 + 2048*b^5*c^3*d*e^7 - 160*b^6*c^2*e^8)*g)*x^2 - 17*(16363*c^8*d^7*e
- 64441*b*c^7*d^6*e^2 + 101913*b^2*c^6*d^5*e^3 - 84195*b^3*c^5*d^4*e^4 + 40200*b^4*c^4*d^3*e^5 - 11568*b^5*c^
3*d^2*e^6 + 1856*b^6*c^2*d*e^7 - 128*b^7*c*e^8)*f - 2*(47067*c^8*d^8 - 246044*b*c^7*d^7*e + 542642*b^2*c^6*d^6
*e^2 - 658380*b^3*c^5*d^5*e^3 + 481275*b^4*c^4*d^4*e^4 - 218352*b^5*c^3*d^3*e^5 + 60624*b^6*c^2*d^2*e^6 - 9472
*b^7*c*d*e^7 + 640*b^8*e^8)*g + (17*(14341*c^8*d^6*e^2 - 21006*b*c^7*d^5*e^3 - 4395*b^2*c^6*d^4*e^4 + 15180*b^
3*c^5*d^3*e^5 - 4920*b^4*c^4*d^2*e^6 + 864*b^5*c^3*d*e^7 - 64*b^6*c^2*e^8)*f - (47067*c^8*d^7*e - 198977*b*c^7
*d^6*e^2 + 343665*b^2*c^6*d^5*e^3 - 314715*b^3*c^5*d^4*e^4 + 166560*b^4*c^4*d^3*e^5 - 51792*b^5*c^3*d^2*e^6 +
8832*b^6*c^2*d*e^7 - 640*b^7*c*e^8)*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)**(3/2)*(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)^(3/2)*(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