3.2182 \(\int \frac{(f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=439 \[ -\frac{256 c^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-13 b e g+16 c d g+10 c e f)}{45045 e^2 (d+e x)^3 (2 c d-b e)^6}+\frac{128 c^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (13 b e g-2 c (8 d g+5 e f))}{15015 e^2 (d+e x)^4 (2 c d-b e)^5}-\frac{32 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-13 b e g+16 c d g+10 c e f)}{3003 e^2 (d+e x)^5 (2 c d-b e)^4}+\frac{16 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (13 b e g-2 c (8 d g+5 e f))}{1287 e^2 (d+e x)^6 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-13 b e g+16 c d g+10 c e f)}{143 e^2 (d+e x)^7 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{13 e^2 (d+e x)^8 (2 c d-b e)} \]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(13*e^2*(2*c*d - b*e)*(d + e*x)^8) - (2*(10*c*e*f
+ 16*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(143*e^2*(2*c*d - b*e)^2*(d + e*x)^7) + (
16*c*(13*b*e*g - 2*c*(5*e*f + 8*d*g))*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(1287*e^2*(2*c*d - b*e)^3*(
d + e*x)^6) - (32*c^2*(10*c*e*f + 16*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(3003*e^2*
(2*c*d - b*e)^4*(d + e*x)^5) + (128*c^3*(13*b*e*g - 2*c*(5*e*f + 8*d*g))*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(3/2))/(15015*e^2*(2*c*d - b*e)^5*(d + e*x)^4) - (256*c^4*(10*c*e*f + 16*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b
*e^2*x - c*e^2*x^2)^(3/2))/(45045*e^2*(2*c*d - b*e)^6*(d + e*x)^3)

________________________________________________________________________________________

Rubi [A]  time = 0.718184, antiderivative size = 439, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.068, Rules used = {792, 658, 650} \[ -\frac{256 c^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-13 b e g+16 c d g+10 c e f)}{45045 e^2 (d+e x)^3 (2 c d-b e)^6}+\frac{128 c^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (13 b e g-2 c (8 d g+5 e f))}{15015 e^2 (d+e x)^4 (2 c d-b e)^5}-\frac{32 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-13 b e g+16 c d g+10 c e f)}{3003 e^2 (d+e x)^5 (2 c d-b e)^4}+\frac{16 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (13 b e g-2 c (8 d g+5 e f))}{1287 e^2 (d+e x)^6 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-13 b e g+16 c d g+10 c e f)}{143 e^2 (d+e x)^7 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{13 e^2 (d+e x)^8 (2 c d-b e)} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^8,x]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(13*e^2*(2*c*d - b*e)*(d + e*x)^8) - (2*(10*c*e*f
+ 16*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(143*e^2*(2*c*d - b*e)^2*(d + e*x)^7) + (
16*c*(13*b*e*g - 2*c*(5*e*f + 8*d*g))*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(1287*e^2*(2*c*d - b*e)^3*(
d + e*x)^6) - (32*c^2*(10*c*e*f + 16*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(3003*e^2*
(2*c*d - b*e)^4*(d + e*x)^5) + (128*c^3*(13*b*e*g - 2*c*(5*e*f + 8*d*g))*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(3/2))/(15015*e^2*(2*c*d - b*e)^5*(d + e*x)^4) - (256*c^4*(10*c*e*f + 16*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b
*e^2*x - c*e^2*x^2)^(3/2))/(45045*e^2*(2*c*d - b*e)^6*(d + e*x)^3)

Rule 792

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d*g - e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/((2*c*d - b*e)*(m + p + 1)), x] + Dist[(m*(g*(c*d - b*e)
+ c*e*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(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] && ((L
tQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]

Rule 658

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

Rule 650

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)), 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 + 2*p + 2, 0]

Rubi steps

\begin{align*} \int \frac{(f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^8} \, dx &=-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{13 e^2 (2 c d-b e) (d+e x)^8}+\frac{(10 c e f+16 c d g-13 b e g) \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^7} \, dx}{13 e (2 c d-b e)}\\ &=-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{13 e^2 (2 c d-b e) (d+e x)^8}-\frac{2 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{143 e^2 (2 c d-b e)^2 (d+e x)^7}+\frac{(8 c (10 c e f+16 c d g-13 b e g)) \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^6} \, dx}{143 e (2 c d-b e)^2}\\ &=-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{13 e^2 (2 c d-b e) (d+e x)^8}-\frac{2 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{143 e^2 (2 c d-b e)^2 (d+e x)^7}-\frac{16 c (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{1287 e^2 (2 c d-b e)^3 (d+e x)^6}+\frac{\left (16 c^2 (10 c e f+16 c d g-13 b e g)\right ) \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^5} \, dx}{429 e (2 c d-b e)^3}\\ &=-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{13 e^2 (2 c d-b e) (d+e x)^8}-\frac{2 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{143 e^2 (2 c d-b e)^2 (d+e x)^7}-\frac{16 c (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{1287 e^2 (2 c d-b e)^3 (d+e x)^6}-\frac{32 c^2 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3003 e^2 (2 c d-b e)^4 (d+e x)^5}+\frac{\left (64 c^3 (10 c e f+16 c d g-13 b e g)\right ) \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^4} \, dx}{3003 e (2 c d-b e)^4}\\ &=-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{13 e^2 (2 c d-b e) (d+e x)^8}-\frac{2 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{143 e^2 (2 c d-b e)^2 (d+e x)^7}-\frac{16 c (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{1287 e^2 (2 c d-b e)^3 (d+e x)^6}-\frac{32 c^2 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3003 e^2 (2 c d-b e)^4 (d+e x)^5}-\frac{128 c^3 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{15015 e^2 (2 c d-b e)^5 (d+e x)^4}+\frac{\left (128 c^4 (10 c e f+16 c d g-13 b e g)\right ) \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^3} \, dx}{15015 e (2 c d-b e)^5}\\ &=-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{13 e^2 (2 c d-b e) (d+e x)^8}-\frac{2 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{143 e^2 (2 c d-b e)^2 (d+e x)^7}-\frac{16 c (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{1287 e^2 (2 c d-b e)^3 (d+e x)^6}-\frac{32 c^2 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3003 e^2 (2 c d-b e)^4 (d+e x)^5}-\frac{128 c^3 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{15015 e^2 (2 c d-b e)^5 (d+e x)^4}-\frac{256 c^4 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{45045 e^2 (2 c d-b e)^6 (d+e x)^3}\\ \end{align*}

Mathematica [A]  time = 0.522882, size = 176, normalized size = 0.4 \[ -\frac{2 ((d+e x) (c (d-e x)-b e))^{3/2} \left (2 (d+e x) \left (8 c (d+e x) \left (2 c (d+e x) \left (4 c (d+e x) (-3 b e+8 c d+2 c e x)+15 (b e-2 c d)^2\right )+35 (2 c d-b e)^3\right )+315 (b e-2 c d)^4\right ) \left (c e (8 d g+5 e f)-\frac{13}{2} b e^2 g\right )-3465 e (b e-2 c d)^5 (e f-d g)\right )}{45045 e^3 (d+e x)^8 (b e-2 c d)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^8,x]

[Out]

(-2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*(-3465*e*(-2*c*d + b*e)^5*(e*f - d*g) + 2*((-13*b*e^2*g)/2 + c*e*
(5*e*f + 8*d*g))*(d + e*x)*(315*(-2*c*d + b*e)^4 + 8*c*(d + e*x)*(35*(2*c*d - b*e)^3 + 2*c*(d + e*x)*(15*(-2*c
*d + b*e)^2 + 4*c*(d + e*x)*(8*c*d - 3*b*e + 2*c*e*x))))))/(45045*e^3*(-2*c*d + b*e)^6*(d + e*x)^8)

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Maple [A]  time = 0.016, size = 782, normalized size = 1.8 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 1664\,b{c}^{4}{e}^{6}g{x}^{5}-2048\,{c}^{5}d{e}^{5}g{x}^{5}-1280\,{c}^{5}{e}^{6}f{x}^{5}-2496\,{b}^{2}{c}^{3}{e}^{6}g{x}^{4}+16384\,b{c}^{4}d{e}^{5}g{x}^{4}+1920\,b{c}^{4}{e}^{6}f{x}^{4}-16384\,{c}^{5}{d}^{2}{e}^{4}g{x}^{4}-10240\,{c}^{5}d{e}^{5}f{x}^{4}+3120\,{b}^{3}{c}^{2}{e}^{6}g{x}^{3}-26304\,{b}^{2}{c}^{3}d{e}^{5}g{x}^{3}-2400\,{b}^{2}{c}^{3}{e}^{6}f{x}^{3}+76736\,b{c}^{4}{d}^{2}{e}^{4}g{x}^{3}+17280\,b{c}^{4}d{e}^{5}f{x}^{3}-60416\,{c}^{5}{d}^{3}{e}^{3}g{x}^{3}-37760\,{c}^{5}{d}^{2}{e}^{4}f{x}^{3}-3640\,{b}^{4}c{e}^{6}g{x}^{2}+35680\,{b}^{3}{c}^{2}d{e}^{5}g{x}^{2}+2800\,{b}^{3}{c}^{2}{e}^{6}f{x}^{2}-134496\,{b}^{2}{c}^{3}{d}^{2}{e}^{4}g{x}^{2}-24000\,{b}^{2}{c}^{3}d{e}^{5}f{x}^{2}+231424\,b{c}^{4}{d}^{3}{e}^{3}g{x}^{2}+73920\,b{c}^{4}{d}^{2}{e}^{4}f{x}^{2}-139264\,{c}^{5}{d}^{4}{e}^{2}g{x}^{2}-87040\,{c}^{5}{d}^{3}{e}^{3}f{x}^{2}+4095\,{b}^{5}{e}^{6}gx-45080\,{b}^{4}cd{e}^{5}gx-3150\,{b}^{4}c{e}^{6}fx+200600\,{b}^{3}{c}^{2}{d}^{2}{e}^{4}gx+30800\,{b}^{3}{c}^{2}d{e}^{5}fx-452064\,{b}^{2}{c}^{3}{d}^{3}{e}^{3}gx-116400\,{b}^{2}{c}^{3}{d}^{2}{e}^{4}fx+516656\,b{c}^{4}{d}^{4}{e}^{2}gx+204480\,b{c}^{4}{d}^{3}{e}^{3}fx-233216\,{c}^{5}{d}^{5}egx-145760\,{c}^{5}{d}^{4}{e}^{2}fx+630\,{b}^{5}d{e}^{5}g+3465\,{b}^{5}{e}^{6}f-6790\,{b}^{4}c{d}^{2}{e}^{4}g-37800\,{b}^{4}cd{e}^{5}f+29440\,{b}^{3}{c}^{2}{d}^{3}{e}^{3}g+166600\,{b}^{3}{c}^{2}{d}^{2}{e}^{4}f-64176\,{b}^{2}{c}^{3}{d}^{4}{e}^{2}g-372000\,{b}^{2}{c}^{3}{d}^{3}{e}^{3}f+70048\,b{c}^{4}{d}^{5}eg+423120\,b{c}^{4}{d}^{4}{e}^{2}f-29152\,{c}^{5}{d}^{6}g-198400\,{c}^{5}{d}^{5}ef \right ) }{45045\, \left ( ex+d \right ) ^{7}{e}^{2} \left ({b}^{6}{e}^{6}-12\,{b}^{5}cd{e}^{5}+60\,{b}^{4}{c}^{2}{d}^{2}{e}^{4}-160\,{b}^{3}{c}^{3}{d}^{3}{e}^{3}+240\,{b}^{2}{c}^{4}{d}^{4}{e}^{2}-192\,b{c}^{5}{d}^{5}e+64\,{c}^{6}{d}^{6} \right ) }\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^8,x)

[Out]

-2/45045*(c*e*x+b*e-c*d)*(1664*b*c^4*e^6*g*x^5-2048*c^5*d*e^5*g*x^5-1280*c^5*e^6*f*x^5-2496*b^2*c^3*e^6*g*x^4+
16384*b*c^4*d*e^5*g*x^4+1920*b*c^4*e^6*f*x^4-16384*c^5*d^2*e^4*g*x^4-10240*c^5*d*e^5*f*x^4+3120*b^3*c^2*e^6*g*
x^3-26304*b^2*c^3*d*e^5*g*x^3-2400*b^2*c^3*e^6*f*x^3+76736*b*c^4*d^2*e^4*g*x^3+17280*b*c^4*d*e^5*f*x^3-60416*c
^5*d^3*e^3*g*x^3-37760*c^5*d^2*e^4*f*x^3-3640*b^4*c*e^6*g*x^2+35680*b^3*c^2*d*e^5*g*x^2+2800*b^3*c^2*e^6*f*x^2
-134496*b^2*c^3*d^2*e^4*g*x^2-24000*b^2*c^3*d*e^5*f*x^2+231424*b*c^4*d^3*e^3*g*x^2+73920*b*c^4*d^2*e^4*f*x^2-1
39264*c^5*d^4*e^2*g*x^2-87040*c^5*d^3*e^3*f*x^2+4095*b^5*e^6*g*x-45080*b^4*c*d*e^5*g*x-3150*b^4*c*e^6*f*x+2006
00*b^3*c^2*d^2*e^4*g*x+30800*b^3*c^2*d*e^5*f*x-452064*b^2*c^3*d^3*e^3*g*x-116400*b^2*c^3*d^2*e^4*f*x+516656*b*
c^4*d^4*e^2*g*x+204480*b*c^4*d^3*e^3*f*x-233216*c^5*d^5*e*g*x-145760*c^5*d^4*e^2*f*x+630*b^5*d*e^5*g+3465*b^5*
e^6*f-6790*b^4*c*d^2*e^4*g-37800*b^4*c*d*e^5*f+29440*b^3*c^2*d^3*e^3*g+166600*b^3*c^2*d^2*e^4*f-64176*b^2*c^3*
d^4*e^2*g-372000*b^2*c^3*d^3*e^3*f+70048*b*c^4*d^5*e*g+423120*b*c^4*d^4*e^2*f-29152*c^5*d^6*g-198400*c^5*d^5*e
*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^7/e^2/(b^6*e^6-12*b^5*c*d*e^5+60*b^4*c^2*d^2*e^4-160*b^3*c^
3*d^3*e^3+240*b^2*c^4*d^4*e^2-192*b*c^5*d^5*e+64*c^6*d^6)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^8,x, algorithm="fricas")

[Out]

Timed out

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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((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2)/(e*x+d)**8,x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^8,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError