3.22.72 \(\int (d+e x)^3 (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx\) [2172]

Optimal. Leaf size=414 \[ \frac {7 (2 c d-b e)^3 (4 c e f+2 c d g-3 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{512 c^5 e}-\frac {7 (2 c d-b e)^2 (4 c e f+2 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{192 c^4 e^2}-\frac {7 (2 c d-b e) (4 c e f+2 c d g-3 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{160 c^3 e^2}-\frac {(4 c e f+2 c d g-3 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{20 c^2 e^2}-\frac {g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 c e^2}+\frac {7 (2 c d-b e)^5 (4 c e f+2 c d g-3 b e g) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{1024 c^{11/2} e^2} \]

[Out]

-7/192*(-b*e+2*c*d)^2*(-3*b*e*g+2*c*d*g+4*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/c^4/e^2-7/160*(-b*e+2*
c*d)*(-3*b*e*g+2*c*d*g+4*c*e*f)*(e*x+d)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/c^3/e^2-1/20*(-3*b*e*g+2*c*d*g+
4*c*e*f)*(e*x+d)^2*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/c^2/e^2-1/6*g*(e*x+d)^3*(d*(-b*e+c*d)-b*e^2*x-c*e^2*
x^2)^(3/2)/c/e^2+7/1024*(-b*e+2*c*d)^5*(-3*b*e*g+2*c*d*g+4*c*e*f)*arctan(1/2*e*(2*c*x+b)/c^(1/2)/(d*(-b*e+c*d)
-b*e^2*x-c*e^2*x^2)^(1/2))/c^(11/2)/e^2+7/512*(-b*e+2*c*d)^3*(-3*b*e*g+2*c*d*g+4*c*e*f)*(2*c*x+b)*(d*(-b*e+c*d
)-b*e^2*x-c*e^2*x^2)^(1/2)/c^5/e

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Rubi [A]
time = 0.50, antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {808, 684, 654, 626, 635, 210} \begin {gather*} \frac {7 (2 c d-b e)^5 \text {ArcTan}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right ) (-3 b e g+2 c d g+4 c e f)}{1024 c^{11/2} e^2}+\frac {7 (b+2 c x) (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-3 b e g+2 c d g+4 c e f)}{512 c^5 e}-\frac {7 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+2 c d g+4 c e f)}{192 c^4 e^2}-\frac {7 (d+e x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+2 c d g+4 c e f)}{160 c^3 e^2}-\frac {(d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+2 c d g+4 c e f)}{20 c^2 e^2}-\frac {g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 c e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*(2*c*d - b*e)^3*(4*c*e*f + 2*c*d*g - 3*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(512*c
^5*e) - (7*(2*c*d - b*e)^2*(4*c*e*f + 2*c*d*g - 3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(192*c^4
*e^2) - (7*(2*c*d - b*e)*(4*c*e*f + 2*c*d*g - 3*b*e*g)*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/
(160*c^3*e^2) - ((4*c*e*f + 2*c*d*g - 3*b*e*g)*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(20*c^
2*e^2) - (g*(d + e*x)^3*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(6*c*e^2) + (7*(2*c*d - b*e)^5*(4*c*e*f +
2*c*d*g - 3*b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(1024*c^(11
/2)*e^2)

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 654

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

Rule 684

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[(m + p)*((2*c*d - b*e)/(c*(m + 2*p + 1))), Int[(d + e
*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rule 808

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])

Rubi steps

\begin {align*} \int (d+e x)^3 (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx &=-\frac {g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 c e^2}-\frac {\left (\frac {3}{2} e \left (-2 c e^2 f+b e^2 g\right )+3 \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right ) \int (d+e x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{6 c e^3}\\ &=-\frac {(4 c e f+2 c d g-3 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{20 c^2 e^2}-\frac {g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 c e^2}+\frac {(7 (2 c d-b e) (4 c e f+2 c d g-3 b e g)) \int (d+e x)^2 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{40 c^2 e}\\ &=-\frac {7 (2 c d-b e) (4 c e f+2 c d g-3 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{160 c^3 e^2}-\frac {(4 c e f+2 c d g-3 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{20 c^2 e^2}-\frac {g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 c e^2}+\frac {\left (7 (2 c d-b e)^2 (4 c e f+2 c d g-3 b e g)\right ) \int (d+e x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{64 c^3 e}\\ &=-\frac {7 (2 c d-b e)^2 (4 c e f+2 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{192 c^4 e^2}-\frac {7 (2 c d-b e) (4 c e f+2 c d g-3 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{160 c^3 e^2}-\frac {(4 c e f+2 c d g-3 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{20 c^2 e^2}-\frac {g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 c e^2}+\frac {\left (7 (2 c d-b e)^3 (4 c e f+2 c d g-3 b e g)\right ) \int \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{128 c^4 e}\\ &=\frac {7 (2 c d-b e)^3 (4 c e f+2 c d g-3 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{512 c^5 e}-\frac {7 (2 c d-b e)^2 (4 c e f+2 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{192 c^4 e^2}-\frac {7 (2 c d-b e) (4 c e f+2 c d g-3 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{160 c^3 e^2}-\frac {(4 c e f+2 c d g-3 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{20 c^2 e^2}-\frac {g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 c e^2}+\frac {\left (7 (2 c d-b e)^5 (4 c e f+2 c d g-3 b e g)\right ) \int \frac {1}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{1024 c^5 e}\\ &=\frac {7 (2 c d-b e)^3 (4 c e f+2 c d g-3 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{512 c^5 e}-\frac {7 (2 c d-b e)^2 (4 c e f+2 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{192 c^4 e^2}-\frac {7 (2 c d-b e) (4 c e f+2 c d g-3 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{160 c^3 e^2}-\frac {(4 c e f+2 c d g-3 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{20 c^2 e^2}-\frac {g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 c e^2}+\frac {\left (7 (2 c d-b e)^5 (4 c e f+2 c d g-3 b e g)\right ) \text {Subst}\left (\int \frac {1}{-4 c e^2-x^2} \, dx,x,\frac {-b e^2-2 c e^2 x}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}\right )}{512 c^5 e}\\ &=\frac {7 (2 c d-b e)^3 (4 c e f+2 c d g-3 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{512 c^5 e}-\frac {7 (2 c d-b e)^2 (4 c e f+2 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{192 c^4 e^2}-\frac {7 (2 c d-b e) (4 c e f+2 c d g-3 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{160 c^3 e^2}-\frac {(4 c e f+2 c d g-3 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{20 c^2 e^2}-\frac {g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 c e^2}+\frac {7 (2 c d-b e)^5 (4 c e f+2 c d g-3 b e g) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{1024 c^{11/2} e^2}\\ \end {align*}

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Mathematica [A]
time = 0.91, size = 448, normalized size = 1.08 \begin {gather*} \frac {(2 c d-b e)^5 \sqrt {(d+e x) (-b e+c (d-e x))} \left (\frac {\sqrt {c} \left (315 b^5 e^5 g-210 b^4 c e^4 (2 e f+14 d g+e g x)+56 b^3 c^2 e^3 \left (193 d^2 g+e^2 x (5 f+3 g x)+d e (65 f+31 g x)\right )-32 c^5 \left (176 d^5 g-36 d e^4 x^3 (5 f+4 g x)-8 e^5 x^4 (6 f+5 g x)-2 d^3 e^2 x (15 f+16 g x)-2 d^2 e^3 x^2 (112 f+85 g x)+d^4 e (272 f+105 g x)\right )+16 b c^4 e \left (1047 d^4 g+4 e^4 x^3 (3 f+2 g x)+4 d e^3 x^2 (23 f+14 g x)+2 d^2 e^2 x (179 f+95 g x)+2 d^3 e (559 f+227 g x)\right )-16 b^2 c^3 e^2 \left (1213 d^3 g+19 d e^2 x (7 f+4 g x)+e^3 x^2 (14 f+9 g x)+d^2 e (749 f+335 g x)\right )\right )}{(2 c d-b e)^5}-\frac {105 (4 c e f+2 c d g-3 b e g) \tan ^{-1}\left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{\sqrt {d+e x} \sqrt {-b e+c (d-e x)}}\right )}{7680 c^{11/2} e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*c*d - b*e)^5*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*((Sqrt[c]*(315*b^5*e^5*g - 210*b^4*c*e^4*(2*e*f + 14*d
*g + e*g*x) + 56*b^3*c^2*e^3*(193*d^2*g + e^2*x*(5*f + 3*g*x) + d*e*(65*f + 31*g*x)) - 32*c^5*(176*d^5*g - 36*
d*e^4*x^3*(5*f + 4*g*x) - 8*e^5*x^4*(6*f + 5*g*x) - 2*d^3*e^2*x*(15*f + 16*g*x) - 2*d^2*e^3*x^2*(112*f + 85*g*
x) + d^4*e*(272*f + 105*g*x)) + 16*b*c^4*e*(1047*d^4*g + 4*e^4*x^3*(3*f + 2*g*x) + 4*d*e^3*x^2*(23*f + 14*g*x)
+ 2*d^2*e^2*x*(179*f + 95*g*x) + 2*d^3*e*(559*f + 227*g*x)) - 16*b^2*c^3*e^2*(1213*d^3*g + 19*d*e^2*x*(7*f +
4*g*x) + e^3*x^2*(14*f + 9*g*x) + d^2*e*(749*f + 335*g*x))))/(2*c*d - b*e)^5 - (105*(4*c*e*f + 2*c*d*g - 3*b*e
*g)*ArcTan[Sqrt[c*d - b*e - c*e*x]/(Sqrt[c]*Sqrt[d + e*x])])/(Sqrt[d + e*x]*Sqrt[-(b*e) + c*(d - e*x)])))/(768
0*c^(11/2)*e^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2485\) vs. \(2(384)=768\).
time = 0.07, size = 2486, normalized size = 6.00

method	result	size
default	\(\text {Expression too large to display}\)	\(2486\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

e^3*g*(-1/6*x^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e^2-3/4*b/c*(-1/5*x^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2
)^(3/2)/c/e^2-7/10*b/c*(-1/4*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e^2-5/8*b/c*(-1/3*(-c*e^2*x^2-b*e^2*x-
b*d*e+c*d^2)^(3/2)/c/e^2-1/2*b/c*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4
*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+
c*d^2)^(1/2))))+1/4*(-b*d*e+c*d^2)/c/e^2*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)
-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*
x-b*d*e+c*d^2)^(1/2))))+2/5*(-b*d*e+c*d^2)/c/e^2*(-1/3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e^2-1/2*b/c*(-
1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^
2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)))))+1/2*(-b*d*e+c*d^2)
/c/e^2*(-1/4*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e^2-5/8*b/c*(-1/3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/
2)/c/e^2-1/2*b/c*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e^2*(-b*d*e+c
*d^2)-b^2*e^4)/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))))+
1/4*(-b*d*e+c*d^2)/c/e^2*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e^2*(
-b*d*e+c*d^2)-b^2*e^4)/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(
1/2)))))+(3*d*e^2*g+e^3*f)*(-1/5*x^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e^2-7/10*b/c*(-1/4*x*(-c*e^2*x^2
-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e^2-5/8*b/c*(-1/3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e^2-1/2*b/c*(-1/4*(-2
*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2/(c*e^
2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))))+1/4*(-b*d*e+c*d^2)/c/e^2*(
-1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e
^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))))+2/5*(-b*d*e+c*d^2)
/c/e^2*(-1/3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e^2-1/2*b/c*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b
*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/
2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)))))+(3*d^2*e*g+3*d*e^2*f)*(-1/4*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^
2)^(3/2)/c/e^2-5/8*b/c*(-1/3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e^2-1/2*b/c*(-1/4*(-2*c*e^2*x-b*e^2)/c/e
^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2/(c*e^2)^(1/2)*arctan((c*
e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))))+1/4*(-b*d*e+c*d^2)/c/e^2*(-1/4*(-2*c*e^2*x-b*
e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2/(c*e^2)^(1/2)*ar
ctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))))+(d^3*g+3*d^2*e*f)*(-1/3*(-c*e^2*x^2-b
*e^2*x-b*d*e+c*d^2)^(3/2)/c/e^2-1/2*b/c*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-
1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x
-b*d*e+c*d^2)^(1/2))))+d^3*f*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e
^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^
2)^(1/2)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(2*c*d-%e*b>0)', see `assume?`
for more det

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Fricas [A]
time = 3.83, size = 1392, normalized size = 3.36 \begin {gather*} \left [-\frac {{\left (105 \, {\left (64 \, c^{6} d^{6} g - {\left (4 \, b^{5} c f - 3 \, b^{6} g\right )} e^{6} + 8 \, {\left (5 \, b^{4} c^{2} d f - 4 \, b^{5} c d g\right )} e^{5} - 20 \, {\left (8 \, b^{3} c^{3} d^{2} f - 7 \, b^{4} c^{2} d^{2} g\right )} e^{4} + 320 \, {\left (b^{2} c^{4} d^{3} f - b^{3} c^{3} d^{3} g\right )} e^{3} - 80 \, {\left (4 \, b c^{5} d^{4} f - 5 \, b^{2} c^{4} d^{4} g\right )} e^{2} + 128 \, {\left (c^{6} d^{5} f - 2 \, b c^{5} d^{5} g\right )} e\right )} \sqrt {-c} \log \left (-4 \, c^{2} d^{2} + 4 \, b c d e - 4 \, \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} {\left (2 \, c x + b\right )} \sqrt {-c} e + {\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2}\right )} e^{2}\right ) + 4 \, {\left (5632 \, c^{6} d^{5} g - {\left (1280 \, c^{6} g x^{5} - 420 \, b^{4} c^{2} f + 315 \, b^{5} c g + 128 \, {\left (12 \, c^{6} f + b c^{5} g\right )} x^{4} + 48 \, {\left (4 \, b c^{5} f - 3 \, b^{2} c^{4} g\right )} x^{3} - 56 \, {\left (4 \, b^{2} c^{4} f - 3 \, b^{3} c^{3} g\right )} x^{2} + 70 \, {\left (4 \, b^{3} c^{3} f - 3 \, b^{4} c^{2} g\right )} x\right )} e^{5} - 4 \, {\left (1152 \, c^{6} d g x^{4} + 910 \, b^{3} c^{3} d f - 735 \, b^{4} c^{2} d g + 32 \, {\left (45 \, c^{6} d f + 7 \, b c^{5} d g\right )} x^{3} + 16 \, {\left (23 \, b c^{5} d f - 19 \, b^{2} c^{4} d g\right )} x^{2} - 14 \, {\left (38 \, b^{2} c^{4} d f - 31 \, b^{3} c^{3} d g\right )} x\right )} e^{4} - 8 \, {\left (680 \, c^{6} d^{2} g x^{3} - 1498 \, b^{2} c^{4} d^{2} f + 1351 \, b^{3} c^{3} d^{2} g + 4 \, {\left (224 \, c^{6} d^{2} f + 95 \, b c^{5} d^{2} g\right )} x^{2} + 2 \, {\left (358 \, b c^{5} d^{2} f - 335 \, b^{2} c^{4} d^{2} g\right )} x\right )} e^{3} - 16 \, {\left (64 \, c^{6} d^{3} g x^{2} + 1118 \, b c^{5} d^{3} f - 1213 \, b^{2} c^{4} d^{3} g + 2 \, {\left (30 \, c^{6} d^{3} f + 227 \, b c^{5} d^{3} g\right )} x\right )} e^{2} + 16 \, {\left (210 \, c^{6} d^{4} g x + 544 \, c^{6} d^{4} f - 1047 \, b c^{5} d^{4} g\right )} e\right )} \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}}\right )} e^{\left (-2\right )}}{30720 \, c^{6}}, -\frac {{\left (105 \, {\left (64 \, c^{6} d^{6} g - {\left (4 \, b^{5} c f - 3 \, b^{6} g\right )} e^{6} + 8 \, {\left (5 \, b^{4} c^{2} d f - 4 \, b^{5} c d g\right )} e^{5} - 20 \, {\left (8 \, b^{3} c^{3} d^{2} f - 7 \, b^{4} c^{2} d^{2} g\right )} e^{4} + 320 \, {\left (b^{2} c^{4} d^{3} f - b^{3} c^{3} d^{3} g\right )} e^{3} - 80 \, {\left (4 \, b c^{5} d^{4} f - 5 \, b^{2} c^{4} d^{4} g\right )} e^{2} + 128 \, {\left (c^{6} d^{5} f - 2 \, b c^{5} d^{5} g\right )} e\right )} \sqrt {c} \arctan \left (-\frac {\sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} {\left (2 \, c x + b\right )} \sqrt {c} e}{2 \, {\left (c^{2} d^{2} - b c d e - {\left (c^{2} x^{2} + b c x\right )} e^{2}\right )}}\right ) + 2 \, {\left (5632 \, c^{6} d^{5} g - {\left (1280 \, c^{6} g x^{5} - 420 \, b^{4} c^{2} f + 315 \, b^{5} c g + 128 \, {\left (12 \, c^{6} f + b c^{5} g\right )} x^{4} + 48 \, {\left (4 \, b c^{5} f - 3 \, b^{2} c^{4} g\right )} x^{3} - 56 \, {\left (4 \, b^{2} c^{4} f - 3 \, b^{3} c^{3} g\right )} x^{2} + 70 \, {\left (4 \, b^{3} c^{3} f - 3 \, b^{4} c^{2} g\right )} x\right )} e^{5} - 4 \, {\left (1152 \, c^{6} d g x^{4} + 910 \, b^{3} c^{3} d f - 735 \, b^{4} c^{2} d g + 32 \, {\left (45 \, c^{6} d f + 7 \, b c^{5} d g\right )} x^{3} + 16 \, {\left (23 \, b c^{5} d f - 19 \, b^{2} c^{4} d g\right )} x^{2} - 14 \, {\left (38 \, b^{2} c^{4} d f - 31 \, b^{3} c^{3} d g\right )} x\right )} e^{4} - 8 \, {\left (680 \, c^{6} d^{2} g x^{3} - 1498 \, b^{2} c^{4} d^{2} f + 1351 \, b^{3} c^{3} d^{2} g + 4 \, {\left (224 \, c^{6} d^{2} f + 95 \, b c^{5} d^{2} g\right )} x^{2} + 2 \, {\left (358 \, b c^{5} d^{2} f - 335 \, b^{2} c^{4} d^{2} g\right )} x\right )} e^{3} - 16 \, {\left (64 \, c^{6} d^{3} g x^{2} + 1118 \, b c^{5} d^{3} f - 1213 \, b^{2} c^{4} d^{3} g + 2 \, {\left (30 \, c^{6} d^{3} f + 227 \, b c^{5} d^{3} g\right )} x\right )} e^{2} + 16 \, {\left (210 \, c^{6} d^{4} g x + 544 \, c^{6} d^{4} f - 1047 \, b c^{5} d^{4} g\right )} e\right )} \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}}\right )} e^{\left (-2\right )}}{15360 \, c^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="fricas")

[Out]

[-1/30720*(105*(64*c^6*d^6*g - (4*b^5*c*f - 3*b^6*g)*e^6 + 8*(5*b^4*c^2*d*f - 4*b^5*c*d*g)*e^5 - 20*(8*b^3*c^3
*d^2*f - 7*b^4*c^2*d^2*g)*e^4 + 320*(b^2*c^4*d^3*f - b^3*c^3*d^3*g)*e^3 - 80*(4*b*c^5*d^4*f - 5*b^2*c^4*d^4*g)
*e^2 + 128*(c^6*d^5*f - 2*b*c^5*d^5*g)*e)*sqrt(-c)*log(-4*c^2*d^2 + 4*b*c*d*e - 4*sqrt(c*d^2 - b*d*e - (c*x^2
+ b*x)*e^2)*(2*c*x + b)*sqrt(-c)*e + (8*c^2*x^2 + 8*b*c*x + b^2)*e^2) + 4*(5632*c^6*d^5*g - (1280*c^6*g*x^5 -
420*b^4*c^2*f + 315*b^5*c*g + 128*(12*c^6*f + b*c^5*g)*x^4 + 48*(4*b*c^5*f - 3*b^2*c^4*g)*x^3 - 56*(4*b^2*c^4*
f - 3*b^3*c^3*g)*x^2 + 70*(4*b^3*c^3*f - 3*b^4*c^2*g)*x)*e^5 - 4*(1152*c^6*d*g*x^4 + 910*b^3*c^3*d*f - 735*b^4
*c^2*d*g + 32*(45*c^6*d*f + 7*b*c^5*d*g)*x^3 + 16*(23*b*c^5*d*f - 19*b^2*c^4*d*g)*x^2 - 14*(38*b^2*c^4*d*f - 3
1*b^3*c^3*d*g)*x)*e^4 - 8*(680*c^6*d^2*g*x^3 - 1498*b^2*c^4*d^2*f + 1351*b^3*c^3*d^2*g + 4*(224*c^6*d^2*f + 95
*b*c^5*d^2*g)*x^2 + 2*(358*b*c^5*d^2*f - 335*b^2*c^4*d^2*g)*x)*e^3 - 16*(64*c^6*d^3*g*x^2 + 1118*b*c^5*d^3*f -
1213*b^2*c^4*d^3*g + 2*(30*c^6*d^3*f + 227*b*c^5*d^3*g)*x)*e^2 + 16*(210*c^6*d^4*g*x + 544*c^6*d^4*f - 1047*b
*c^5*d^4*g)*e)*sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2))*e^(-2)/c^6, -1/15360*(105*(64*c^6*d^6*g - (4*b^5*c*f -
3*b^6*g)*e^6 + 8*(5*b^4*c^2*d*f - 4*b^5*c*d*g)*e^5 - 20*(8*b^3*c^3*d^2*f - 7*b^4*c^2*d^2*g)*e^4 + 320*(b^2*c^
4*d^3*f - b^3*c^3*d^3*g)*e^3 - 80*(4*b*c^5*d^4*f - 5*b^2*c^4*d^4*g)*e^2 + 128*(c^6*d^5*f - 2*b*c^5*d^5*g)*e)*s
qrt(c)*arctan(-1/2*sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2)*(2*c*x + b)*sqrt(c)*e/(c^2*d^2 - b*c*d*e - (c^2*x^2
+ b*c*x)*e^2)) + 2*(5632*c^6*d^5*g - (1280*c^6*g*x^5 - 420*b^4*c^2*f + 315*b^5*c*g + 128*(12*c^6*f + b*c^5*g)
*x^4 + 48*(4*b*c^5*f - 3*b^2*c^4*g)*x^3 - 56*(4*b^2*c^4*f - 3*b^3*c^3*g)*x^2 + 70*(4*b^3*c^3*f - 3*b^4*c^2*g)*
x)*e^5 - 4*(1152*c^6*d*g*x^4 + 910*b^3*c^3*d*f - 735*b^4*c^2*d*g + 32*(45*c^6*d*f + 7*b*c^5*d*g)*x^3 + 16*(23*
b*c^5*d*f - 19*b^2*c^4*d*g)*x^2 - 14*(38*b^2*c^4*d*f - 31*b^3*c^3*d*g)*x)*e^4 - 8*(680*c^6*d^2*g*x^3 - 1498*b^
2*c^4*d^2*f + 1351*b^3*c^3*d^2*g + 4*(224*c^6*d^2*f + 95*b*c^5*d^2*g)*x^2 + 2*(358*b*c^5*d^2*f - 335*b^2*c^4*d
^2*g)*x)*e^3 - 16*(64*c^6*d^3*g*x^2 + 1118*b*c^5*d^3*f - 1213*b^2*c^4*d^3*g + 2*(30*c^6*d^3*f + 227*b*c^5*d^3*
g)*x)*e^2 + 16*(210*c^6*d^4*g*x + 544*c^6*d^4*f - 1047*b*c^5*d^4*g)*e)*sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2)
)*e^(-2)/c^6]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (d + e x\right )^{3} \left (f + g x\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)

[Out]

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

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Giac [A]
time = 1.26, size = 697, normalized size = 1.68 \begin {gather*} \frac {1}{7680} \, \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, g x e^{3} + \frac {{\left (36 \, c^{5} d g e^{10} + 12 \, c^{5} f e^{11} + b c^{4} g e^{11}\right )} e^{\left (-8\right )}}{c^{5}}\right )} x + \frac {{\left (340 \, c^{5} d^{2} g e^{9} + 360 \, c^{5} d f e^{10} + 56 \, b c^{4} d g e^{10} + 12 \, b c^{4} f e^{11} - 9 \, b^{2} c^{3} g e^{11}\right )} e^{\left (-8\right )}}{c^{5}}\right )} x + \frac {{\left (128 \, c^{5} d^{3} g e^{8} + 896 \, c^{5} d^{2} f e^{9} + 380 \, b c^{4} d^{2} g e^{9} + 184 \, b c^{4} d f e^{10} - 152 \, b^{2} c^{3} d g e^{10} - 28 \, b^{2} c^{3} f e^{11} + 21 \, b^{3} c^{2} g e^{11}\right )} e^{\left (-8\right )}}{c^{5}}\right )} x - \frac {{\left (1680 \, c^{5} d^{4} g e^{7} - 480 \, c^{5} d^{3} f e^{8} - 3632 \, b c^{4} d^{3} g e^{8} - 2864 \, b c^{4} d^{2} f e^{9} + 2680 \, b^{2} c^{3} d^{2} g e^{9} + 1064 \, b^{2} c^{3} d f e^{10} - 868 \, b^{3} c^{2} d g e^{10} - 140 \, b^{3} c^{2} f e^{11} + 105 \, b^{4} c g e^{11}\right )} e^{\left (-8\right )}}{c^{5}}\right )} x - \frac {{\left (5632 \, c^{5} d^{5} g e^{6} + 8704 \, c^{5} d^{4} f e^{7} - 16752 \, b c^{4} d^{4} g e^{7} - 17888 \, b c^{4} d^{3} f e^{8} + 19408 \, b^{2} c^{3} d^{3} g e^{8} + 11984 \, b^{2} c^{3} d^{2} f e^{9} - 10808 \, b^{3} c^{2} d^{2} g e^{9} - 3640 \, b^{3} c^{2} d f e^{10} + 2940 \, b^{4} c d g e^{10} + 420 \, b^{4} c f e^{11} - 315 \, b^{5} g e^{11}\right )} e^{\left (-8\right )}}{c^{5}}\right )} + \frac {7 \, {\left (64 \, c^{6} d^{6} g + 128 \, c^{6} d^{5} f e - 256 \, b c^{5} d^{5} g e - 320 \, b c^{5} d^{4} f e^{2} + 400 \, b^{2} c^{4} d^{4} g e^{2} + 320 \, b^{2} c^{4} d^{3} f e^{3} - 320 \, b^{3} c^{3} d^{3} g e^{3} - 160 \, b^{3} c^{3} d^{2} f e^{4} + 140 \, b^{4} c^{2} d^{2} g e^{4} + 40 \, b^{4} c^{2} d f e^{5} - 32 \, b^{5} c d g e^{5} - 4 \, b^{5} c f e^{6} + 3 \, b^{6} g e^{6}\right )} \sqrt {-c} e^{\left (-2\right )} \log \left ({\left | -b \sqrt {-c} e - 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} c \right |}\right )}{1024 \, c^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="giac")

[Out]

1/7680*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(2*(4*(2*(8*(10*g*x*e^3 + (36*c^5*d*g*e^10 + 12*c^5*f*e^11 +
b*c^4*g*e^11)*e^(-8)/c^5)*x + (340*c^5*d^2*g*e^9 + 360*c^5*d*f*e^10 + 56*b*c^4*d*g*e^10 + 12*b*c^4*f*e^11 - 9
*b^2*c^3*g*e^11)*e^(-8)/c^5)*x + (128*c^5*d^3*g*e^8 + 896*c^5*d^2*f*e^9 + 380*b*c^4*d^2*g*e^9 + 184*b*c^4*d*f*
e^10 - 152*b^2*c^3*d*g*e^10 - 28*b^2*c^3*f*e^11 + 21*b^3*c^2*g*e^11)*e^(-8)/c^5)*x - (1680*c^5*d^4*g*e^7 - 480
*c^5*d^3*f*e^8 - 3632*b*c^4*d^3*g*e^8 - 2864*b*c^4*d^2*f*e^9 + 2680*b^2*c^3*d^2*g*e^9 + 1064*b^2*c^3*d*f*e^10
- 868*b^3*c^2*d*g*e^10 - 140*b^3*c^2*f*e^11 + 105*b^4*c*g*e^11)*e^(-8)/c^5)*x - (5632*c^5*d^5*g*e^6 + 8704*c^5
*d^4*f*e^7 - 16752*b*c^4*d^4*g*e^7 - 17888*b*c^4*d^3*f*e^8 + 19408*b^2*c^3*d^3*g*e^8 + 11984*b^2*c^3*d^2*f*e^9
- 10808*b^3*c^2*d^2*g*e^9 - 3640*b^3*c^2*d*f*e^10 + 2940*b^4*c*d*g*e^10 + 420*b^4*c*f*e^11 - 315*b^5*g*e^11)*
e^(-8)/c^5) + 7/1024*(64*c^6*d^6*g + 128*c^6*d^5*f*e - 256*b*c^5*d^5*g*e - 320*b*c^5*d^4*f*e^2 + 400*b^2*c^4*d
^4*g*e^2 + 320*b^2*c^4*d^3*f*e^3 - 320*b^3*c^3*d^3*g*e^3 - 160*b^3*c^3*d^2*f*e^4 + 140*b^4*c^2*d^2*g*e^4 + 40*
b^4*c^2*d*f*e^5 - 32*b^5*c*d*g*e^5 - 4*b^5*c*f*e^6 + 3*b^6*g*e^6)*sqrt(-c)*e^(-2)*log(abs(-b*sqrt(-c)*e - 2*(s
qrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))*c))/c^6

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Mupad [B]
time = 8.14, size = 2500, normalized size = 6.04 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f + g*x)*(d + e*x)^3*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2),x)

[Out]

d^3*f*(x/2 + b/(4*c))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) + (3*d*f*(c*d^2 - b*d*e)*((x/2 + b/(4*c))*(c
*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) - (log(b*e^2 - 2*(-c*e^2)^(1/2)*(-(d + e*x)*(b*e - c*d + c*e*x))^(1/
2) + 2*c*e^2*x)*((b^2*e^4)/4 + c*e^2*(c*d^2 - b*d*e)))/(2*(-c*e^2)^(3/2))))/(4*c) - (7*b*e^3*f*((5*b*((log(b*e
^2 - 2*(-c*e^2)^(1/2)*(-(d + e*x)*(b*e - c*d + c*e*x))^(1/2) + 2*c*e^2*x)*(b^3*e^6 + 4*b*c*e^4*(c*d^2 - b*d*e)
))/(16*(-c*e^2)^(5/2)) - ((8*c*e^2*(c*e^2*x^2 - c*d^2 + b*d*e) - 3*b^2*e^4 + 2*b*c*e^4*x)*(c*d^2 - c*e^2*x^2 -
b*d*e - b*e^2*x)^(1/2))/(24*c^2*e^4)))/(8*c) + ((c*d^2 - b*d*e)*((x/2 + b/(4*c))*(c*d^2 - c*e^2*x^2 - b*d*e -
b*e^2*x)^(1/2) - (log(b*e^2 - 2*(-c*e^2)^(1/2)*(-(d + e*x)*(b*e - c*d + c*e*x))^(1/2) + 2*c*e^2*x)*((b^2*e^4)
/4 + c*e^2*(c*d^2 - b*d*e)))/(2*(-c*e^2)^(3/2))))/(4*c*e^2) - (x*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/
(4*c*e^2)))/(10*c) + (3*b*e^3*g*((7*b*((5*b*((log(b*e^2 - 2*(-c*e^2)^(1/2)*(-(d + e*x)*(b*e - c*d + c*e*x))^(1
/2) + 2*c*e^2*x)*(b^3*e^6 + 4*b*c*e^4*(c*d^2 - b*d*e)))/(16*(-c*e^2)^(5/2)) - ((8*c*e^2*(c*e^2*x^2 - c*d^2 + b
*d*e) - 3*b^2*e^4 + 2*b*c*e^4*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(24*c^2*e^4)))/(8*c) + ((c*d^2 -
b*d*e)*((x/2 + b/(4*c))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) - (log(b*e^2 - 2*(-c*e^2)^(1/2)*(-(d + e*
x)*(b*e - c*d + c*e*x))^(1/2) + 2*c*e^2*x)*((b^2*e^4)/4 + c*e^2*(c*d^2 - b*d*e)))/(2*(-c*e^2)^(3/2))))/(4*c*e^
2) - (x*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(4*c*e^2)))/(10*c) + ((2*c*d^2 - 2*b*d*e)*((log(b*e^2 - 2
*(-c*e^2)^(1/2)*(-(d + e*x)*(b*e - c*d + c*e*x))^(1/2) + 2*c*e^2*x)*(b^3*e^6 + 4*b*c*e^4*(c*d^2 - b*d*e)))/(16
*(-c*e^2)^(5/2)) - ((8*c*e^2*(c*e^2*x^2 - c*d^2 + b*d*e) - 3*b^2*e^4 + 2*b*c*e^4*x)*(c*d^2 - c*e^2*x^2 - b*d*e
- b*e^2*x)^(1/2))/(24*c^2*e^4)))/(5*c*e^2) + (x^2*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(5*c*e^2)))/(4
*c) - (d^3*f*log(b*e^2 - 2*(-c*e^2)^(1/2)*(-(d + e*x)*(b*e - c*d + c*e*x))^(1/2) + 2*c*e^2*x)*((b^2*e^4)/4 + c
*e^2*(c*d^2 - b*d*e)))/(2*(-c*e^2)^(3/2)) - (3*d*g*(2*c*d^2 - 2*b*d*e)*((log(b*e^2 - 2*(-c*e^2)^(1/2)*(-(d + e
*x)*(b*e - c*d + c*e*x))^(1/2) + 2*c*e^2*x)*(b^3*e^6 + 4*b*c*e^4*(c*d^2 - b*d*e)))/(16*(-c*e^2)^(5/2)) - ((8*c
*e^2*(c*e^2*x^2 - c*d^2 + b*d*e) - 3*b^2*e^4 + 2*b*c*e^4*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(24*c
^2*e^4)))/(5*c) - (e*f*(2*c*d^2 - 2*b*d*e)*((log(b*e^2 - 2*(-c*e^2)^(1/2)*(-(d + e*x)*(b*e - c*d + c*e*x))^(1/
2) + 2*c*e^2*x)*(b^3*e^6 + 4*b*c*e^4*(c*d^2 - b*d*e)))/(16*(-c*e^2)^(5/2)) - ((8*c*e^2*(c*e^2*x^2 - c*d^2 + b*
d*e) - 3*b^2*e^4 + 2*b*c*e^4*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(24*c^2*e^4)))/(5*c) - (3*d*f*x*(
c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(4*c) - (d^3*g*log(b*e^2 - 2*(-c*e^2)^(1/2)*(-(d + e*x)*(b*e - c*d
+ c*e*x))^(1/2) + 2*c*e^2*x)*(b^3*e^6 + 4*b*c*e^4*(c*d^2 - b*d*e)))/(16*(-c*e^2)^(5/2)) + (e*g*(3*c*d^2 - 3*b
*d*e)*((5*b*((log(b*e^2 - 2*(-c*e^2)^(1/2)*(-(d + e*x)*(b*e - c*d + c*e*x))^(1/2) + 2*c*e^2*x)*(b^3*e^6 + 4*b*
c*e^4*(c*d^2 - b*d*e)))/(16*(-c*e^2)^(5/2)) - ((8*c*e^2*(c*e^2*x^2 - c*d^2 + b*d*e) - 3*b^2*e^4 + 2*b*c*e^4*x)
*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(24*c^2*e^4)))/(8*c) + ((c*d^2 - b*d*e)*((x/2 + b/(4*c))*(c*d^2
- c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) - (log(b*e^2 - 2*(-c*e^2)^(1/2)*(-(d + e*x)*(b*e - c*d + c*e*x))^(1/2) +
2*c*e^2*x)*((b^2*e^4)/4 + c*e^2*(c*d^2 - b*d*e)))/(2*(-c*e^2)^(3/2))))/(4*c*e^2) - (x*(c*d^2 - c*e^2*x^2 - b*d
*e - b*e^2*x)^(3/2))/(4*c*e^2)))/(6*c) - (3*d*g*x^2*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(5*c) - (e*f*
x^2*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(5*c) - (e*g*x^3*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))
/(6*c) + (15*b*d*e^2*f*((log(b*e^2 - 2*(-c*e^2)^(1/2)*(-(d + e*x)*(b*e - c*d + c*e*x))^(1/2) + 2*c*e^2*x)*(b^3
*e^6 + 4*b*c*e^4*(c*d^2 - b*d*e)))/(16*(-c*e^2)^(5/2)) - ((8*c*e^2*(c*e^2*x^2 - c*d^2 + b*d*e) - 3*b^2*e^4 + 2
*b*c*e^4*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(24*c^2*e^4)))/(8*c) + (15*b*d^2*e*g*((log(b*e^2 - 2*
(-c*e^2)^(1/2)*(-(d + e*x)*(b*e - c*d + c*e*x))^(1/2) + 2*c*e^2*x)*(b^3*e^6 + 4*b*c*e^4*(c*d^2 - b*d*e)))/(16*
(-c*e^2)^(5/2)) - ((8*c*e^2*(c*e^2*x^2 - c*d^2 + b*d*e) - 3*b^2*e^4 + 2*b*c*e^4*x)*(c*d^2 - c*e^2*x^2 - b*d*e
- b*e^2*x)^(1/2))/(24*c^2*e^4)))/(8*c) - (3*d^2*e*f*log(b*e^2 - 2*(-c*e^2)^(1/2)*(-(d + e*x)*(b*e - c*d + c*e*
x))^(1/2) + 2*c*e^2*x)*(b^3*e^6 + 4*b*c*e^4*(c*d^2 - b*d*e)))/(16*(-c*e^2)^(5/2)) + (d^2*f*(8*c*e^2*(c*e^2*x^2
- c*d^2 + b*d*e) - 3*b^2*e^4 + 2*b*c*e^4*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(8*c^2*e^3) + (d^3*g
*(8*c*e^2*(c*e^2*x^2 - c*d^2 + b*d*e) - 3*b^2*e^4 + 2*b*c*e^4*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/
(24*c^2*e^4) + (3*d^2*g*(c*d^2 - b*d*e)*((x/2 + b/(4*c))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) - (log(b*
e^2 - 2*(-c*e^2)^(1/2)*(-(d + e*x)*(b*e - c*d + c*e*x))^(1/2) + 2*c*e^2*x)*((b^2*e^4)/4 + c*e^2*(c*d^2 - b*d*e
)))/(2*(-c*e^2)^(3/2))))/(4*c*e) - (21*b*d*e^2*g*((5*b*((log(b*e^2 - 2*(-c*e^2)^(1/2)*(-(d + e*x)*(b*e - c*d +
c*e*x))^(1/2) + 2*c*e^2*x)*(b^3*e^6 + 4*b*c*e^...

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