3.22.83 \(\int (d+e x)^3 (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2} \, dx\) [2183]

Optimal. Leaf size=488 \[ \frac {9 (2 c d-b e)^5 (16 c e f+6 c d g-11 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{16384 c^6 e}+\frac {3 (2 c d-b e)^3 (16 c e f+6 c d g-11 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2048 c^5 e}-\frac {3 (2 c d-b e)^2 (16 c e f+6 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{640 c^4 e^2}-\frac {3 (2 c d-b e) (16 c e f+6 c d g-11 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{448 c^3 e^2}-\frac {(16 c e f+6 c d g-11 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{112 c^2 e^2}-\frac {g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{8 c e^2}+\frac {9 (2 c d-b e)^7 (16 c e f+6 c d g-11 b e g) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{32768 c^{13/2} e^2} \]

[Out]

3/2048*(-b*e+2*c*d)^3*(-11*b*e*g+6*c*d*g+16*c*e*f)*(2*c*x+b)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/c^5/e-3/64
0*(-b*e+2*c*d)^2*(-11*b*e*g+6*c*d*g+16*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/c^4/e^2-3/448*(-b*e+2*c*d
)*(-11*b*e*g+6*c*d*g+16*c*e*f)*(e*x+d)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/c^3/e^2-1/112*(-11*b*e*g+6*c*d*g
+16*c*e*f)*(e*x+d)^2*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/c^2/e^2-1/8*g*(e*x+d)^3*(d*(-b*e+c*d)-b*e^2*x-c*e^
2*x^2)^(5/2)/c/e^2+9/32768*(-b*e+2*c*d)^7*(-11*b*e*g+6*c*d*g+16*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^(13/2)/e^2+9/16384*(-b*e+2*c*d)^5*(-11*b*e*g+6*c*d*g+16*c*e*f)*(2*c*x+b)*(d*
(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/c^6/e

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Rubi [A]
time = 0.61, antiderivative size = 488, normalized size of antiderivative = 1.00, number of steps used = 8, 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 {9 (2 c d-b e)^7 \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 ) (-11 b e g+6 c d g+16 c e f)}{32768 c^{13/2} e^2}+\frac {9 (b+2 c x) (2 c d-b e)^5 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-11 b e g+6 c d g+16 c e f)}{16384 c^6 e}+\frac {3 (b+2 c x) (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+6 c d g+16 c e f)}{2048 c^5 e}-\frac {3 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+6 c d g+16 c e f)}{640 c^4 e^2}-\frac {3 (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} (-11 b e g+6 c d g+16 c e f)}{448 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 )^{5/2} (-11 b e g+6 c d g+16 c e f)}{112 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 )^{5/2}}{8 c e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(9*(2*c*d - b*e)^5*(16*c*e*f + 6*c*d*g - 11*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(163
84*c^6*e) + (3*(2*c*d - b*e)^3*(16*c*e*f + 6*c*d*g - 11*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^
2)^(3/2))/(2048*c^5*e) - (3*(2*c*d - b*e)^2*(16*c*e*f + 6*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x
^2)^(5/2))/(640*c^4*e^2) - (3*(2*c*d - b*e)*(16*c*e*f + 6*c*d*g - 11*b*e*g)*(d + e*x)*(d*(c*d - b*e) - b*e^2*x
- c*e^2*x^2)^(5/2))/(448*c^3*e^2) - ((16*c*e*f + 6*c*d*g - 11*b*e*g)*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c
*e^2*x^2)^(5/2))/(112*c^2*e^2) - (g*(d + e*x)^3*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(8*c*e^2) + (9*(2
*c*d - b*e)^7*(16*c*e*f + 6*c*d*g - 11*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])])/(32768*c^(13/2)*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) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx &=-\frac {g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{8 c e^2}-\frac {\left (\frac {5}{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 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{8 c e^3}\\ &=-\frac {(16 c e f+6 c d g-11 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{112 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 )^{5/2}}{8 c e^2}+\frac {(9 (2 c d-b e) (16 c e f+6 c d g-11 b e g)) \int (d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{224 c^2 e}\\ &=-\frac {3 (2 c d-b e) (16 c e f+6 c d g-11 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{448 c^3 e^2}-\frac {(16 c e f+6 c d g-11 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{112 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 )^{5/2}}{8 c e^2}+\frac {\left (3 (2 c d-b e)^2 (16 c e f+6 c d g-11 b e g)\right ) \int (d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{128 c^3 e}\\ &=-\frac {3 (2 c d-b e)^2 (16 c e f+6 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{640 c^4 e^2}-\frac {3 (2 c d-b e) (16 c e f+6 c d g-11 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{448 c^3 e^2}-\frac {(16 c e f+6 c d g-11 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{112 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 )^{5/2}}{8 c e^2}+\frac {\left (3 (2 c d-b e)^3 (16 c e f+6 c d g-11 b e g)\right ) \int \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{256 c^4 e}\\ &=\frac {3 (2 c d-b e)^3 (16 c e f+6 c d g-11 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2048 c^5 e}-\frac {3 (2 c d-b e)^2 (16 c e f+6 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{640 c^4 e^2}-\frac {3 (2 c d-b e) (16 c e f+6 c d g-11 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{448 c^3 e^2}-\frac {(16 c e f+6 c d g-11 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{112 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 )^{5/2}}{8 c e^2}+\frac {\left (9 (2 c d-b e)^5 (16 c e f+6 c d g-11 b e g)\right ) \int \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{4096 c^5 e}\\ &=\frac {9 (2 c d-b e)^5 (16 c e f+6 c d g-11 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{16384 c^6 e}+\frac {3 (2 c d-b e)^3 (16 c e f+6 c d g-11 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2048 c^5 e}-\frac {3 (2 c d-b e)^2 (16 c e f+6 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{640 c^4 e^2}-\frac {3 (2 c d-b e) (16 c e f+6 c d g-11 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{448 c^3 e^2}-\frac {(16 c e f+6 c d g-11 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{112 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 )^{5/2}}{8 c e^2}+\frac {\left (9 (2 c d-b e)^7 (16 c e f+6 c d g-11 b e g)\right ) \int \frac {1}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{32768 c^6 e}\\ &=\frac {9 (2 c d-b e)^5 (16 c e f+6 c d g-11 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{16384 c^6 e}+\frac {3 (2 c d-b e)^3 (16 c e f+6 c d g-11 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2048 c^5 e}-\frac {3 (2 c d-b e)^2 (16 c e f+6 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{640 c^4 e^2}-\frac {3 (2 c d-b e) (16 c e f+6 c d g-11 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{448 c^3 e^2}-\frac {(16 c e f+6 c d g-11 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{112 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 )^{5/2}}{8 c e^2}+\frac {\left (9 (2 c d-b e)^7 (16 c e f+6 c d g-11 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 )}{16384 c^6 e}\\ &=\frac {9 (2 c d-b e)^5 (16 c e f+6 c d g-11 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{16384 c^6 e}+\frac {3 (2 c d-b e)^3 (16 c e f+6 c d g-11 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2048 c^5 e}-\frac {3 (2 c d-b e)^2 (16 c e f+6 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{640 c^4 e^2}-\frac {3 (2 c d-b e) (16 c e f+6 c d g-11 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{448 c^3 e^2}-\frac {(16 c e f+6 c d g-11 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{112 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 )^{5/2}}{8 c e^2}+\frac {9 (2 c d-b e)^7 (16 c e f+6 c d g-11 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 )}{32768 c^{13/2} e^2}\\ \end {align*}

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Mathematica [A]
time = 2.19, size = 733, normalized size = 1.50 \begin {gather*} \frac {(2 c d-b e)^7 ((d+e x) (-b e+c (d-e x)))^{3/2} \left (-\frac {\sqrt {c} \left (-3465 b^7 e^7 g+210 b^6 c e^6 (24 e f+218 d g+11 e g x)-84 b^5 c^2 e^5 \left (3057 d^2 g+2 e^2 x (20 f+11 g x)+d e (760 f+334 g x)\right )+128 c^7 \left (1664 d^7 g+320 d e^6 x^5 (7 f+6 g x)+80 e^7 x^6 (8 f+7 g x)-16 d^3 e^4 x^3 (175 f+136 g x)+8 d^2 e^5 x^4 (208 f+175 g x)-8 d^5 e^2 x (245 f+176 g x)+d^6 e (2944 f+945 g x)-2 d^4 e^3 x^2 (2624 f+1925 g x)\right )+24 b^4 c^3 e^4 \left (32924 d^3 g+2 e^3 x^2 (56 f+33 g x)+8 d e^2 x (203 f+107 g x)+3 d^2 e (4704 f+1963 g x)\right )+64 b c^6 e \left (-13647 d^6 g+80 e^6 x^5 (20 f+17 g x)+6 d^4 e^2 x (-116 f+123 g x)+48 d e^5 x^4 (164 f+135 g x)+8 d^3 e^3 x^2 (1574 f+1187 g x)+8 d^2 e^4 x^3 (1882 f+1483 g x)-2 d^5 e (9812 f+3263 g x)\right )-16 b^3 c^4 e^3 \left (89587 d^4 g+8 e^4 x^3 (18 f+11 g x)+8 d e^3 x^2 (222 f+125 g x)+12 d^2 e^2 x (960 f+479 g x)+4 d^3 e (15072 f+5887 g x)\right )+32 b^2 c^5 e^2 \left (47490 d^5 g+8 e^5 x^4 (8 f+5 g x)+16 d e^4 x^3 (43 f+25 g x)+12 d^2 e^3 x^2 (308 f+163 g x)+8 d^3 e^2 x (1748 f+809 g x)+d^4 e (48712 f+17401 g x)\right )\right )}{(2 c d-b e)^7 (d+e x) (-b e+c (d-e x))}-\frac {315 (16 c e f+6 c d g-11 b e g) \tan ^{-1}\left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{(d+e x)^{3/2} (-b e+c (d-e x))^{3/2}}\right )}{573440 c^{13/2} e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*(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)^7*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*(-((Sqrt[c]*(-3465*b^7*e^7*g + 210*b^6*c*e^6*(24*e*f
+ 218*d*g + 11*e*g*x) - 84*b^5*c^2*e^5*(3057*d^2*g + 2*e^2*x*(20*f + 11*g*x) + d*e*(760*f + 334*g*x)) + 128*c
^7*(1664*d^7*g + 320*d*e^6*x^5*(7*f + 6*g*x) + 80*e^7*x^6*(8*f + 7*g*x) - 16*d^3*e^4*x^3*(175*f + 136*g*x) + 8
*d^2*e^5*x^4*(208*f + 175*g*x) - 8*d^5*e^2*x*(245*f + 176*g*x) + d^6*e*(2944*f + 945*g*x) - 2*d^4*e^3*x^2*(262
4*f + 1925*g*x)) + 24*b^4*c^3*e^4*(32924*d^3*g + 2*e^3*x^2*(56*f + 33*g*x) + 8*d*e^2*x*(203*f + 107*g*x) + 3*d
^2*e*(4704*f + 1963*g*x)) + 64*b*c^6*e*(-13647*d^6*g + 80*e^6*x^5*(20*f + 17*g*x) + 6*d^4*e^2*x*(-116*f + 123*
g*x) + 48*d*e^5*x^4*(164*f + 135*g*x) + 8*d^3*e^3*x^2*(1574*f + 1187*g*x) + 8*d^2*e^4*x^3*(1882*f + 1483*g*x)
- 2*d^5*e*(9812*f + 3263*g*x)) - 16*b^3*c^4*e^3*(89587*d^4*g + 8*e^4*x^3*(18*f + 11*g*x) + 8*d*e^3*x^2*(222*f
+ 125*g*x) + 12*d^2*e^2*x*(960*f + 479*g*x) + 4*d^3*e*(15072*f + 5887*g*x)) + 32*b^2*c^5*e^2*(47490*d^5*g + 8*
e^5*x^4*(8*f + 5*g*x) + 16*d*e^4*x^3*(43*f + 25*g*x) + 12*d^2*e^3*x^2*(308*f + 163*g*x) + 8*d^3*e^2*x*(1748*f
+ 809*g*x) + d^4*e*(48712*f + 17401*g*x))))/((2*c*d - b*e)^7*(d + e*x)*(-(b*e) + c*(d - e*x)))) - (315*(16*c*e
*f + 6*c*d*g - 11*b*e*g)*ArcTan[Sqrt[c*d - b*e - c*e*x]/(Sqrt[c]*Sqrt[d + e*x])])/((d + e*x)^(3/2)*(-(b*e) + c
*(d - e*x))^(3/2))))/(573440*c^(13/2)*e^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3517\) vs. \(2(454)=908\).
time = 0.06, size = 3518, normalized size = 7.21

method	result	size
default	\(\text {Expression too large to display}\)	\(3518\)

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)^(3/2),x,method=_RETURNVERBOSE)

[Out]

e^3*g*(-1/8*x^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c/e^2-11/16*b/c*(-1/7*x^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d
^2)^(5/2)/c/e^2-9/14*b/c*(-1/6*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c/e^2-7/12*b/c*(-1/5*(-c*e^2*x^2-b*e^2
*x-b*d*e+c*d^2)^(5/2)/c/e^2-1/2*b/c*(-1/8*(-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)^(3/2)-3/16
*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/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)))))+1/6*(-b*d*e+c*d^2)/c/e^2*(-1/8*(-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)^(3/2)-3/16*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/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)))))+2/7*(-b*d*e+c*d^2)/c/e^2*(-1/5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c
/e^2-1/2*b/c*(-1/8*(-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)^(3/2)-3/16*(-4*c*e^2*(-b*d*e+c*d^
2)-b^2*e^4)/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))))
))+3/8*(-b*d*e+c*d^2)/c/e^2*(-1/6*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c/e^2-7/12*b/c*(-1/5*(-c*e^2*x^2-b*
e^2*x-b*d*e+c*d^2)^(5/2)/c/e^2-1/2*b/c*(-1/8*(-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)^(3/2)-3
/16*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/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)))))+1/6*(-b*d*e+c*d^2)/c/e^2*(-1/8*(-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)^(3/2)-3/16*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/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)*(-1/7*x^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/
c/e^2-9/14*b/c*(-1/6*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c/e^2-7/12*b/c*(-1/5*(-c*e^2*x^2-b*e^2*x-b*d*e+c
*d^2)^(5/2)/c/e^2-1/2*b/c*(-1/8*(-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)^(3/2)-3/16*(-4*c*e^2
*(-b*d*e+c*d^2)-b^2*e^4)/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)))))+1/6*(-b*d*e+c*d^2)/c/e^2*(-1/8*(-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)^(3/2)-
3/16*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/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)))))+2/7*(-b*d*e+c*d^2)/c/e^2*(-1/5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c/e^2-1/2*b
/c*(-1/8*(-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)^(3/2)-3/16*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4
)/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^2*
e*g+3*d*e^2*f)*(-1/6*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c/e^2-7/12*b/c*(-1/5*(-c*e^2*x^2-b*e^2*x-b*d*e+c
*d^2)^(5/2)/c/e^2-1/2*b/c*(-1/8*(-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)^(3/2)-3/16*(-4*c*e^2
*(-b*d*e+c*d^2)-b^2*e^4)/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)))))+1/6*(-b*d*e+c*d^2)/c/e^2*(-1/8*(-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)^(3/2)-
3/16*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/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)))))+(d^3*g+3*d^2*e*f)*(-1/5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c/e^2-1/2*b/c*(-1/
8*(-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)^(3/2)-3/16*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/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)))))+d^3*f*(-1/8*(-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)^(3/2)-3/16*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/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))))

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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)^(3/2),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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1123 vs. \(2 (461) = 922\).
time = 3.58, size = 2254, normalized size = 4.62 \begin {gather*} \text {Too large to display} \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)^(3/2),x, algorithm="fricas")

[Out]

[-1/2293760*(315*(768*c^8*d^8*g - (16*b^7*c*f - 11*b^8*g)*e^8 + 32*(7*b^6*c^2*d*f - 5*b^7*c*d*g)*e^7 - 336*(4*
b^5*c^3*d^2*f - 3*b^6*c^2*d^2*g)*e^6 + 896*(5*b^4*c^4*d^3*f - 4*b^5*c^3*d^3*g)*e^5 - 1120*(8*b^3*c^5*d^4*f - 7
*b^4*c^4*d^4*g)*e^4 + 10752*(b^2*c^6*d^5*f - b^3*c^5*d^5*g)*e^3 - 1792*(4*b*c^7*d^6*f - 5*b^2*c^6*d^6*g)*e^2 +
2048*(c^8*d^7*f - 2*b*c^7*d^7*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*(212992*c^8*d^7*g + (71680*c^8*g*x^7 + 50
40*b^6*c^2*f - 3465*b^7*c*g + 5120*(16*c^8*f + 17*b*c^7*g)*x^6 + 1280*(80*b*c^7*f + b^2*c^6*g)*x^5 + 128*(16*b
^2*c^6*f - 11*b^3*c^5*g)*x^4 - 144*(16*b^3*c^5*f - 11*b^4*c^4*g)*x^3 + 168*(16*b^4*c^4*f - 11*b^5*c^3*g)*x^2 -
210*(16*b^5*c^3*f - 11*b^6*c^2*g)*x)*e^7 + 4*(61440*c^8*d*g*x^6 - 15960*b^5*c^3*d*f + 11445*b^6*c^2*d*g + 128
0*(56*c^8*d*f + 81*b*c^7*d*g)*x^5 + 128*(984*b*c^7*d*f + 25*b^2*c^6*d*g)*x^4 + 32*(172*b^2*c^6*d*f - 125*b^3*c
^5*d*g)*x^3 - 48*(148*b^3*c^5*d*f - 107*b^4*c^4*d*g)*x^2 + 42*(232*b^4*c^4*d*f - 167*b^5*c^3*d*g)*x)*e^6 + 4*(
44800*c^8*d^2*g*x^5 + 84672*b^4*c^4*d^2*f - 64197*b^5*c^3*d^2*g + 128*(416*c^8*d^2*f + 1483*b*c^7*d^2*g)*x^4 +
32*(7528*b*c^7*d^2*f + 489*b^2*c^6*d^2*g)*x^3 + 48*(616*b^2*c^6*d^2*f - 479*b^3*c^5*d^2*g)*x^2 - 18*(2560*b^3
*c^5*d^2*f - 1963*b^4*c^4*d^2*g)*x)*e^5 - 32*(8704*c^8*d^3*g*x^4 + 30144*b^3*c^5*d^3*f - 24693*b^4*c^4*d^3*g +
16*(700*c^8*d^3*f - 1187*b*c^7*d^3*g)*x^3 - 8*(3148*b*c^7*d^3*f + 809*b^2*c^6*d^3*g)*x^2 - 2*(6992*b^2*c^6*d^
3*f - 5887*b^3*c^5*d^3*g)*x)*e^4 - 16*(30800*c^8*d^4*g*x^3 - 97424*b^2*c^6*d^4*f + 89587*b^3*c^5*d^4*g + 328*(
128*c^8*d^4*f - 9*b*c^7*d^4*g)*x^2 + 2*(1392*b*c^7*d^4*f - 17401*b^2*c^6*d^4*g)*x)*e^3 - 64*(2816*c^8*d^5*g*x^
2 + 19624*b*c^7*d^5*f - 23745*b^2*c^6*d^5*g + 2*(1960*c^8*d^5*f + 3263*b*c^7*d^5*g)*x)*e^2 + 64*(1890*c^8*d^6*
g*x + 5888*c^8*d^6*f - 13647*b*c^7*d^6*g)*e)*sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2))*e^(-2)/c^7, -1/1146880*(
315*(768*c^8*d^8*g - (16*b^7*c*f - 11*b^8*g)*e^8 + 32*(7*b^6*c^2*d*f - 5*b^7*c*d*g)*e^7 - 336*(4*b^5*c^3*d^2*f
- 3*b^6*c^2*d^2*g)*e^6 + 896*(5*b^4*c^4*d^3*f - 4*b^5*c^3*d^3*g)*e^5 - 1120*(8*b^3*c^5*d^4*f - 7*b^4*c^4*d^4*
g)*e^4 + 10752*(b^2*c^6*d^5*f - b^3*c^5*d^5*g)*e^3 - 1792*(4*b*c^7*d^6*f - 5*b^2*c^6*d^6*g)*e^2 + 2048*(c^8*d^
7*f - 2*b*c^7*d^7*g)*e)*sqrt(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*(212992*c^8*d^7*g + (71680*c^8*g*x^7 + 5040*b^6*c^2*f - 3465*b^7*
c*g + 5120*(16*c^8*f + 17*b*c^7*g)*x^6 + 1280*(80*b*c^7*f + b^2*c^6*g)*x^5 + 128*(16*b^2*c^6*f - 11*b^3*c^5*g)
*x^4 - 144*(16*b^3*c^5*f - 11*b^4*c^4*g)*x^3 + 168*(16*b^4*c^4*f - 11*b^5*c^3*g)*x^2 - 210*(16*b^5*c^3*f - 11*
b^6*c^2*g)*x)*e^7 + 4*(61440*c^8*d*g*x^6 - 15960*b^5*c^3*d*f + 11445*b^6*c^2*d*g + 1280*(56*c^8*d*f + 81*b*c^7
*d*g)*x^5 + 128*(984*b*c^7*d*f + 25*b^2*c^6*d*g)*x^4 + 32*(172*b^2*c^6*d*f - 125*b^3*c^5*d*g)*x^3 - 48*(148*b^
3*c^5*d*f - 107*b^4*c^4*d*g)*x^2 + 42*(232*b^4*c^4*d*f - 167*b^5*c^3*d*g)*x)*e^6 + 4*(44800*c^8*d^2*g*x^5 + 84
672*b^4*c^4*d^2*f - 64197*b^5*c^3*d^2*g + 128*(416*c^8*d^2*f + 1483*b*c^7*d^2*g)*x^4 + 32*(7528*b*c^7*d^2*f +
489*b^2*c^6*d^2*g)*x^3 + 48*(616*b^2*c^6*d^2*f - 479*b^3*c^5*d^2*g)*x^2 - 18*(2560*b^3*c^5*d^2*f - 1963*b^4*c^
4*d^2*g)*x)*e^5 - 32*(8704*c^8*d^3*g*x^4 + 30144*b^3*c^5*d^3*f - 24693*b^4*c^4*d^3*g + 16*(700*c^8*d^3*f - 118
7*b*c^7*d^3*g)*x^3 - 8*(3148*b*c^7*d^3*f + 809*b^2*c^6*d^3*g)*x^2 - 2*(6992*b^2*c^6*d^3*f - 5887*b^3*c^5*d^3*g
)*x)*e^4 - 16*(30800*c^8*d^4*g*x^3 - 97424*b^2*c^6*d^4*f + 89587*b^3*c^5*d^4*g + 328*(128*c^8*d^4*f - 9*b*c^7*
d^4*g)*x^2 + 2*(1392*b*c^7*d^4*f - 17401*b^2*c^6*d^4*g)*x)*e^3 - 64*(2816*c^8*d^5*g*x^2 + 19624*b*c^7*d^5*f -
23745*b^2*c^6*d^5*g + 2*(1960*c^8*d^5*f + 3263*b*c^7*d^5*g)*x)*e^2 + 64*(1890*c^8*d^6*g*x + 5888*c^8*d^6*f - 1
3647*b*c^7*d^6*g)*e)*sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2))*e^(-2)/c^7]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 )^{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)**(3/2),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1136 vs. \(2 (461) = 922\).
time = 1.44, size = 1136, normalized size = 2.33 \begin {gather*} -\frac {1}{573440} \, \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 \, {\left (4 \, {\left (14 \, c g x e^{5} + \frac {{\left (48 \, c^{8} d g e^{16} + 16 \, c^{8} f e^{17} + 17 \, b c^{7} g e^{17}\right )} e^{\left (-12\right )}}{c^{7}}\right )} x + \frac {{\left (140 \, c^{8} d^{2} g e^{15} + 224 \, c^{8} d f e^{16} + 324 \, b c^{7} d g e^{16} + 80 \, b c^{7} f e^{17} + b^{2} c^{6} g e^{17}\right )} e^{\left (-12\right )}}{c^{7}}\right )} x - \frac {{\left (2176 \, c^{8} d^{3} g e^{14} - 1664 \, c^{8} d^{2} f e^{15} - 5932 \, b c^{7} d^{2} g e^{15} - 3936 \, b c^{7} d f e^{16} - 100 \, b^{2} c^{6} d g e^{16} - 16 \, b^{2} c^{6} f e^{17} + 11 \, b^{3} c^{5} g e^{17}\right )} e^{\left (-12\right )}}{c^{7}}\right )} x - \frac {{\left (30800 \, c^{8} d^{4} g e^{13} + 22400 \, c^{8} d^{3} f e^{14} - 37984 \, b c^{7} d^{3} g e^{14} - 60224 \, b c^{7} d^{2} f e^{15} - 3912 \, b^{2} c^{6} d^{2} g e^{15} - 1376 \, b^{2} c^{6} d f e^{16} + 1000 \, b^{3} c^{5} d g e^{16} + 144 \, b^{3} c^{5} f e^{17} - 99 \, b^{4} c^{4} g e^{17}\right )} e^{\left (-12\right )}}{c^{7}}\right )} x - \frac {{\left (22528 \, c^{8} d^{5} g e^{12} + 83968 \, c^{8} d^{4} f e^{13} - 5904 \, b c^{7} d^{4} g e^{13} - 100736 \, b c^{7} d^{3} f e^{14} - 25888 \, b^{2} c^{6} d^{3} g e^{14} - 14784 \, b^{2} c^{6} d^{2} f e^{15} + 11496 \, b^{3} c^{5} d^{2} g e^{15} + 3552 \, b^{3} c^{5} d f e^{16} - 2568 \, b^{4} c^{4} d g e^{16} - 336 \, b^{4} c^{4} f e^{17} + 231 \, b^{5} c^{3} g e^{17}\right )} e^{\left (-12\right )}}{c^{7}}\right )} x + \frac {{\left (60480 \, c^{8} d^{6} g e^{11} - 125440 \, c^{8} d^{5} f e^{12} - 208832 \, b c^{7} d^{5} g e^{12} - 22272 \, b c^{7} d^{4} f e^{13} + 278416 \, b^{2} c^{6} d^{4} g e^{13} + 223744 \, b^{2} c^{6} d^{3} f e^{14} - 188384 \, b^{3} c^{5} d^{3} g e^{14} - 92160 \, b^{3} c^{5} d^{2} f e^{15} + 70668 \, b^{4} c^{4} d^{2} g e^{15} + 19488 \, b^{4} c^{4} d f e^{16} - 14028 \, b^{5} c^{3} d g e^{16} - 1680 \, b^{5} c^{3} f e^{17} + 1155 \, b^{6} c^{2} g e^{17}\right )} e^{\left (-12\right )}}{c^{7}}\right )} x + \frac {{\left (212992 \, c^{8} d^{7} g e^{10} + 376832 \, c^{8} d^{6} f e^{11} - 873408 \, b c^{7} d^{6} g e^{11} - 1255936 \, b c^{7} d^{5} f e^{12} + 1519680 \, b^{2} c^{6} d^{5} g e^{12} + 1558784 \, b^{2} c^{6} d^{4} f e^{13} - 1433392 \, b^{3} c^{5} d^{4} g e^{13} - 964608 \, b^{3} c^{5} d^{3} f e^{14} + 790176 \, b^{4} c^{4} d^{3} g e^{14} + 338688 \, b^{4} c^{4} d^{2} f e^{15} - 256788 \, b^{5} c^{3} d^{2} g e^{15} - 63840 \, b^{5} c^{3} d f e^{16} + 45780 \, b^{6} c^{2} d g e^{16} + 5040 \, b^{6} c^{2} f e^{17} - 3465 \, b^{7} c g e^{17}\right )} e^{\left (-12\right )}}{c^{7}}\right )} + \frac {9 \, {\left (768 \, c^{8} d^{8} g + 2048 \, c^{8} d^{7} f e - 4096 \, b c^{7} d^{7} g e - 7168 \, b c^{7} d^{6} f e^{2} + 8960 \, b^{2} c^{6} d^{6} g e^{2} + 10752 \, b^{2} c^{6} d^{5} f e^{3} - 10752 \, b^{3} c^{5} d^{5} g e^{3} - 8960 \, b^{3} c^{5} d^{4} f e^{4} + 7840 \, b^{4} c^{4} d^{4} g e^{4} + 4480 \, b^{4} c^{4} d^{3} f e^{5} - 3584 \, b^{5} c^{3} d^{3} g e^{5} - 1344 \, b^{5} c^{3} d^{2} f e^{6} + 1008 \, b^{6} c^{2} d^{2} g e^{6} + 224 \, b^{6} c^{2} d f e^{7} - 160 \, b^{7} c d g e^{7} - 16 \, b^{7} c f e^{8} + 11 \, b^{8} g e^{8}\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 )}{32768 \, c^{7}} \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)^(3/2),x, algorithm="giac")

[Out]

-1/573440*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(2*(4*(2*(8*(10*(4*(14*c*g*x*e^5 + (48*c^8*d*g*e^16 + 16*
c^8*f*e^17 + 17*b*c^7*g*e^17)*e^(-12)/c^7)*x + (140*c^8*d^2*g*e^15 + 224*c^8*d*f*e^16 + 324*b*c^7*d*g*e^16 + 8
0*b*c^7*f*e^17 + b^2*c^6*g*e^17)*e^(-12)/c^7)*x - (2176*c^8*d^3*g*e^14 - 1664*c^8*d^2*f*e^15 - 5932*b*c^7*d^2*
g*e^15 - 3936*b*c^7*d*f*e^16 - 100*b^2*c^6*d*g*e^16 - 16*b^2*c^6*f*e^17 + 11*b^3*c^5*g*e^17)*e^(-12)/c^7)*x -
(30800*c^8*d^4*g*e^13 + 22400*c^8*d^3*f*e^14 - 37984*b*c^7*d^3*g*e^14 - 60224*b*c^7*d^2*f*e^15 - 3912*b^2*c^6*
d^2*g*e^15 - 1376*b^2*c^6*d*f*e^16 + 1000*b^3*c^5*d*g*e^16 + 144*b^3*c^5*f*e^17 - 99*b^4*c^4*g*e^17)*e^(-12)/c
^7)*x - (22528*c^8*d^5*g*e^12 + 83968*c^8*d^4*f*e^13 - 5904*b*c^7*d^4*g*e^13 - 100736*b*c^7*d^3*f*e^14 - 25888
*b^2*c^6*d^3*g*e^14 - 14784*b^2*c^6*d^2*f*e^15 + 11496*b^3*c^5*d^2*g*e^15 + 3552*b^3*c^5*d*f*e^16 - 2568*b^4*c
^4*d*g*e^16 - 336*b^4*c^4*f*e^17 + 231*b^5*c^3*g*e^17)*e^(-12)/c^7)*x + (60480*c^8*d^6*g*e^11 - 125440*c^8*d^5
*f*e^12 - 208832*b*c^7*d^5*g*e^12 - 22272*b*c^7*d^4*f*e^13 + 278416*b^2*c^6*d^4*g*e^13 + 223744*b^2*c^6*d^3*f*
e^14 - 188384*b^3*c^5*d^3*g*e^14 - 92160*b^3*c^5*d^2*f*e^15 + 70668*b^4*c^4*d^2*g*e^15 + 19488*b^4*c^4*d*f*e^1
6 - 14028*b^5*c^3*d*g*e^16 - 1680*b^5*c^3*f*e^17 + 1155*b^6*c^2*g*e^17)*e^(-12)/c^7)*x + (212992*c^8*d^7*g*e^1
0 + 376832*c^8*d^6*f*e^11 - 873408*b*c^7*d^6*g*e^11 - 1255936*b*c^7*d^5*f*e^12 + 1519680*b^2*c^6*d^5*g*e^12 +
1558784*b^2*c^6*d^4*f*e^13 - 1433392*b^3*c^5*d^4*g*e^13 - 964608*b^3*c^5*d^3*f*e^14 + 790176*b^4*c^4*d^3*g*e^1
4 + 338688*b^4*c^4*d^2*f*e^15 - 256788*b^5*c^3*d^2*g*e^15 - 63840*b^5*c^3*d*f*e^16 + 45780*b^6*c^2*d*g*e^16 +
5040*b^6*c^2*f*e^17 - 3465*b^7*c*g*e^17)*e^(-12)/c^7) + 9/32768*(768*c^8*d^8*g + 2048*c^8*d^7*f*e - 4096*b*c^7
*d^7*g*e - 7168*b*c^7*d^6*f*e^2 + 8960*b^2*c^6*d^6*g*e^2 + 10752*b^2*c^6*d^5*f*e^3 - 10752*b^3*c^5*d^5*g*e^3 -
8960*b^3*c^5*d^4*f*e^4 + 7840*b^4*c^4*d^4*g*e^4 + 4480*b^4*c^4*d^3*f*e^5 - 3584*b^5*c^3*d^3*g*e^5 - 1344*b^5*
c^3*d^2*f*e^6 + 1008*b^6*c^2*d^2*g*e^6 + 224*b^6*c^2*d*f*e^7 - 160*b^7*c*d*g*e^7 - 16*b^7*c*f*e^8 + 11*b^8*g*e
^8)*sqrt(-c)*e^(-2)*log(abs(-b*sqrt(-c)*e - 2*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))*c)
)/c^7

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (f+g\,x\right )\,{\left (d+e\,x\right )}^3\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2} \,d x \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)^(3/2),x)

[Out]

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

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