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

Optimal. Leaf size=413 \[ \frac {(2 c d-b e)^6 (-9 b e g+4 c d g+14 c e f) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2048 c^{11/2} e^2}+\frac {(b+2 c x) (2 c d-b e)^4 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+4 c d g+14 c e f)}{1024 c^5 e}+\frac {(b+2 c x) (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-9 b e g+4 c d g+14 c e f)}{384 c^4 e}-\frac {(2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+4 c d g+14 c e f)}{120 c^3 e^2}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+4 c d g+14 c e f)}{84 c^2 e^2}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2} \]

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Rubi [A]  time = 0.60, antiderivative size = 413, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.159, Rules used = {1638, 12, 670, 640, 612, 621, 204} \begin {gather*} \frac {(b+2 c x) (2 c d-b e)^4 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+4 c d g+14 c e f)}{1024 c^5 e}+\frac {(b+2 c x) (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-9 b e g+4 c d g+14 c e f)}{384 c^4 e}-\frac {(2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+4 c d g+14 c e f)}{120 c^3 e^2}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+4 c d g+14 c e f)}{84 c^2 e^2}+\frac {(2 c d-b e)^6 (-9 b e g+4 c d g+14 c e f) \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 )}{2048 c^{11/2} e^2}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*c*d - b*e)^4*(14*c*e*f + 4*c*d*g - 9*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(1024*c
^5*e) + ((2*c*d - b*e)^2*(14*c*e*f + 4*c*d*g - 9*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2
))/(384*c^4*e) - ((2*c*d - b*e)*(14*c*e*f + 4*c*d*g - 9*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(1
20*c^3*e^2) - ((14*c*e*f + 4*c*d*g - 9*b*e*g)*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(84*c^2*e
^2) - (g*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(7*c*e^2) + ((2*c*d - b*e)^6*(14*c*e*f + 4*c
*d*g - 9*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])])/(2048*c^(11/2)*
e^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 204

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

Rule 612

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 621

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 640

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 670

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 1638

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq
, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*e^(q - 1)*(
m + q + 2*p + 1)), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q + e*f*(m + p + q)*(d + e*x)^(q - 2)*(b*d - 2*a*e +
(2*c*d - b*e)*x), x], x], x] /; NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] &&
NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rubi steps

\begin {align*} \int (d+e x)^2 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx &=-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}-\frac {\int -\frac {1}{2} e^2 (14 c e f+4 c d g-9 b e g) (d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{7 c e^3}\\ &=-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}+\frac {(14 c e f+4 c d g-9 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}{14 c e}\\ &=-\frac {(14 c e f+4 c d g-9 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{84 c^2 e^2}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}+\frac {((2 c d-b e) (14 c e f+4 c d g-9 b e g)) \int (d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{24 c^2 e}\\ &=-\frac {(2 c d-b e) (14 c e f+4 c d g-9 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{120 c^3 e^2}-\frac {(14 c e f+4 c d g-9 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{84 c^2 e^2}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}+\frac {\left ((2 c d-b e)^2 (14 c e f+4 c d g-9 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}{48 c^3 e}\\ &=\frac {(2 c d-b e)^2 (14 c e f+4 c d g-9 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}}{384 c^4 e}-\frac {(2 c d-b e) (14 c e f+4 c d g-9 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{120 c^3 e^2}-\frac {(14 c e f+4 c d g-9 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{84 c^2 e^2}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}+\frac {\left ((2 c d-b e)^4 (14 c e f+4 c d g-9 b e g)\right ) \int \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{256 c^4 e}\\ &=\frac {(2 c d-b e)^4 (14 c e f+4 c d g-9 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{1024 c^5 e}+\frac {(2 c d-b e)^2 (14 c e f+4 c d g-9 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}}{384 c^4 e}-\frac {(2 c d-b e) (14 c e f+4 c d g-9 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{120 c^3 e^2}-\frac {(14 c e f+4 c d g-9 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{84 c^2 e^2}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}+\frac {\left ((2 c d-b e)^6 (14 c e f+4 c d g-9 b e g)\right ) \int \frac {1}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{2048 c^5 e}\\ &=\frac {(2 c d-b e)^4 (14 c e f+4 c d g-9 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{1024 c^5 e}+\frac {(2 c d-b e)^2 (14 c e f+4 c d g-9 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}}{384 c^4 e}-\frac {(2 c d-b e) (14 c e f+4 c d g-9 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{120 c^3 e^2}-\frac {(14 c e f+4 c d g-9 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{84 c^2 e^2}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}+\frac {\left ((2 c d-b e)^6 (14 c e f+4 c d g-9 b e g)\right ) \operatorname {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 )}{1024 c^5 e}\\ &=\frac {(2 c d-b e)^4 (14 c e f+4 c d g-9 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{1024 c^5 e}+\frac {(2 c d-b e)^2 (14 c e f+4 c d g-9 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}}{384 c^4 e}-\frac {(2 c d-b e) (14 c e f+4 c d g-9 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{120 c^3 e^2}-\frac {(14 c e f+4 c d g-9 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{84 c^2 e^2}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}+\frac {(2 c d-b e)^6 (14 c e f+4 c d g-9 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 )}{2048 c^{11/2} e^2}\\ \end {align*}

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Mathematica [B]  time = 6.30, size = 1329, normalized size = 3.22 \begin {gather*} -\frac {g (c d-b e-c e x) ((d+e x) (c (d-e x)-b e))^{3/2} (d+e x)^3}{7 c e^2}-\frac {2 (c d e+(c d-b e) e) \left (-7 c f e^2-\left (\frac {9}{2} e (c d-b e)-\frac {5 c d e}{2}\right ) g\right ) ((d+e x) (c (d-e x)-b e))^{3/2} \left (1-\frac {c e^2 (d+e x)}{(c d e+(c d-b e) e) \left (\frac {c d e^2}{c d e+(c d-b e) e}+\frac {(c d-b e) e^2}{c d e+(c d-b e) e}\right )}\right )^{5/2} \left (\frac {63 (c d e+(c d-b e) e)^5 \left (-\frac {32 c^4 (d+e x)^4 e^8}{35 (c d e+(c d-b e) e)^4 \left (\frac {c d e^2}{c d e+(c d-b e) e}+\frac {(c d-b e) e^2}{c d e+(c d-b e) e}\right )^4}-\frac {16 c^3 (d+e x)^3 e^6}{15 (c d e+(c d-b e) e)^3 \left (\frac {c d e^2}{c d e+(c d-b e) e}+\frac {(c d-b e) e^2}{c d e+(c d-b e) e}\right )^3}-\frac {4 c^2 (d+e x)^2 e^4}{3 (c d e+(c d-b e) e)^2 \left (\frac {c d e^2}{c d e+(c d-b e) e}+\frac {(c d-b e) e^2}{c d e+(c d-b e) e}\right )^2}-\frac {2 c (d+e x) e^2}{(c d e+(c d-b e) e) \left (\frac {c d e^2}{c d e+(c d-b e) e}+\frac {(c d-b e) e^2}{c d e+(c d-b e) e}\right )}+\frac {2 \sqrt {c} \sqrt {d+e x} \sin ^{-1}\left (\frac {\sqrt {c} e \sqrt {d+e x}}{\sqrt {c d e+(c d-b e) e} \sqrt {\frac {c d e^2}{c d e+(c d-b e) e}+\frac {(c d-b e) e^2}{c d e+(c d-b e) e}}}\right ) e}{\sqrt {c d e+(c d-b e) e} \sqrt {\frac {c d e^2}{c d e+(c d-b e) e}+\frac {(c d-b e) e^2}{c d e+(c d-b e) e}} \sqrt {1-\frac {c e^2 (d+e x)}{(c d e+(c d-b e) e) \left (\frac {c d e^2}{c d e+(c d-b e) e}+\frac {(c d-b e) e^2}{c d e+(c d-b e) e}\right )}}}\right ) \left (\frac {c d e^2}{c d e+(c d-b e) e}+\frac {(c d-b e) e^2}{c d e+(c d-b e) e}\right )^5}{2048 c^5 e^{10} (d+e x)^5 \left (1-\frac {c e^2 (d+e x)}{(c d e+(c d-b e) e) \left (\frac {c d e^2}{c d e+(c d-b e) e}+\frac {(c d-b e) e^2}{c d e+(c d-b e) e}\right )}\right )^2}+\frac {3}{4} \left (\frac {1}{1-\frac {c e^2 (d+e x)}{(c d e+(c d-b e) e) \left (\frac {c d e^2}{c d e+(c d-b e) e}+\frac {(c d-b e) e^2}{c d e+(c d-b e) e}\right )}}+\frac {3}{10 \left (1-\frac {c e^2 (d+e x)}{(c d e+(c d-b e) e) \left (\frac {c d e^2}{c d e+(c d-b e) e}+\frac {(c d-b e) e^2}{c d e+(c d-b e) e}\right )}\right )^2}\right )\right ) (d+e x)^3}{63 c e^4 \left (\frac {e}{\frac {c d e^2}{c d e+(c d-b e) e}+\frac {(c d-b e) e^2}{c d e+(c d-b e) e}}\right )^{3/2} (c d-b e-c e x) \sqrt {\frac {e (c d-b e-c e x)}{c d e+(c d-b e) e}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/7*(g*(d + e*x)^3*(c*d - b*e - c*e*x)*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2))/(c*e^2) - (2*(c*d*e + e*(c*d
- b*e))*(-7*c*e^2*f - ((-5*c*d*e)/2 + (9*e*(c*d - b*e))/2)*g)*(d + e*x)^3*((d + e*x)*(-(b*e) + c*(d - e*x)))^
(3/2)*(1 - (c*e^2*(d + e*x))/((c*d*e + e*(c*d - b*e))*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(
c*d*e + e*(c*d - b*e)))))^(5/2)*((3*(3/(10*(1 - (c*e^2*(d + e*x))/((c*d*e + e*(c*d - b*e))*((c*d*e^2)/(c*d*e +
e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e)))))^2) + (1 - (c*e^2*(d + e*x))/((c*d*e + e*(c*d -
b*e))*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e)))))^(-1)))/4 + (63*(c*d*e
+ e*(c*d - b*e))^5*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e)))^5*((-2*c*e^
2*(d + e*x))/((c*d*e + e*(c*d - b*e))*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d -
b*e)))) - (4*c^2*e^4*(d + e*x)^2)/(3*(c*d*e + e*(c*d - b*e))^2*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d
- b*e))/(c*d*e + e*(c*d - b*e)))^2) - (16*c^3*e^6*(d + e*x)^3)/(15*(c*d*e + e*(c*d - b*e))^3*((c*d*e^2)/(c*d*
e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e)))^3) - (32*c^4*e^8*(d + e*x)^4)/(35*(c*d*e + e*(
c*d - b*e))^4*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e)))^4) + (2*Sqrt[c]*
e*Sqrt[d + e*x]*ArcSin[(Sqrt[c]*e*Sqrt[d + e*x])/(Sqrt[c*d*e + e*(c*d - b*e)]*Sqrt[(c*d*e^2)/(c*d*e + e*(c*d -
b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e))])])/(Sqrt[c*d*e + e*(c*d - b*e)]*Sqrt[(c*d*e^2)/(c*d*e + e*
(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e))]*Sqrt[1 - (c*e^2*(d + e*x))/((c*d*e + e*(c*d - b*e))*
((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e))))])))/(2048*c^5*e^10*(d + e*x)^
5*(1 - (c*e^2*(d + e*x))/((c*d*e + e*(c*d - b*e))*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*
e + e*(c*d - b*e)))))^2)))/(63*c*e^4*(e/((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d
- b*e))))^(3/2)*(c*d - b*e - c*e*x)*Sqrt[(e*(c*d - b*e - c*e*x))/(c*d*e + e*(c*d - b*e))])

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IntegrateAlgebraic [F]  time = 180.22, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

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fricas [B]  time = 3.35, size = 1877, normalized size = 4.54

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/430080*(105*(14*(64*c^7*d^6*e - 192*b*c^6*d^5*e^2 + 240*b^2*c^5*d^4*e^3 - 160*b^3*c^4*d^3*e^4 + 60*b^4*c^3*
d^2*e^5 - 12*b^5*c^2*d*e^6 + b^6*c*e^7)*f + (256*c^7*d^7 - 1344*b*c^6*d^6*e + 2688*b^2*c^5*d^5*e^2 - 2800*b^3*
c^4*d^4*e^3 + 1680*b^4*c^3*d^3*e^4 - 588*b^5*c^2*d^2*e^5 + 112*b^6*c*d*e^6 - 9*b^7*e^7)*g)*sqrt(-c)*log(8*c^2*
e^2*x^2 + 8*b*c*e^2*x - 4*c^2*d^2 + 4*b*c*d*e + b^2*e^2 + 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*
x + b*e)*sqrt(-c)) - 4*(15360*c^7*e^6*g*x^6 + 1280*(14*c^7*e^6*f + (28*c^7*d*e^5 + 15*b*c^6*e^6)*g)*x^5 + 128*
(14*(24*c^7*d*e^5 + 13*b*c^6*e^6)*f - (24*c^7*d^2*e^4 - 556*b*c^6*d*e^5 - 3*b^2*c^5*e^6)*g)*x^4 - 16*(14*(20*c
^7*d^2*e^4 - 404*b*c^6*d*e^5 - 3*b^2*c^5*e^6)*f + (3920*c^7*d^3*e^3 - 5636*b*c^6*d^2*e^4 - 216*b^2*c^5*d*e^5 +
27*b^3*c^4*e^6)*g)*x^3 - 8*(14*(768*c^7*d^3*e^3 - 1092*b*c^6*d^2*e^4 - 60*b^2*c^5*d*e^5 + 7*b^3*c^4*e^6)*f +
(4992*c^7*d^4*e^2 - 3600*b*c^6*d^3*e^3 - 1932*b^2*c^5*d^2*e^4 + 568*b^3*c^4*d*e^5 - 63*b^4*c^3*e^6)*g)*x^2 + 1
4*(3072*c^7*d^5*e - 9840*b*c^6*d^4*e^2 + 10464*b^2*c^5*d^3*e^3 - 4816*b^3*c^4*d^2*e^4 + 1120*b^4*c^3*d*e^5 - 1
05*b^5*c^2*e^6)*f + (27648*c^7*d^6 - 97728*b*c^6*d^5*e + 145776*b^2*c^5*d^4*e^2 - 113440*b^3*c^4*d^3*e^3 + 478
24*b^4*c^3*d^2*e^4 - 10500*b^5*c^2*d*e^5 + 945*b^6*c*e^6)*g - 2*(14*(2160*c^7*d^4*e^2 - 1248*b*c^6*d^3*e^3 - 1
248*b^2*c^5*d^2*e^4 + 336*b^3*c^4*d*e^5 - 35*b^4*c^3*e^6)*f - (6720*c^7*d^5*e - 21648*b*c^6*d^4*e^2 + 24480*b^
2*c^5*d^3*e^3 - 12576*b^3*c^4*d^2*e^4 + 3164*b^4*c^3*d*e^5 - 315*b^5*c^2*e^6)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x
+ c*d^2 - b*d*e))/(c^6*e^2), -1/215040*(105*(14*(64*c^7*d^6*e - 192*b*c^6*d^5*e^2 + 240*b^2*c^5*d^4*e^3 - 160*
b^3*c^4*d^3*e^4 + 60*b^4*c^3*d^2*e^5 - 12*b^5*c^2*d*e^6 + b^6*c*e^7)*f + (256*c^7*d^7 - 1344*b*c^6*d^6*e + 268
8*b^2*c^5*d^5*e^2 - 2800*b^3*c^4*d^4*e^3 + 1680*b^4*c^3*d^3*e^4 - 588*b^5*c^2*d^2*e^5 + 112*b^6*c*d*e^6 - 9*b^
7*e^7)*g)*sqrt(c)*arctan(1/2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(c)/(c^2*e^2*x^2 +
b*c*e^2*x - c^2*d^2 + b*c*d*e)) + 2*(15360*c^7*e^6*g*x^6 + 1280*(14*c^7*e^6*f + (28*c^7*d*e^5 + 15*b*c^6*e^6)
*g)*x^5 + 128*(14*(24*c^7*d*e^5 + 13*b*c^6*e^6)*f - (24*c^7*d^2*e^4 - 556*b*c^6*d*e^5 - 3*b^2*c^5*e^6)*g)*x^4
- 16*(14*(20*c^7*d^2*e^4 - 404*b*c^6*d*e^5 - 3*b^2*c^5*e^6)*f + (3920*c^7*d^3*e^3 - 5636*b*c^6*d^2*e^4 - 216*b
^2*c^5*d*e^5 + 27*b^3*c^4*e^6)*g)*x^3 - 8*(14*(768*c^7*d^3*e^3 - 1092*b*c^6*d^2*e^4 - 60*b^2*c^5*d*e^5 + 7*b^3
*c^4*e^6)*f + (4992*c^7*d^4*e^2 - 3600*b*c^6*d^3*e^3 - 1932*b^2*c^5*d^2*e^4 + 568*b^3*c^4*d*e^5 - 63*b^4*c^3*e
^6)*g)*x^2 + 14*(3072*c^7*d^5*e - 9840*b*c^6*d^4*e^2 + 10464*b^2*c^5*d^3*e^3 - 4816*b^3*c^4*d^2*e^4 + 1120*b^4
*c^3*d*e^5 - 105*b^5*c^2*e^6)*f + (27648*c^7*d^6 - 97728*b*c^6*d^5*e + 145776*b^2*c^5*d^4*e^2 - 113440*b^3*c^4
*d^3*e^3 + 47824*b^4*c^3*d^2*e^4 - 10500*b^5*c^2*d*e^5 + 945*b^6*c*e^6)*g - 2*(14*(2160*c^7*d^4*e^2 - 1248*b*c
^6*d^3*e^3 - 1248*b^2*c^5*d^2*e^4 + 336*b^3*c^4*d*e^5 - 35*b^4*c^3*e^6)*f - (6720*c^7*d^5*e - 21648*b*c^6*d^4*
e^2 + 24480*b^2*c^5*d^3*e^3 - 12576*b^3*c^4*d^2*e^4 + 3164*b^4*c^3*d*e^5 - 315*b^5*c^2*e^6)*g)*x)*sqrt(-c*e^2*
x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^6*e^2)]

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giac [B]  time = 0.47, size = 910, normalized size = 2.20 \begin {gather*} -\frac {1}{107520} \, \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 (12 \, c g x e^{4} + \frac {{\left (28 \, c^{7} d g e^{13} + 14 \, c^{7} f e^{14} + 15 \, b c^{6} g e^{14}\right )} e^{\left (-10\right )}}{c^{6}}\right )} x - \frac {{\left (24 \, c^{7} d^{2} g e^{12} - 336 \, c^{7} d f e^{13} - 556 \, b c^{6} d g e^{13} - 182 \, b c^{6} f e^{14} - 3 \, b^{2} c^{5} g e^{14}\right )} e^{\left (-10\right )}}{c^{6}}\right )} x - \frac {{\left (3920 \, c^{7} d^{3} g e^{11} + 280 \, c^{7} d^{2} f e^{12} - 5636 \, b c^{6} d^{2} g e^{12} - 5656 \, b c^{6} d f e^{13} - 216 \, b^{2} c^{5} d g e^{13} - 42 \, b^{2} c^{5} f e^{14} + 27 \, b^{3} c^{4} g e^{14}\right )} e^{\left (-10\right )}}{c^{6}}\right )} x - \frac {{\left (4992 \, c^{7} d^{4} g e^{10} + 10752 \, c^{7} d^{3} f e^{11} - 3600 \, b c^{6} d^{3} g e^{11} - 15288 \, b c^{6} d^{2} f e^{12} - 1932 \, b^{2} c^{5} d^{2} g e^{12} - 840 \, b^{2} c^{5} d f e^{13} + 568 \, b^{3} c^{4} d g e^{13} + 98 \, b^{3} c^{4} f e^{14} - 63 \, b^{4} c^{3} g e^{14}\right )} e^{\left (-10\right )}}{c^{6}}\right )} x + \frac {{\left (6720 \, c^{7} d^{5} g e^{9} - 30240 \, c^{7} d^{4} f e^{10} - 21648 \, b c^{6} d^{4} g e^{10} + 17472 \, b c^{6} d^{3} f e^{11} + 24480 \, b^{2} c^{5} d^{3} g e^{11} + 17472 \, b^{2} c^{5} d^{2} f e^{12} - 12576 \, b^{3} c^{4} d^{2} g e^{12} - 4704 \, b^{3} c^{4} d f e^{13} + 3164 \, b^{4} c^{3} d g e^{13} + 490 \, b^{4} c^{3} f e^{14} - 315 \, b^{5} c^{2} g e^{14}\right )} e^{\left (-10\right )}}{c^{6}}\right )} x + \frac {{\left (27648 \, c^{7} d^{6} g e^{8} + 43008 \, c^{7} d^{5} f e^{9} - 97728 \, b c^{6} d^{5} g e^{9} - 137760 \, b c^{6} d^{4} f e^{10} + 145776 \, b^{2} c^{5} d^{4} g e^{10} + 146496 \, b^{2} c^{5} d^{3} f e^{11} - 113440 \, b^{3} c^{4} d^{3} g e^{11} - 67424 \, b^{3} c^{4} d^{2} f e^{12} + 47824 \, b^{4} c^{3} d^{2} g e^{12} + 15680 \, b^{4} c^{3} d f e^{13} - 10500 \, b^{5} c^{2} d g e^{13} - 1470 \, b^{5} c^{2} f e^{14} + 945 \, b^{6} c g e^{14}\right )} e^{\left (-10\right )}}{c^{6}}\right )} + \frac {{\left (256 \, c^{7} d^{7} g + 896 \, c^{7} d^{6} f e - 1344 \, b c^{6} d^{6} g e - 2688 \, b c^{6} d^{5} f e^{2} + 2688 \, b^{2} c^{5} d^{5} g e^{2} + 3360 \, b^{2} c^{5} d^{4} f e^{3} - 2800 \, b^{3} c^{4} d^{4} g e^{3} - 2240 \, b^{3} c^{4} d^{3} f e^{4} + 1680 \, b^{4} c^{3} d^{3} g e^{4} + 840 \, b^{4} c^{3} d^{2} f e^{5} - 588 \, b^{5} c^{2} d^{2} g e^{5} - 168 \, b^{5} c^{2} d f e^{6} + 112 \, b^{6} c d g e^{6} + 14 \, b^{6} c f e^{7} - 9 \, b^{7} g e^{7}\right )} \sqrt {-c e^{2}} e^{\left (-3\right )} \log \left ({\left | -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 - \sqrt {-c e^{2}} b \right |}\right )}{2048 \, c^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/107520*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(2*(4*(2*(8*(10*(12*c*g*x*e^4 + (28*c^7*d*g*e^13 + 14*c^7
*f*e^14 + 15*b*c^6*g*e^14)*e^(-10)/c^6)*x - (24*c^7*d^2*g*e^12 - 336*c^7*d*f*e^13 - 556*b*c^6*d*g*e^13 - 182*b
*c^6*f*e^14 - 3*b^2*c^5*g*e^14)*e^(-10)/c^6)*x - (3920*c^7*d^3*g*e^11 + 280*c^7*d^2*f*e^12 - 5636*b*c^6*d^2*g*
e^12 - 5656*b*c^6*d*f*e^13 - 216*b^2*c^5*d*g*e^13 - 42*b^2*c^5*f*e^14 + 27*b^3*c^4*g*e^14)*e^(-10)/c^6)*x - (4
992*c^7*d^4*g*e^10 + 10752*c^7*d^3*f*e^11 - 3600*b*c^6*d^3*g*e^11 - 15288*b*c^6*d^2*f*e^12 - 1932*b^2*c^5*d^2*
g*e^12 - 840*b^2*c^5*d*f*e^13 + 568*b^3*c^4*d*g*e^13 + 98*b^3*c^4*f*e^14 - 63*b^4*c^3*g*e^14)*e^(-10)/c^6)*x +
(6720*c^7*d^5*g*e^9 - 30240*c^7*d^4*f*e^10 - 21648*b*c^6*d^4*g*e^10 + 17472*b*c^6*d^3*f*e^11 + 24480*b^2*c^5*
d^3*g*e^11 + 17472*b^2*c^5*d^2*f*e^12 - 12576*b^3*c^4*d^2*g*e^12 - 4704*b^3*c^4*d*f*e^13 + 3164*b^4*c^3*d*g*e^
13 + 490*b^4*c^3*f*e^14 - 315*b^5*c^2*g*e^14)*e^(-10)/c^6)*x + (27648*c^7*d^6*g*e^8 + 43008*c^7*d^5*f*e^9 - 97
728*b*c^6*d^5*g*e^9 - 137760*b*c^6*d^4*f*e^10 + 145776*b^2*c^5*d^4*g*e^10 + 146496*b^2*c^5*d^3*f*e^11 - 113440
*b^3*c^4*d^3*g*e^11 - 67424*b^3*c^4*d^2*f*e^12 + 47824*b^4*c^3*d^2*g*e^12 + 15680*b^4*c^3*d*f*e^13 - 10500*b^5
*c^2*d*g*e^13 - 1470*b^5*c^2*f*e^14 + 945*b^6*c*g*e^14)*e^(-10)/c^6) + 1/2048*(256*c^7*d^7*g + 896*c^7*d^6*f*e
- 1344*b*c^6*d^6*g*e - 2688*b*c^6*d^5*f*e^2 + 2688*b^2*c^5*d^5*g*e^2 + 3360*b^2*c^5*d^4*f*e^3 - 2800*b^3*c^4*
d^4*g*e^3 - 2240*b^3*c^4*d^3*f*e^4 + 1680*b^4*c^3*d^3*g*e^4 + 840*b^4*c^3*d^2*f*e^5 - 588*b^5*c^2*d^2*g*e^5 -
168*b^5*c^2*d*f*e^6 + 112*b^6*c*d*g*e^6 + 14*b^6*c*f*e^7 - 9*b^7*g*e^7)*sqrt(-c*e^2)*e^(-3)*log(abs(-2*(sqrt(-
c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))*c - sqrt(-c*e^2)*b))/c^6

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maple [B]  time = 0.07, size = 2799, normalized size = 6.78

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c*f+7/60*b/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*f+7/32*(-c
*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^4*f-21/256*b^5/c^3*e^5/(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*f+19/128*e^3*g*b^4/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d+5/
24*e*g*b^2/c^2*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d+105/128*e^3*g*b^4/c^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^3-175/128*e^2*g*b^3/c/(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^4-147/512*e^4*g*b^5/c^3/(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^2+7/128*e^5*g*b^6/c^4/(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-31/64*e^2*g*b^3/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+
c*d^2)^(1/2)*x*d^2+7/24*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2*f-3/40*g*b^2/c^3*(-c*e^2*x^2-b*e^2*x-b*d*
e+c*d^2)^(5/2)-1/7*g*x^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c-3/128*e^2*g*b^4/c^4*(-c*e^2*x^2-b*e^2*x-b*d*
e+c*d^2)^(3/2)-9/1024*e^4*g*b^6/c^5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)+105/256*b^4/c^2*e^4/(c*e^2)^(1/2)*a
rctan((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^2*f-35/32*b^3/c*e^3/(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*f-7/32*b^3/c^2*e^3*(-c*e^2*x^2-b*e^
2*x-b*d*e+c*d^2)^(1/2)*x*d*f-21/16*e*c/(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))*b*d^5*f-7/24*e/c*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b*d*f+21/32*b^2/c*(-c*e^2*x^2-b*e^2*
x-b*d*e+c*d^2)^(1/2)*x*d^2*e^2*f+3/4*e*g*b^2/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^3-7/64*b^4/c^3*e^3*(
-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*f+7/1024*b^6/c^4*e^6/(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))*f+7/256*b^4/c^3*e^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*f+105/64*b^
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^4*e^2*f-7/8*e*(-c*e
^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b*d^3*f-1/3*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c/e*d*g-21/32*g*b*c/(
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^6-13/48*g*b/c*x*(-c*e^
2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2-3/64*e^2*g*b^3/c^3*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)+21/16*e*g*b^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^5+3/8*e*g*b^3/c^2*(-
c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^3+5/48*e*g*b^3/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d-9/35/e^2*g/
c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*d^2-2/5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c/e*d*f-17/64*g*b^2/c*
(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^4+3/28*g*b/c^2*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)-13/96*g*b^2/c
^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2-17/32*g*b*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^4+7/16*c*(-
c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^4*f+7/16*c^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^6*f+7/48/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b*d^2*f+7/512*b^5/c^4*e^4*(
-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*f+7/192*b^3/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*e^2*f+1/16/e*(-c*
e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^5*g+1/12/e*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^3*g-31/128*e^2*g*
b^4/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2+19/256*e^3*g*b^5/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)
*d-9/2048*e^6*g*b^7/c^5/(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))
-9/512*e^4*g*b^5/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x+61/210/e*g/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^
(5/2)*b*d+7/96*b^2/c^2*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*e^2*f+21/64*b^3/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+
c*d^2)^(1/2)*d^2*e^2*f+1/8/e*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^5*g-7/16*e/c*(-c*e^2*x^2-b*e^2*x-b*d
*e+c*d^2)^(1/2)*b^2*d^3*f+1/8/e*c^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^7*g-7/48*e/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b^2*d*f+1/24/e/c*(-c*e^2*x^2-b*e^2*x-b*d*
e+c*d^2)^(3/2)*b*d^3*g

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

[Out]

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(b*e-2*c*d>0)', see `assume?` f
or more details)Is b*e-2*c*d zero or nonzero?

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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 )}^2\,{\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)^2*(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)^2*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2), x)

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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 )^{2} \left (f + g x\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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