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

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

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Rubi [A]  time = 1.07, 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 = {794, 670, 640, 612, 621, 204} \begin {gather*} \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 {9 (2 c d-b e)^7 (-11 b e g+6 c d g+16 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 )}{32768 c^{13/2} 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 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 794

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)
*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g
, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 0])

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 ) \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 )}{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 [B]  time = 6.48, size = 1418, normalized size = 2.91 \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)^4}{8 c e^2}-\frac {(c d e+(c d-b e) e) \left (-8 c f e^2-\left (\frac {11}{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 {99 (c d e+(c d-b e) e)^6 \left (-\frac {256 c^5 (d+e x)^5 e^{10}}{315 (c d e+(c d-b e) e)^5 \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}-\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 )^6}{4096 c^6 e^{12} (d+e x)^6 \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 {11}{14} \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 {1}{4 \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)^4}{44 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)^3*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]

[Out]

-1/8*(g*(d + e*x)^4*(c*d - b*e - c*e*x)*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2))/(c*e^2) - ((c*d*e + e*(c*d -
b*e))*(-8*c*e^2*f - ((-5*c*d*e)/2 + (11*e*(c*d - b*e))/2)*g)*(d + e*x)^4*((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)*((11*(1/(4*(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)))/14 + (99*(c*d*e
+ e*(c*d - b*e))^6*((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)))^6*((-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) - (256*c^5*e^
10*(d + e*x)^5)/(315*(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*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)
)))])))/(4096*c^6*e^12*(d + e*x)^6*(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)))/(44*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 = 180.12, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(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]

$Aborted

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fricas [B]  time = 6.77, size = 2337, normalized size = 4.79

result too large to display

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*(16*(128*c^8*d^7*e - 448*b*c^7*d^6*e^2 + 672*b^2*c^6*d^5*e^3 - 560*b^3*c^5*d^4*e^4 + 280*b^4*
c^4*d^3*e^5 - 84*b^5*c^3*d^2*e^6 + 14*b^6*c^2*d*e^7 - b^7*c*e^8)*f + (768*c^8*d^8 - 4096*b*c^7*d^7*e + 8960*b^
2*c^6*d^6*e^2 - 10752*b^3*c^5*d^5*e^3 + 7840*b^4*c^4*d^4*e^4 - 3584*b^5*c^3*d^3*e^5 + 1008*b^6*c^2*d^2*e^6 - 1
60*b^7*c*d*e^7 + 11*b^8*e^8)*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*(71680*c^8*e^7*g*x^7 + 5120*(16*c^8*
e^7*f + (48*c^8*d*e^6 + 17*b*c^7*e^7)*g)*x^6 + 1280*(16*(14*c^8*d*e^6 + 5*b*c^7*e^7)*f + (140*c^8*d^2*e^5 + 32
4*b*c^7*d*e^6 + b^2*c^6*e^7)*g)*x^5 + 128*(16*(104*c^8*d^2*e^5 + 246*b*c^7*d*e^6 + b^2*c^6*e^7)*f - (2176*c^8*
d^3*e^4 - 5932*b*c^7*d^2*e^5 - 100*b^2*c^6*d*e^6 + 11*b^3*c^5*e^7)*g)*x^4 - 16*(16*(1400*c^8*d^3*e^4 - 3764*b*
c^7*d^2*e^5 - 86*b^2*c^6*d*e^6 + 9*b^3*c^5*e^7)*f + (30800*c^8*d^4*e^3 - 37984*b*c^7*d^3*e^4 - 3912*b^2*c^6*d^
2*e^5 + 1000*b^3*c^5*d*e^6 - 99*b^4*c^4*e^7)*g)*x^3 - 8*(16*(5248*c^8*d^4*e^3 - 6296*b*c^7*d^3*e^4 - 924*b^2*c
^6*d^2*e^5 + 222*b^3*c^5*d*e^6 - 21*b^4*c^4*e^7)*f + (22528*c^8*d^5*e^2 - 5904*b*c^7*d^4*e^3 - 25888*b^2*c^6*d
^3*e^4 + 11496*b^3*c^5*d^2*e^5 - 2568*b^4*c^4*d*e^6 + 231*b^5*c^3*e^7)*g)*x^2 + 16*(23552*c^8*d^6*e - 78496*b*
c^7*d^5*e^2 + 97424*b^2*c^6*d^4*e^3 - 60288*b^3*c^5*d^3*e^4 + 21168*b^4*c^4*d^2*e^5 - 3990*b^5*c^3*d*e^6 + 315
*b^6*c^2*e^7)*f + (212992*c^8*d^7 - 873408*b*c^7*d^6*e + 1519680*b^2*c^6*d^5*e^2 - 1433392*b^3*c^5*d^4*e^3 + 7
90176*b^4*c^4*d^3*e^4 - 256788*b^5*c^3*d^2*e^5 + 45780*b^6*c^2*d*e^6 - 3465*b^7*c*e^7)*g - 2*(16*(7840*c^8*d^5
*e^2 + 1392*b*c^7*d^4*e^3 - 13984*b^2*c^6*d^3*e^4 + 5760*b^3*c^5*d^2*e^5 - 1218*b^4*c^4*d*e^6 + 105*b^5*c^3*e^
7)*f - (60480*c^8*d^6*e - 208832*b*c^7*d^5*e^2 + 278416*b^2*c^6*d^4*e^3 - 188384*b^3*c^5*d^3*e^4 + 70668*b^4*c
^4*d^2*e^5 - 14028*b^5*c^3*d*e^6 + 1155*b^6*c^2*e^7)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^7*e^
2), -1/1146880*(315*(16*(128*c^8*d^7*e - 448*b*c^7*d^6*e^2 + 672*b^2*c^6*d^5*e^3 - 560*b^3*c^5*d^4*e^4 + 280*b
^4*c^4*d^3*e^5 - 84*b^5*c^3*d^2*e^6 + 14*b^6*c^2*d*e^7 - b^7*c*e^8)*f + (768*c^8*d^8 - 4096*b*c^7*d^7*e + 8960
*b^2*c^6*d^6*e^2 - 10752*b^3*c^5*d^5*e^3 + 7840*b^4*c^4*d^4*e^4 - 3584*b^5*c^3*d^3*e^5 + 1008*b^6*c^2*d^2*e^6
- 160*b^7*c*d*e^7 + 11*b^8*e^8)*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*(71680*c^8*e^7*g*x^7 + 5120*(16*c^8*e^7*f + (48*
c^8*d*e^6 + 17*b*c^7*e^7)*g)*x^6 + 1280*(16*(14*c^8*d*e^6 + 5*b*c^7*e^7)*f + (140*c^8*d^2*e^5 + 324*b*c^7*d*e^
6 + b^2*c^6*e^7)*g)*x^5 + 128*(16*(104*c^8*d^2*e^5 + 246*b*c^7*d*e^6 + b^2*c^6*e^7)*f - (2176*c^8*d^3*e^4 - 59
32*b*c^7*d^2*e^5 - 100*b^2*c^6*d*e^6 + 11*b^3*c^5*e^7)*g)*x^4 - 16*(16*(1400*c^8*d^3*e^4 - 3764*b*c^7*d^2*e^5
- 86*b^2*c^6*d*e^6 + 9*b^3*c^5*e^7)*f + (30800*c^8*d^4*e^3 - 37984*b*c^7*d^3*e^4 - 3912*b^2*c^6*d^2*e^5 + 1000
*b^3*c^5*d*e^6 - 99*b^4*c^4*e^7)*g)*x^3 - 8*(16*(5248*c^8*d^4*e^3 - 6296*b*c^7*d^3*e^4 - 924*b^2*c^6*d^2*e^5 +
222*b^3*c^5*d*e^6 - 21*b^4*c^4*e^7)*f + (22528*c^8*d^5*e^2 - 5904*b*c^7*d^4*e^3 - 25888*b^2*c^6*d^3*e^4 + 114
96*b^3*c^5*d^2*e^5 - 2568*b^4*c^4*d*e^6 + 231*b^5*c^3*e^7)*g)*x^2 + 16*(23552*c^8*d^6*e - 78496*b*c^7*d^5*e^2
+ 97424*b^2*c^6*d^4*e^3 - 60288*b^3*c^5*d^3*e^4 + 21168*b^4*c^4*d^2*e^5 - 3990*b^5*c^3*d*e^6 + 315*b^6*c^2*e^7
)*f + (212992*c^8*d^7 - 873408*b*c^7*d^6*e + 1519680*b^2*c^6*d^5*e^2 - 1433392*b^3*c^5*d^4*e^3 + 790176*b^4*c^
4*d^3*e^4 - 256788*b^5*c^3*d^2*e^5 + 45780*b^6*c^2*d*e^6 - 3465*b^7*c*e^7)*g - 2*(16*(7840*c^8*d^5*e^2 + 1392*
b*c^7*d^4*e^3 - 13984*b^2*c^6*d^3*e^4 + 5760*b^3*c^5*d^2*e^5 - 1218*b^4*c^4*d*e^6 + 105*b^5*c^3*e^7)*f - (6048
0*c^8*d^6*e - 208832*b*c^7*d^5*e^2 + 278416*b^2*c^6*d^4*e^3 - 188384*b^3*c^5*d^3*e^4 + 70668*b^4*c^4*d^2*e^5 -
14028*b^5*c^3*d*e^6 + 1155*b^6*c^2*e^7)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^7*e^2)]

________________________________________________________________________________________

giac [B]  time = 0.64, size = 1138, normalized size = 2.33

result too large to display

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)*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^7

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maple [B]  time = 0.08, size = 3576, normalized size = 7.33 \begin {gather*} \text {output too large to display} \end {gather*}

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)

[Out]

9/32*b^2/c^2*e^2*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d*f+45/256*b^4/c^3*e^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d
^2)^(1/2)*x*d*f+63/1024*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))*d*f+315/256*b^4/c^2*e^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^3*f-189/512*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^2*f-63/32*b*c*e/(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+45/32*b^2/c*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^3*f+3/8*x*(-c*e^2*x^2-b*e^2*x-b*d
*e+c*d^2)^(3/2)*d^3*f+9/32*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^5*f+3/16/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d
^2)^(3/2)*b*d^3*f+567/2048*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^2-63/64*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^3-45/64*b^3/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^2*e^3*f+63/128*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^2+765/512*e*g*b^2/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^4-27/12
8*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)*d-189/64*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^5-315/256*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^3+1125/2048*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^2+2205/1024*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^4-315/128*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*e^3*f-261/2048*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*d-45/1024*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))*d+33/2048*e^3*g*b^5/c^5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)-1
/8*e*g*x^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c+9/64/e*g*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^4+27/2
56/e*g*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^6-69/224*g*b^2/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*d-
117/128*g*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^5*b-117/256*g/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^
2*d^5-13/35/e^2/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*d^3*g-23/35/e/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2
)*d^2*f-1/7*x^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c*e*f-3/128*b^4/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3
/2)*e^3*f-15/64*g/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b^2*d^3+9/16*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(
1/2)*x*d^5*f+9/16*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
^7*f-1/2*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c*d*f-3/40*b^2/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*e*
f-9/1024*b^6/c^5*e^5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*f-3/7*x^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c
*d*g+57/140/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*b*d*f+33/640*e*g*b^3/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^
2)^(5/2)+99/16384*e^5*g*b^7/c^6*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-9/16*b/c*e*x*(-c*e^2*x^2-b*e^2*x-b*d*e+
c*d^2)^(3/2)*d^2*f-261/4096*e^4*g*b^6/c^5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d+99/32768*e^7*g*b^8/c^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))+99/8192*e^5*g*b^6/c^5*(-c*e
^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x-33/448*e*g*b^2/c^3*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)+11/112*e*g*b/c
^2*x^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)+45/64*b^3/c^2*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^3*f+7
65/1024*e*g*b^3/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^4-315/512*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^3+677/1120/e*g*b/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*d^2-45/32*b*e*(-c*e^2*x^2-b*e^2*x-
b*d*e+c*d^2)^(1/2)*x*d^4*f-3/64*b^3/c^3*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*e^3*f-45/128*b^4/c^3*(-c*e^2*
x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2*e^3*f-9/32*b^2/c^2*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2*f-45/64*b^2
/c*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^4*f-27/256*e^2*g*b^4/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*
d+9/64*b^3/c^3*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d*f-9/16/e*g/c*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5
/2)*d^2+43/112*g/c^2*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*b*d-9/8*g*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^7*b-15/32*g/c*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^
3*b+9/128/e*g/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b*d^4+27/128/e*g*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/
2)*x*d^6+27/128/e*g*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^8+189/64*b^2*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^5
*f+3/28*b/c^2*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*e*f-9/512*b^5/c^4*e^5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^
(1/2)*x*f+45/512*b^5/c^4*e^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*f-9/2048*b^7/c^5*e^7/(c*e^2)^(1/2)*arcta
n((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+63/256*e*g*b^3/c^3*(-c*e^2*x^2-b*e^2*x-b
*d*e+c*d^2)^(3/2)*d^2+33/1024*e^3*g*b^4/c^4*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)+315/128*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^6+1125/4096*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^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(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 )}^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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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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