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

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

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Rubi [A]  time = 0.83, 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 = {794, 670, 640, 612, 621, 204} \begin {gather*} \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 {7 (2 c d-b e)^5 (-3 b e g+2 c d g+4 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 )}{1024 c^{11/2} 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 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) \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 ) \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 )}{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 = 5.93, size = 527, normalized size = 1.27 \begin {gather*} \frac {(d+e x)^3 ((d+e x) (c (d-e x)-b e))^{3/2} \left (\frac {9 (-3 b e g+2 c d g+4 c e f) \left (-384 c^5 e^{12} (d+e x)^5 \sqrt {e (2 c d-b e)} (b e-2 c d) \sqrt {\frac {b e-c d+c e x}{b e-2 c d}}-48 c^4 e^{12} (d+e x)^4 \sqrt {e (2 c d-b e)} (b e-2 c d)^2 \sqrt {\frac {b e-c d+c e x}{b e-2 c d}}+56 c^3 e^{12} (d+e x)^3 \sqrt {e (2 c d-b e)} (b e-2 c d)^3 \sqrt {\frac {b e-c d+c e x}{b e-2 c d}}-70 c^2 e^{12} (d+e x)^2 \sqrt {e (2 c d-b e)} (b e-2 c d)^4 \sqrt {\frac {b e-c d+c e x}{b e-2 c d}}+105 \sqrt {c} e^{25/2} \sqrt {d+e x} (b e-2 c d)^6 \sin ^{-1}\left (\frac {\sqrt {c} \sqrt {e} \sqrt {d+e x}}{\sqrt {e (2 c d-b e)}}\right )+105 c e^{12} (d+e x) \sqrt {e (2 c d-b e)} (b e-2 c d)^5 \sqrt {\frac {b e-c d+c e x}{b e-2 c d}}\right )}{1280 c^5 e^{11} (d+e x)^5 \sqrt {e (2 c d-b e)} (b e-2 c d)^2 \left (\frac {b e-c d+c e x}{b e-2 c d}\right )^{3/2}}-9 e g\right )}{54 c e^3} \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]

((d + e*x)^3*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*(-9*e*g + (9*(4*c*e*f + 2*c*d*g - 3*b*e*g)*(105*c*e^12*S
qrt[e*(2*c*d - b*e)]*(-2*c*d + b*e)^5*(d + e*x)*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)] - 70*c^2*e^12*Sqrt
[e*(2*c*d - b*e)]*(-2*c*d + b*e)^4*(d + e*x)^2*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)] + 56*c^3*e^12*Sqrt[
e*(2*c*d - b*e)]*(-2*c*d + b*e)^3*(d + e*x)^3*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)] - 48*c^4*e^12*Sqrt[e
*(2*c*d - b*e)]*(-2*c*d + b*e)^2*(d + e*x)^4*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)] - 384*c^5*e^12*Sqrt[e
*(2*c*d - b*e)]*(-2*c*d + b*e)*(d + e*x)^5*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)] + 105*Sqrt[c]*e^(25/2)*
(-2*c*d + b*e)^6*Sqrt[d + e*x]*ArcSin[(Sqrt[c]*Sqrt[e]*Sqrt[d + e*x])/Sqrt[e*(2*c*d - b*e)]]))/(1280*c^5*e^11*
Sqrt[e*(2*c*d - b*e)]*(-2*c*d + b*e)^2*(d + e*x)^5*((-(c*d) + b*e + c*e*x)/(-2*c*d + b*e))^(3/2))))/(54*c*e^3)

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IntegrateAlgebraic [F]  time = 180.06, 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)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2],x]

[Out]

$Aborted

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fricas [A]  time = 1.59, size = 1469, normalized size = 3.55

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)^(1/2),x, algorithm="fricas")

[Out]

[-1/30720*(105*(4*(32*c^6*d^5*e - 80*b*c^5*d^4*e^2 + 80*b^2*c^4*d^3*e^3 - 40*b^3*c^3*d^2*e^4 + 10*b^4*c^2*d*e^
5 - b^5*c*e^6)*f + (64*c^6*d^6 - 256*b*c^5*d^5*e + 400*b^2*c^4*d^4*e^2 - 320*b^3*c^3*d^3*e^3 + 140*b^4*c^2*d^2
*e^4 - 32*b^5*c*d*e^5 + 3*b^6*e^6)*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*(1280*c^6*e^5*g*x^5 + 128*(12*
c^6*e^5*f + (36*c^6*d*e^4 + b*c^5*e^5)*g)*x^4 + 16*(12*(30*c^6*d*e^4 + b*c^5*e^5)*f + (340*c^6*d^2*e^3 + 56*b*
c^5*d*e^4 - 9*b^2*c^4*e^5)*g)*x^3 + 8*(4*(224*c^6*d^2*e^3 + 46*b*c^5*d*e^4 - 7*b^2*c^4*e^5)*f + (128*c^6*d^3*e
^2 + 380*b*c^5*d^2*e^3 - 152*b^2*c^4*d*e^4 + 21*b^3*c^3*e^5)*g)*x^2 - 4*(2176*c^6*d^4*e - 4472*b*c^5*d^3*e^2 +
2996*b^2*c^4*d^2*e^3 - 910*b^3*c^3*d*e^4 + 105*b^4*c^2*e^5)*f - (5632*c^6*d^5 - 16752*b*c^5*d^4*e + 19408*b^2
*c^4*d^3*e^2 - 10808*b^3*c^3*d^2*e^3 + 2940*b^4*c^2*d*e^4 - 315*b^5*c*e^5)*g + 2*(4*(120*c^6*d^3*e^2 + 716*b*c
^5*d^2*e^3 - 266*b^2*c^4*d*e^4 + 35*b^3*c^3*e^5)*f - (1680*c^6*d^4*e - 3632*b*c^5*d^3*e^2 + 2680*b^2*c^4*d^2*e
^3 - 868*b^3*c^3*d*e^4 + 105*b^4*c^2*e^5)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^6*e^2), -1/1536
0*(105*(4*(32*c^6*d^5*e - 80*b*c^5*d^4*e^2 + 80*b^2*c^4*d^3*e^3 - 40*b^3*c^3*d^2*e^4 + 10*b^4*c^2*d*e^5 - b^5*
c*e^6)*f + (64*c^6*d^6 - 256*b*c^5*d^5*e + 400*b^2*c^4*d^4*e^2 - 320*b^3*c^3*d^3*e^3 + 140*b^4*c^2*d^2*e^4 - 3
2*b^5*c*d*e^5 + 3*b^6*e^6)*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)*sq
rt(c)/(c^2*e^2*x^2 + b*c*e^2*x - c^2*d^2 + b*c*d*e)) - 2*(1280*c^6*e^5*g*x^5 + 128*(12*c^6*e^5*f + (36*c^6*d*e
^4 + b*c^5*e^5)*g)*x^4 + 16*(12*(30*c^6*d*e^4 + b*c^5*e^5)*f + (340*c^6*d^2*e^3 + 56*b*c^5*d*e^4 - 9*b^2*c^4*e
^5)*g)*x^3 + 8*(4*(224*c^6*d^2*e^3 + 46*b*c^5*d*e^4 - 7*b^2*c^4*e^5)*f + (128*c^6*d^3*e^2 + 380*b*c^5*d^2*e^3
- 152*b^2*c^4*d*e^4 + 21*b^3*c^3*e^5)*g)*x^2 - 4*(2176*c^6*d^4*e - 4472*b*c^5*d^3*e^2 + 2996*b^2*c^4*d^2*e^3 -
910*b^3*c^3*d*e^4 + 105*b^4*c^2*e^5)*f - (5632*c^6*d^5 - 16752*b*c^5*d^4*e + 19408*b^2*c^4*d^3*e^2 - 10808*b^
3*c^3*d^2*e^3 + 2940*b^4*c^2*d*e^4 - 315*b^5*c*e^5)*g + 2*(4*(120*c^6*d^3*e^2 + 716*b*c^5*d^2*e^3 - 266*b^2*c^
4*d*e^4 + 35*b^3*c^3*e^5)*f - (1680*c^6*d^4*e - 3632*b*c^5*d^3*e^2 + 2680*b^2*c^4*d^2*e^3 - 868*b^3*c^3*d*e^4
+ 105*b^4*c^2*e^5)*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 [A]  time = 0.47, size = 699, normalized size = 1.69 \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^{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 )}{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)*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 = 0.11, size = 2217, normalized size = 5.36

result too large to display

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)

[Out]

-21/16*b/c*e*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2*f-35/64*e^2*g*b^3/c^3*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*
d^2)^(1/2)*d+35/128*b^4/c^3*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*f+21/32*b^2/c^2*e^2*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*f-7/32*e^4*g*b^5/c^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+21/16*e*g*b^2/c^2*x*(-c*e^2*x^2-b*e^
2*x-b*d*e+c*d^2)^(1/2)*d^2-35/16*e^2*g*b^3/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+35/16*b^2/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-35/32*b^3/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^2*e^3*f+175/64*e*g*b^2/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+245/256*e^3*g*b^4/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+7/8*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^3*f-3/4*x*(-c*e^2*x^2-b*e^2*x-b*d*
e+c*d^2)^(3/2)/c*d*f+7/16/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^3*f+7/8*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*f-77/120*g*b^2/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d
^2)^(3/2)*d-21/32*g/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^2*d^3-1/6*e*g*x^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c
*d^2)^(3/2)/c+7/16/e*g*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^4+91/120/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2
)^(3/2)*b*d*f-3/5*x^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c*d*g-1/5*x^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3
/2)/c*e*f-7/48*b^2/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*e*f-7/128*b^4/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^
2)^(1/2)*e^3*f-7/4*g/(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+7/64*e*g*b^3/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)-11/15/e^2/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)
*d^3*g-17/15/e/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2*f+21/512*e^3*g*b^5/c^5*(-c*e^2*x^2-b*e^2*x-b*d*e+c
*d^2)^(1/2)-7/8/e*g/c*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2+7/40*b/c^2*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^
2)^(3/2)*e*f-7/64*b^3/c^3*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*e^3*f-21/32*b^2/c^2*e*(-c*e^2*x^2-b*e^2*x-b
*d*e+c*d^2)^(1/2)*d^2*f-35/16*b*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^4*f+21/64*b^3/c^3*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*f-7/256*b^5/c^4*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))*f-21/16*g/c*x*(-c*e^2*x^2-b*e^2*x-b*d
*e+c*d^2)^(1/2)*b*d^3+13/20*g/c^2*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b*d+3/20*e*g*b/c^2*x^2*(-c*e^2*x^2-
b*e^2*x-b*d*e+c*d^2)^(3/2)-21/160*e*g*b^2/c^3*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)+21/1024*e^5*g*b^6/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))+21/256*e^3*g*b^4/c^4*x*(
-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-35/128*e^2*g*b^4/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d+21/32*e*g*
b^3/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2+59/48/e*g*b/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2+
7/32/e*g/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^4+7/16/e*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^6

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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(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 [B]  time = 8.14, size = 3311, normalized size = 8.00

result too large to display

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^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*d^2*g*x*(c*d^2 - c*e^2
*x^2 - b*d*e - b*e^2*x)^(3/2))/(4*c*e)

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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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