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

Optimal. Leaf size=339 \[ \frac {(2 c d-b e)^4 (-7 b e g+4 c d g+10 c e f) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{256 c^{9/2} e^2}+\frac {(b+2 c x) (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+4 c d g+10 c e f)}{128 c^4 e}-\frac {(2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-7 b e g+4 c d g+10 c e f)}{48 c^3 e^2}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-7 b e g+4 c d g+10 c e f)}{40 c^2 e^2}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e^2} \]

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Rubi [A]  time = 0.46, antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 7, 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)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+4 c d g+10 c e f)}{128 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 )^{3/2} (-7 b e g+4 c d g+10 c e f)}{48 c^3 e^2}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-7 b e g+4 c d g+10 c e f)}{40 c^2 e^2}+\frac {(2 c d-b e)^4 (-7 b e g+4 c d g+10 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 )}{256 c^{9/2} e^2}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*c*d - b*e)^2*(10*c*e*f + 4*c*d*g - 7*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(128*c^
4*e) - ((2*c*d - b*e)*(10*c*e*f + 4*c*d*g - 7*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(48*c^3*e^2)
- ((10*c*e*f + 4*c*d*g - 7*b*e*g)*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(40*c^2*e^2) - (g*(d
+ e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(5*c*e^2) + ((2*c*d - b*e)^4*(10*c*e*f + 4*c*d*g - 7*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])])/(256*c^(9/2)*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) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx &=-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e^2}-\frac {\int -\frac {1}{2} e^2 (10 c e f+4 c d g-7 b e g) (d+e x)^2 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{5 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 )^{3/2}}{5 c e^2}+\frac {(10 c e f+4 c d g-7 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}{10 c e}\\ &=-\frac {(10 c e f+4 c d g-7 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{40 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 )^{3/2}}{5 c e^2}+\frac {((2 c d-b e) (10 c e f+4 c d g-7 b e g)) \int (d+e x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{16 c^2 e}\\ &=-\frac {(2 c d-b e) (10 c e f+4 c d g-7 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{48 c^3 e^2}-\frac {(10 c e f+4 c d g-7 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{40 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 )^{3/2}}{5 c e^2}+\frac {\left ((2 c d-b e)^2 (10 c e f+4 c d g-7 b e g)\right ) \int \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{32 c^3 e}\\ &=\frac {(2 c d-b e)^2 (10 c e f+4 c d g-7 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{128 c^4 e}-\frac {(2 c d-b e) (10 c e f+4 c d g-7 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{48 c^3 e^2}-\frac {(10 c e f+4 c d g-7 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{40 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 )^{3/2}}{5 c e^2}+\frac {\left ((2 c d-b e)^4 (10 c e f+4 c d g-7 b e g)\right ) \int \frac {1}{\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)^2 (10 c e f+4 c d g-7 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{128 c^4 e}-\frac {(2 c d-b e) (10 c e f+4 c d g-7 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{48 c^3 e^2}-\frac {(10 c e f+4 c d g-7 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{40 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 )^{3/2}}{5 c e^2}+\frac {\left ((2 c d-b e)^4 (10 c e f+4 c d g-7 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 )}{128 c^4 e}\\ &=\frac {(2 c d-b e)^2 (10 c e f+4 c d g-7 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{128 c^4 e}-\frac {(2 c d-b e) (10 c e f+4 c d g-7 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{48 c^3 e^2}-\frac {(10 c e f+4 c d g-7 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{40 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 )^{3/2}}{5 c e^2}+\frac {(2 c d-b e)^4 (10 c e f+4 c d g-7 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 )}{256 c^{9/2} e^2}\\ \end {align*}

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Mathematica [A]  time = 3.93, size = 464, normalized size = 1.37 \begin {gather*} \frac {(d+e x)^2 ((d+e x) (c (d-e x)-b e))^{3/2} \left (\frac {7 (-7 b e g+4 c d g+10 c e f) \left (-48 c^4 e^{10} (d+e x)^4 \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}}-e^4 (b e-2 c d)^2 \left (8 c^3 e^6 (d+e x)^3 \sqrt {e (2 c d-b e)} \sqrt {\frac {b e-c d+c e x}{b e-2 c d}}-10 c^2 e^6 (d+e x)^2 \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}}+15 \sqrt {c} e^{13/2} \sqrt {d+e x} (b e-2 c d)^3 \sin ^{-1}\left (\frac {\sqrt {c} \sqrt {e} \sqrt {d+e x}}{\sqrt {e (2 c d-b e)}}\right )+15 c e^6 (d+e x) \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}}\right )\right )}{384 c^4 e^9 (d+e x)^4 \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}}-7 e g\right )}{35 c e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*(-7*e*g + (7*(10*c*e*f + 4*c*d*g - 7*b*e*g)*(-48*c^4*e^1
0*Sqrt[e*(2*c*d - b*e)]*(-2*c*d + b*e)*(d + e*x)^4*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)] - e^4*(-2*c*d +
b*e)^2*(15*c*e^6*Sqrt[e*(2*c*d - b*e)]*(-2*c*d + b*e)^2*(d + e*x)*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)]
- 10*c^2*e^6*Sqrt[e*(2*c*d - b*e)]*(-2*c*d + b*e)*(d + e*x)^2*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)] + 8
*c^3*e^6*Sqrt[e*(2*c*d - b*e)]*(d + e*x)^3*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)] + 15*Sqrt[c]*e^(13/2)*(
-2*c*d + b*e)^3*Sqrt[d + e*x]*ArcSin[(Sqrt[c]*Sqrt[e]*Sqrt[d + e*x])/Sqrt[e*(2*c*d - b*e)]])))/(384*c^4*e^9*Sq
rt[e*(2*c*d - b*e)]*(-2*c*d + b*e)^2*(d + e*x)^4*((-(c*d) + b*e + c*e*x)/(-2*c*d + b*e))^(3/2))))/(35*c*e^3)

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IntegrateAlgebraic [F]  time = 180.03, 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)*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 = 0.90, size = 1117, normalized size = 3.29 \begin {gather*} \left [\frac {15 \, {\left (10 \, {\left (16 \, c^{5} d^{4} e - 32 \, b c^{4} d^{3} e^{2} + 24 \, b^{2} c^{3} d^{2} e^{3} - 8 \, b^{3} c^{2} d e^{4} + b^{4} c e^{5}\right )} f + {\left (64 \, c^{5} d^{5} - 240 \, b c^{4} d^{4} e + 320 \, b^{2} c^{3} d^{3} e^{2} - 200 \, b^{3} c^{2} d^{2} e^{3} + 60 \, b^{4} c d e^{4} - 7 \, b^{5} e^{5}\right )} g\right )} \sqrt {-c} \log \left (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} {\left (2 \, c e x + b e\right )} \sqrt {-c}\right ) + 4 \, {\left (384 \, c^{5} e^{4} g x^{4} + 48 \, {\left (10 \, c^{5} e^{4} f + {\left (20 \, c^{5} d e^{3} + b c^{4} e^{4}\right )} g\right )} x^{3} + 8 \, {\left (10 \, {\left (16 \, c^{5} d e^{3} + b c^{4} e^{4}\right )} f + {\left (64 \, c^{5} d^{2} e^{2} + 36 \, b c^{4} d e^{3} - 7 \, b^{2} c^{3} e^{4}\right )} g\right )} x^{2} - 10 \, {\left (128 \, c^{5} d^{3} e - 228 \, b c^{4} d^{2} e^{2} + 100 \, b^{2} c^{3} d e^{3} - 15 \, b^{3} c^{2} e^{4}\right )} f - {\left (896 \, c^{5} d^{4} - 2192 \, b c^{4} d^{3} e + 1996 \, b^{2} c^{3} d^{2} e^{2} - 760 \, b^{3} c^{2} d e^{3} + 105 \, b^{4} c e^{4}\right )} g + 2 \, {\left (10 \, {\left (36 \, c^{5} d^{2} e^{2} + 28 \, b c^{4} d e^{3} - 5 \, b^{2} c^{3} e^{4}\right )} f - {\left (240 \, c^{5} d^{3} e - 436 \, b c^{4} d^{2} e^{2} + 216 \, b^{2} c^{3} d e^{3} - 35 \, b^{3} c^{2} e^{4}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{7680 \, c^{5} e^{2}}, -\frac {15 \, {\left (10 \, {\left (16 \, c^{5} d^{4} e - 32 \, b c^{4} d^{3} e^{2} + 24 \, b^{2} c^{3} d^{2} e^{3} - 8 \, b^{3} c^{2} d e^{4} + b^{4} c e^{5}\right )} f + {\left (64 \, c^{5} d^{5} - 240 \, b c^{4} d^{4} e + 320 \, b^{2} c^{3} d^{3} e^{2} - 200 \, b^{3} c^{2} d^{2} e^{3} + 60 \, b^{4} c d e^{4} - 7 \, b^{5} e^{5}\right )} g\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {c}}{2 \, {\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) - 2 \, {\left (384 \, c^{5} e^{4} g x^{4} + 48 \, {\left (10 \, c^{5} e^{4} f + {\left (20 \, c^{5} d e^{3} + b c^{4} e^{4}\right )} g\right )} x^{3} + 8 \, {\left (10 \, {\left (16 \, c^{5} d e^{3} + b c^{4} e^{4}\right )} f + {\left (64 \, c^{5} d^{2} e^{2} + 36 \, b c^{4} d e^{3} - 7 \, b^{2} c^{3} e^{4}\right )} g\right )} x^{2} - 10 \, {\left (128 \, c^{5} d^{3} e - 228 \, b c^{4} d^{2} e^{2} + 100 \, b^{2} c^{3} d e^{3} - 15 \, b^{3} c^{2} e^{4}\right )} f - {\left (896 \, c^{5} d^{4} - 2192 \, b c^{4} d^{3} e + 1996 \, b^{2} c^{3} d^{2} e^{2} - 760 \, b^{3} c^{2} d e^{3} + 105 \, b^{4} c e^{4}\right )} g + 2 \, {\left (10 \, {\left (36 \, c^{5} d^{2} e^{2} + 28 \, b c^{4} d e^{3} - 5 \, b^{2} c^{3} e^{4}\right )} f - {\left (240 \, c^{5} d^{3} e - 436 \, b c^{4} d^{2} e^{2} + 216 \, b^{2} c^{3} d e^{3} - 35 \, b^{3} c^{2} e^{4}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{3840 \, c^{5} e^{2}}\right ] \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)^(1/2),x, algorithm="fricas")

[Out]

[1/7680*(15*(10*(16*c^5*d^4*e - 32*b*c^4*d^3*e^2 + 24*b^2*c^3*d^2*e^3 - 8*b^3*c^2*d*e^4 + b^4*c*e^5)*f + (64*c
^5*d^5 - 240*b*c^4*d^4*e + 320*b^2*c^3*d^3*e^2 - 200*b^3*c^2*d^2*e^3 + 60*b^4*c*d*e^4 - 7*b^5*e^5)*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*(384*c^5*e^4*g*x^4 + 48*(10*c^5*e^4*f + (20*c^5*d*e^3 + b*c^4*e^4)*g)*x^3 +
8*(10*(16*c^5*d*e^3 + b*c^4*e^4)*f + (64*c^5*d^2*e^2 + 36*b*c^4*d*e^3 - 7*b^2*c^3*e^4)*g)*x^2 - 10*(128*c^5*d^
3*e - 228*b*c^4*d^2*e^2 + 100*b^2*c^3*d*e^3 - 15*b^3*c^2*e^4)*f - (896*c^5*d^4 - 2192*b*c^4*d^3*e + 1996*b^2*c
^3*d^2*e^2 - 760*b^3*c^2*d*e^3 + 105*b^4*c*e^4)*g + 2*(10*(36*c^5*d^2*e^2 + 28*b*c^4*d*e^3 - 5*b^2*c^3*e^4)*f
- (240*c^5*d^3*e - 436*b*c^4*d^2*e^2 + 216*b^2*c^3*d*e^3 - 35*b^3*c^2*e^4)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c
*d^2 - b*d*e))/(c^5*e^2), -1/3840*(15*(10*(16*c^5*d^4*e - 32*b*c^4*d^3*e^2 + 24*b^2*c^3*d^2*e^3 - 8*b^3*c^2*d*
e^4 + b^4*c*e^5)*f + (64*c^5*d^5 - 240*b*c^4*d^4*e + 320*b^2*c^3*d^3*e^2 - 200*b^3*c^2*d^2*e^3 + 60*b^4*c*d*e^
4 - 7*b^5*e^5)*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*(384*c^5*e^4*g*x^4 + 48*(10*c^5*e^4*f + (20*c^5*d*e^3 + b*c^4*e^4
)*g)*x^3 + 8*(10*(16*c^5*d*e^3 + b*c^4*e^4)*f + (64*c^5*d^2*e^2 + 36*b*c^4*d*e^3 - 7*b^2*c^3*e^4)*g)*x^2 - 10*
(128*c^5*d^3*e - 228*b*c^4*d^2*e^2 + 100*b^2*c^3*d*e^3 - 15*b^3*c^2*e^4)*f - (896*c^5*d^4 - 2192*b*c^4*d^3*e +
1996*b^2*c^3*d^2*e^2 - 760*b^3*c^2*d*e^3 + 105*b^4*c*e^4)*g + 2*(10*(36*c^5*d^2*e^2 + 28*b*c^4*d*e^3 - 5*b^2*
c^3*e^4)*f - (240*c^5*d^3*e - 436*b*c^4*d^2*e^2 + 216*b^2*c^3*d*e^3 - 35*b^3*c^2*e^4)*g)*x)*sqrt(-c*e^2*x^2 -
b*e^2*x + c*d^2 - b*d*e))/(c^5*e^2)]

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giac [A]  time = 0.35, size = 526, normalized size = 1.55 \begin {gather*} \frac {1}{1920} \, \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, g x e^{2} + \frac {{\left (20 \, c^{4} d g e^{7} + 10 \, c^{4} f e^{8} + b c^{3} g e^{8}\right )} e^{\left (-6\right )}}{c^{4}}\right )} x + \frac {{\left (64 \, c^{4} d^{2} g e^{6} + 160 \, c^{4} d f e^{7} + 36 \, b c^{3} d g e^{7} + 10 \, b c^{3} f e^{8} - 7 \, b^{2} c^{2} g e^{8}\right )} e^{\left (-6\right )}}{c^{4}}\right )} x - \frac {{\left (240 \, c^{4} d^{3} g e^{5} - 360 \, c^{4} d^{2} f e^{6} - 436 \, b c^{3} d^{2} g e^{6} - 280 \, b c^{3} d f e^{7} + 216 \, b^{2} c^{2} d g e^{7} + 50 \, b^{2} c^{2} f e^{8} - 35 \, b^{3} c g e^{8}\right )} e^{\left (-6\right )}}{c^{4}}\right )} x - \frac {{\left (896 \, c^{4} d^{4} g e^{4} + 1280 \, c^{4} d^{3} f e^{5} - 2192 \, b c^{3} d^{3} g e^{5} - 2280 \, b c^{3} d^{2} f e^{6} + 1996 \, b^{2} c^{2} d^{2} g e^{6} + 1000 \, b^{2} c^{2} d f e^{7} - 760 \, b^{3} c d g e^{7} - 150 \, b^{3} c f e^{8} + 105 \, b^{4} g e^{8}\right )} e^{\left (-6\right )}}{c^{4}}\right )} + \frac {{\left (64 \, c^{5} d^{5} g + 160 \, c^{5} d^{4} f e - 240 \, b c^{4} d^{4} g e - 320 \, b c^{4} d^{3} f e^{2} + 320 \, b^{2} c^{3} d^{3} g e^{2} + 240 \, b^{2} c^{3} d^{2} f e^{3} - 200 \, b^{3} c^{2} d^{2} g e^{3} - 80 \, b^{3} c^{2} d f e^{4} + 60 \, b^{4} c d g e^{4} + 10 \, b^{4} c f e^{5} - 7 \, b^{5} g e^{5}\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 )}{256 \, c^{5}} \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)^(1/2),x, algorithm="giac")

[Out]

1/1920*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(2*(4*(6*(8*g*x*e^2 + (20*c^4*d*g*e^7 + 10*c^4*f*e^8 + b*c^3
*g*e^8)*e^(-6)/c^4)*x + (64*c^4*d^2*g*e^6 + 160*c^4*d*f*e^7 + 36*b*c^3*d*g*e^7 + 10*b*c^3*f*e^8 - 7*b^2*c^2*g*
e^8)*e^(-6)/c^4)*x - (240*c^4*d^3*g*e^5 - 360*c^4*d^2*f*e^6 - 436*b*c^3*d^2*g*e^6 - 280*b*c^3*d*f*e^7 + 216*b^
2*c^2*d*g*e^7 + 50*b^2*c^2*f*e^8 - 35*b^3*c*g*e^8)*e^(-6)/c^4)*x - (896*c^4*d^4*g*e^4 + 1280*c^4*d^3*f*e^5 - 2
192*b*c^3*d^3*g*e^5 - 2280*b*c^3*d^2*f*e^6 + 1996*b^2*c^2*d^2*g*e^6 + 1000*b^2*c^2*d*f*e^7 - 760*b^3*c*d*g*e^7
- 150*b^3*c*f*e^8 + 105*b^4*g*e^8)*e^(-6)/c^4) + 1/256*(64*c^5*d^5*g + 160*c^5*d^4*f*e - 240*b*c^4*d^4*g*e -
320*b*c^4*d^3*f*e^2 + 320*b^2*c^3*d^3*g*e^2 + 240*b^2*c^3*d^2*f*e^3 - 200*b^3*c^2*d^2*g*e^3 - 80*b^3*c^2*d*f*e
^4 + 60*b^4*c*d*g*e^4 + 10*b^4*c*f*e^5 - 7*b^5*g*e^5)*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^5

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maple [B]  time = 0.07, size = 1618, normalized size = 4.77

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)^(1/2),x)

[Out]

15/64*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+7
/40*g*b/c^2*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)-7/128*e^2*g*b^4/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2
)-5/4*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))*b*d^3*f+1/4/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))*d^5*g+5/128*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))*f-5/16*e/c^2*(-c*e^2*
x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^2*d*f+1/8/e/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^3*g-1/2*x*(-c*e^2*x^
2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e*d*g+15/16*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^2*e^2*f-5/8*e/c*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d*f-5/16*b^3/c^2*e^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*f-25/32*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^2+1/2*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+5/4*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^3-1/5*g*x^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c-7/48*g*b^2/c^3*(-c*e^2*x^
2-b*e^2*x-b*d*e+c*d^2)^(3/2)-1/4*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c*f+5/24*b/c^2*(-c*e^2*x^2-b*e^2*x-b
*d*e+c*d^2)^(3/2)*f+5/8*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2*f+11/20/e*g/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e
+c*d^2)^(3/2)*b*d-11/16*g*b/c*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2+1/4*e*g*b^3/c^3*(-c*e^2*x^2-b*e^2*x
-b*d*e+c*d^2)^(1/2)*d-7/64*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)-7/256*e^4*g*b^5/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))+5/32*b^2/c^2*x*(-c*e^2*x^2-b*e^
2*x-b*d*e+c*d^2)^(1/2)*e^2*f-11/32*g*b^2/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2-15/16*g*b/(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^4-7/15/e^2*g/c*(-c*e^2*x^2-b*e^2*
x-b*d*e+c*d^2)^(3/2)*d^2-2/3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e*d*f+5/16/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c
*d^2)^(1/2)*b*d^2*f+5/8*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^4*f+5/64*b^3/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*e^2*f+1/4/e*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1
/2)*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)^(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 = 5.55, size = 1732, normalized size = 5.11

result too large to display

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)^(1/2),x)

[Out]

d^2*f*(x/2 + b/(4*c))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) - (f*x*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)
^(3/2))/(4*c) - (g*x^2*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(5*c) + (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) - (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) - (7*b*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) - (d^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^2*e^4)/4 + c*e^2*(c*d^2
- b*d*e)))/(2*(-c*e^2)^(3/2)) + (5*b*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) - (d^2*g*lo
g(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*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) - (d*g*x*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(2*c*e)
+ (5*b*d*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)))/(4*c) - (d*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)))/(8*(-c*e^2)^(5/2)) + (
d*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
))/(12*c^2*e^3) + (d*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))))/(2*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 )^{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)**(1/2),x)

[Out]

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

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