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

Optimal. Leaf size=223 \[ \frac {(2 c d-b e)^3 (-5 b e g+2 c d g+8 c e f) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{128 c^{7/2} e^2}+\frac {(b+2 c x) (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+2 c d g+8 c e f)}{64 c^3 e}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (5 b e g-8 c (d g+e f)-6 c e g x)}{24 c^2 e^2} \]

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Rubi [A]  time = 0.28, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {779, 612, 621, 204} \begin {gather*} \frac {(b+2 c x) (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+2 c d g+8 c e f)}{64 c^3 e}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (5 b e g-8 c (d g+e f)-6 c e g x)}{24 c^2 e^2}+\frac {(2 c d-b e)^3 (-5 b e g+2 c d g+8 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 )}{128 c^{7/2} e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*(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)*(8*c*e*f + 2*c*d*g - 5*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(64*c^3*e)
+ ((5*b*e*g - 8*c*(e*f + d*g) - 6*c*e*g*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(24*c^2*e^2) + ((2*c*
d - b*e)^3*(8*c*e*f + 2*c*d*g - 5*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])])/(128*c^(7/2)*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 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rubi steps

\begin {align*} \int (d+e x) (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx &=\frac {(5 b e g-8 c (e f+d g)-6 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 c^2 e^2}+\frac {((2 c d-b e) (8 c e f+2 c d g-5 b e g)) \int \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) (8 c e f+2 c d g-5 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{64 c^3 e}+\frac {(5 b e g-8 c (e f+d g)-6 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 c^2 e^2}+\frac {\left ((2 c d-b e)^3 (8 c e f+2 c d g-5 b e g)\right ) \int \frac {1}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{128 c^3 e}\\ &=\frac {(2 c d-b e) (8 c e f+2 c d g-5 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{64 c^3 e}+\frac {(5 b e g-8 c (e f+d g)-6 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 c^2 e^2}+\frac {\left ((2 c d-b e)^3 (8 c e f+2 c d g-5 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 )}{64 c^3 e}\\ &=\frac {(2 c d-b e) (8 c e f+2 c d g-5 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{64 c^3 e}+\frac {(5 b e g-8 c (e f+d g)-6 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 c^2 e^2}+\frac {(2 c d-b e)^3 (8 c e f+2 c d g-5 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 )}{128 c^{7/2} e^2}\\ \end {align*}

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Mathematica [A]  time = 2.44, size = 391, normalized size = 1.75 \begin {gather*} \frac {(d+e x) ((d+e x) (c (d-e x)-b e))^{3/2} \left (\frac {5 (-5 b e g+2 c d g+8 c e f) \left (-8 c^3 e^8 (d+e x)^3 \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}}-2 c^2 e^8 (d+e x)^2 \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}}+3 \sqrt {c} e^{17/2} \sqrt {d+e x} (b e-2 c d)^4 \sin ^{-1}\left (\frac {\sqrt {c} \sqrt {e} \sqrt {d+e x}}{\sqrt {e (2 c d-b e)}}\right )+3 c e^8 (d+e x) \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}}\right )}{48 c^3 e^7 (d+e x)^3 \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}}-5 e g\right )}{20 c e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*(-5*e*g + (5*(8*c*e*f + 2*c*d*g - 5*b*e*g)*(3*c*e^8*Sqrt[e
*(2*c*d - b*e)]*(-2*c*d + b*e)^3*(d + e*x)*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)] - 2*c^2*e^8*Sqrt[e*(2*c
*d - b*e)]*(-2*c*d + b*e)^2*(d + e*x)^2*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)] - 8*c^3*e^8*Sqrt[e*(2*c*d
- b*e)]*(-2*c*d + b*e)*(d + e*x)^3*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)] + 3*Sqrt[c]*e^(17/2)*(-2*c*d +
b*e)^4*Sqrt[d + e*x]*ArcSin[(Sqrt[c]*Sqrt[e]*Sqrt[d + e*x])/Sqrt[e*(2*c*d - b*e)]]))/(48*c^3*e^7*Sqrt[e*(2*c*d
- b*e)]*(-2*c*d + b*e)^2*(d + e*x)^3*((-(c*d) + b*e + c*e*x)/(-2*c*d + b*e))^(3/2))))/(20*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)*(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 = 0.60, size = 825, normalized size = 3.70 \begin {gather*} \left [-\frac {3 \, {\left (8 \, {\left (8 \, c^{4} d^{3} e - 12 \, b c^{3} d^{2} e^{2} + 6 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} f + {\left (16 \, c^{4} d^{4} - 64 \, b c^{3} d^{3} e + 72 \, b^{2} c^{2} d^{2} e^{2} - 32 \, b^{3} c d e^{3} + 5 \, b^{4} e^{4}\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 (48 \, c^{4} e^{3} g x^{3} + 8 \, {\left (8 \, c^{4} e^{3} f + {\left (8 \, c^{4} d e^{2} + b c^{3} e^{3}\right )} g\right )} x^{2} - 8 \, {\left (8 \, c^{4} d^{2} e - 14 \, b c^{3} d e^{2} + 3 \, b^{2} c^{2} e^{3}\right )} f - {\left (64 \, c^{4} d^{3} - 116 \, b c^{3} d^{2} e + 76 \, b^{2} c^{2} d e^{2} - 15 \, b^{3} c e^{3}\right )} g + 2 \, {\left (8 \, {\left (6 \, c^{4} d e^{2} + b c^{3} e^{3}\right )} f - {\left (12 \, c^{4} d^{2} e - 20 \, b c^{3} d e^{2} + 5 \, b^{2} c^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{768 \, c^{4} e^{2}}, -\frac {3 \, {\left (8 \, {\left (8 \, c^{4} d^{3} e - 12 \, b c^{3} d^{2} e^{2} + 6 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} f + {\left (16 \, c^{4} d^{4} - 64 \, b c^{3} d^{3} e + 72 \, b^{2} c^{2} d^{2} e^{2} - 32 \, b^{3} c d e^{3} + 5 \, b^{4} e^{4}\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 (48 \, c^{4} e^{3} g x^{3} + 8 \, {\left (8 \, c^{4} e^{3} f + {\left (8 \, c^{4} d e^{2} + b c^{3} e^{3}\right )} g\right )} x^{2} - 8 \, {\left (8 \, c^{4} d^{2} e - 14 \, b c^{3} d e^{2} + 3 \, b^{2} c^{2} e^{3}\right )} f - {\left (64 \, c^{4} d^{3} - 116 \, b c^{3} d^{2} e + 76 \, b^{2} c^{2} d e^{2} - 15 \, b^{3} c e^{3}\right )} g + 2 \, {\left (8 \, {\left (6 \, c^{4} d e^{2} + b c^{3} e^{3}\right )} f - {\left (12 \, c^{4} d^{2} e - 20 \, b c^{3} d e^{2} + 5 \, b^{2} c^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{384 \, c^{4} e^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="fricas")

[Out]

[-1/768*(3*(8*(8*c^4*d^3*e - 12*b*c^3*d^2*e^2 + 6*b^2*c^2*d*e^3 - b^3*c*e^4)*f + (16*c^4*d^4 - 64*b*c^3*d^3*e
+ 72*b^2*c^2*d^2*e^2 - 32*b^3*c*d*e^3 + 5*b^4*e^4)*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*(48*c^4*e^3*g*
x^3 + 8*(8*c^4*e^3*f + (8*c^4*d*e^2 + b*c^3*e^3)*g)*x^2 - 8*(8*c^4*d^2*e - 14*b*c^3*d*e^2 + 3*b^2*c^2*e^3)*f -
(64*c^4*d^3 - 116*b*c^3*d^2*e + 76*b^2*c^2*d*e^2 - 15*b^3*c*e^3)*g + 2*(8*(6*c^4*d*e^2 + b*c^3*e^3)*f - (12*c
^4*d^2*e - 20*b*c^3*d*e^2 + 5*b^2*c^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^4*e^2), -1/384
*(3*(8*(8*c^4*d^3*e - 12*b*c^3*d^2*e^2 + 6*b^2*c^2*d*e^3 - b^3*c*e^4)*f + (16*c^4*d^4 - 64*b*c^3*d^3*e + 72*b^
2*c^2*d^2*e^2 - 32*b^3*c*d*e^3 + 5*b^4*e^4)*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*(48*c^4*e^3*g*x^3 + 8*(8*c^4*e^3*f +
(8*c^4*d*e^2 + b*c^3*e^3)*g)*x^2 - 8*(8*c^4*d^2*e - 14*b*c^3*d*e^2 + 3*b^2*c^2*e^3)*f - (64*c^4*d^3 - 116*b*c
^3*d^2*e + 76*b^2*c^2*d*e^2 - 15*b^3*c*e^3)*g + 2*(8*(6*c^4*d*e^2 + b*c^3*e^3)*f - (12*c^4*d^2*e - 20*b*c^3*d*
e^2 + 5*b^2*c^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^4*e^2)]

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giac [A]  time = 0.39, size = 381, normalized size = 1.71 \begin {gather*} \frac {1}{192} \, \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e} {\left (2 \, {\left (4 \, {\left (6 \, g x e + \frac {{\left (8 \, c^{3} d g e^{4} + 8 \, c^{3} f e^{5} + b c^{2} g e^{5}\right )} e^{\left (-4\right )}}{c^{3}}\right )} x - \frac {{\left (12 \, c^{3} d^{2} g e^{3} - 48 \, c^{3} d f e^{4} - 20 \, b c^{2} d g e^{4} - 8 \, b c^{2} f e^{5} + 5 \, b^{2} c g e^{5}\right )} e^{\left (-4\right )}}{c^{3}}\right )} x - \frac {{\left (64 \, c^{3} d^{3} g e^{2} + 64 \, c^{3} d^{2} f e^{3} - 116 \, b c^{2} d^{2} g e^{3} - 112 \, b c^{2} d f e^{4} + 76 \, b^{2} c d g e^{4} + 24 \, b^{2} c f e^{5} - 15 \, b^{3} g e^{5}\right )} e^{\left (-4\right )}}{c^{3}}\right )} + \frac {{\left (16 \, c^{4} d^{4} g + 64 \, c^{4} d^{3} f e - 64 \, b c^{3} d^{3} g e - 96 \, b c^{3} d^{2} f e^{2} + 72 \, b^{2} c^{2} d^{2} g e^{2} + 48 \, b^{2} c^{2} d f e^{3} - 32 \, b^{3} c d g e^{3} - 8 \, b^{3} c f e^{4} + 5 \, b^{4} g e^{4}\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 )}{128 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="giac")

[Out]

1/192*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(2*(4*(6*g*x*e + (8*c^3*d*g*e^4 + 8*c^3*f*e^5 + b*c^2*g*e^5)*
e^(-4)/c^3)*x - (12*c^3*d^2*g*e^3 - 48*c^3*d*f*e^4 - 20*b*c^2*d*g*e^4 - 8*b*c^2*f*e^5 + 5*b^2*c*g*e^5)*e^(-4)/
c^3)*x - (64*c^3*d^3*g*e^2 + 64*c^3*d^2*f*e^3 - 116*b*c^2*d^2*g*e^3 - 112*b*c^2*d*f*e^4 + 76*b^2*c*d*g*e^4 + 2
4*b^2*c*f*e^5 - 15*b^3*g*e^5)*e^(-4)/c^3) + 1/128*(16*c^4*d^4*g + 64*c^4*d^3*f*e - 64*b*c^3*d^3*g*e - 96*b*c^3
*d^2*f*e^2 + 72*b^2*c^2*d^2*g*e^2 + 48*b^2*c^2*d*f*e^3 - 32*b^3*c*d*g*e^3 - 8*b^3*c*f*e^4 + 5*b^4*g*e^4)*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
^4

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maple [B]  time = 0.07, size = 1114, normalized size = 5.00 \begin {gather*} \frac {5 b^{4} e^{3} g \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{128 \sqrt {c \,e^{2}}\, c^{3}}-\frac {b^{3} d \,e^{2} g \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{4 \sqrt {c \,e^{2}}\, c^{2}}-\frac {b^{3} e^{3} f \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{16 \sqrt {c \,e^{2}}\, c^{2}}+\frac {9 b^{2} d^{2} e g \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{16 \sqrt {c \,e^{2}}\, c}+\frac {3 b^{2} d \,e^{2} f \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{8 \sqrt {c \,e^{2}}\, c}-\frac {b \,d^{3} g \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{2 \sqrt {c \,e^{2}}}-\frac {3 b \,d^{2} e f \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{4 \sqrt {c \,e^{2}}}+\frac {c \,d^{4} g \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{8 \sqrt {c \,e^{2}}\, e}+\frac {c \,d^{3} f \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{2 \sqrt {c \,e^{2}}}+\frac {5 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, b^{2} e g x}{32 c^{2}}-\frac {3 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, b d g x}{8 c}-\frac {\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, b e f x}{4 c}+\frac {\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, d^{2} g x}{8 e}+\frac {\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, d f x}{2}+\frac {5 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, b^{3} e g}{64 c^{3}}-\frac {3 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, b^{2} d g}{16 c^{2}}-\frac {\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, b^{2} e f}{8 c^{2}}+\frac {\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, b \,d^{2} g}{16 c e}+\frac {\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, b d f}{4 c}-\frac {\left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}} g x}{4 c e}+\frac {5 \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}} b g}{24 c^{2} e}-\frac {\left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}} d g}{3 c \,e^{2}}-\frac {\left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}} f}{3 c e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)

[Out]

-1/4*b/c*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*e*f-1/16*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))*f-3/4*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^2*e*f+3/8*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*f-1/4*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+9/16*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^2+5/32*e*g*b^2/c^2*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)+5/128*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))-3/8*g/c*x*(-c*e^2*x^2-b*e^
2*x-b*d*e+c*d^2)^(1/2)*b*d+1/16/e*g/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^2+1/8/e*g*c/(c*e^2)^(1/2)*arc
tan((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+1/2*d*f*x*(-c*e^2*x^2-b*e^2*x-b*d*e+
c*d^2)^(1/2)-1/3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e*f+1/2*d^3*f*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))+1/8/e*g*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2-1/3*(-
c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e^2*d*g-1/8*b^2/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*e*f+1/4*d*f/
c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b+5/64*e*g*b^3/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/4/e*g*x*(
-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c+5/24/e*g*b/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)-3/16*g/c^2*(-c*e
^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^2*d-1/2*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^3

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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)*(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 = 4.16, size = 801, normalized size = 3.59 \begin {gather*} d\,f\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}+\frac {5\,b\,e\,g\,\left (\frac {\ln \left (b\,e^2-2\,\sqrt {-c\,e^2}\,\sqrt {-\left (d+e\,x\right )\,\left (b\,e-c\,d+c\,e\,x\right )}+2\,c\,e^2\,x\right )\,\left (b^3\,e^6+4\,b\,c\,e^4\,\left (c\,d^2-b\,d\,e\right )\right )}{16\,{\left (-c\,e^2\right )}^{5/2}}-\frac {\left (8\,c\,e^2\,\left (-c\,d^2+b\,d\,e+c\,e^2\,x^2\right )-3\,b^2\,e^4+2\,b\,c\,e^4\,x\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{24\,c^2\,e^4}\right )}{8\,c}-\frac {d\,g\,\ln \left (b\,e^2-2\,\sqrt {-c\,e^2}\,\sqrt {-\left (d+e\,x\right )\,\left (b\,e-c\,d+c\,e\,x\right )}+2\,c\,e^2\,x\right )\,\left (b^3\,e^6+4\,b\,c\,e^4\,\left (c\,d^2-b\,d\,e\right )\right )}{16\,{\left (-c\,e^2\right )}^{5/2}}-\frac {e\,f\,\ln \left (b\,e^2-2\,\sqrt {-c\,e^2}\,\sqrt {-\left (d+e\,x\right )\,\left (b\,e-c\,d+c\,e\,x\right )}+2\,c\,e^2\,x\right )\,\left (b^3\,e^6+4\,b\,c\,e^4\,\left (c\,d^2-b\,d\,e\right )\right )}{16\,{\left (-c\,e^2\right )}^{5/2}}+\frac {f\,\left (8\,c\,e^2\,\left (-c\,d^2+b\,d\,e+c\,e^2\,x^2\right )-3\,b^2\,e^4+2\,b\,c\,e^4\,x\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{24\,c^2\,e^3}+\frac {g\,\left (c\,d^2-b\,d\,e\right )\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}-\frac {\ln \left (b\,e^2-2\,\sqrt {-c\,e^2}\,\sqrt {-\left (d+e\,x\right )\,\left (b\,e-c\,d+c\,e\,x\right )}+2\,c\,e^2\,x\right )\,\left (\frac {b^2\,e^4}{4}+c\,e^2\,\left (c\,d^2-b\,d\,e\right )\right )}{2\,{\left (-c\,e^2\right )}^{3/2}}\right )}{4\,c\,e}-\frac {d\,f\,\ln \left (b\,e^2-2\,\sqrt {-c\,e^2}\,\sqrt {-\left (d+e\,x\right )\,\left (b\,e-c\,d+c\,e\,x\right )}+2\,c\,e^2\,x\right )\,\left (\frac {b^2\,e^4}{4}+c\,e^2\,\left (c\,d^2-b\,d\,e\right )\right )}{2\,{\left (-c\,e^2\right )}^{3/2}}-\frac {g\,x\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2}}{4\,c\,e}+\frac {d\,g\,\left (8\,c\,e^2\,\left (-c\,d^2+b\,d\,e+c\,e^2\,x^2\right )-3\,b^2\,e^4+2\,b\,c\,e^4\,x\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{24\,c^2\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f + g*x)*(d + e*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2),x)

[Out]

d*f*(x/2 + b/(4*c))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) + (5*b*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) - (d*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*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)) + (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))/(24*c^2*e^3)
+ (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) - (d*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)) - (g*x*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(4*c*e)
+ (d*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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(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)*(f + g*x), x)

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