3.2174 \(\int (d+e x) (f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx\)

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

[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)

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Rubi [A]  time = 0.281498, antiderivative size = 223, normalized size of antiderivative = 1., 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} \[ \frac{(b+2 c x) (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+2 c d g+8 c e f)}{64 c^3 e}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (5 b e g-8 c (d g+e f)-6 c e g x)}{24 c^2 e^2}+\frac{(2 c d-b e)^3 (-5 b e g+2 c d g+8 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{128 c^{7/2} e^2} \]

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 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]

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 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])

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*}

Mathematica [A]  time = 2.881, size = 391, normalized size = 1.75 \[ \frac{(d+e x) ((d+e x) (c (d-e x)-b e))^{3/2} \left (\frac{5 (-5 b e g+2 c d g+8 c e f) \left (-8 c^3 e^8 (d+e x)^3 \sqrt{e (2 c d-b e)} (b e-2 c d) \sqrt{\frac{b e-c d+c e x}{b e-2 c d}}-2 c^2 e^8 (d+e x)^2 \sqrt{e (2 c d-b e)} (b e-2 c d)^2 \sqrt{\frac{b e-c d+c e x}{b e-2 c d}}+3 c e^8 (d+e x) \sqrt{e (2 c d-b e)} (b e-2 c d)^3 \sqrt{\frac{b e-c d+c e x}{b e-2 c d}}+3 \sqrt{c} e^{17/2} \sqrt{d+e x} (b e-2 c d)^4 \sin ^{-1}\left (\frac{\sqrt{c} \sqrt{e} \sqrt{d+e x}}{\sqrt{e (2 c d-b e)}}\right )\right )}{48 c^3 e^7 (d+e x)^3 \sqrt{e (2 c d-b e)} (b e-2 c d)^2 \left (\frac{b e-c d+c e x}{b e-2 c d}\right )^{3/2}}-5 e g\right )}{20 c e^3} \]

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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Maple [B]  time = 0.01, size = 1114, normalized size = 5. \begin{align*} \text{result too large to display} \end{align*}

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*d*f/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b+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*b^2/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*e*f-1/3*(-c*e^2*x^2
-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e^2*d*g-1/4/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)+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)+1/8/e*g*(-c*e^2*x^2-b
*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^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)*ar
ctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*b*d^3+3/8*b^2/c*e^2/(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))*d*f+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^2-1/4*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-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-1/4*b/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*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+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)*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+5/32*e*g*b^2/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x+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*(-c*e
^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b*d+1/2*d*f*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x-1/3*(-c*e^2*x^2-b*e^2
*x-b*d*e+c*d^2)^(3/2)/c/e*f

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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

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Fricas [A]  time = 2.51058, size = 1748, normalized size = 7.84 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}

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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Giac [A]  time = 1.25165, size = 514, normalized size = 2.3 \begin{align*} \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{align*}

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