3.2175 \(\int \frac{(f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{d+e x} \, dx\)

Optimal. Leaf size=192 \[ \frac{(2 c d-b e) (-b e g-2 c d g+4 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{3/2} e^2}+\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-b e g-2 c d g+4 c e f)}{4 c e^2}-\frac{g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 c e^2 (d+e x)} \]

[Out]

((4*c*e*f - 2*c*d*g - b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*c*e^2) - (g*(d*(c*d - b*e) - b*e^2*
x - c*e^2*x^2)^(3/2))/(2*c*e^2*(d + e*x)) + ((2*c*d - b*e)*(4*c*e*f - 2*c*d*g - 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])])/(8*c^(3/2)*e^2)

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Rubi [A]  time = 0.219909, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {794, 664, 621, 204} \[ \frac{(2 c d-b e) (-b e g-2 c d g+4 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{3/2} e^2}+\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-b e g-2 c d g+4 c e f)}{4 c e^2}-\frac{g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 c e^2 (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*c*e*f - 2*c*d*g - b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*c*e^2) - (g*(d*(c*d - b*e) - b*e^2*
x - c*e^2*x^2)^(3/2))/(2*c*e^2*(d + e*x)) + ((2*c*d - b*e)*(4*c*e*f - 2*c*d*g - 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])])/(8*c^(3/2)*e^2)

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])

Rule 664

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 2*p + 1)), x] - Dist[(p*(2*c*d - b*e))/(e^2*(m + 2*p + 1)), Int[(d + e*x)^(m + 1)*
(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a
*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[m + 2*p + 1, 0] && IntegerQ[2*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 \frac{(f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{d+e x} \, dx &=-\frac{g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 c e^2 (d+e x)}-\frac{\left (c e^3 f-\left (-c d e^2+b e^3\right ) g+\frac{3}{2} e \left (-2 c e^2 f+b e^2 g\right )\right ) \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{d+e x} \, dx}{2 c e^3}\\ &=\frac{(4 c e f-2 c d g-b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2}-\frac{g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 c e^2 (d+e x)}+\frac{((2 c d-b e) (4 c e f-2 c d g-b e g)) \int \frac{1}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{8 c e}\\ &=\frac{(4 c e f-2 c d g-b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2}-\frac{g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 c e^2 (d+e x)}+\frac{((2 c d-b e) (4 c e f-2 c d g-b e g)) \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 )}{4 c e}\\ &=\frac{(4 c e f-2 c d g-b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2}-\frac{g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 c e^2 (d+e x)}+\frac{(2 c d-b e) (4 c e f-2 c d g-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 )}{8 c^{3/2} e^2}\\ \end{align*}

Mathematica [A]  time = 0.55782, size = 174, normalized size = 0.91 \[ \frac{\sqrt{(d+e x) (c (d-e x)-b e)} \left (\sqrt{c} \sqrt{e} (b e g+2 c (-2 d g+2 e f+e g x))+\frac{\sqrt{e (2 c d-b e)} (-b e g-2 c d g+4 c e f) \sin ^{-1}\left (\frac{\sqrt{c} \sqrt{e} \sqrt{d+e x}}{\sqrt{e (2 c d-b e)}}\right )}{\sqrt{d+e x} \sqrt{\frac{b e-c d+c e x}{b e-2 c d}}}\right )}{4 c^{3/2} e^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(Sqrt[c]*Sqrt[e]*(b*e*g + 2*c*(2*e*f - 2*d*g + e*g*x)) + (Sqrt[e*(2*c*
d - b*e)]*(4*c*e*f - 2*c*d*g - b*e*g)*ArcSin[(Sqrt[c]*Sqrt[e]*Sqrt[d + e*x])/Sqrt[e*(2*c*d - b*e)]])/(Sqrt[d +
e*x]*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)])))/(4*c^(3/2)*e^(5/2))

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Maple [B]  time = 0.011, size = 697, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*g/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x+1/4*g/e/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b+1/8*g*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))*b^2-1/2*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+1/2*g/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^2-1/e^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*
(x+d/e))^(1/2)*d*g+1/e*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*f+1/2/(c*e^2)^(1/2)*arctan((c*e^2)^(1
/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*b*d*g-1/2*e/(c*e^2)^
(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2
))*b*f-1/e/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c
*d*e)*(x+d/e))^(1/2))*c*d^2*g+1/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e
)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*c*d*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((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.75545, size = 837, normalized size = 4.36 \begin{align*} \left [-\frac{{\left (4 \,{\left (2 \, c^{2} d e - b c e^{2}\right )} f -{\left (4 \, c^{2} d^{2} - b^{2} e^{2}\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 (2 \, c^{2} e g x + 4 \, c^{2} e f -{\left (4 \, c^{2} d - b c e\right )} g\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{16 \, c^{2} e^{2}}, -\frac{{\left (4 \,{\left (2 \, c^{2} d e - b c e^{2}\right )} f -{\left (4 \, c^{2} d^{2} - b^{2} e^{2}\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 (2 \, c^{2} e g x + 4 \, c^{2} e f -{\left (4 \, c^{2} d - b c e\right )} g\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{8 \, c^{2} e^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/16*((4*(2*c^2*d*e - b*c*e^2)*f - (4*c^2*d^2 - b^2*e^2)*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*(2*c^2
*e*g*x + 4*c^2*e*f - (4*c^2*d - b*c*e)*g)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^2*e^2), -1/8*((4*(2*c
^2*d*e - b*c*e^2)*f - (4*c^2*d^2 - b^2*e^2)*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*(2*c^2*e*g*x + 4*c^2*e*f - (4*c^2*d
- b*c*e)*g)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^2*e^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{d + e x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError