3.2219 \(\int \frac{(d+e x) (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=129 \[ \frac{2 (d+e x) (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{g \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{c^{3/2} e^2} \]

[Out]

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

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Rubi [A]  time = 0.172781, antiderivative size = 148, normalized size of antiderivative = 1.15, number of steps used = 3, number of rules used = 3, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {777, 621, 204} \[ \frac{2 (e x (2 c d-b e)+d (2 c d-b e)) (-b e g+c d g+c e f)}{c e^2 (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{g \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{c^{3/2} e^2} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]

[Out]

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

Rule 777

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

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{(d+e x) (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx &=\frac{2 (c e f+c d g-b e g) (d (2 c d-b e)+e (2 c d-b e) x)}{c e^2 (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{g \int \frac{1}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{c e}\\ &=\frac{2 (c e f+c d g-b e g) (d (2 c d-b e)+e (2 c d-b e) x)}{c e^2 (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{(2 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 )}{c e}\\ &=\frac{2 (c e f+c d g-b e g) (d (2 c d-b e)+e (2 c d-b e) x)}{c e^2 (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{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 )}{c^{3/2} e^2}\\ \end{align*}

Mathematica [A]  time = 0.501974, size = 173, normalized size = 1.34 \[ -\frac{2 \left (\sqrt{c} \sqrt{e} (d+e x) (-b e g+c d g+c e f)+g \sqrt{d+e x} \sqrt{e (2 c d-b e)} (b e-2 c d) \sqrt{\frac{b e-c d+c e x}{b e-2 c d}} \sin ^{-1}\left (\frac{\sqrt{c} \sqrt{e} \sqrt{d+e x}}{\sqrt{e (2 c d-b e)}}\right )\right )}{c^{3/2} e^{5/2} (b e-2 c d) \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]

[Out]

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

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Maple [B]  time = 0.009, size = 710, normalized size = 5.5 \begin{align*}{\frac{gx}{ce}{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}}-{\frac{gb}{2\,e{c}^{2}}{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}}-{\frac{{e}^{3}g{b}^{2}x}{c \left ( -{b}^{2}{e}^{4}+4\,bcd{e}^{3}-4\,{c}^{2}{d}^{2}{e}^{2} \right ) }{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}}-{\frac{{e}^{3}g{b}^{3}}{2\,{c}^{2} \left ( -{b}^{2}{e}^{4}+4\,bcd{e}^{3}-4\,{c}^{2}{d}^{2}{e}^{2} \right ) }{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}}-{\frac{g}{ce}\arctan \left ({\sqrt{c{e}^{2}} \left ( x+{\frac{b}{2\,c}} \right ){\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}} \right ){\frac{1}{\sqrt{c{e}^{2}}}}}+{\frac{dg}{c{e}^{2}}{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}}+{\frac{f}{ce}{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}}+2\,{\frac{bd{e}^{2}gx}{ \left ( -{b}^{2}{e}^{4}+4\,bcd{e}^{3}-4\,{c}^{2}{d}^{2}{e}^{2} \right ) \sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}+2\,{\frac{b{e}^{3}fx}{ \left ( -{b}^{2}{e}^{4}+4\,bcd{e}^{3}-4\,{c}^{2}{d}^{2}{e}^{2} \right ) \sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}+{\frac{{b}^{2}d{e}^{2}g}{c \left ( -{b}^{2}{e}^{4}+4\,bcd{e}^{3}-4\,{c}^{2}{d}^{2}{e}^{2} \right ) }{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}}+{\frac{{b}^{2}{e}^{3}f}{c \left ( -{b}^{2}{e}^{4}+4\,bcd{e}^{3}-4\,{c}^{2}{d}^{2}{e}^{2} \right ) }{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}}+2\,{\frac{df \left ( -2\,c{e}^{2}x-b{e}^{2} \right ) }{ \left ( -4\,c{e}^{2} \left ( -bde+c{d}^{2} \right ) -{b}^{2}{e}^{4} \right ) \sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}} \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)^(3/2),x)

[Out]

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

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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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 4.06937, size = 991, normalized size = 7.68 \begin{align*} \left [\frac{{\left ({\left (2 \, c^{2} d e - b c e^{2}\right )} g x -{\left (2 \, c^{2} d^{2} - 3 \, b c d e + 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 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (c^{2} e f +{\left (c^{2} d - b c e\right )} g\right )}}{2 \,{\left (2 \, c^{4} d^{2} e^{2} - 3 \, b c^{3} d e^{3} + b^{2} c^{2} e^{4} -{\left (2 \, c^{4} d e^{3} - b c^{3} e^{4}\right )} x\right )}}, -\frac{{\left ({\left (2 \, c^{2} d e - b c e^{2}\right )} g x -{\left (2 \, c^{2} d^{2} - 3 \, b c d e + 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 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (c^{2} e f +{\left (c^{2} d - b c e\right )} g\right )}}{2 \, c^{4} d^{2} e^{2} - 3 \, b c^{3} d e^{3} + b^{2} c^{2} e^{4} -{\left (2 \, c^{4} d e^{3} - b c^{3} e^{4}\right )} x}\right ] \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)^(3/2),x, algorithm="fricas")

[Out]

[1/2*(((2*c^2*d*e - b*c*e^2)*g*x - (2*c^2*d^2 - 3*b*c*d*e + 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*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(c^2*e*f + (c^2*d - b*c*e)*g))/(2*c^4*d^2*e^2 - 3*b*c^3*d*e^3 +
b^2*c^2*e^4 - (2*c^4*d*e^3 - b*c^3*e^4)*x), -(((2*c^2*d*e - b*c*e^2)*g*x - (2*c^2*d^2 - 3*b*c*d*e + 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*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(c^2*e*f + (c^2*d - b*c*e)*g))/(2*c^4
*d^2*e^2 - 3*b*c^3*d*e^3 + b^2*c^2*e^4 - (2*c^4*d*e^3 - b*c^3*e^4)*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right ) \left (f + g x\right )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac{3}{2}}}\, 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)**(3/2),x)

[Out]

Integral((d + e*x)*(f + g*x)/(-(d + e*x)*(b*e - c*d + c*e*x))**(3/2), x)

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Giac [B]  time = 1.30074, size = 381, normalized size = 2.95 \begin{align*} -\frac{\sqrt{-c e^{2}} g 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 )}{c^{2}} - \frac{2 \, \sqrt{-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}{\left (\frac{{\left (2 \, c^{2} d^{2} g e^{2} + 2 \, c^{2} d f e^{3} - 3 \, b c d g e^{3} - b c f e^{4} + b^{2} g e^{4}\right )} x}{4 \, c^{3} d^{2} e^{3} - 4 \, b c^{2} d e^{4} + b^{2} c e^{5}} + \frac{2 \, c^{2} d^{3} g e + 2 \, c^{2} d^{2} f e^{2} - 3 \, b c d^{2} g e^{2} - b c d f e^{3} + b^{2} d g e^{3}}{4 \, c^{3} d^{2} e^{3} - 4 \, b c^{2} d e^{4} + b^{2} c e^{5}}\right )}}{c x^{2} e^{2} - c d^{2} + b x e^{2} + b d e} \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)^(3/2),x, algorithm="giac")

[Out]

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