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

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*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)

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Rubi [A]  time = 0.369888, 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 \[ \frac{2 (d+e x) (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{g \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{c^{3/2} e^2} \]

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 + 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*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)

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Rubi in Sympy [A]  time = 36.2207, size = 119, normalized size = 0.92 \[ \frac{2 \left (d + e x\right ) \left (b e g - c d g - c e f\right )}{c e^{2} \left (b e - 2 c d\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} - \frac{g \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{c^{\frac{3}{2}} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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]

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

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Mathematica [C]  time = 0.464465, size = 155, normalized size = 1.2 \[ \frac{-2 \sqrt{c} (d+e x) (-b e g+c d g+c e f)+i g \sqrt{d+e x} (2 c d-b e) \sqrt{c (d-e x)-b e} \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{c^{3/2} e^2 (b e-2 c d) \sqrt{(d+e x) (c (d-e x)-b e)}} \]

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

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Maple [B]  time = 0.013, size = 710, normalized size = 5.5 \[ 2\,{\frac{fd \left ( -2\,c{e}^{2}x-b{e}^{2} \right ) }{ \left ( -4\,c{e}^{2} \left ( -bde+c{d}^{2} \right ) -{b}^{2}{e}^{4} \right ) \sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}+{\frac{dg}{c{e}^{2}}{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}}+{\frac{f}{ce}{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}}+2\,{\frac{bd{e}^{2}gx}{ \left ( -{b}^{2}{e}^{4}+4\,bd{e}^{3}c-4\,{d}^{2}{e}^{2}{c}^{2} \right ) \sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}+2\,{\frac{b{e}^{3}fx}{ \left ( -{b}^{2}{e}^{4}+4\,bd{e}^{3}c-4\,{d}^{2}{e}^{2}{c}^{2} \right ) \sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}+{\frac{{b}^{2}d{e}^{2}g}{c \left ( -{b}^{2}{e}^{4}+4\,bd{e}^{3}c-4\,{d}^{2}{e}^{2}{c}^{2} \right ) }{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}}+{\frac{{b}^{2}{e}^{3}f}{c \left ( -{b}^{2}{e}^{4}+4\,bd{e}^{3}c-4\,{d}^{2}{e}^{2}{c}^{2} \right ) }{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}}+{\frac{gx}{ce}{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}}-{\frac{bg}{2\,e{c}^{2}}{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}}-{\frac{{e}^{3}g{b}^{2}x}{c \left ( -{b}^{2}{e}^{4}+4\,bd{e}^{3}c-4\,{d}^{2}{e}^{2}{c}^{2} \right ) }{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}}-{\frac{{e}^{3}g{b}^{3}}{2\,{c}^{2} \left ( -{b}^{2}{e}^{4}+4\,bd{e}^{3}c-4\,{d}^{2}{e}^{2}{c}^{2} \right ) }{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}}-{\frac{g}{ce}\arctan \left ({1\sqrt{c{e}^{2}} \left ( x+{\frac{b}{2\,c}} \right ){\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}} \right ){\frac{1}{\sqrt{c{e}^{2}}}}} \]

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]

2*f*d*(-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)+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+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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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 + c*d^2 - b*d*e)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.991623, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (c e f +{\left (c d - b e\right )} g\right )} \sqrt{-c} -{\left ({\left (2 \, c^{2} d e - b c e^{2}\right )} g x -{\left (2 \, c^{2} d^{2} - 3 \, b c d e + b^{2} e^{2}\right )} g\right )} \log \left (-4 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \, c^{2} e x + b c e\right )} +{\left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2}\right )} \sqrt{-c}\right )}{2 \,{\left (2 \, c^{3} d^{2} e^{2} - 3 \, b c^{2} d e^{3} + b^{2} c e^{4} -{\left (2 \, c^{3} d e^{3} - b c^{2} e^{4}\right )} x\right )} \sqrt{-c}}, \frac{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (c e f +{\left (c d - b e\right )} g\right )} \sqrt{c} +{\left ({\left (2 \, c^{2} d e - b c e^{2}\right )} g x -{\left (2 \, c^{2} d^{2} - 3 \, b c d e + b^{2} e^{2}\right )} g\right )} \arctan \left (\frac{2 \, c e x + b e}{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{c}}\right )}{{\left (2 \, c^{3} d^{2} e^{2} - 3 \, b c^{2} d e^{3} + b^{2} c e^{4} -{\left (2 \, c^{3} d e^{3} - b c^{2} e^{4}\right )} x\right )} \sqrt{c}}\right ] \]

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 + c*d^2 - b*d*e)^(3/2),x, algorithm="fricas")

[Out]

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

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

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/XCAS [A]  time = 0.330024, size = 381, normalized size = 2.95 \[ -\frac{\sqrt{-c e^{2}} g e^{\left (-3\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{-c e^{2}} x - \sqrt{-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} c - \sqrt{-c e^{2}} b \right |}\right )}{c^{2}} - \frac{2 \, \sqrt{-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}{\left (\frac{{\left (2 \, c^{2} d^{2} g e^{2} + 2 \, c^{2} d f e^{3} - 3 \, b c d g e^{3} - b c f e^{4} + b^{2} g e^{4}\right )} x}{4 \, c^{3} d^{2} e^{3} - 4 \, b c^{2} d e^{4} + b^{2} c e^{5}} + \frac{2 \, c^{2} d^{3} g e + 2 \, c^{2} d^{2} f e^{2} - 3 \, b c d^{2} g e^{2} - b c d f e^{3} + b^{2} d g e^{3}}{4 \, c^{3} d^{2} e^{3} - 4 \, b c^{2} d e^{4} + b^{2} c e^{5}}\right )}}{c x^{2} e^{2} - c d^{2} + b x e^{2} + b d e} \]

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 + c*d^2 - b*d*e)^(3/2),x, algorithm="giac")

[Out]

-sqrt(-c*e^2)*g*e^(-3)*ln(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)