3.2218 \(\int \frac{(d+e x)^2 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=213 \[ -\frac{(-3 b e g+4 c d g+2 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 c^{5/2} e^2}+\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-3 b e g+4 c d g+2 c e f)}{c^2 e^2 (2 c d-b e)}+\frac{2 (d+e x)^2 (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.689251, antiderivative size = 213, 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 \[ -\frac{(-3 b e g+4 c d g+2 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 c^{5/2} e^2}+\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-3 b e g+4 c d g+2 c e f)}{c^2 e^2 (2 c d-b e)}+\frac{2 (d+e x)^2 (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

Antiderivative was successfully verified.

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

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Rubi in Sympy [A]  time = 75.1553, size = 206, normalized size = 0.97 \[ \frac{2 \left (d + e x\right )^{2} \left (b e g - c d g - c e f\right )}{c e^{2} \left (b e - 2 c d\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} + \frac{2 \left (\frac{3 b e g}{2} - 2 c d g - c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{c^{2} e^{2} \left (b e - 2 c d\right )} + \frac{2 \left (\frac{3 b e g}{4} - c d g - \frac{c e f}{2}\right ) \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{c^{\frac{5}{2}} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**2*(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)**2*(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))) + 2*(3*b*e*g/2 - 2*c*d*g - c*e*f)*sqrt(-b*e**2*x -
c*e**2*x**2 + d*(-b*e + c*d))/(c**2*e**2*(b*e - 2*c*d)) + 2*(3*b*e*g/4 - c*d*g -
c*e*f/2)*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**(5/2)*e**2)

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Mathematica [C]  time = 0.458992, size = 162, normalized size = 0.76 \[ \frac{2 \sqrt{c} (d+e x) (c (3 d g+2 e f-e g x)-3 b e g)-i \sqrt{d+e x} \sqrt{c (d-e x)-b e} (-3 b e g+4 c d g+2 c e f) \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{2 c^{5/2} e^2 \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

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

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Maple [B]  time = 0.017, size = 1320, normalized size = 6.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-g*x^2/c/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)+3/4*g*b^2/c^3/(-c*e^2*x^2-b*e^2*
x-b*d*e+c*d^2)^(1/2)-3/2*g*b/c^2*x/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)+3/2*g*
b/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))+3/e^2*g/c/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2-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*e^4*f-3*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
)*e^3*d*g-1/2*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)*e^4*f+3/4*e^4*g*b^4/c^3/(-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/c/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)
*f-1/2*b/c^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*f-1/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))*f+3/2*e^4*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)
*x-2/c/e/(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*g+4*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*d*f+2*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*d*f+6*e^2*g*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*d^2+3*e^2*g/c*b^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)*d^2+2/c/e/(-c*e^
2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*f-6*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*e^3*d*g-3*b/c^2/e/(-c*e^2*x^2-b*e^2*x
-b*d*e+c*d^2)^(1/2)*d*g+2*x/c/e/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*g+2*d^2
*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]  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)^2*(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 = 0.803471, size = 1, normalized size = 0. \[ \left [\frac{4 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (c e g x - 2 \, c e f - 3 \,{\left (c d - b e\right )} g\right )} \sqrt{-c} +{\left (2 \,{\left (c^{2} d e - b c e^{2}\right )} f +{\left (4 \, c^{2} d^{2} - 7 \, b c d e + 3 \, b^{2} e^{2}\right )} g -{\left (2 \, c^{2} e^{2} f +{\left (4 \, c^{2} d e - 3 \, b c e^{2}\right )} g\right )} x\right )} \log \left (4 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \, c^{2} e x + b c e\right )} +{\left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2}\right )} \sqrt{-c}\right )}{4 \,{\left (c^{3} e^{3} x - c^{3} d e^{2} + b c^{2} e^{3}\right )} \sqrt{-c}}, \frac{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (c e g x - 2 \, c e f - 3 \,{\left (c d - b e\right )} g\right )} \sqrt{c} +{\left (2 \,{\left (c^{2} d e - b c e^{2}\right )} f +{\left (4 \, c^{2} d^{2} - 7 \, b c d e + 3 \, b^{2} e^{2}\right )} g -{\left (2 \, c^{2} e^{2} f +{\left (4 \, c^{2} d e - 3 \, b c e^{2}\right )} g\right )} x\right )} \arctan \left (\frac{2 \, c e x + b e}{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{c}}\right )}{2 \,{\left (c^{3} e^{3} x - c^{3} d e^{2} + b c^{2} e^{3}\right )} \sqrt{c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2*(g*x + f)/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2),x, algorithm="fricas")

[Out]

[1/4*(4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(c*e*g*x - 2*c*e*f - 3*(c*d -
b*e)*g)*sqrt(-c) + (2*(c^2*d*e - b*c*e^2)*f + (4*c^2*d^2 - 7*b*c*d*e + 3*b^2*e^
2)*g - (2*c^2*e^2*f + (4*c^2*d*e - 3*b*c*e^2)*g)*x)*log(4*sqrt(-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)))/((c^3*e^3*x - c^3*d*e^2 + b*c^2*e^3)*sqrt(
-c)), 1/2*(2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(c*e*g*x - 2*c*e*f - 3*(
c*d - b*e)*g)*sqrt(c) + (2*(c^2*d*e - b*c*e^2)*f + (4*c^2*d^2 - 7*b*c*d*e + 3*b^
2*e^2)*g - (2*c^2*e^2*f + (4*c^2*d*e - 3*b*c*e^2)*g)*x)*arctan(1/2*(2*c*e*x + b*
e)/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(c))))/((c^3*e^3*x - c^3*d*e^
2 + b*c^2*e^3)*sqrt(c))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

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GIAC/XCAS [A]  time = 0.32844, size = 576, normalized size = 2.7 \[ \frac{\sqrt{-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}{\left ({\left (\frac{{\left (4 \, c^{3} d^{2} g e^{3} - 4 \, b c^{2} d g e^{4} + b^{2} c g e^{5}\right )} x}{4 \, c^{4} d^{2} e^{3} - 4 \, b c^{3} d e^{4} + b^{2} c^{2} e^{5}} - \frac{8 \, c^{3} d^{3} g e^{2} + 8 \, c^{3} d^{2} f e^{3} - 20 \, b c^{2} d^{2} g e^{3} - 8 \, b c^{2} d f e^{4} + 14 \, b^{2} c d g e^{4} + 2 \, b^{2} c f e^{5} - 3 \, b^{3} g e^{5}}{4 \, c^{4} d^{2} e^{3} - 4 \, b c^{3} d e^{4} + b^{2} c^{2} e^{5}}\right )} x - \frac{12 \, c^{3} d^{4} g e + 8 \, c^{3} d^{3} f e^{2} - 24 \, b c^{2} d^{3} g e^{2} - 8 \, b c^{2} d^{2} f e^{3} + 15 \, b^{2} c d^{2} g e^{3} + 2 \, b^{2} c d f e^{4} - 3 \, b^{3} d g e^{4}}{4 \, c^{4} d^{2} e^{3} - 4 \, b c^{3} d e^{4} + b^{2} c^{2} e^{5}}\right )}}{c x^{2} e^{2} - c d^{2} + b x e^{2} + b d e} - \frac{{\left (4 \, c d g + 2 \, c f e - 3 \, b g e\right )} \sqrt{-c e^{2}} e^{\left (-3\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{-c e^{2}} x - \sqrt{-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} c - \sqrt{-c e^{2}} b \right |}\right )}{2 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2*(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*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(((4*c^3*d^2*g*e^3 - 4*b*c^2*d*g*e^4
+ b^2*c*g*e^5)*x/(4*c^4*d^2*e^3 - 4*b*c^3*d*e^4 + b^2*c^2*e^5) - (8*c^3*d^3*g*e^
2 + 8*c^3*d^2*f*e^3 - 20*b*c^2*d^2*g*e^3 - 8*b*c^2*d*f*e^4 + 14*b^2*c*d*g*e^4 +
2*b^2*c*f*e^5 - 3*b^3*g*e^5)/(4*c^4*d^2*e^3 - 4*b*c^3*d*e^4 + b^2*c^2*e^5))*x -
(12*c^3*d^4*g*e + 8*c^3*d^3*f*e^2 - 24*b*c^2*d^3*g*e^2 - 8*b*c^2*d^2*f*e^3 + 15*
b^2*c*d^2*g*e^3 + 2*b^2*c*d*f*e^4 - 3*b^3*d*g*e^4)/(4*c^4*d^2*e^3 - 4*b*c^3*d*e^
4 + b^2*c^2*e^5))/(c*x^2*e^2 - c*d^2 + b*x*e^2 + b*d*e) - 1/2*(4*c*d*g + 2*c*f*e
- 3*b*g*e)*sqrt(-c*e^2)*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^3