3.2374 \(\int \frac{d+e x}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=45 \[ -\frac{2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]

[Out]

(-2*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.0450085, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)/(a + b*x + c*x^2)^(3/2),x]

[Out]

(-2*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 7.20804, size = 42, normalized size = 0.93 \[ \frac{4 a e - 2 b d + x \left (2 b e - 4 c d\right )}{\left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)/(c*x**2+b*x+a)**(3/2),x)

[Out]

(4*a*e - 2*b*d + x*(2*b*e - 4*c*d))/((-4*a*c + b**2)*sqrt(a + b*x + c*x**2))

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Mathematica [A]  time = 0.057332, size = 43, normalized size = 0.96 \[ \frac{4 a e-2 b d+2 b e x-4 c d x}{\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)/(a + b*x + c*x^2)^(3/2),x]

[Out]

(-2*b*d + 4*a*e - 4*c*d*x + 2*b*e*x)/((b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)])

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Maple [A]  time = 0.007, size = 45, normalized size = 1. \[ -2\,{\frac{bex-2\,cdx+2\,ae-bd}{\sqrt{c{x}^{2}+bx+a} \left ( 4\,ac-{b}^{2} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(c*x^2+b*x+a)^(1/2)*(b*e*x-2*c*d*x+2*a*e-b*d)/(4*a*c-b^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)/(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.273531, size = 100, normalized size = 2.22 \[ -\frac{2 \, \sqrt{c x^{2} + b x + a}{\left (b d - 2 \, a e +{\left (2 \, c d - b e\right )} x\right )}}{a b^{2} - 4 \, a^{2} c +{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (b^{3} - 4 \, a b c\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x)/(a + b*x + c*x**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.216246, size = 77, normalized size = 1.71 \[ -\frac{2 \,{\left (\frac{{\left (2 \, c d - b e\right )} x}{b^{2} - 4 \, a c} + \frac{b d - 2 \, a e}{b^{2} - 4 \, a c}\right )}}{\sqrt{c x^{2} + b x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(c*x^2 + b*x + a)^(3/2),x, algorithm="giac")

[Out]

-2*((2*c*d - b*e)*x/(b^2 - 4*a*c) + (b*d - 2*a*e)/(b^2 - 4*a*c))/sqrt(c*x^2 + b*
x + a)