3.2475 \(\int \frac{A+B x}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0457454, 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 \[ -\frac{2 (-2 a B-x (b B-2 A c)+A b)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.43817, size = 44, normalized size = 0.98 \[ - \frac{2 A b - 4 B a + x \left (4 A c - 2 B b\right )}{\left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*A*b - 4*B*a + x*(4*A*c - 2*B*b))/((-4*a*c + b**2)*sqrt(a + b*x + c*x**2))

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Mathematica [A]  time = 0.0621045, size = 44, normalized size = 0.98 \[ \frac{2 B (2 a+b x)-2 A (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 45, normalized size = 1. \[ 2\,{\frac{2\,Acx-Bbx+Ab-2\,aB}{\sqrt{c{x}^{2}+bx+a} \left ( 4\,ac-{b}^{2} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(c*x^2+b*x+a)^(1/2)*(2*A*c*x-B*b*x+A*b-2*B*a)/(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((B*x + A)/(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.330098, size = 100, normalized size = 2.22 \[ \frac{2 \, \sqrt{c x^{2} + b x + a}{\left (2 \, B a - A b +{\left (B b - 2 \, A c\right )} x\right )}}{a b^{2} - 4 \, a^{2} c +{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (b^{3} - 4 \, a b c\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(c*x^2 + b*x + a)*(2*B*a - A*b + (B*b - 2*A*c)*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. \[ \int \frac{A + B x}{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)/(a + b*x + c*x**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.272656, size = 74, normalized size = 1.64 \[ \frac{2 \,{\left (\frac{{\left (B b - 2 \, A c\right )} x}{b^{2} - 4 \, a c} + \frac{2 \, B a - A b}{b^{2} - 4 \, a c}\right )}}{\sqrt{c x^{2} + b x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*((B*b - 2*A*c)*x/(b^2 - 4*a*c) + (2*B*a - A*b)/(b^2 - 4*a*c))/sqrt(c*x^2 + b*x
+ a)