3.1592 \(\int \frac{b+2 c x}{\left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=18 \[ -\frac{2}{3 \left (a+b x+c x^2\right )^{3/2}} \]

[Out]

-2/(3*(a + b*x + c*x^2)^(3/2))

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Rubi [A]  time = 0.00955405, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ -\frac{2}{3 \left (a+b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(b + 2*c*x)/(a + b*x + c*x^2)^(5/2),x]

[Out]

-2/(3*(a + b*x + c*x^2)^(3/2))

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Rubi in Sympy [A]  time = 3.90503, size = 17, normalized size = 0.94 \[ - \frac{2}{3 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*x+b)/(c*x**2+b*x+a)**(5/2),x)

[Out]

-2/(3*(a + b*x + c*x**2)**(3/2))

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Mathematica [A]  time = 0.0185337, size = 17, normalized size = 0.94 \[ -\frac{2}{3 (a+x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b + 2*c*x)/(a + b*x + c*x^2)^(5/2),x]

[Out]

-2/(3*(a + x*(b + c*x))^(3/2))

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Maple [A]  time = 0.007, size = 15, normalized size = 0.8 \[ -{\frac{2}{3} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)/(c*x^2+b*x+a)^(5/2),x)

[Out]

-2/3/(c*x^2+b*x+a)^(3/2)

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Maxima [A]  time = 0.717001, size = 19, normalized size = 1.06 \[ -\frac{2}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)/(c*x^2 + b*x + a)^(5/2),x, algorithm="maxima")

[Out]

-2/3/(c*x^2 + b*x + a)^(3/2)

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Fricas [A]  time = 0.377747, size = 69, normalized size = 3.83 \[ -\frac{2 \, \sqrt{c x^{2} + b x + a}}{3 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)/(c*x^2 + b*x + a)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.2824, size = 58, normalized size = 3.22 \[ - \frac{2}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x+b)/(c*x**2+b*x+a)**(5/2),x)

[Out]

-2/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x**2*sqrt(a
+ b*x + c*x**2))

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GIAC/XCAS [A]  time = 0.279946, size = 19, normalized size = 1.06 \[ -\frac{2}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)/(c*x^2 + b*x + a)^(5/2),x, algorithm="giac")

[Out]

-2/3/(c*x^2 + b*x + a)^(3/2)