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

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 = 0.006047, 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, Rules used = {629} \[ -\frac{2}{3 \left (a+b x+c x^2\right )^{3/2}} \]

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))

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{b+2 c x}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2}{3 \left (a+b x+c x^2\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.009291, 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 = 0.006, size = 15, normalized size = 0.8 \begin{align*} -{\frac{2}{3} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

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 = 0.98824, size = 19, normalized size = 1.06 \begin{align*} -\frac{2}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \end{align*}

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 [B]  time = 2.84585, size = 116, normalized size = 6.44 \begin{align*} -\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 )}} \end{align*}

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 [B]  time = 1.49032, size = 58, normalized size = 3.22 \begin{align*} - \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}}} \end{align*}

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 [A]  time = 1.41217, size = 19, normalized size = 1.06 \begin{align*} -\frac{2}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \end{align*}

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)