3.2038 \(\int \frac{a+b x}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=27 \[ -\frac{1}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \]

[Out]

-1/(3*b*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))

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Rubi [A]  time = 0.0079978, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {629} \[ -\frac{1}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(3*b*(a^2 + 2*a*b*x + b^2*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{a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=-\frac{1}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0079313, size = 18, normalized size = 0.67 \[ -\frac{1}{3 b \left ((a+b x)^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(3*b*((a + b*x)^2)^(3/2))

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Maple [A]  time = 0.005, size = 22, normalized size = 0.8 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{2}}{3\,b} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

-1/3*(b*x+a)^2/b/((b*x+a)^2)^(5/2)

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Maxima [A]  time = 0.948407, size = 31, normalized size = 1.15 \begin{align*} -\frac{1}{3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b)

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Fricas [A]  time = 1.52885, size = 70, normalized size = 2.59 \begin{align*} -\frac{1}{3 \,{\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2*x + a^3*b)

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Sympy [A]  time = 1.31193, size = 97, normalized size = 3.59 \begin{align*} \begin{cases} - \frac{1}{3 a^{2} b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} + 6 a b^{2} x \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} + 3 b^{3} x^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} & \text{for}\: b \neq 0 \\\frac{a x}{\left (a^{2}\right )^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Piecewise((-1/(3*a**2*b*sqrt(a**2 + 2*a*b*x + b**2*x**2) + 6*a*b**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2) + 3*b**
3*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)), Ne(b, 0)), (a*x/(a**2)**(5/2), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b x + a}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2), x)