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

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

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {1}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.67 \begin {gather*} -\frac {1}{3 b \left ((a+b x)^2\right )^{3/2}} \end {gather*}

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*1/(b*((a + b*x)^2)^(3/2))

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IntegrateAlgebraic [A]  time = 0.03, size = 18, normalized size = 0.67 \begin {gather*} -\frac {1}{3 b \left ((a+b x)^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*1/(b*((a + b*x)^2)^(3/2))

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fricas [A]  time = 0.43, size = 35, normalized size = 1.30 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.20, size = 23, normalized size = 0.85 \begin {gather*} -\frac {1}{3 \, {\left (a^{2} + {\left (b x^{2} + 2 \, a x\right )} b\right )}^{\frac {3}{2}} b} \end {gather*}

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]

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

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maple [A]  time = 0.05, size = 22, normalized size = 0.81 \begin {gather*} -\frac {\left (b x +a \right )^{2}}{3 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b} \end {gather*}

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 = 0.59, size = 23, normalized size = 0.85 \begin {gather*} -\frac {1}{3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b} \end {gather*}

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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mupad [B]  time = 2.09, size = 30, normalized size = 1.11 \begin {gather*} -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,b\,{\left (a+b\,x\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.57, size = 97, normalized size = 3.59 \begin {gather*} \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 {gather*}

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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