3.14.12 \(\int \frac {1}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=34 \[ -\frac {1}{4 b (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \]

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Rubi [A]  time = 0.00, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {607} \begin {gather*} -\frac {1}{4 b (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 607

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(2*(a + b*x + c*x^2)^(p + 1))/((2*p + 1)*(b + 2
*c*x)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=-\frac {1}{4 b (a+b x) \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 = 23, normalized size = 0.68 \begin {gather*} -\frac {a+b x}{4 b \left ((a+b x)^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(a + b*x)/(b*((a + b*x)^2)^(5/2))

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IntegrateAlgebraic [B]  time = 0.00, size = 197, normalized size = 5.79 \begin {gather*} \frac {-2 \left (b^5 \left (-x^4\right )-a^4 b\right )-2 \sqrt {b^2} \sqrt {a^2+2 a b x+b^2 x^2} \left (-a^3+a^2 b x-a b^2 x^2+b^3 x^3\right )}{b x^4 \sqrt {a^2+2 a b x+b^2 x^2} \left (-8 a^3 b^5-24 a^2 b^6 x-24 a b^7 x^2-8 b^8 x^3\right )+b \sqrt {b^2} x^4 \left (8 a^4 b^4+32 a^3 b^5 x+48 a^2 b^6 x^2+32 a b^7 x^3+8 b^8 x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[b^2]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-a^3 + a^2*b*x - a*b^2*x^2 + b^3*x^3) - 2*(-(a^4*b) - b^5*x^4))/(
b*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-8*a^3*b^5 - 24*a^2*b^6*x - 24*a*b^7*x^2 - 8*b^8*x^3) + b*Sqrt[b^2]*x^4*(
8*a^4*b^4 + 32*a^3*b^5*x + 48*a^2*b^6*x^2 + 32*a*b^7*x^3 + 8*b^8*x^4))

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fricas [A]  time = 0.40, size = 46, normalized size = 1.35 \begin {gather*} -\frac {1}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

-1/4/(b^5*x^4 + 4*a*b^4*x^3 + 6*a^2*b^3*x^2 + 4*a^3*b^2*x + a^4*b)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 0.05, size = 20, normalized size = 0.59 \begin {gather*} -\frac {b x +a}{4 \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(1/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

-1/4*(b*x+a)/b/((b*x+a)^2)^(5/2)

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maxima [A]  time = 0.96, size = 14, normalized size = 0.41 \begin {gather*} -\frac {1}{4 \, b^{5} {\left (x + \frac {a}{b}\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

-1/4/(b^5*(x + a/b)^4)

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mupad [B]  time = 0.62, size = 30, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,b\,{\left (a+b\,x\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(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)/(4*b*(a + b*x)^5)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral((a**2 + 2*a*b*x + b**2*x**2)**(-5/2), x)

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