3.7.24 \(\int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^3} \, dx\)

Optimal. Leaf size=33 \[ -\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 a b (a+b x)^3} \]

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Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {651} \begin {gather*} -\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 a b (a+b x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a^2 - b^2*x^2]/(a + b*x)^3,x]

[Out]

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

Rule 651

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

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 39, normalized size = 1.18 \begin {gather*} -\frac {(a-b x) \sqrt {a^2-b^2 x^2}}{3 a b (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a^2 - b^2*x^2]/(a + b*x)^3,x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.38, size = 40, normalized size = 1.21 \begin {gather*} \frac {(b x-a) \sqrt {a^2-b^2 x^2}}{3 a b (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[a^2 - b^2*x^2]/(a + b*x)^3,x]

[Out]

((-a + b*x)*Sqrt[a^2 - b^2*x^2])/(3*a*b*(a + b*x)^2)

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fricas [B]  time = 0.41, size = 66, normalized size = 2.00 \begin {gather*} -\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - \sqrt {-b^{2} x^{2} + a^{2}} {\left (b x - a\right )}}{3 \, {\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b^2*x^2+a^2)^(1/2)/(b*x+a)^3,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.23, size = 74, normalized size = 2.24 \begin {gather*} \frac {2 \, {\left (\frac {3 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + 1\right )}}{3 \, a {\left (\frac {a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}}{b^{2} x} + 1\right )}^{3} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b^2*x^2+a^2)^(1/2)/(b*x+a)^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 36, normalized size = 1.09 \begin {gather*} -\frac {\left (-b x +a \right ) \sqrt {-b^{2} x^{2}+a^{2}}}{3 \left (b x +a \right )^{2} a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.34, size = 69, normalized size = 2.09 \begin {gather*} -\frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{3 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} + \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{3 \, {\left (a b^{2} x + a^{2} b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b^2*x^2+a^2)^(1/2)/(b*x+a)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.53, size = 35, normalized size = 1.06 \begin {gather*} -\frac {\sqrt {a^2-b^2\,x^2}\,\left (a-b\,x\right )}{3\,a\,b\,{\left (a+b\,x\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (- a + b x\right ) \left (a + b x\right )}}{\left (a + b x\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-(-a + b*x)*(a + b*x))/(a + b*x)**3, x)

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