### 3.783 $$\int \frac{\sqrt{a^2-b^2 x^2}}{(a+b x)^3} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0094301, antiderivative size = 33, normalized size of antiderivative = 1., 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} $-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{3 a b (a+b x)^3}$

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*}

Mathematica [A]  time = 0.0325069, size = 39, normalized size = 1.18 $-\frac{(a-b x) \sqrt{a^2-b^2 x^2}}{3 a b (a+b x)^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[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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Maple [A]  time = 0.043, size = 36, normalized size = 1.1 \begin{align*} -{\frac{-bx+a}{3\, \left ( bx+a \right ) ^{2}ba}\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}} \end{align*}

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [B]  time = 1.7758, size = 134, normalized size = 4.06 \begin{align*} -\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{align*}

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

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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Giac [B]  time = 1.28214, size = 100, normalized size = 3.03 \begin{align*} \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{align*}

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))