3.786 \(\int \frac{\sqrt{a^2-b^2 x^2}}{(a+b x)^6} \, dx\)

Optimal. Leaf size=133 \[ -\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{315 a^4 b (a+b x)^3}-\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{105 a^3 b (a+b x)^4}-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{21 a^2 b (a+b x)^5}-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{9 a b (a+b x)^6} \]

[Out]

-(a^2 - b^2*x^2)^(3/2)/(9*a*b*(a + b*x)^6) - (a^2 - b^2*x^2)^(3/2)/(21*a^2*b*(a + b*x)^5) - (2*(a^2 - b^2*x^2)
^(3/2))/(105*a^3*b*(a + b*x)^4) - (2*(a^2 - b^2*x^2)^(3/2))/(315*a^4*b*(a + b*x)^3)

________________________________________________________________________________________

Rubi [A]  time = 0.0533064, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {659, 651} \[ -\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{315 a^4 b (a+b x)^3}-\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{105 a^3 b (a+b x)^4}-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{21 a^2 b (a+b x)^5}-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{9 a b (a+b x)^6} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a^2 - b^2*x^2]/(a + b*x)^6,x]

[Out]

-(a^2 - b^2*x^2)^(3/2)/(9*a*b*(a + b*x)^6) - (a^2 - b^2*x^2)^(3/2)/(21*a^2*b*(a + b*x)^5) - (2*(a^2 - b^2*x^2)
^(3/2))/(105*a^3*b*(a + b*x)^4) - (2*(a^2 - b^2*x^2)^(3/2))/(315*a^4*b*(a + b*x)^3)

Rule 659

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*(m + p + 1)), x] + Dist[Simplify[m + 2*p + 2]/(2*d*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p,
x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2
], 0]

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)^6} \, dx &=-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{9 a b (a+b x)^6}+\frac{\int \frac{\sqrt{a^2-b^2 x^2}}{(a+b x)^5} \, dx}{3 a}\\ &=-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{9 a b (a+b x)^6}-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{21 a^2 b (a+b x)^5}+\frac{2 \int \frac{\sqrt{a^2-b^2 x^2}}{(a+b x)^4} \, dx}{21 a^2}\\ &=-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{9 a b (a+b x)^6}-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{21 a^2 b (a+b x)^5}-\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{105 a^3 b (a+b x)^4}+\frac{2 \int \frac{\sqrt{a^2-b^2 x^2}}{(a+b x)^3} \, dx}{105 a^3}\\ &=-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{9 a b (a+b x)^6}-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{21 a^2 b (a+b x)^5}-\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{105 a^3 b (a+b x)^4}-\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{315 a^4 b (a+b x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0449516, size = 74, normalized size = 0.56 \[ \frac{\sqrt{a^2-b^2 x^2} \left (21 a^2 b^2 x^2+25 a^3 b x-58 a^4+10 a b^3 x^3+2 b^4 x^4\right )}{315 a^4 b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a^2 - b^2*x^2]/(a + b*x)^6,x]

[Out]

(Sqrt[a^2 - b^2*x^2]*(-58*a^4 + 25*a^3*b*x + 21*a^2*b^2*x^2 + 10*a*b^3*x^3 + 2*b^4*x^4))/(315*a^4*b*(a + b*x)^
5)

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 66, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,{b}^{3}{x}^{3}+12\,a{b}^{2}{x}^{2}+33\,x{a}^{2}b+58\,{a}^{3} \right ) \left ( -bx+a \right ) }{315\, \left ( bx+a \right ) ^{5}{a}^{4}b}\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)^6,x)

[Out]

-1/315*(-b*x+a)*(2*b^3*x^3+12*a*b^2*x^2+33*a^2*b*x+58*a^3)*(-b^2*x^2+a^2)^(1/2)/(b*x+a)^5/a^4/b

________________________________________________________________________________________

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)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.07224, size = 366, normalized size = 2.75 \begin{align*} -\frac{58 \, b^{5} x^{5} + 290 \, a b^{4} x^{4} + 580 \, a^{2} b^{3} x^{3} + 580 \, a^{3} b^{2} x^{2} + 290 \, a^{4} b x + 58 \, a^{5} -{\left (2 \, b^{4} x^{4} + 10 \, a b^{3} x^{3} + 21 \, a^{2} b^{2} x^{2} + 25 \, a^{3} b x - 58 \, a^{4}\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{315 \,{\left (a^{4} b^{6} x^{5} + 5 \, a^{5} b^{5} x^{4} + 10 \, a^{6} b^{4} x^{3} + 10 \, a^{7} b^{3} x^{2} + 5 \, a^{8} b^{2} x + a^{9} 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)^6,x, algorithm="fricas")

[Out]

-1/315*(58*b^5*x^5 + 290*a*b^4*x^4 + 580*a^2*b^3*x^3 + 580*a^3*b^2*x^2 + 290*a^4*b*x + 58*a^5 - (2*b^4*x^4 + 1
0*a*b^3*x^3 + 21*a^2*b^2*x^2 + 25*a^3*b*x - 58*a^4)*sqrt(-b^2*x^2 + a^2))/(a^4*b^6*x^5 + 5*a^5*b^5*x^4 + 10*a^
6*b^4*x^3 + 10*a^7*b^3*x^2 + 5*a^8*b^2*x + a^9*b)

________________________________________________________________________________________

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 )^{6}}\, 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)**6,x)

[Out]

Integral(sqrt(-(-a + b*x)*(a + b*x))/(a + b*x)**6, x)

________________________________________________________________________________________

Giac [B]  time = 1.2051, size = 390, normalized size = 2.93 \begin{align*} \frac{2 \,{\left (\frac{207 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}}{b^{2} x} + \frac{1143 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + \frac{2247 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{3}}{b^{6} x^{3}} + \frac{3843 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{4}}{b^{8} x^{4}} + \frac{3465 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{5}}{b^{10} x^{5}} + \frac{2625 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{6}}{b^{12} x^{6}} + \frac{945 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{7}}{b^{14} x^{7}} + \frac{315 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{8}}{b^{16} x^{8}} + 58\right )}}{315 \, a^{4}{\left (\frac{a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}}{b^{2} x} + 1\right )}^{9}{\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)^6,x, algorithm="giac")

[Out]

2/315*(207*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 1143*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^2/(b^4*x^2)
+ 2247*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^3/(b^6*x^3) + 3843*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^4/(b^8*x^4)
+ 3465*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^5/(b^10*x^5) + 2625*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^6/(b^12*x^6
) + 945*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^7/(b^14*x^7) + 315*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^8/(b^16*x^8
) + 58)/(a^4*((a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 1)^9*abs(b))