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

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

[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 = 0.164131, 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 \[ -\frac{\left (a^2-b^2 x^2\right )^{3/2}}{21 a^2 b (a+b x)^5}-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{9 a b (a+b x)^6}-\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{315 a^4 b (a+b x)^3}-\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{105 a^3 b (a+b x)^4} \]

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)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.2882, size = 110, normalized size = 0.83 \[ - \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{9 a b \left (a + b x\right )^{6}} - \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{21 a^{2} b \left (a + b x\right )^{5}} - \frac{2 \left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{105 a^{3} b \left (a + b x\right )^{4}} - \frac{2 \left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{315 a^{4} b \left (a + b x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b**2*x**2+a**2)**(1/2)/(b*x+a)**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)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0468845, size = 74, normalized size = 0.56 \[ \frac{\sqrt{a^2-b^2 x^2} \left (-58 a^4+25 a^3 b x+21 a^2 b^2 x^2+10 a b^3 x^3+2 b^4 x^4\right )}{315 a^4 b (a+b x)^5} \]

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 = 0.01, size = 66, normalized size = 0.5 \[ -{\frac{ \left ( 2\,{b}^{3}{x}^{3}+12\,a{b}^{2}{x}^{2}+33\,{a}^{2}bx+58\,{a}^{3} \right ) \left ( -bx+a \right ) }{315\, \left ( bx+a \right ) ^{5}{a}^{4}b}\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}} \]

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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b^2*x^2 + a^2)/(b*x + a)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.240255, size = 512, normalized size = 3.85 \[ -\frac{56 \, b^{8} x^{9} + 522 \, a b^{7} x^{8} + 1089 \, a^{2} b^{6} x^{7} - 924 \, a^{3} b^{5} x^{6} - 5607 \, a^{4} b^{4} x^{5} - 6300 \, a^{5} b^{3} x^{4} + 420 \, a^{6} b^{2} x^{3} + 7560 \, a^{7} b x^{2} + 5040 \, a^{8} x - 3 \,{\left (20 \, b^{7} x^{8} + 6 \, a b^{6} x^{7} - 413 \, a^{2} b^{5} x^{6} - 1169 \, a^{3} b^{4} x^{5} - 840 \, a^{4} b^{3} x^{4} + 980 \, a^{5} b^{2} x^{3} + 2520 \, a^{6} b x^{2} + 1680 \, a^{7} x\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{315 \,{\left (a^{4} b^{9} x^{9} + 9 \, a^{5} b^{8} x^{8} + 18 \, a^{6} b^{7} x^{7} - 18 \, a^{7} b^{6} x^{6} - 99 \, a^{8} b^{5} x^{5} - 99 \, a^{9} b^{4} x^{4} + 24 \, a^{10} b^{3} x^{3} + 108 \, a^{11} b^{2} x^{2} + 72 \, a^{12} b x + 16 \, a^{13} -{\left (a^{4} b^{8} x^{8} - 22 \, a^{6} b^{6} x^{6} - 60 \, a^{7} b^{5} x^{5} - 39 \, a^{8} b^{4} x^{4} + 60 \, a^{9} b^{3} x^{3} + 116 \, a^{10} b^{2} x^{2} + 72 \, a^{11} b x + 16 \, a^{12}\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b^2*x^2 + a^2)/(b*x + a)^6,x, algorithm="fricas")

[Out]

-1/315*(56*b^8*x^9 + 522*a*b^7*x^8 + 1089*a^2*b^6*x^7 - 924*a^3*b^5*x^6 - 5607*a
^4*b^4*x^5 - 6300*a^5*b^3*x^4 + 420*a^6*b^2*x^3 + 7560*a^7*b*x^2 + 5040*a^8*x -
3*(20*b^7*x^8 + 6*a*b^6*x^7 - 413*a^2*b^5*x^6 - 1169*a^3*b^4*x^5 - 840*a^4*b^3*x
^4 + 980*a^5*b^2*x^3 + 2520*a^6*b*x^2 + 1680*a^7*x)*sqrt(-b^2*x^2 + a^2))/(a^4*b
^9*x^9 + 9*a^5*b^8*x^8 + 18*a^6*b^7*x^7 - 18*a^7*b^6*x^6 - 99*a^8*b^5*x^5 - 99*a
^9*b^4*x^4 + 24*a^10*b^3*x^3 + 108*a^11*b^2*x^2 + 72*a^12*b*x + 16*a^13 - (a^4*b
^8*x^8 - 22*a^6*b^6*x^6 - 60*a^7*b^5*x^5 - 39*a^8*b^4*x^4 + 60*a^9*b^3*x^3 + 116
*a^10*b^2*x^2 + 72*a^11*b*x + 16*a^12)*sqrt(-b^2*x^2 + a^2))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (- a + b x\right ) \left (a + b x\right )}}{\left (a + b x\right )^{6}}\, dx \]

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/XCAS [A]  time = 0.241347, size = 390, normalized size = 2.93 \[ \frac{2 \,{\left (\frac{207 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}}{b^{2} x} + \frac{1143 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + \frac{2247 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{3}}{b^{6} x^{3}} + \frac{3843 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{4}}{b^{8} x^{4}} + \frac{3465 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{5}}{b^{10} x^{5}} + \frac{2625 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{6}}{b^{12} x^{6}} + \frac{945 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{7}}{b^{14} x^{7}} + \frac{315 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{8}}{b^{16} x^{8}} + 58\right )}}{315 \, a^{4}{\left (\frac{a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}}{b^{2} x} + 1\right )}^{9}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b^2*x^2 + a^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))