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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]