3.774 \(\int \frac{\sqrt{a^2-b^2 x^2}}{(a+b x)^5} \, dx\)

Optimal. Leaf size=100 \[ -\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{35 a^2 b (a+b x)^4}-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{7 a b (a+b x)^5}-\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{105 a^3 b (a+b x)^3} \]

[Out]

-(a^2 - b^2*x^2)^(3/2)/(7*a*b*(a + b*x)^5) - (2*(a^2 - b^2*x^2)^(3/2))/(35*a^2*b
*(a + b*x)^4) - (2*(a^2 - b^2*x^2)^(3/2))/(105*a^3*b*(a + b*x)^3)

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Rubi [A]  time = 0.116576, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{35 a^2 b (a+b x)^4}-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{7 a b (a+b x)^5}-\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{105 a^3 b (a+b x)^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a^2 - b^2*x^2]/(a + b*x)^5,x]

[Out]

-(a^2 - b^2*x^2)^(3/2)/(7*a*b*(a + b*x)^5) - (2*(a^2 - b^2*x^2)^(3/2))/(35*a^2*b
*(a + b*x)^4) - (2*(a^2 - b^2*x^2)^(3/2))/(105*a^3*b*(a + b*x)^3)

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Rubi in Sympy [A]  time = 12.8157, size = 83, normalized size = 0.83 \[ - \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{7 a b \left (a + b x\right )^{5}} - \frac{2 \left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{35 a^{2} b \left (a + b x\right )^{4}} - \frac{2 \left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{105 a^{3} b \left (a + b x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b**2*x**2+a**2)**(1/2)/(b*x+a)**5,x)

[Out]

-(a**2 - b**2*x**2)**(3/2)/(7*a*b*(a + b*x)**5) - 2*(a**2 - b**2*x**2)**(3/2)/(3
5*a**2*b*(a + b*x)**4) - 2*(a**2 - b**2*x**2)**(3/2)/(105*a**3*b*(a + b*x)**3)

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Mathematica [A]  time = 0.0448501, size = 63, normalized size = 0.63 \[ \frac{\sqrt{a^2-b^2 x^2} \left (-23 a^3+13 a^2 b x+8 a b^2 x^2+2 b^3 x^3\right )}{105 a^3 b (a+b x)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a^2 - b^2*x^2]/(a + b*x)^5,x]

[Out]

(Sqrt[a^2 - b^2*x^2]*(-23*a^3 + 13*a^2*b*x + 8*a*b^2*x^2 + 2*b^3*x^3))/(105*a^3*
b*(a + b*x)^4)

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Maple [A]  time = 0.008, size = 55, normalized size = 0.6 \[ -{\frac{ \left ( 2\,{b}^{2}{x}^{2}+10\,abx+23\,{a}^{2} \right ) \left ( -bx+a \right ) }{105\, \left ( bx+a \right ) ^{4}{a}^{3}b}\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b^2*x^2+a^2)^(1/2)/(b*x+a)^5,x)

[Out]

-1/105*(-b*x+a)*(2*b^2*x^2+10*a*b*x+23*a^2)*(-b^2*x^2+a^2)^(1/2)/(b*x+a)^4/a^3/b

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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)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224252, size = 392, normalized size = 3.92 \[ -\frac{25 \, b^{6} x^{7} + 14 \, a b^{5} x^{6} - 301 \, a^{2} b^{4} x^{5} - 700 \, a^{3} b^{3} x^{4} - 350 \, a^{4} b^{2} x^{3} + 840 \, a^{5} b x^{2} + 840 \, a^{6} x + 7 \,{\left (3 \, b^{5} x^{6} + 23 \, a b^{4} x^{5} + 40 \, a^{2} b^{3} x^{4} - 10 \, a^{3} b^{2} x^{3} - 120 \, a^{4} b x^{2} - 120 \, a^{5} x\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{105 \,{\left (a^{3} b^{7} x^{7} - 14 \, a^{5} b^{5} x^{5} - 28 \, a^{6} b^{4} x^{4} - 7 \, a^{7} b^{3} x^{3} + 28 \, a^{8} b^{2} x^{2} + 28 \, a^{9} b x + 8 \, a^{10} +{\left (a^{3} b^{6} x^{6} + 7 \, a^{4} b^{5} x^{5} + 11 \, a^{5} b^{4} x^{4} - 7 \, a^{6} b^{3} x^{3} - 32 \, a^{7} b^{2} x^{2} - 28 \, a^{8} b x - 8 \, a^{9}\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b^2*x^2 + a^2)/(b*x + a)^5,x, algorithm="fricas")

[Out]

-1/105*(25*b^6*x^7 + 14*a*b^5*x^6 - 301*a^2*b^4*x^5 - 700*a^3*b^3*x^4 - 350*a^4*
b^2*x^3 + 840*a^5*b*x^2 + 840*a^6*x + 7*(3*b^5*x^6 + 23*a*b^4*x^5 + 40*a^2*b^3*x
^4 - 10*a^3*b^2*x^3 - 120*a^4*b*x^2 - 120*a^5*x)*sqrt(-b^2*x^2 + a^2))/(a^3*b^7*
x^7 - 14*a^5*b^5*x^5 - 28*a^6*b^4*x^4 - 7*a^7*b^3*x^3 + 28*a^8*b^2*x^2 + 28*a^9*
b*x + 8*a^10 + (a^3*b^6*x^6 + 7*a^4*b^5*x^5 + 11*a^5*b^4*x^4 - 7*a^6*b^3*x^3 - 3
2*a^7*b^2*x^2 - 28*a^8*b*x - 8*a^9)*sqrt(-b^2*x^2 + a^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (- a + b x\right ) \left (a + b x\right )}}{\left (a + b x\right )^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b**2*x**2+a**2)**(1/2)/(b*x+a)**5,x)

[Out]

Integral(sqrt(-(-a + b*x)*(a + b*x))/(a + b*x)**5, x)

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GIAC/XCAS [A]  time = 0.235492, size = 146, normalized size = 1.46 \[ -\frac{1}{420} \,{\left (\frac{8 \, i{\rm sign}\left (\frac{1}{b x + a}\right ){\rm sign}\left (b\right )}{a^{3} b^{2}} + \frac{{\left (15 \, a^{12} b^{12}{\left (\frac{2 \, a}{b x + a} - 1\right )}^{\frac{7}{2}} + 42 \, a^{12} b^{12}{\left (\frac{2 \, a}{b x + a} - 1\right )}^{\frac{5}{2}} + 35 \, a^{12} b^{12}{\left (\frac{2 \, a}{b x + a} - 1\right )}^{\frac{3}{2}}\right )}{\rm sign}\left (\frac{1}{b x + a}\right ){\rm sign}\left (b\right )}{a^{15} b^{14}}\right )}{\left | b \right |} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b^2*x^2 + a^2)/(b*x + a)^5,x, algorithm="giac")

[Out]

-1/420*(8*i*sign(1/(b*x + a))*sign(b)/(a^3*b^2) + (15*a^12*b^12*(2*a/(b*x + a) -
1)^(7/2) + 42*a^12*b^12*(2*a/(b*x + a) - 1)^(5/2) + 35*a^12*b^12*(2*a/(b*x + a)
- 1)^(3/2))*sign(1/(b*x + a))*sign(b)/(a^15*b^14))*abs(b)