3.785 \(\int \frac{\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^6} \, dx\)

Optimal. Leaf size=67 \[ -\frac{\left (a^2-b^2 x^2\right )^{5/2}}{35 a^2 b (a+b x)^5}-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{7 a b (a+b x)^6} \]

[Out]

-(a^2 - b^2*x^2)^(5/2)/(7*a*b*(a + b*x)^6) - (a^2 - b^2*x^2)^(5/2)/(35*a^2*b*(a
+ b*x)^5)

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Rubi [A]  time = 0.0743916, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{\left (a^2-b^2 x^2\right )^{5/2}}{35 a^2 b (a+b x)^5}-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{7 a b (a+b x)^6} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^6,x]

[Out]

-(a^2 - b^2*x^2)^(5/2)/(7*a*b*(a + b*x)^6) - (a^2 - b^2*x^2)^(5/2)/(35*a^2*b*(a
+ b*x)^5)

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Rubi in Sympy [A]  time = 8.25169, size = 53, normalized size = 0.79 \[ - \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{5}{2}}}{7 a b \left (a + b x\right )^{6}} - \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{5}{2}}}{35 a^{2} b \left (a + b x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b**2*x**2+a**2)**(3/2)/(b*x+a)**6,x)

[Out]

-(a**2 - b**2*x**2)**(5/2)/(7*a*b*(a + b*x)**6) - (a**2 - b**2*x**2)**(5/2)/(35*
a**2*b*(a + b*x)**5)

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Mathematica [A]  time = 0.0501737, size = 48, normalized size = 0.72 \[ -\frac{(a-b x)^2 (6 a+b x) \sqrt{a^2-b^2 x^2}}{35 a^2 b (a+b x)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^6,x]

[Out]

-((a - b*x)^2*(6*a + b*x)*Sqrt[a^2 - b^2*x^2])/(35*a^2*b*(a + b*x)^4)

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Maple [A]  time = 0.011, size = 43, normalized size = 0.6 \[ -{\frac{ \left ( bx+6\,a \right ) \left ( -bx+a \right ) }{35\, \left ( bx+a \right ) ^{5}{a}^{2}b} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b^2*x^2+a^2)^(3/2)/(b*x+a)^6,x)

[Out]

-1/35*(-b*x+a)*(b*x+6*a)*(-b^2*x^2+a^2)^(3/2)/(b*x+a)^5/a^2/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((-b^2*x^2 + a^2)^(3/2)/(b*x + a)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224385, size = 375, normalized size = 5.6 \[ -\frac{5 \, b^{6} x^{7} - 7 \, a b^{5} x^{6} - 77 \, a^{2} b^{4} x^{5} - 105 \, a^{3} b^{3} x^{4} - 140 \, a^{4} b^{2} x^{3} + 140 \, a^{5} b x^{2} + 280 \, a^{6} x + 7 \,{\left (b^{5} x^{6} + 6 \, a b^{4} x^{5} + 5 \, a^{2} b^{3} x^{4} - 20 \, a^{4} b x^{2} - 40 \, a^{5} x\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{35 \,{\left (a^{2} b^{7} x^{7} - 14 \, a^{4} b^{5} x^{5} - 28 \, a^{5} b^{4} x^{4} - 7 \, a^{6} b^{3} x^{3} + 28 \, a^{7} b^{2} x^{2} + 28 \, a^{8} b x + 8 \, a^{9} +{\left (a^{2} b^{6} x^{6} + 7 \, a^{3} b^{5} x^{5} + 11 \, a^{4} b^{4} x^{4} - 7 \, a^{5} b^{3} x^{3} - 32 \, a^{6} b^{2} x^{2} - 28 \, a^{7} b x - 8 \, a^{8}\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b^2*x^2 + a^2)^(3/2)/(b*x + a)^6,x, algorithm="fricas")

[Out]

-1/35*(5*b^6*x^7 - 7*a*b^5*x^6 - 77*a^2*b^4*x^5 - 105*a^3*b^3*x^4 - 140*a^4*b^2*
x^3 + 140*a^5*b*x^2 + 280*a^6*x + 7*(b^5*x^6 + 6*a*b^4*x^5 + 5*a^2*b^3*x^4 - 20*
a^4*b*x^2 - 40*a^5*x)*sqrt(-b^2*x^2 + a^2))/(a^2*b^7*x^7 - 14*a^4*b^5*x^5 - 28*a
^5*b^4*x^4 - 7*a^6*b^3*x^3 + 28*a^7*b^2*x^2 + 28*a^8*b*x + 8*a^9 + (a^2*b^6*x^6
+ 7*a^3*b^5*x^5 + 11*a^4*b^4*x^4 - 7*a^5*b^3*x^3 - 32*a^6*b^2*x^2 - 28*a^7*b*x -
8*a^8)*sqrt(-b^2*x^2 + a^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- \left (- a + b x\right ) \left (a + b x\right )\right )^{\frac{3}{2}}}{\left (a + b x\right )^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b**2*x**2+a**2)**(3/2)/(b*x+a)**6,x)

[Out]

Integral((-(-a + b*x)*(a + b*x))**(3/2)/(a + b*x)**6, x)

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GIAC/XCAS [A]  time = 0.234091, size = 306, normalized size = 4.57 \[ \frac{2 \,{\left (\frac{7 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}}{b^{2} x} + \frac{91 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + \frac{70 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{3}}{b^{6} x^{3}} + \frac{140 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{4}}{b^{8} x^{4}} + \frac{35 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{5}}{b^{10} x^{5}} + \frac{35 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{6}}{b^{12} x^{6}} + 6\right )}}{35 \, a^{2}{\left (\frac{a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}}{b^{2} x} + 1\right )}^{7}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b^2*x^2 + a^2)^(3/2)/(b*x + a)^6,x, algorithm="giac")

[Out]

2/35*(7*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 91*(a*b + sqrt(-b^2*x^2 +
a^2)*abs(b))^2/(b^4*x^2) + 70*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^3/(b^6*x^3) +
140*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^4/(b^8*x^4) + 35*(a*b + sqrt(-b^2*x^2 +
a^2)*abs(b))^5/(b^10*x^5) + 35*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^6/(b^12*x^6)
+ 6)/(a^2*((a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 1)^7*abs(b))