3.566 \(\int \frac{A+B x^2}{x^6 \sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=84 \[ -\frac{2 b \sqrt{a+b x^2} (4 A b-5 a B)}{15 a^3 x}+\frac{\sqrt{a+b x^2} (4 A b-5 a B)}{15 a^2 x^3}-\frac{A \sqrt{a+b x^2}}{5 a x^5} \]

[Out]

-(A*Sqrt[a + b*x^2])/(5*a*x^5) + ((4*A*b - 5*a*B)*Sqrt[a + b*x^2])/(15*a^2*x^3)
- (2*b*(4*A*b - 5*a*B)*Sqrt[a + b*x^2])/(15*a^3*x)

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Rubi [A]  time = 0.117464, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{2 b \sqrt{a+b x^2} (4 A b-5 a B)}{15 a^3 x}+\frac{\sqrt{a+b x^2} (4 A b-5 a B)}{15 a^2 x^3}-\frac{A \sqrt{a+b x^2}}{5 a x^5} \]

Antiderivative was successfully verified.

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

[Out]

-(A*Sqrt[a + b*x^2])/(5*a*x^5) + ((4*A*b - 5*a*B)*Sqrt[a + b*x^2])/(15*a^2*x^3)
- (2*b*(4*A*b - 5*a*B)*Sqrt[a + b*x^2])/(15*a^3*x)

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Rubi in Sympy [A]  time = 12.8754, size = 76, normalized size = 0.9 \[ - \frac{A \sqrt{a + b x^{2}}}{5 a x^{5}} + \frac{\sqrt{a + b x^{2}} \left (4 A b - 5 B a\right )}{15 a^{2} x^{3}} - \frac{2 b \sqrt{a + b x^{2}} \left (4 A b - 5 B a\right )}{15 a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**6/(b*x**2+a)**(1/2),x)

[Out]

-A*sqrt(a + b*x**2)/(5*a*x**5) + sqrt(a + b*x**2)*(4*A*b - 5*B*a)/(15*a**2*x**3)
- 2*b*sqrt(a + b*x**2)*(4*A*b - 5*B*a)/(15*a**3*x)

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Mathematica [A]  time = 0.0558745, size = 63, normalized size = 0.75 \[ \sqrt{a+b x^2} \left (\frac{2 b (5 a B-4 A b)}{15 a^3 x}+\frac{4 A b-5 a B}{15 a^2 x^3}-\frac{A}{5 a x^5}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-A/(5*a*x^5) + (4*A*b - 5*a*B)/(15*a^2*x^3) + (2*b*(-4*A*b + 5*a*B))/(15*a^3*x)
)*Sqrt[a + b*x^2]

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Maple [A]  time = 0.008, size = 59, normalized size = 0.7 \[ -{\frac{8\,A{b}^{2}{x}^{4}-10\,Bab{x}^{4}-4\,aAb{x}^{2}+5\,B{a}^{2}{x}^{2}+3\,A{a}^{2}}{15\,{x}^{5}{a}^{3}}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^6/(b*x^2+a)^(1/2),x)

[Out]

-1/15*(b*x^2+a)^(1/2)*(8*A*b^2*x^4-10*B*a*b*x^4-4*A*a*b*x^2+5*B*a^2*x^2+3*A*a^2)
/x^5/a^3

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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*x^2 + A)/(sqrt(b*x^2 + a)*x^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23782, size = 78, normalized size = 0.93 \[ \frac{{\left (2 \,{\left (5 \, B a b - 4 \, A b^{2}\right )} x^{4} - 3 \, A a^{2} -{\left (5 \, B a^{2} - 4 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{15 \, a^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(2*(5*B*a*b - 4*A*b^2)*x^4 - 3*A*a^2 - (5*B*a^2 - 4*A*a*b)*x^2)*sqrt(b*x^2
+ a)/(a^3*x^5)

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Sympy [A]  time = 7.16687, size = 355, normalized size = 4.23 \[ - \frac{3 A a^{4} b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac{2 A a^{3} b^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac{3 A a^{2} b^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac{12 A a b^{\frac{15}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac{8 A b^{\frac{17}{2}} x^{8} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac{B \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a x^{2}} + \frac{2 B b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)/x**6/(b*x**2+a)**(1/2),x)

[Out]

-3*A*a**4*b**(9/2)*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 +
15*a**3*b**6*x**8) - 2*A*a**3*b**(11/2)*x**2*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4
*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) - 3*A*a**2*b**(13/2)*x**4*sqrt(a/
(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) - 12*A
*a*b**(15/2)*x**6*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 +
15*a**3*b**6*x**8) - 8*A*b**(17/2)*x**8*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4
+ 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) - B*sqrt(b)*sqrt(a/(b*x**2) + 1)/(3*a*x
**2) + 2*B*b**(3/2)*sqrt(a/(b*x**2) + 1)/(3*a**2)

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GIAC/XCAS [A]  time = 0.243855, size = 238, normalized size = 2.83 \[ \frac{4 \,{\left (15 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B b^{\frac{3}{2}} - 35 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a b^{\frac{3}{2}} + 40 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A b^{\frac{5}{2}} + 25 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{2} b^{\frac{3}{2}} - 20 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a b^{\frac{5}{2}} - 5 \, B a^{3} b^{\frac{3}{2}} + 4 \, A a^{2} b^{\frac{5}{2}}\right )}}{15 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/15*(15*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*b^(3/2) - 35*(sqrt(b)*x - sqrt(b*x^2
+ a))^4*B*a*b^(3/2) + 40*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*b^(5/2) + 25*(sqrt(b)
*x - sqrt(b*x^2 + a))^2*B*a^2*b^(3/2) - 20*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a*b
^(5/2) - 5*B*a^3*b^(3/2) + 4*A*a^2*b^(5/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a
)^5