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

Optimal. Leaf size=53 \[ \frac{\left (a+b x^2\right )^{3/2} (2 A b-5 a B)}{15 a^2 x^3}-\frac{A \left (a+b x^2\right )^{3/2}}{5 a x^5} \]

[Out]

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

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Rubi [A]  time = 0.0819841, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\left (a+b x^2\right )^{3/2} (2 A b-5 a B)}{15 a^2 x^3}-\frac{A \left (a+b x^2\right )^{3/2}}{5 a x^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.2663, size = 46, normalized size = 0.87 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{3}{2}}}{5 a x^{5}} + \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \left (2 A b - 5 B a\right )}{15 a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0498783, size = 40, normalized size = 0.75 \[ -\frac{\left (a+b x^2\right )^{3/2} \left (3 a A+5 a B x^2-2 A b x^2\right )}{15 a^2 x^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 37, normalized size = 0.7 \[ -{\frac{-2\,Ab{x}^{2}+5\,Ba{x}^{2}+3\,Aa}{15\,{x}^{5}{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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 = 0.220597, size = 74, normalized size = 1.4 \[ -\frac{{\left ({\left (5 \, B a b - 2 \, A b^{2}\right )} x^{4} + 3 \, A a^{2} +{\left (5 \, B a^{2} + A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{15 \, a^{2} x^{5}} \]

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

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Sympy [A]  time = 5.89713, size = 119, normalized size = 2.25 \[ - \frac{A \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{5 x^{4}} - \frac{A b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a x^{2}} + \frac{2 A b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{2}} - \frac{B \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{3 x^{2}} - \frac{B b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.231074, size = 313, normalized size = 5.91 \[ \frac{2 \,{\left (15 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} B b^{\frac{3}{2}} - 30 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B a b^{\frac{3}{2}} + 30 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} A b^{\frac{5}{2}} + 20 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{2} b^{\frac{3}{2}} + 10 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a b^{\frac{5}{2}} - 10 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{3} b^{\frac{3}{2}} + 10 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a^{2} b^{\frac{5}{2}} + 5 \, B a^{4} b^{\frac{3}{2}} - 2 \, A a^{3} b^{\frac{5}{2}}\right )}}{15 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{5}} \]

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]

2/15*(15*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*b^(3/2) - 30*(sqrt(b)*x - sqrt(b*x^2
+ a))^6*B*a*b^(3/2) + 30*(sqrt(b)*x - sqrt(b*x^2 + a))^6*A*b^(5/2) + 20*(sqrt(b)
*x - sqrt(b*x^2 + a))^4*B*a^2*b^(3/2) + 10*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a*b
^(5/2) - 10*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^3*b^(3/2) + 10*(sqrt(b)*x - sqrt
(b*x^2 + a))^2*A*a^2*b^(5/2) + 5*B*a^4*b^(3/2) - 2*A*a^3*b^(5/2))/((sqrt(b)*x -
sqrt(b*x^2 + a))^2 - a)^5