3.75 \(\int \frac{x^5 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=82 \[ -\frac{a^2 (A b-a B)}{2 b^4 \left (a+b x^2\right )}-\frac{a (2 A b-3 a B) \log \left (a+b x^2\right )}{2 b^4}+\frac{x^2 (A b-2 a B)}{2 b^3}+\frac{B x^4}{4 b^2} \]

[Out]

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

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Rubi [A]  time = 0.214515, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{a^2 (A b-a B)}{2 b^4 \left (a+b x^2\right )}-\frac{a (2 A b-3 a B) \log \left (a+b x^2\right )}{2 b^4}+\frac{x^2 (A b-2 a B)}{2 b^3}+\frac{B x^4}{4 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B \int ^{x^{2}} x\, dx}{2 b^{2}} - \frac{a^{2} \left (A b - B a\right )}{2 b^{4} \left (a + b x^{2}\right )} - \frac{a \left (2 A b - 3 B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{4}} + \left (\frac{A b}{2} - B a\right ) \int ^{x^{2}} \frac{1}{b^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(B*x**2+A)/(b*x**2+a)**2,x)

[Out]

B*Integral(x, (x, x**2))/(2*b**2) - a**2*(A*b - B*a)/(2*b**4*(a + b*x**2)) - a*(
2*A*b - 3*B*a)*log(a + b*x**2)/(2*b**4) + (A*b/2 - B*a)*Integral(b**(-3), (x, x*
*2))

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Mathematica [A]  time = 0.113879, size = 72, normalized size = 0.88 \[ \frac{\frac{2 a^2 (a B-A b)}{a+b x^2}+2 b x^2 (A b-2 a B)+2 a (3 a B-2 A b) \log \left (a+b x^2\right )+b^2 B x^4}{4 b^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 98, normalized size = 1.2 \[{\frac{B{x}^{4}}{4\,{b}^{2}}}+{\frac{A{x}^{2}}{2\,{b}^{2}}}-{\frac{B{x}^{2}a}{{b}^{3}}}-{\frac{a\ln \left ( b{x}^{2}+a \right ) A}{{b}^{3}}}+{\frac{3\,{a}^{2}\ln \left ( b{x}^{2}+a \right ) B}{2\,{b}^{4}}}-{\frac{{a}^{2}A}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{B{a}^{3}}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.351, size = 111, normalized size = 1.35 \[ \frac{B a^{3} - A a^{2} b}{2 \,{\left (b^{5} x^{2} + a b^{4}\right )}} + \frac{B b x^{4} - 2 \,{\left (2 \, B a - A b\right )} x^{2}}{4 \, b^{3}} + \frac{{\left (3 \, B a^{2} - 2 \, A a b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^5/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.233355, size = 163, normalized size = 1.99 \[ \frac{B b^{3} x^{6} -{\left (3 \, B a b^{2} - 2 \, A b^{3}\right )} x^{4} + 2 \, B a^{3} - 2 \, A a^{2} b - 2 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} x^{2} + 2 \,{\left (3 \, B a^{3} - 2 \, A a^{2} b +{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{4 \,{\left (b^{5} x^{2} + a b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(B*b^3*x^6 - (3*B*a*b^2 - 2*A*b^3)*x^4 + 2*B*a^3 - 2*A*a^2*b - 2*(2*B*a^2*b
- A*a*b^2)*x^2 + 2*(3*B*a^3 - 2*A*a^2*b + (3*B*a^2*b - 2*A*a*b^2)*x^2)*log(b*x^2
+ a))/(b^5*x^2 + a*b^4)

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Sympy [A]  time = 3.20477, size = 78, normalized size = 0.95 \[ \frac{B x^{4}}{4 b^{2}} + \frac{a \left (- 2 A b + 3 B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{4}} + \frac{- A a^{2} b + B a^{3}}{2 a b^{4} + 2 b^{5} x^{2}} - \frac{x^{2} \left (- A b + 2 B a\right )}{2 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(B*x**2+A)/(b*x**2+a)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.240037, size = 143, normalized size = 1.74 \[ \frac{{\left (3 \, B a^{2} - 2 \, A a b\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} + \frac{B b^{2} x^{4} - 4 \, B a b x^{2} + 2 \, A b^{2} x^{2}}{4 \, b^{4}} - \frac{3 \, B a^{2} b x^{2} - 2 \, A a b^{2} x^{2} + 2 \, B a^{3} - A a^{2} b}{2 \,{\left (b x^{2} + a\right )} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(3*B*a^2 - 2*A*a*b)*ln(abs(b*x^2 + a))/b^4 + 1/4*(B*b^2*x^4 - 4*B*a*b*x^2 +
2*A*b^2*x^2)/b^4 - 1/2*(3*B*a^2*b*x^2 - 2*A*a*b^2*x^2 + 2*B*a^3 - A*a^2*b)/((b*x
^2 + a)*b^4)