3.71 \(\int \frac{x^9 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=126 \[ -\frac{a^4 (A b-a B)}{2 b^6 \left (a+b x^2\right )}-\frac{a^3 (4 A b-5 a B) \log \left (a+b x^2\right )}{2 b^6}+\frac{a^2 x^2 (3 A b-4 a B)}{2 b^5}-\frac{a x^4 (2 A b-3 a B)}{4 b^4}+\frac{x^6 (A b-2 a B)}{6 b^3}+\frac{B x^8}{8 b^2} \]

[Out]

(a^2*(3*A*b - 4*a*B)*x^2)/(2*b^5) - (a*(2*A*b - 3*a*B)*x^4)/(4*b^4) + ((A*b - 2*
a*B)*x^6)/(6*b^3) + (B*x^8)/(8*b^2) - (a^4*(A*b - a*B))/(2*b^6*(a + b*x^2)) - (a
^3*(4*A*b - 5*a*B)*Log[a + b*x^2])/(2*b^6)

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Rubi [A]  time = 0.373282, antiderivative size = 126, 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 \[ -\frac{a^4 (A b-a B)}{2 b^6 \left (a+b x^2\right )}-\frac{a^3 (4 A b-5 a B) \log \left (a+b x^2\right )}{2 b^6}+\frac{a^2 x^2 (3 A b-4 a B)}{2 b^5}-\frac{a x^4 (2 A b-3 a B)}{4 b^4}+\frac{x^6 (A b-2 a B)}{6 b^3}+\frac{B x^8}{8 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^9*(A + B*x^2))/(a + b*x^2)^2,x]

[Out]

(a^2*(3*A*b - 4*a*B)*x^2)/(2*b^5) - (a*(2*A*b - 3*a*B)*x^4)/(4*b^4) + ((A*b - 2*
a*B)*x^6)/(6*b^3) + (B*x^8)/(8*b^2) - (a^4*(A*b - a*B))/(2*b^6*(a + b*x^2)) - (a
^3*(4*A*b - 5*a*B)*Log[a + b*x^2])/(2*b^6)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B x^{8}}{8 b^{2}} - \frac{a^{4} \left (A b - B a\right )}{2 b^{6} \left (a + b x^{2}\right )} - \frac{a^{3} \left (4 A b - 5 B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{6}} - \frac{a \left (2 A b - 3 B a\right ) \int ^{x^{2}} x\, dx}{2 b^{4}} + \frac{x^{6} \left (A b - 2 B a\right )}{6 b^{3}} + \frac{\left (3 A b - 4 B a\right ) \int ^{x^{2}} a^{2}\, dx}{2 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9*(B*x**2+A)/(b*x**2+a)**2,x)

[Out]

B*x**8/(8*b**2) - a**4*(A*b - B*a)/(2*b**6*(a + b*x**2)) - a**3*(4*A*b - 5*B*a)*
log(a + b*x**2)/(2*b**6) - a*(2*A*b - 3*B*a)*Integral(x, (x, x**2))/(2*b**4) + x
**6*(A*b - 2*B*a)/(6*b**3) + (3*A*b - 4*B*a)*Integral(a**2, (x, x**2))/(2*b**5)

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Mathematica [A]  time = 0.135669, size = 113, normalized size = 0.9 \[ \frac{\frac{12 a^4 (a B-A b)}{a+b x^2}+12 a^3 (5 a B-4 A b) \log \left (a+b x^2\right )-12 a^2 b x^2 (4 a B-3 A b)+4 b^3 x^6 (A b-2 a B)+6 a b^2 x^4 (3 a B-2 A b)+3 b^4 B x^8}{24 b^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^9*(A + B*x^2))/(a + b*x^2)^2,x]

[Out]

(-12*a^2*b*(-3*A*b + 4*a*B)*x^2 + 6*a*b^2*(-2*A*b + 3*a*B)*x^4 + 4*b^3*(A*b - 2*
a*B)*x^6 + 3*b^4*B*x^8 + (12*a^4*(-(A*b) + a*B))/(a + b*x^2) + 12*a^3*(-4*A*b +
5*a*B)*Log[a + b*x^2])/(24*b^6)

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Maple [A]  time = 0.016, size = 146, normalized size = 1.2 \[{\frac{B{x}^{8}}{8\,{b}^{2}}}+{\frac{{x}^{6}A}{6\,{b}^{2}}}-{\frac{{x}^{6}Ba}{3\,{b}^{3}}}-{\frac{{x}^{4}Aa}{2\,{b}^{3}}}+{\frac{3\,{x}^{4}B{a}^{2}}{4\,{b}^{4}}}+{\frac{3\,{a}^{2}A{x}^{2}}{2\,{b}^{4}}}-2\,{\frac{B{x}^{2}{a}^{3}}{{b}^{5}}}-2\,{\frac{{a}^{3}\ln \left ( b{x}^{2}+a \right ) A}{{b}^{5}}}+{\frac{5\,{a}^{4}\ln \left ( b{x}^{2}+a \right ) B}{2\,{b}^{6}}}-{\frac{{a}^{4}A}{2\,{b}^{5} \left ( b{x}^{2}+a \right ) }}+{\frac{{a}^{5}B}{2\,{b}^{6} \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9*(B*x^2+A)/(b*x^2+a)^2,x)

[Out]

1/8*B*x^8/b^2+1/6/b^2*x^6*A-1/3/b^3*x^6*B*a-1/2/b^3*x^4*A*a+3/4/b^4*x^4*B*a^2+3/
2/b^4*x^2*A*a^2-2/b^5*x^2*B*a^3-2*a^3/b^5*ln(b*x^2+a)*A+5/2*a^4/b^6*ln(b*x^2+a)*
B-1/2*a^4/b^5/(b*x^2+a)*A+1/2*a^5/b^6/(b*x^2+a)*B

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Maxima [A]  time = 1.35179, size = 177, normalized size = 1.4 \[ \frac{B a^{5} - A a^{4} b}{2 \,{\left (b^{7} x^{2} + a b^{6}\right )}} + \frac{3 \, B b^{3} x^{8} - 4 \,{\left (2 \, B a b^{2} - A b^{3}\right )} x^{6} + 6 \,{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{4} - 12 \,{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} x^{2}}{24 \, b^{5}} + \frac{{\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^9/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

1/2*(B*a^5 - A*a^4*b)/(b^7*x^2 + a*b^6) + 1/24*(3*B*b^3*x^8 - 4*(2*B*a*b^2 - A*b
^3)*x^6 + 6*(3*B*a^2*b - 2*A*a*b^2)*x^4 - 12*(4*B*a^3 - 3*A*a^2*b)*x^2)/b^5 + 1/
2*(5*B*a^4 - 4*A*a^3*b)*log(b*x^2 + a)/b^6

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Fricas [A]  time = 0.233262, size = 232, normalized size = 1.84 \[ \frac{3 \, B b^{5} x^{10} -{\left (5 \, B a b^{4} - 4 \, A b^{5}\right )} x^{8} + 2 \,{\left (5 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} x^{6} + 12 \, B a^{5} - 12 \, A a^{4} b - 6 \,{\left (5 \, B a^{3} b^{2} - 4 \, A a^{2} b^{3}\right )} x^{4} - 12 \,{\left (4 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x^{2} + 12 \,{\left (5 \, B a^{5} - 4 \, A a^{4} b +{\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{24 \,{\left (b^{7} x^{2} + a b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^9/(b*x^2 + a)^2,x, algorithm="fricas")

[Out]

1/24*(3*B*b^5*x^10 - (5*B*a*b^4 - 4*A*b^5)*x^8 + 2*(5*B*a^2*b^3 - 4*A*a*b^4)*x^6
+ 12*B*a^5 - 12*A*a^4*b - 6*(5*B*a^3*b^2 - 4*A*a^2*b^3)*x^4 - 12*(4*B*a^4*b - 3
*A*a^3*b^2)*x^2 + 12*(5*B*a^5 - 4*A*a^4*b + (5*B*a^4*b - 4*A*a^3*b^2)*x^2)*log(b
*x^2 + a))/(b^7*x^2 + a*b^6)

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Sympy [A]  time = 3.82095, size = 126, normalized size = 1. \[ \frac{B x^{8}}{8 b^{2}} + \frac{a^{3} \left (- 4 A b + 5 B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{6}} + \frac{- A a^{4} b + B a^{5}}{2 a b^{6} + 2 b^{7} x^{2}} - \frac{x^{6} \left (- A b + 2 B a\right )}{6 b^{3}} + \frac{x^{4} \left (- 2 A a b + 3 B a^{2}\right )}{4 b^{4}} - \frac{x^{2} \left (- 3 A a^{2} b + 4 B a^{3}\right )}{2 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9*(B*x**2+A)/(b*x**2+a)**2,x)

[Out]

B*x**8/(8*b**2) + a**3*(-4*A*b + 5*B*a)*log(a + b*x**2)/(2*b**6) + (-A*a**4*b +
B*a**5)/(2*a*b**6 + 2*b**7*x**2) - x**6*(-A*b + 2*B*a)/(6*b**3) + x**4*(-2*A*a*b
+ 3*B*a**2)/(4*b**4) - x**2*(-3*A*a**2*b + 4*B*a**3)/(2*b**5)

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GIAC/XCAS [A]  time = 0.240052, size = 215, normalized size = 1.71 \[ \frac{{\left (5 \, B a^{4} - 4 \, A a^{3} b\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{6}} - \frac{5 \, B a^{4} b x^{2} - 4 \, A a^{3} b^{2} x^{2} + 4 \, B a^{5} - 3 \, A a^{4} b}{2 \,{\left (b x^{2} + a\right )} b^{6}} + \frac{3 \, B b^{6} x^{8} - 8 \, B a b^{5} x^{6} + 4 \, A b^{6} x^{6} + 18 \, B a^{2} b^{4} x^{4} - 12 \, A a b^{5} x^{4} - 48 \, B a^{3} b^{3} x^{2} + 36 \, A a^{2} b^{4} x^{2}}{24 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^9/(b*x^2 + a)^2,x, algorithm="giac")

[Out]

1/2*(5*B*a^4 - 4*A*a^3*b)*ln(abs(b*x^2 + a))/b^6 - 1/2*(5*B*a^4*b*x^2 - 4*A*a^3*
b^2*x^2 + 4*B*a^5 - 3*A*a^4*b)/((b*x^2 + a)*b^6) + 1/24*(3*B*b^6*x^8 - 8*B*a*b^5
*x^6 + 4*A*b^6*x^6 + 18*B*a^2*b^4*x^4 - 12*A*a*b^5*x^4 - 48*B*a^3*b^3*x^2 + 36*A
*a^2*b^4*x^2)/b^8