3.89 \(\int \frac{x^9 \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx\)

Optimal. Leaf size=128 \[ -\frac{a^4 (A b-a B)}{4 b^6 \left (a+b x^2\right )^2}+\frac{a^3 (4 A b-5 a B)}{2 b^6 \left (a+b x^2\right )}+\frac{a^2 (3 A b-5 a B) \log \left (a+b x^2\right )}{b^6}-\frac{3 a x^2 (A b-2 a B)}{2 b^5}+\frac{x^4 (A b-3 a B)}{4 b^4}+\frac{B x^6}{6 b^3} \]

[Out]

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

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Rubi [A]  time = 0.371073, antiderivative size = 128, 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 \[ -\frac{a^4 (A b-a B)}{4 b^6 \left (a+b x^2\right )^2}+\frac{a^3 (4 A b-5 a B)}{2 b^6 \left (a+b x^2\right )}+\frac{a^2 (3 A b-5 a B) \log \left (a+b x^2\right )}{b^6}-\frac{3 a x^2 (A b-2 a B)}{2 b^5}+\frac{x^4 (A b-3 a B)}{4 b^4}+\frac{B x^6}{6 b^3} \]

Antiderivative was successfully verified.

[In]  Int[(x^9*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9*(B*x**2+A)/(b*x**2+a)**3,x)

[Out]

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

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Mathematica [A]  time = 0.136892, size = 116, normalized size = 0.91 \[ \frac{\frac{3 a^4 (a B-A b)}{\left (a+b x^2\right )^2}+\frac{6 a^3 (4 A b-5 a B)}{a+b x^2}+12 a^2 (3 A b-5 a B) \log \left (a+b x^2\right )+3 b^2 x^4 (A b-3 a B)+18 a b x^2 (2 a B-A b)+2 b^3 B x^6}{12 b^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^9*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

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

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Maple [A]  time = 0.018, size = 158, normalized size = 1.2 \[{\frac{B{x}^{6}}{6\,{b}^{3}}}+{\frac{A{x}^{4}}{4\,{b}^{3}}}-{\frac{3\,B{x}^{4}a}{4\,{b}^{4}}}-{\frac{3\,aA{x}^{2}}{2\,{b}^{4}}}+3\,{\frac{B{x}^{2}{a}^{2}}{{b}^{5}}}-{\frac{{a}^{4}A}{4\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{{a}^{5}B}{4\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}+3\,{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ) A}{{b}^{5}}}-5\,{\frac{{a}^{3}\ln \left ( b{x}^{2}+a \right ) B}{{b}^{6}}}+2\,{\frac{{a}^{3}A}{{b}^{5} \left ( b{x}^{2}+a \right ) }}-{\frac{5\,{a}^{4}B}{2\,{b}^{6} \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9*(B*x^2+A)/(b*x^2+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.3449, size = 190, normalized size = 1.48 \[ -\frac{9 \, B a^{5} - 7 \, A a^{4} b + 2 \,{\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} x^{2}}{4 \,{\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}} + \frac{2 \, B b^{2} x^{6} - 3 \,{\left (3 \, B a b - A b^{2}\right )} x^{4} + 18 \,{\left (2 \, B a^{2} - A a b\right )} x^{2}}{12 \, b^{5}} - \frac{{\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} \log \left (b x^{2} + a\right )}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^9/(b*x^2 + a)^3,x, algorithm="maxima")

[Out]

-1/4*(9*B*a^5 - 7*A*a^4*b + 2*(5*B*a^4*b - 4*A*a^3*b^2)*x^2)/(b^8*x^4 + 2*a*b^7*
x^2 + a^2*b^6) + 1/12*(2*B*b^2*x^6 - 3*(3*B*a*b - A*b^2)*x^4 + 18*(2*B*a^2 - A*a
*b)*x^2)/b^5 - (5*B*a^3 - 3*A*a^2*b)*log(b*x^2 + a)/b^6

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Fricas [A]  time = 0.220257, size = 277, normalized size = 2.16 \[ \frac{2 \, B b^{5} x^{10} -{\left (5 \, B a b^{4} - 3 \, A b^{5}\right )} x^{8} + 4 \,{\left (5 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{6} - 27 \, B a^{5} + 21 \, A a^{4} b + 3 \,{\left (21 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{4} + 6 \,{\left (B a^{4} b + A a^{3} b^{2}\right )} x^{2} - 12 \,{\left (5 \, B a^{5} - 3 \, A a^{4} b +{\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{4} + 2 \,{\left (5 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{12 \,{\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^9/(b*x^2 + a)^3,x, algorithm="fricas")

[Out]

1/12*(2*B*b^5*x^10 - (5*B*a*b^4 - 3*A*b^5)*x^8 + 4*(5*B*a^2*b^3 - 3*A*a*b^4)*x^6
- 27*B*a^5 + 21*A*a^4*b + 3*(21*B*a^3*b^2 - 11*A*a^2*b^3)*x^4 + 6*(B*a^4*b + A*
a^3*b^2)*x^2 - 12*(5*B*a^5 - 3*A*a^4*b + (5*B*a^3*b^2 - 3*A*a^2*b^3)*x^4 + 2*(5*
B*a^4*b - 3*A*a^3*b^2)*x^2)*log(b*x^2 + a))/(b^8*x^4 + 2*a*b^7*x^2 + a^2*b^6)

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Sympy [A]  time = 6.70035, size = 138, normalized size = 1.08 \[ \frac{B x^{6}}{6 b^{3}} - \frac{a^{2} \left (- 3 A b + 5 B a\right ) \log{\left (a + b x^{2} \right )}}{b^{6}} - \frac{- 7 A a^{4} b + 9 B a^{5} + x^{2} \left (- 8 A a^{3} b^{2} + 10 B a^{4} b\right )}{4 a^{2} b^{6} + 8 a b^{7} x^{2} + 4 b^{8} x^{4}} - \frac{x^{4} \left (- A b + 3 B a\right )}{4 b^{4}} + \frac{x^{2} \left (- 3 A a b + 6 B a^{2}\right )}{2 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9*(B*x**2+A)/(b*x**2+a)**3,x)

[Out]

B*x**6/(6*b**3) - a**2*(-3*A*b + 5*B*a)*log(a + b*x**2)/b**6 - (-7*A*a**4*b + 9*
B*a**5 + x**2*(-8*A*a**3*b**2 + 10*B*a**4*b))/(4*a**2*b**6 + 8*a*b**7*x**2 + 4*b
**8*x**4) - x**4*(-A*b + 3*B*a)/(4*b**4) + x**2*(-3*A*a*b + 6*B*a**2)/(2*b**5)

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GIAC/XCAS [A]  time = 0.233176, size = 215, normalized size = 1.68 \[ -\frac{{\left (5 \, B a^{3} - 3 \, A a^{2} b\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{b^{6}} + \frac{30 \, B a^{3} b^{2} x^{4} - 18 \, A a^{2} b^{3} x^{4} + 50 \, B a^{4} b x^{2} - 28 \, A a^{3} b^{2} x^{2} + 21 \, B a^{5} - 11 \, A a^{4} b}{4 \,{\left (b x^{2} + a\right )}^{2} b^{6}} + \frac{2 \, B b^{6} x^{6} - 9 \, B a b^{5} x^{4} + 3 \, A b^{6} x^{4} + 36 \, B a^{2} b^{4} x^{2} - 18 \, A a b^{5} x^{2}}{12 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^9/(b*x^2 + a)^3,x, algorithm="giac")

[Out]

-(5*B*a^3 - 3*A*a^2*b)*ln(abs(b*x^2 + a))/b^6 + 1/4*(30*B*a^3*b^2*x^4 - 18*A*a^2
*b^3*x^4 + 50*B*a^4*b*x^2 - 28*A*a^3*b^2*x^2 + 21*B*a^5 - 11*A*a^4*b)/((b*x^2 +
a)^2*b^6) + 1/12*(2*B*b^6*x^6 - 9*B*a*b^5*x^4 + 3*A*b^6*x^4 + 36*B*a^2*b^4*x^2 -
18*A*a*b^5*x^2)/b^9