3.72 \(\int \frac{x^8 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=131 \[ -\frac{a^{5/2} (7 A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{11/2}}+\frac{a^3 x (A b-a B)}{2 b^5 \left (a+b x^2\right )}+\frac{a^2 x (3 A b-4 a B)}{b^5}-\frac{a x^3 (2 A b-3 a B)}{3 b^4}+\frac{x^5 (A b-2 a B)}{5 b^3}+\frac{B x^7}{7 b^2} \]

[Out]

(a^2*(3*A*b - 4*a*B)*x)/b^5 - (a*(2*A*b - 3*a*B)*x^3)/(3*b^4) + ((A*b - 2*a*B)*x
^5)/(5*b^3) + (B*x^7)/(7*b^2) + (a^3*(A*b - a*B)*x)/(2*b^5*(a + b*x^2)) - (a^(5/
2)*(7*A*b - 9*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(11/2))

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Rubi [A]  time = 0.287295, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{a^{5/2} (7 A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{11/2}}+\frac{a^3 x (A b-a B)}{2 b^5 \left (a+b x^2\right )}+\frac{a^2 x (3 A b-4 a B)}{b^5}-\frac{a x^3 (2 A b-3 a B)}{3 b^4}+\frac{x^5 (A b-2 a B)}{5 b^3}+\frac{B x^7}{7 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^8*(A + B*x^2))/(a + b*x^2)^2,x]

[Out]

(a^2*(3*A*b - 4*a*B)*x)/b^5 - (a*(2*A*b - 3*a*B)*x^3)/(3*b^4) + ((A*b - 2*a*B)*x
^5)/(5*b^3) + (B*x^7)/(7*b^2) + (a^3*(A*b - a*B)*x)/(2*b^5*(a + b*x^2)) - (a^(5/
2)*(7*A*b - 9*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(11/2))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B x^{7}}{7 b^{2}} - \frac{a^{\frac{5}{2}} \left (7 A b - 9 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 b^{\frac{11}{2}}} + \frac{a^{3} x \left (A b - B a\right )}{2 b^{5} \left (a + b x^{2}\right )} - \frac{a x^{3} \left (2 A b - 3 B a\right )}{3 b^{4}} + \frac{x^{5} \left (A b - 2 B a\right )}{5 b^{3}} + \frac{\left (3 A b - 4 B a\right ) \int a^{2}\, dx}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8*(B*x**2+A)/(b*x**2+a)**2,x)

[Out]

B*x**7/(7*b**2) - a**(5/2)*(7*A*b - 9*B*a)*atan(sqrt(b)*x/sqrt(a))/(2*b**(11/2))
+ a**3*x*(A*b - B*a)/(2*b**5*(a + b*x**2)) - a*x**3*(2*A*b - 3*B*a)/(3*b**4) +
x**5*(A*b - 2*B*a)/(5*b**3) + (3*A*b - 4*B*a)*Integral(a**2, x)/b**5

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Mathematica [A]  time = 0.179153, size = 134, normalized size = 1.02 \[ \frac{a^{5/2} (9 a B-7 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{11/2}}-\frac{a^2 x (4 a B-3 A b)}{b^5}+\frac{x \left (a^3 A b-a^4 B\right )}{2 b^5 \left (a+b x^2\right )}+\frac{a x^3 (3 a B-2 A b)}{3 b^4}+\frac{x^5 (A b-2 a B)}{5 b^3}+\frac{B x^7}{7 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^8*(A + B*x^2))/(a + b*x^2)^2,x]

[Out]

-((a^2*(-3*A*b + 4*a*B)*x)/b^5) + (a*(-2*A*b + 3*a*B)*x^3)/(3*b^4) + ((A*b - 2*a
*B)*x^5)/(5*b^3) + (B*x^7)/(7*b^2) + ((a^3*A*b - a^4*B)*x)/(2*b^5*(a + b*x^2)) +
(a^(5/2)*(-7*A*b + 9*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(11/2))

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Maple [A]  time = 0.013, size = 155, normalized size = 1.2 \[{\frac{B{x}^{7}}{7\,{b}^{2}}}+{\frac{A{x}^{5}}{5\,{b}^{2}}}-{\frac{2\,B{x}^{5}a}{5\,{b}^{3}}}-{\frac{2\,aA{x}^{3}}{3\,{b}^{3}}}+{\frac{B{x}^{3}{a}^{2}}{{b}^{4}}}+3\,{\frac{{a}^{2}Ax}{{b}^{4}}}-4\,{\frac{B{a}^{3}x}{{b}^{5}}}+{\frac{{a}^{3}xA}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{{a}^{4}xB}{2\,{b}^{5} \left ( b{x}^{2}+a \right ) }}-{\frac{7\,A{a}^{3}}{2\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{9\,B{a}^{4}}{2\,{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(B*x^2+A)/(b*x^2+a)^2,x)

[Out]

1/7*B*x^7/b^2+1/5/b^2*A*x^5-2/5/b^3*B*x^5*a-2/3/b^3*A*x^3*a+1/b^4*B*x^3*a^2+3/b^
4*A*a^2*x-4/b^5*B*a^3*x+1/2*a^3/b^4*x/(b*x^2+a)*A-1/2*a^4/b^5*x/(b*x^2+a)*B-7/2*
a^3/b^4/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*A+9/2*a^4/b^5/(a*b)^(1/2)*arctan(x*b
/(a*b)^(1/2))*B

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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)*x^8/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.248265, size = 1, normalized size = 0.01 \[ \left [\frac{60 \, B b^{4} x^{9} - 12 \,{\left (9 \, B a b^{3} - 7 \, A b^{4}\right )} x^{7} + 28 \,{\left (9 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{5} - 140 \,{\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{3} - 105 \,{\left (9 \, B a^{4} - 7 \, A a^{3} b +{\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) - 210 \,{\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} x}{420 \,{\left (b^{6} x^{2} + a b^{5}\right )}}, \frac{30 \, B b^{4} x^{9} - 6 \,{\left (9 \, B a b^{3} - 7 \, A b^{4}\right )} x^{7} + 14 \,{\left (9 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{5} - 70 \,{\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{3} + 105 \,{\left (9 \, B a^{4} - 7 \, A a^{3} b +{\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) - 105 \,{\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} x}{210 \,{\left (b^{6} x^{2} + a b^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^8/(b*x^2 + a)^2,x, algorithm="fricas")

[Out]

[1/420*(60*B*b^4*x^9 - 12*(9*B*a*b^3 - 7*A*b^4)*x^7 + 28*(9*B*a^2*b^2 - 7*A*a*b^
3)*x^5 - 140*(9*B*a^3*b - 7*A*a^2*b^2)*x^3 - 105*(9*B*a^4 - 7*A*a^3*b + (9*B*a^3
*b - 7*A*a^2*b^2)*x^2)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)
) - 210*(9*B*a^4 - 7*A*a^3*b)*x)/(b^6*x^2 + a*b^5), 1/210*(30*B*b^4*x^9 - 6*(9*B
*a*b^3 - 7*A*b^4)*x^7 + 14*(9*B*a^2*b^2 - 7*A*a*b^3)*x^5 - 70*(9*B*a^3*b - 7*A*a
^2*b^2)*x^3 + 105*(9*B*a^4 - 7*A*a^3*b + (9*B*a^3*b - 7*A*a^2*b^2)*x^2)*sqrt(a/b
)*arctan(x/sqrt(a/b)) - 105*(9*B*a^4 - 7*A*a^3*b)*x)/(b^6*x^2 + a*b^5)]

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Sympy [A]  time = 3.52406, size = 233, normalized size = 1.78 \[ \frac{B x^{7}}{7 b^{2}} - \frac{x \left (- A a^{3} b + B a^{4}\right )}{2 a b^{5} + 2 b^{6} x^{2}} - \frac{\sqrt{- \frac{a^{5}}{b^{11}}} \left (- 7 A b + 9 B a\right ) \log{\left (- \frac{b^{5} \sqrt{- \frac{a^{5}}{b^{11}}} \left (- 7 A b + 9 B a\right )}{- 7 A a^{2} b + 9 B a^{3}} + x \right )}}{4} + \frac{\sqrt{- \frac{a^{5}}{b^{11}}} \left (- 7 A b + 9 B a\right ) \log{\left (\frac{b^{5} \sqrt{- \frac{a^{5}}{b^{11}}} \left (- 7 A b + 9 B a\right )}{- 7 A a^{2} b + 9 B a^{3}} + x \right )}}{4} - \frac{x^{5} \left (- A b + 2 B a\right )}{5 b^{3}} + \frac{x^{3} \left (- 2 A a b + 3 B a^{2}\right )}{3 b^{4}} - \frac{x \left (- 3 A a^{2} b + 4 B a^{3}\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8*(B*x**2+A)/(b*x**2+a)**2,x)

[Out]

B*x**7/(7*b**2) - x*(-A*a**3*b + B*a**4)/(2*a*b**5 + 2*b**6*x**2) - sqrt(-a**5/b
**11)*(-7*A*b + 9*B*a)*log(-b**5*sqrt(-a**5/b**11)*(-7*A*b + 9*B*a)/(-7*A*a**2*b
+ 9*B*a**3) + x)/4 + sqrt(-a**5/b**11)*(-7*A*b + 9*B*a)*log(b**5*sqrt(-a**5/b**
11)*(-7*A*b + 9*B*a)/(-7*A*a**2*b + 9*B*a**3) + x)/4 - x**5*(-A*b + 2*B*a)/(5*b*
*3) + x**3*(-2*A*a*b + 3*B*a**2)/(3*b**4) - x*(-3*A*a**2*b + 4*B*a**3)/b**5

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GIAC/XCAS [A]  time = 0.231291, size = 188, normalized size = 1.44 \[ \frac{{\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{5}} - \frac{B a^{4} x - A a^{3} b x}{2 \,{\left (b x^{2} + a\right )} b^{5}} + \frac{15 \, B b^{12} x^{7} - 42 \, B a b^{11} x^{5} + 21 \, A b^{12} x^{5} + 105 \, B a^{2} b^{10} x^{3} - 70 \, A a b^{11} x^{3} - 420 \, B a^{3} b^{9} x + 315 \, A a^{2} b^{10} x}{105 \, b^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^8/(b*x^2 + a)^2,x, algorithm="giac")

[Out]

1/2*(9*B*a^4 - 7*A*a^3*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^5) - 1/2*(B*a^4*x -
A*a^3*b*x)/((b*x^2 + a)*b^5) + 1/105*(15*B*b^12*x^7 - 42*B*a*b^11*x^5 + 21*A*b^
12*x^5 + 105*B*a^2*b^10*x^3 - 70*A*a*b^11*x^3 - 420*B*a^3*b^9*x + 315*A*a^2*b^10
*x)/b^14