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

Optimal. Leaf size=110 \[ \frac{a^{3/2} (5 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{9/2}}-\frac{a^2 x (A b-a B)}{2 b^4 \left (a+b x^2\right )}-\frac{a x (2 A b-3 a B)}{b^4}+\frac{x^3 (A b-2 a B)}{3 b^3}+\frac{B x^5}{5 b^2} \]

[Out]

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

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Rubi [A]  time = 0.231389, antiderivative size = 110, 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 \[ \frac{a^{3/2} (5 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{9/2}}-\frac{a^2 x (A b-a B)}{2 b^4 \left (a+b x^2\right )}-\frac{a x (2 A b-3 a B)}{b^4}+\frac{x^3 (A b-2 a B)}{3 b^3}+\frac{B x^5}{5 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 59.4279, size = 102, normalized size = 0.93 \[ \frac{B x^{5}}{5 b^{2}} + \frac{a^{\frac{3}{2}} \left (5 A b - 7 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 b^{\frac{9}{2}}} - \frac{a^{2} x \left (A b - B a\right )}{2 b^{4} \left (a + b x^{2}\right )} - \frac{a x \left (2 A b - 3 B a\right )}{b^{4}} + \frac{x^{3} \left (A b - 2 B a\right )}{3 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.148634, size = 111, normalized size = 1.01 \[ -\frac{a^{3/2} (7 a B-5 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{9/2}}-\frac{x \left (a^2 A b-a^3 B\right )}{2 b^4 \left (a+b x^2\right )}+\frac{a x (3 a B-2 A b)}{b^4}+\frac{x^3 (A b-2 a B)}{3 b^3}+\frac{B x^5}{5 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 132, normalized size = 1.2 \[{\frac{B{x}^{5}}{5\,{b}^{2}}}+{\frac{A{x}^{3}}{3\,{b}^{2}}}-{\frac{2\,B{x}^{3}a}{3\,{b}^{3}}}-2\,{\frac{aAx}{{b}^{3}}}+3\,{\frac{Bx{a}^{2}}{{b}^{4}}}-{\frac{{a}^{2}Ax}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{{a}^{3}xB}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{5\,A{a}^{2}}{2\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{7\,B{a}^{3}}{2\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*B*x^5/b^2+1/3/b^2*A*x^3-2/3/b^3*B*x^3*a-2/b^3*A*x*a+3/b^4*B*x*a^2-1/2*a^2/b^
3*x/(b*x^2+a)*A+1/2*a^3/b^4*x/(b*x^2+a)*B+5/2*a^2/b^3/(a*b)^(1/2)*arctan(x*b/(a*
b)^(1/2))*A-7/2*a^3/b^4/(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^6/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237218, size = 1, normalized size = 0.01 \[ \left [\frac{12 \, B b^{3} x^{7} - 4 \,{\left (7 \, B a b^{2} - 5 \, A b^{3}\right )} x^{5} + 20 \,{\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{3} - 15 \,{\left (7 \, B a^{3} - 5 \, A a^{2} b +{\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 30 \,{\left (7 \, B a^{3} - 5 \, A a^{2} b\right )} x}{60 \,{\left (b^{5} x^{2} + a b^{4}\right )}}, \frac{6 \, B b^{3} x^{7} - 2 \,{\left (7 \, B a b^{2} - 5 \, A b^{3}\right )} x^{5} + 10 \,{\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{3} - 15 \,{\left (7 \, B a^{3} - 5 \, A a^{2} b +{\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) + 15 \,{\left (7 \, B a^{3} - 5 \, A a^{2} b\right )} x}{30 \,{\left (b^{5} x^{2} + a b^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^6/(b*x^2 + a)^2,x, algorithm="fricas")

[Out]

[1/60*(12*B*b^3*x^7 - 4*(7*B*a*b^2 - 5*A*b^3)*x^5 + 20*(7*B*a^2*b - 5*A*a*b^2)*x
^3 - 15*(7*B*a^3 - 5*A*a^2*b + (7*B*a^2*b - 5*A*a*b^2)*x^2)*sqrt(-a/b)*log((b*x^
2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 30*(7*B*a^3 - 5*A*a^2*b)*x)/(b^5*x^2 +
a*b^4), 1/30*(6*B*b^3*x^7 - 2*(7*B*a*b^2 - 5*A*b^3)*x^5 + 10*(7*B*a^2*b - 5*A*a*
b^2)*x^3 - 15*(7*B*a^3 - 5*A*a^2*b + (7*B*a^2*b - 5*A*a*b^2)*x^2)*sqrt(a/b)*arct
an(x/sqrt(a/b)) + 15*(7*B*a^3 - 5*A*a^2*b)*x)/(b^5*x^2 + a*b^4)]

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Sympy [A]  time = 3.32524, size = 206, normalized size = 1.87 \[ \frac{B x^{5}}{5 b^{2}} + \frac{x \left (- A a^{2} b + B a^{3}\right )}{2 a b^{4} + 2 b^{5} x^{2}} + \frac{\sqrt{- \frac{a^{3}}{b^{9}}} \left (- 5 A b + 7 B a\right ) \log{\left (- \frac{b^{4} \sqrt{- \frac{a^{3}}{b^{9}}} \left (- 5 A b + 7 B a\right )}{- 5 A a b + 7 B a^{2}} + x \right )}}{4} - \frac{\sqrt{- \frac{a^{3}}{b^{9}}} \left (- 5 A b + 7 B a\right ) \log{\left (\frac{b^{4} \sqrt{- \frac{a^{3}}{b^{9}}} \left (- 5 A b + 7 B a\right )}{- 5 A a b + 7 B a^{2}} + x \right )}}{4} - \frac{x^{3} \left (- A b + 2 B a\right )}{3 b^{3}} + \frac{x \left (- 2 A a b + 3 B a^{2}\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.226564, size = 155, normalized size = 1.41 \[ -\frac{{\left (7 \, B a^{3} - 5 \, A a^{2} b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{4}} + \frac{B a^{3} x - A a^{2} b x}{2 \,{\left (b x^{2} + a\right )} b^{4}} + \frac{3 \, B b^{8} x^{5} - 10 \, B a b^{7} x^{3} + 5 \, A b^{8} x^{3} + 45 \, B a^{2} b^{6} x - 30 \, A a b^{7} x}{15 \, b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^6/(b*x^2 + a)^2,x, algorithm="giac")

[Out]

-1/2*(7*B*a^3 - 5*A*a^2*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^4) + 1/2*(B*a^3*x
- A*a^2*b*x)/((b*x^2 + a)*b^4) + 1/15*(3*B*b^8*x^5 - 10*B*a*b^7*x^3 + 5*A*b^8*x^
3 + 45*B*a^2*b^6*x - 30*A*a*b^7*x)/b^10