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

Optimal. Leaf size=98 \[ -\frac{a^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{9/2}}+\frac{a^2 x (A b-a B)}{b^4}-\frac{a x^3 (A b-a B)}{3 b^3}+\frac{x^5 (A b-a B)}{5 b^2}+\frac{B x^7}{7 b} \]

[Out]

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

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Rubi [A]  time = 0.170725, antiderivative size = 98, 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 \[ -\frac{a^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{9/2}}+\frac{a^2 x (A b-a B)}{b^4}-\frac{a x^3 (A b-a B)}{3 b^3}+\frac{x^5 (A b-a B)}{5 b^2}+\frac{B x^7}{7 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.113186, size = 98, normalized size = 1. \[ \frac{a^{5/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{9/2}}-\frac{a^2 x (a B-A b)}{b^4}+\frac{a x^3 (a B-A b)}{3 b^3}+\frac{x^5 (A b-a B)}{5 b^2}+\frac{B x^7}{7 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 116, normalized size = 1.2 \[{\frac{B{x}^{7}}{7\,b}}+{\frac{A{x}^{5}}{5\,b}}-{\frac{B{x}^{5}a}{5\,{b}^{2}}}-{\frac{aA{x}^{3}}{3\,{b}^{2}}}+{\frac{B{x}^{3}{a}^{2}}{3\,{b}^{3}}}+{\frac{{a}^{2}Ax}{{b}^{3}}}-{\frac{B{a}^{3}x}{{b}^{4}}}-{\frac{{a}^{3}A}{{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{B{a}^{4}}{{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(B*x^2+A)/(b*x^2+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238728, size = 1, normalized size = 0.01 \[ \left [\frac{30 \, B b^{3} x^{7} - 42 \,{\left (B a b^{2} - A b^{3}\right )} x^{5} + 70 \,{\left (B a^{2} b - A a b^{2}\right )} x^{3} - 105 \,{\left (B a^{3} - A a^{2} b\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) - 210 \,{\left (B a^{3} - A a^{2} b\right )} x}{210 \, b^{4}}, \frac{15 \, B b^{3} x^{7} - 21 \,{\left (B a b^{2} - A b^{3}\right )} x^{5} + 35 \,{\left (B a^{2} b - A a b^{2}\right )} x^{3} + 105 \,{\left (B a^{3} - A a^{2} b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) - 105 \,{\left (B a^{3} - A a^{2} b\right )} x}{105 \, b^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/210*(30*B*b^3*x^7 - 42*(B*a*b^2 - A*b^3)*x^5 + 70*(B*a^2*b - A*a*b^2)*x^3 - 1
05*(B*a^3 - A*a^2*b)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a))
- 210*(B*a^3 - A*a^2*b)*x)/b^4, 1/105*(15*B*b^3*x^7 - 21*(B*a*b^2 - A*b^3)*x^5 +
35*(B*a^2*b - A*a*b^2)*x^3 + 105*(B*a^3 - A*a^2*b)*sqrt(a/b)*arctan(x/sqrt(a/b)
) - 105*(B*a^3 - A*a^2*b)*x)/b^4]

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Sympy [A]  time = 2.02436, size = 173, normalized size = 1.77 \[ \frac{B x^{7}}{7 b} - \frac{\sqrt{- \frac{a^{5}}{b^{9}}} \left (- A b + B a\right ) \log{\left (- \frac{b^{4} \sqrt{- \frac{a^{5}}{b^{9}}} \left (- A b + B a\right )}{- A a^{2} b + B a^{3}} + x \right )}}{2} + \frac{\sqrt{- \frac{a^{5}}{b^{9}}} \left (- A b + B a\right ) \log{\left (\frac{b^{4} \sqrt{- \frac{a^{5}}{b^{9}}} \left (- A b + B a\right )}{- A a^{2} b + B a^{3}} + x \right )}}{2} - \frac{x^{5} \left (- A b + B a\right )}{5 b^{2}} + \frac{x^{3} \left (- A a b + B a^{2}\right )}{3 b^{3}} - \frac{x \left (- A a^{2} b + B a^{3}\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.229673, size = 146, normalized size = 1.49 \[ \frac{{\left (B a^{4} - A a^{3} b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{4}} + \frac{15 \, B b^{6} x^{7} - 21 \, B a b^{5} x^{5} + 21 \, A b^{6} x^{5} + 35 \, B a^{2} b^{4} x^{3} - 35 \, A a b^{5} x^{3} - 105 \, B a^{3} b^{3} x + 105 \, A a^{2} b^{4} x}{105 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(B*a^4 - A*a^3*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^4) + 1/105*(15*B*b^6*x^7 -
21*B*a*b^5*x^5 + 21*A*b^6*x^5 + 35*B*a^2*b^4*x^3 - 35*A*a*b^5*x^3 - 105*B*a^3*b^
3*x + 105*A*a^2*b^4*x)/b^7