Optimal. Leaf size=116 \[ -\frac{a^2 x (A b-a B)}{4 b^4 \left (a+b x^2\right )^2}-\frac{5 \sqrt{a} (3 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{9/2}}+\frac{a x (9 A b-13 a B)}{8 b^4 \left (a+b x^2\right )}+\frac{x (A b-3 a B)}{b^4}+\frac{B x^3}{3 b^3} \]
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Rubi [A] time = 0.312935, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{a^2 x (A b-a B)}{4 b^4 \left (a+b x^2\right )^2}-\frac{5 \sqrt{a} (3 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{9/2}}+\frac{a x (9 A b-13 a B)}{8 b^4 \left (a+b x^2\right )}+\frac{x (A b-3 a B)}{b^4}+\frac{B x^3}{3 b^3} \]
Antiderivative was successfully verified.
[In] Int[(x^6*(A + B*x^2))/(a + b*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 88.7129, size = 110, normalized size = 0.95 \[ \frac{B x^{3}}{3 b^{3}} - \frac{5 \sqrt{a} \left (3 A b - 7 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{9}{2}}} - \frac{a^{2} x \left (A b - B a\right )}{4 b^{4} \left (a + b x^{2}\right )^{2}} + \frac{a x \left (9 A b - 13 B a\right )}{8 b^{4} \left (a + b x^{2}\right )} + \frac{x \left (A b - 3 B a\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**6*(B*x**2+A)/(b*x**2+a)**3,x)
[Out]
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Mathematica [A] time = 0.150343, size = 113, normalized size = 0.97 \[ \frac{-105 a^3 B x+5 a^2 b x \left (9 A-35 B x^2\right )+a b^2 x^3 \left (75 A-56 B x^2\right )+8 b^3 x^5 \left (3 A+B x^2\right )}{24 b^4 \left (a+b x^2\right )^2}+\frac{5 \sqrt{a} (7 a B-3 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^6*(A + B*x^2))/(a + b*x^2)^3,x]
[Out]
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Maple [A] time = 0.016, size = 147, normalized size = 1.3 \[{\frac{B{x}^{3}}{3\,{b}^{3}}}+{\frac{Ax}{{b}^{3}}}-3\,{\frac{Bxa}{{b}^{4}}}+{\frac{9\,aA{x}^{3}}{8\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{13\,{a}^{2}B{x}^{3}}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{7\,{a}^{2}Ax}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{11\,B{a}^{3}x}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{15\,Aa}{8\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{35\,{a}^{2}B}{8\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^6*(B*x^2+A)/(b*x^2+a)^3,x)
[Out]
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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)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.233512, size = 1, normalized size = 0.01 \[ \left [\frac{16 \, B b^{3} x^{7} - 16 \,{\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} x^{5} - 50 \,{\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{3} - 15 \,{\left ({\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} x^{4} + 7 \, B a^{3} - 3 \, A a^{2} b + 2 \,{\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) - 30 \,{\left (7 \, B a^{3} - 3 \, A a^{2} b\right )} x}{48 \,{\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}}, \frac{8 \, B b^{3} x^{7} - 8 \,{\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} x^{5} - 25 \,{\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{3} + 15 \,{\left ({\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} x^{4} + 7 \, B a^{3} - 3 \, A a^{2} b + 2 \,{\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) - 15 \,{\left (7 \, B a^{3} - 3 \, A a^{2} b\right )} x}{24 \,{\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^6/(b*x^2 + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.87337, size = 212, normalized size = 1.83 \[ \frac{B x^{3}}{3 b^{3}} - \frac{5 \sqrt{- \frac{a}{b^{9}}} \left (- 3 A b + 7 B a\right ) \log{\left (- \frac{5 b^{4} \sqrt{- \frac{a}{b^{9}}} \left (- 3 A b + 7 B a\right )}{- 15 A b + 35 B a} + x \right )}}{16} + \frac{5 \sqrt{- \frac{a}{b^{9}}} \left (- 3 A b + 7 B a\right ) \log{\left (\frac{5 b^{4} \sqrt{- \frac{a}{b^{9}}} \left (- 3 A b + 7 B a\right )}{- 15 A b + 35 B a} + x \right )}}{16} - \frac{x^{3} \left (- 9 A a b^{2} + 13 B a^{2} b\right ) + x \left (- 7 A a^{2} b + 11 B a^{3}\right )}{8 a^{2} b^{4} + 16 a b^{5} x^{2} + 8 b^{6} x^{4}} - \frac{x \left (- A b + 3 B a\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**6*(B*x**2+A)/(b*x**2+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.224981, size = 150, normalized size = 1.29 \[ \frac{5 \,{\left (7 \, B a^{2} - 3 \, A a b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} b^{4}} - \frac{13 \, B a^{2} b x^{3} - 9 \, A a b^{2} x^{3} + 11 \, B a^{3} x - 7 \, A a^{2} b x}{8 \,{\left (b x^{2} + a\right )}^{2} b^{4}} + \frac{B b^{6} x^{3} - 9 \, B a b^{5} x + 3 \, A b^{6} x}{3 \, b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^6/(b*x^2 + a)^3,x, algorithm="giac")
[Out]