Optimal. Leaf size=109 \[ \frac{a^3 (A b-a B)}{4 b^5 \left (a+b x^2\right )^2}-\frac{a^2 (3 A b-4 a B)}{2 b^5 \left (a+b x^2\right )}-\frac{3 a (A b-2 a B) \log \left (a+b x^2\right )}{2 b^5}+\frac{x^2 (A b-3 a B)}{2 b^4}+\frac{B x^4}{4 b^3} \]
[Out]
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Rubi [A] time = 0.29629, antiderivative size = 109, 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 \[ \frac{a^3 (A b-a B)}{4 b^5 \left (a+b x^2\right )^2}-\frac{a^2 (3 A b-4 a B)}{2 b^5 \left (a+b x^2\right )}-\frac{3 a (A b-2 a B) \log \left (a+b x^2\right )}{2 b^5}+\frac{x^2 (A b-3 a B)}{2 b^4}+\frac{B x^4}{4 b^3} \]
Antiderivative was successfully verified.
[In] Int[(x^7*(A + B*x^2))/(a + b*x^2)^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{B \int ^{x^{2}} x\, dx}{2 b^{3}} + \frac{a^{3} \left (A b - B a\right )}{4 b^{5} \left (a + b x^{2}\right )^{2}} - \frac{a^{2} \left (3 A b - 4 B a\right )}{2 b^{5} \left (a + b x^{2}\right )} - \frac{3 a \left (A b - 2 B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{5}} + \left (\frac{A b}{2} - \frac{3 B a}{2}\right ) \int ^{x^{2}} \frac{1}{b^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**7*(B*x**2+A)/(b*x**2+a)**3,x)
[Out]
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Mathematica [A] time = 0.108668, size = 94, normalized size = 0.86 \[ \frac{\frac{a^3 (A b-a B)}{\left (a+b x^2\right )^2}+\frac{2 a^2 (4 a B-3 A b)}{a+b x^2}+2 b x^2 (A b-3 a B)+6 a (2 a B-A b) \log \left (a+b x^2\right )+b^2 B x^4}{4 b^5} \]
Antiderivative was successfully verified.
[In] Integrate[(x^7*(A + B*x^2))/(a + b*x^2)^3,x]
[Out]
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Maple [A] time = 0.016, size = 134, normalized size = 1.2 \[{\frac{B{x}^{4}}{4\,{b}^{3}}}-{\frac{3\,B{x}^{2}a}{2\,{b}^{4}}}+{\frac{A{x}^{2}}{2\,{b}^{3}}}+{\frac{{a}^{3}A}{4\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{B{a}^{4}}{4\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{3\,a\ln \left ( b{x}^{2}+a \right ) A}{2\,{b}^{4}}}+3\,{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ) B}{{b}^{5}}}-{\frac{3\,A{a}^{2}}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}+2\,{\frac{B{a}^{3}}{{b}^{5} \left ( b{x}^{2}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^7*(B*x^2+A)/(b*x^2+a)^3,x)
[Out]
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Maxima [A] time = 1.35077, size = 157, normalized size = 1.44 \[ \frac{7 \, B a^{4} - 5 \, A a^{3} b + 2 \,{\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{2}}{4 \,{\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}} + \frac{B b x^{4} - 2 \,{\left (3 \, B a - A b\right )} x^{2}}{4 \, b^{4}} + \frac{3 \,{\left (2 \, B a^{2} - A a b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^7/(b*x^2 + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.216152, size = 242, normalized size = 2.22 \[ \frac{B b^{4} x^{8} - 2 \,{\left (2 \, B a b^{3} - A b^{4}\right )} x^{6} + 7 \, B a^{4} - 5 \, A a^{3} b -{\left (11 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{4} + 2 \,{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2} + 6 \,{\left (2 \, B a^{4} - A a^{3} b +{\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} x^{4} + 2 \,{\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{4 \,{\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^7/(b*x^2 + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.31504, size = 116, normalized size = 1.06 \[ \frac{B x^{4}}{4 b^{3}} + \frac{3 a \left (- A b + 2 B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{5}} + \frac{- 5 A a^{3} b + 7 B a^{4} + x^{2} \left (- 6 A a^{2} b^{2} + 8 B a^{3} b\right )}{4 a^{2} b^{5} + 8 a b^{6} x^{2} + 4 b^{7} x^{4}} - \frac{x^{2} \left (- A b + 3 B a\right )}{2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**7*(B*x**2+A)/(b*x**2+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.238793, size = 178, normalized size = 1.63 \[ \frac{3 \,{\left (2 \, B a^{2} - A a b\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{5}} + \frac{B b^{3} x^{4} - 6 \, B a b^{2} x^{2} + 2 \, A b^{3} x^{2}}{4 \, b^{6}} - \frac{18 \, B a^{2} b^{2} x^{4} - 9 \, A a b^{3} x^{4} + 28 \, B a^{3} b x^{2} - 12 \, A a^{2} b^{2} x^{2} + 11 \, B a^{4} - 4 \, A a^{3} b}{4 \,{\left (b x^{2} + a\right )}^{2} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^7/(b*x^2 + a)^3,x, algorithm="giac")
[Out]