Optimal. Leaf size=104 \[ \frac{c^{3/2} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{d^{9/2}}-\frac{c x (b c-a d)^2}{d^4}+\frac{x^3 (b c-a d)^2}{3 d^3}-\frac{b x^5 (b c-2 a d)}{5 d^2}+\frac{b^2 x^7}{7 d} \]
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Rubi [A] time = 0.177235, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{c^{3/2} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{d^{9/2}}-\frac{c x (b c-a d)^2}{d^4}+\frac{x^3 (b c-a d)^2}{3 d^3}-\frac{b x^5 (b c-2 a d)}{5 d^2}+\frac{b^2 x^7}{7 d} \]
Antiderivative was successfully verified.
[In] Int[(x^4*(a + b*x^2)^2)/(c + d*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{b^{2} x^{7}}{7 d} + \frac{b x^{5} \left (2 a d - b c\right )}{5 d^{2}} + \frac{c^{\frac{3}{2}} \left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{d^{\frac{9}{2}}} + \frac{x^{3} \left (a d - b c\right )^{2}}{3 d^{3}} - \frac{\left (a d - b c\right )^{2} \int c\, dx}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(b*x**2+a)**2/(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.157526, size = 104, normalized size = 1. \[ \frac{c^{3/2} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{d^{9/2}}-\frac{c x (b c-a d)^2}{d^4}+\frac{x^3 (a d-b c)^2}{3 d^3}-\frac{b x^5 (b c-2 a d)}{5 d^2}+\frac{b^2 x^7}{7 d} \]
Antiderivative was successfully verified.
[In] Integrate[(x^4*(a + b*x^2)^2)/(c + d*x^2),x]
[Out]
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Maple [A] time = 0.004, size = 176, normalized size = 1.7 \[{\frac{{b}^{2}{x}^{7}}{7\,d}}+{\frac{2\,{x}^{5}ab}{5\,d}}-{\frac{{b}^{2}{x}^{5}c}{5\,{d}^{2}}}+{\frac{{x}^{3}{a}^{2}}{3\,d}}-{\frac{2\,ab{x}^{3}c}{3\,{d}^{2}}}+{\frac{{x}^{3}{b}^{2}{c}^{2}}{3\,{d}^{3}}}-{\frac{{a}^{2}cx}{{d}^{2}}}+2\,{\frac{xab{c}^{2}}{{d}^{3}}}-{\frac{x{b}^{2}{c}^{3}}{{d}^{4}}}+{\frac{{a}^{2}{c}^{2}}{{d}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-2\,{\frac{ab{c}^{3}}{{d}^{3}\sqrt{cd}}\arctan \left ({\frac{dx}{\sqrt{cd}}} \right ) }+{\frac{{b}^{2}{c}^{4}}{{d}^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(b*x^2+a)^2/(d*x^2+c),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)^2*x^4/(d*x^2 + c),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.244786, size = 1, normalized size = 0.01 \[ \left [\frac{30 \, b^{2} d^{3} x^{7} - 42 \,{\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{5} + 70 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} + 105 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{2} + 2 \, d x \sqrt{-\frac{c}{d}} - c}{d x^{2} + c}\right ) - 210 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} x}{210 \, d^{4}}, \frac{15 \, b^{2} d^{3} x^{7} - 21 \,{\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{5} + 35 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} + 105 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt{\frac{c}{d}} \arctan \left (\frac{x}{\sqrt{\frac{c}{d}}}\right ) - 105 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} x}{105 \, d^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x^4/(d*x^2 + c),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.72646, size = 240, normalized size = 2.31 \[ \frac{b^{2} x^{7}}{7 d} - \frac{\sqrt{- \frac{c^{3}}{d^{9}}} \left (a d - b c\right )^{2} \log{\left (- \frac{d^{4} \sqrt{- \frac{c^{3}}{d^{9}}} \left (a d - b c\right )^{2}}{a^{2} c d^{2} - 2 a b c^{2} d + b^{2} c^{3}} + x \right )}}{2} + \frac{\sqrt{- \frac{c^{3}}{d^{9}}} \left (a d - b c\right )^{2} \log{\left (\frac{d^{4} \sqrt{- \frac{c^{3}}{d^{9}}} \left (a d - b c\right )^{2}}{a^{2} c d^{2} - 2 a b c^{2} d + b^{2} c^{3}} + x \right )}}{2} + \frac{x^{5} \left (2 a b d - b^{2} c\right )}{5 d^{2}} + \frac{x^{3} \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{3 d^{3}} - \frac{x \left (a^{2} c d^{2} - 2 a b c^{2} d + b^{2} c^{3}\right )}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(b*x**2+a)**2/(d*x**2+c),x)
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GIAC/XCAS [A] time = 0.221318, size = 207, normalized size = 1.99 \[ \frac{{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{\sqrt{c d} d^{4}} + \frac{15 \, b^{2} d^{6} x^{7} - 21 \, b^{2} c d^{5} x^{5} + 42 \, a b d^{6} x^{5} + 35 \, b^{2} c^{2} d^{4} x^{3} - 70 \, a b c d^{5} x^{3} + 35 \, a^{2} d^{6} x^{3} - 105 \, b^{2} c^{3} d^{3} x + 210 \, a b c^{2} d^{4} x - 105 \, a^{2} c d^{5} x}{105 \, d^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x^4/(d*x^2 + c),x, algorithm="giac")
[Out]