Optimal. Leaf size=163 \[ \frac{\left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 \sqrt{c} d^{9/2}}-\frac{x \left (a^2 d^2-10 a b c d+13 b^2 c^2\right )}{4 c d^4}-\frac{x (b c-a d) (9 b c-a d)}{8 d^4 \left (c+d x^2\right )}+\frac{x^5 (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}+\frac{b^2 x^3}{3 d^3} \]
[Out]
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Rubi [A] time = 0.392188, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 \sqrt{c} d^{9/2}}-\frac{x \left (a^2 d^2-10 a b c d+13 b^2 c^2\right )}{4 c d^4}-\frac{x (b c-a d) (9 b c-a d)}{8 d^4 \left (c+d x^2\right )}+\frac{x^5 (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}+\frac{b^2 x^3}{3 d^3} \]
Antiderivative was successfully verified.
[In] Int[(x^4*(a + b*x^2)^2)/(c + d*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 106.856, size = 150, normalized size = 0.92 \[ \frac{b^{2} x^{3}}{3 d^{3}} - \frac{x \left (a d - 9 b c\right ) \left (a d - b c\right )}{8 d^{4} \left (c + d x^{2}\right )} + \frac{x^{5} \left (a d - b c\right )^{2}}{4 c d^{2} \left (c + d x^{2}\right )^{2}} - \frac{x \left (a^{2} d^{2} - 10 a b c d + 13 b^{2} c^{2}\right )}{4 c d^{4}} + \frac{\left (3 a^{2} d^{2} - 30 a b c d + 35 b^{2} c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 \sqrt{c} d^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(b*x**2+a)**2/(d*x**2+c)**3,x)
[Out]
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Mathematica [A] time = 0.153554, size = 148, normalized size = 0.91 \[ \frac{\left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 \sqrt{c} d^{9/2}}-\frac{x \left (5 a^2 d^2-18 a b c d+13 b^2 c^2\right )}{8 d^4 \left (c+d x^2\right )}+\frac{c x (b c-a d)^2}{4 d^4 \left (c+d x^2\right )^2}-\frac{b x (3 b c-2 a d)}{d^4}+\frac{b^2 x^3}{3 d^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x^4*(a + b*x^2)^2)/(c + d*x^2)^3,x]
[Out]
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Maple [A] time = 0.016, size = 223, normalized size = 1.4 \[{\frac{{b}^{2}{x}^{3}}{3\,{d}^{3}}}+2\,{\frac{abx}{{d}^{3}}}-3\,{\frac{x{b}^{2}c}{{d}^{4}}}-{\frac{5\,{x}^{3}{a}^{2}}{8\,d \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{9\,ab{x}^{3}c}{4\,{d}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{13\,{x}^{3}{b}^{2}{c}^{2}}{8\,{d}^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{3\,{a}^{2}cx}{8\,{d}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{7\,xab{c}^{2}}{4\,{d}^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{11\,x{b}^{2}{c}^{3}}{8\,{d}^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,{a}^{2}}{8\,{d}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{15\,abc}{4\,{d}^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{35\,{b}^{2}{c}^{2}}{8\,{d}^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(b*x^2+a)^2/(d*x^2+c)^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)^2*x^4/(d*x^2 + c)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.241915, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (35 \, b^{2} c^{4} - 30 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} +{\left (35 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{4} + 2 \,{\left (35 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x^{2}\right )} \log \left (\frac{2 \, c d x +{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right ) + 2 \,{\left (8 \, b^{2} d^{3} x^{7} - 8 \,{\left (7 \, b^{2} c d^{2} - 6 \, a b d^{3}\right )} x^{5} - 5 \,{\left (35 \, b^{2} c^{2} d - 30 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} x^{3} - 3 \,{\left (35 \, b^{2} c^{3} - 30 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x\right )} \sqrt{-c d}}{48 \,{\left (d^{6} x^{4} + 2 \, c d^{5} x^{2} + c^{2} d^{4}\right )} \sqrt{-c d}}, \frac{3 \,{\left (35 \, b^{2} c^{4} - 30 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} +{\left (35 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{4} + 2 \,{\left (35 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) +{\left (8 \, b^{2} d^{3} x^{7} - 8 \,{\left (7 \, b^{2} c d^{2} - 6 \, a b d^{3}\right )} x^{5} - 5 \,{\left (35 \, b^{2} c^{2} d - 30 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} x^{3} - 3 \,{\left (35 \, b^{2} c^{3} - 30 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x\right )} \sqrt{c d}}{24 \,{\left (d^{6} x^{4} + 2 \, c d^{5} x^{2} + c^{2} d^{4}\right )} \sqrt{c d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x^4/(d*x^2 + c)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.47404, size = 238, normalized size = 1.46 \[ \frac{b^{2} x^{3}}{3 d^{3}} - \frac{\sqrt{- \frac{1}{c d^{9}}} \left (3 a^{2} d^{2} - 30 a b c d + 35 b^{2} c^{2}\right ) \log{\left (- c d^{4} \sqrt{- \frac{1}{c d^{9}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{c d^{9}}} \left (3 a^{2} d^{2} - 30 a b c d + 35 b^{2} c^{2}\right ) \log{\left (c d^{4} \sqrt{- \frac{1}{c d^{9}}} + x \right )}}{16} - \frac{x^{3} \left (5 a^{2} d^{3} - 18 a b c d^{2} + 13 b^{2} c^{2} d\right ) + x \left (3 a^{2} c d^{2} - 14 a b c^{2} d + 11 b^{2} c^{3}\right )}{8 c^{2} d^{4} + 16 c d^{5} x^{2} + 8 d^{6} x^{4}} + \frac{x \left (2 a b d - 3 b^{2} c\right )}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(b*x**2+a)**2/(d*x**2+c)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.227829, size = 208, normalized size = 1.28 \[ \frac{{\left (35 \, b^{2} c^{2} - 30 \, a b c d + 3 \, a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \, \sqrt{c d} d^{4}} - \frac{13 \, b^{2} c^{2} d x^{3} - 18 \, a b c d^{2} x^{3} + 5 \, a^{2} d^{3} x^{3} + 11 \, b^{2} c^{3} x - 14 \, a b c^{2} d x + 3 \, a^{2} c d^{2} x}{8 \,{\left (d x^{2} + c\right )}^{2} d^{4}} + \frac{b^{2} d^{6} x^{3} - 9 \, b^{2} c d^{5} x + 6 \, a b d^{6} x}{3 \, d^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x^4/(d*x^2 + c)^3,x, algorithm="giac")
[Out]