Optimal. Leaf size=111 \[ -\frac{8 d x \sqrt{c+\frac{d}{x^2}} (5 b c-6 a d)}{15 c^4}+\frac{4 d x (5 b c-6 a d)}{15 c^3 \sqrt{c+\frac{d}{x^2}}}+\frac{x^3 (5 b c-6 a d)}{15 c^2 \sqrt{c+\frac{d}{x^2}}}+\frac{a x^5}{5 c \sqrt{c+\frac{d}{x^2}}} \]
[Out]
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Rubi [A] time = 0.16344, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{8 d x \sqrt{c+\frac{d}{x^2}} (5 b c-6 a d)}{15 c^4}+\frac{4 d x (5 b c-6 a d)}{15 c^3 \sqrt{c+\frac{d}{x^2}}}+\frac{x^3 (5 b c-6 a d)}{15 c^2 \sqrt{c+\frac{d}{x^2}}}+\frac{a x^5}{5 c \sqrt{c+\frac{d}{x^2}}} \]
Antiderivative was successfully verified.
[In] Int[((a + b/x^2)*x^4)/(c + d/x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 11.9217, size = 107, normalized size = 0.96 \[ \frac{a x^{5}}{5 c \sqrt{c + \frac{d}{x^{2}}}} - \frac{x^{3} \left (6 a d - 5 b c\right )}{15 c^{2} \sqrt{c + \frac{d}{x^{2}}}} - \frac{4 d x \left (6 a d - 5 b c\right )}{15 c^{3} \sqrt{c + \frac{d}{x^{2}}}} + \frac{8 d x \sqrt{c + \frac{d}{x^{2}}} \left (6 a d - 5 b c\right )}{15 c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)*x**4/(c+d/x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0881316, size = 80, normalized size = 0.72 \[ \frac{3 a \left (c^3 x^6-2 c^2 d x^4+8 c d^2 x^2+16 d^3\right )+5 b c \left (c^2 x^4-4 c d x^2-8 d^2\right )}{15 c^4 x \sqrt{c+\frac{d}{x^2}}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b/x^2)*x^4)/(c + d/x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.012, size = 91, normalized size = 0.8 \[{\frac{ \left ( 3\,a{x}^{6}{c}^{3}-6\,a{c}^{2}d{x}^{4}+5\,b{c}^{3}{x}^{4}+24\,ac{d}^{2}{x}^{2}-20\,b{c}^{2}d{x}^{2}+48\,a{d}^{3}-40\,bc{d}^{2} \right ) \left ( c{x}^{2}+d \right ) }{15\,{x}^{3}{c}^{4}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)*x^4/(c+d/x^2)^(3/2),x)
[Out]
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Maxima [A] time = 1.39, size = 173, normalized size = 1.56 \[ \frac{1}{3} \, b{\left (\frac{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} x^{3} - 6 \, \sqrt{c + \frac{d}{x^{2}}} d x}{c^{3}} - \frac{3 \, d^{2}}{\sqrt{c + \frac{d}{x^{2}}} c^{3} x}\right )} + \frac{1}{5} \, a{\left (\frac{5 \, d^{3}}{\sqrt{c + \frac{d}{x^{2}}} c^{4} x} + \frac{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} x^{5} - 5 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} d x^{3} + 15 \, \sqrt{c + \frac{d}{x^{2}}} d^{2} x}{c^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)*x^4/(c + d/x^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.231258, size = 128, normalized size = 1.15 \[ \frac{{\left (3 \, a c^{3} x^{7} +{\left (5 \, b c^{3} - 6 \, a c^{2} d\right )} x^{5} - 4 \,{\left (5 \, b c^{2} d - 6 \, a c d^{2}\right )} x^{3} - 8 \,{\left (5 \, b c d^{2} - 6 \, a d^{3}\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{15 \,{\left (c^{5} x^{2} + c^{4} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)*x^4/(c + d/x^2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 16.4018, size = 561, normalized size = 5.05 \[ a \left (\frac{c^{5} d^{\frac{19}{2}} x^{10} \sqrt{\frac{c x^{2}}{d} + 1}}{5 c^{7} d^{9} x^{6} + 15 c^{6} d^{10} x^{4} + 15 c^{5} d^{11} x^{2} + 5 c^{4} d^{12}} + \frac{5 c^{3} d^{\frac{23}{2}} x^{6} \sqrt{\frac{c x^{2}}{d} + 1}}{5 c^{7} d^{9} x^{6} + 15 c^{6} d^{10} x^{4} + 15 c^{5} d^{11} x^{2} + 5 c^{4} d^{12}} + \frac{30 c^{2} d^{\frac{25}{2}} x^{4} \sqrt{\frac{c x^{2}}{d} + 1}}{5 c^{7} d^{9} x^{6} + 15 c^{6} d^{10} x^{4} + 15 c^{5} d^{11} x^{2} + 5 c^{4} d^{12}} + \frac{40 c d^{\frac{27}{2}} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{5 c^{7} d^{9} x^{6} + 15 c^{6} d^{10} x^{4} + 15 c^{5} d^{11} x^{2} + 5 c^{4} d^{12}} + \frac{16 d^{\frac{29}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{5 c^{7} d^{9} x^{6} + 15 c^{6} d^{10} x^{4} + 15 c^{5} d^{11} x^{2} + 5 c^{4} d^{12}}\right ) + b \left (\frac{c^{3} d^{\frac{9}{2}} x^{6} \sqrt{\frac{c x^{2}}{d} + 1}}{3 c^{5} d^{4} x^{4} + 6 c^{4} d^{5} x^{2} + 3 c^{3} d^{6}} - \frac{3 c^{2} d^{\frac{11}{2}} x^{4} \sqrt{\frac{c x^{2}}{d} + 1}}{3 c^{5} d^{4} x^{4} + 6 c^{4} d^{5} x^{2} + 3 c^{3} d^{6}} - \frac{12 c d^{\frac{13}{2}} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{3 c^{5} d^{4} x^{4} + 6 c^{4} d^{5} x^{2} + 3 c^{3} d^{6}} - \frac{8 d^{\frac{15}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{3 c^{5} d^{4} x^{4} + 6 c^{4} d^{5} x^{2} + 3 c^{3} d^{6}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)*x**4/(c+d/x**2)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (a + \frac{b}{x^{2}}\right )} x^{4}}{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)*x^4/(c + d/x^2)^(3/2),x, algorithm="giac")
[Out]