Optimal. Leaf size=104 \[ \frac{c^2 \left (c+\frac{d}{x^2}\right )^{3/2} (b c-a d)}{3 d^4}+\frac{\left (c+\frac{d}{x^2}\right )^{7/2} (3 b c-a d)}{7 d^4}-\frac{c \left (c+\frac{d}{x^2}\right )^{5/2} (3 b c-2 a d)}{5 d^4}-\frac{b \left (c+\frac{d}{x^2}\right )^{9/2}}{9 d^4} \]
[Out]
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Rubi [A] time = 0.234944, 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 \[ \frac{c^2 \left (c+\frac{d}{x^2}\right )^{3/2} (b c-a d)}{3 d^4}+\frac{\left (c+\frac{d}{x^2}\right )^{7/2} (3 b c-a d)}{7 d^4}-\frac{c \left (c+\frac{d}{x^2}\right )^{5/2} (3 b c-2 a d)}{5 d^4}-\frac{b \left (c+\frac{d}{x^2}\right )^{9/2}}{9 d^4} \]
Antiderivative was successfully verified.
[In] Int[((a + b/x^2)*Sqrt[c + d/x^2])/x^7,x]
[Out]
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Rubi in Sympy [A] time = 22.8243, size = 92, normalized size = 0.88 \[ - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{9}{2}}}{9 d^{4}} - \frac{c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (a d - b c\right )}{3 d^{4}} + \frac{c \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}} \left (2 a d - 3 b c\right )}{5 d^{4}} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}} \left (a d - 3 b c\right )}{7 d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)*(c+d/x**2)**(1/2)/x**7,x)
[Out]
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Mathematica [A] time = 0.104687, size = 91, normalized size = 0.88 \[ -\frac{\sqrt{c+\frac{d}{x^2}} \left (c x^2+d\right ) \left (3 a d x^2 \left (8 c^2 x^4-12 c d x^2+15 d^2\right )+b \left (-16 c^3 x^6+24 c^2 d x^4-30 c d^2 x^2+35 d^3\right )\right )}{315 d^4 x^8} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b/x^2)*Sqrt[c + d/x^2])/x^7,x]
[Out]
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Maple [A] time = 0.011, size = 94, normalized size = 0.9 \[ -{\frac{ \left ( 24\,a{c}^{2}d{x}^{6}-16\,b{c}^{3}{x}^{6}-36\,ac{d}^{2}{x}^{4}+24\,b{c}^{2}d{x}^{4}+45\,a{d}^{3}{x}^{2}-30\,bc{d}^{2}{x}^{2}+35\,b{d}^{3} \right ) \left ( c{x}^{2}+d \right ) }{315\,{d}^{4}{x}^{8}}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)*(c+d/x^2)^(1/2)/x^7,x)
[Out]
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Maxima [A] time = 1.39348, size = 159, normalized size = 1.53 \[ -\frac{1}{315} \, b{\left (\frac{35 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{9}{2}}}{d^{4}} - \frac{135 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} c}{d^{4}} + \frac{189 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} c^{2}}{d^{4}} - \frac{105 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c^{3}}{d^{4}}\right )} - \frac{1}{105} \, a{\left (\frac{15 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}}}{d^{3}} - \frac{42 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} c}{d^{3}} + \frac{35 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c^{2}}{d^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)*sqrt(c + d/x^2)/x^7,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.330539, size = 147, normalized size = 1.41 \[ \frac{{\left (8 \,{\left (2 \, b c^{4} - 3 \, a c^{3} d\right )} x^{8} - 4 \,{\left (2 \, b c^{3} d - 3 \, a c^{2} d^{2}\right )} x^{6} - 35 \, b d^{4} + 3 \,{\left (2 \, b c^{2} d^{2} - 3 \, a c d^{3}\right )} x^{4} - 5 \,{\left (b c d^{3} + 9 \, a d^{4}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{315 \, d^{4} x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)*sqrt(c + d/x^2)/x^7,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.75093, size = 112, normalized size = 1.08 \[ - \frac{a \left (\frac{c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} - \frac{2 c \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7}\right )}{d^{3}} - \frac{b \left (- \frac{c^{3} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} + \frac{3 c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5} - \frac{3 c \left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{9}{2}}}{9}\right )}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)*(c+d/x**2)**(1/2)/x**7,x)
[Out]
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GIAC/XCAS [A] time = 0.819908, size = 500, normalized size = 4.81 \[ \frac{16 \,{\left (210 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{12} a c^{\frac{7}{2}}{\rm sign}\left (x\right ) + 630 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{10} b c^{\frac{9}{2}}{\rm sign}\left (x\right ) - 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{10} a c^{\frac{7}{2}} d{\rm sign}\left (x\right ) + 378 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} b c^{\frac{9}{2}} d{\rm sign}\left (x\right ) + 63 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} a c^{\frac{7}{2}} d^{2}{\rm sign}\left (x\right ) + 168 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} b c^{\frac{9}{2}} d^{2}{\rm sign}\left (x\right ) - 42 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} a c^{\frac{7}{2}} d^{3}{\rm sign}\left (x\right ) - 72 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} b c^{\frac{9}{2}} d^{3}{\rm sign}\left (x\right ) + 108 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} a c^{\frac{7}{2}} d^{4}{\rm sign}\left (x\right ) + 18 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} b c^{\frac{9}{2}} d^{4}{\rm sign}\left (x\right ) - 27 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} a c^{\frac{7}{2}} d^{5}{\rm sign}\left (x\right ) - 2 \, b c^{\frac{9}{2}} d^{5}{\rm sign}\left (x\right ) + 3 \, a c^{\frac{7}{2}} d^{6}{\rm sign}\left (x\right )\right )}}{315 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} - d\right )}^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)*sqrt(c + d/x^2)/x^7,x, algorithm="giac")
[Out]