Optimal. Leaf size=192 \[ -\frac{a^2}{6 c x^6 \sqrt{c+d x^2}}+\frac{d \left (24 b^2 c^2-5 a d (12 b c-7 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{16 c^{9/2}}-\frac{\sqrt{c+d x^2} \left (24 b^2 c^2-5 a d (12 b c-7 a d)\right )}{16 c^4 x^2}+\frac{24 b^2 c^2-5 a d (12 b c-7 a d)}{24 c^3 x^2 \sqrt{c+d x^2}}-\frac{a (12 b c-7 a d)}{24 c^2 x^4 \sqrt{c+d x^2}} \]
[Out]
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Rubi [A] time = 0.526352, antiderivative size = 193, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{35 a^2 d^2-60 a b c d+24 b^2 c^2}{24 c^3 x^2 \sqrt{c+d x^2}}-\frac{a^2}{6 c x^6 \sqrt{c+d x^2}}+\frac{d \left (24 b^2 c^2-5 a d (12 b c-7 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{16 c^{9/2}}-\frac{\sqrt{c+d x^2} \left (24 b^2 c^2-5 a d (12 b c-7 a d)\right )}{16 c^4 x^2}-\frac{a (12 b c-7 a d)}{24 c^2 x^4 \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^2/(x^7*(c + d*x^2)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 35.4833, size = 184, normalized size = 0.96 \[ - \frac{a^{2}}{6 c x^{6} \sqrt{c + d x^{2}}} + \frac{a \left (7 a d - 12 b c\right )}{24 c^{2} x^{4} \sqrt{c + d x^{2}}} + \frac{5 a d \left (7 a d - 12 b c\right ) + 24 b^{2} c^{2}}{24 c^{3} x^{2} \sqrt{c + d x^{2}}} - \frac{\sqrt{c + d x^{2}} \left (5 a d \left (7 a d - 12 b c\right ) + 24 b^{2} c^{2}\right )}{16 c^{4} x^{2}} + \frac{d \left (5 a d \left (7 a d - 12 b c\right ) + 24 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{16 c^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2/x**7/(d*x**2+c)**(3/2),x)
[Out]
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Mathematica [A] time = 0.558826, size = 190, normalized size = 0.99 \[ \frac{3 d \left (35 a^2 d^2-60 a b c d+24 b^2 c^2\right ) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )-3 d \log (x) \left (35 a^2 d^2-60 a b c d+24 b^2 c^2\right )-\frac{\sqrt{c} \left (a^2 \left (8 c^3-14 c^2 d x^2+35 c d^2 x^4+105 d^3 x^6\right )+12 a b c x^2 \left (2 c^2-5 c d x^2-15 d^2 x^4\right )+24 b^2 c^2 x^4 \left (c+3 d x^2\right )\right )}{x^6 \sqrt{c+d x^2}}}{48 c^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^2/(x^7*(c + d*x^2)^(3/2)),x]
[Out]
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Maple [A] time = 0.02, size = 281, normalized size = 1.5 \[ -{\frac{{a}^{2}}{6\,c{x}^{6}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{7\,{a}^{2}d}{24\,{c}^{2}{x}^{4}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{35\,{a}^{2}{d}^{2}}{48\,{c}^{3}{x}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{35\,{a}^{2}{d}^{3}}{16\,{c}^{4}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{35\,{a}^{2}{d}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{9}{2}}}}-{\frac{{b}^{2}}{2\,c{x}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{3\,{b}^{2}d}{2\,{c}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{3\,{b}^{2}d}{2}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{5}{2}}}}-{\frac{ab}{2\,c{x}^{4}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{5\,abd}{4\,{c}^{2}{x}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{15\,ab{d}^{2}}{4\,{c}^{3}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{15\,ab{d}^{2}}{4}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2/x^7/(d*x^2+c)^(3/2),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/((d*x^2 + c)^(3/2)*x^7),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.247037, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (3 \,{\left (24 \, b^{2} c^{2} d - 60 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x^{6} + 8 \, a^{2} c^{3} +{\left (24 \, b^{2} c^{3} - 60 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} x^{4} + 2 \,{\left (12 \, a b c^{3} - 7 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{c} - 3 \,{\left ({\left (24 \, b^{2} c^{2} d^{2} - 60 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{8} +{\left (24 \, b^{2} c^{3} d - 60 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{6}\right )} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right )}{96 \,{\left (c^{4} d x^{8} + c^{5} x^{6}\right )} \sqrt{c}}, -\frac{{\left (3 \,{\left (24 \, b^{2} c^{2} d - 60 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x^{6} + 8 \, a^{2} c^{3} +{\left (24 \, b^{2} c^{3} - 60 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} x^{4} + 2 \,{\left (12 \, a b c^{3} - 7 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-c} - 3 \,{\left ({\left (24 \, b^{2} c^{2} d^{2} - 60 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{8} +{\left (24 \, b^{2} c^{3} d - 60 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{6}\right )} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right )}{48 \,{\left (c^{4} d x^{8} + c^{5} x^{6}\right )} \sqrt{-c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*x^7),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2}\right )^{2}}{x^{7} \left (c + d x^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2/x**7/(d*x**2+c)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.240611, size = 360, normalized size = 1.88 \[ -\frac{{\left (24 \, b^{2} c^{2} d - 60 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{16 \, \sqrt{-c} c^{4}} - \frac{b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}}{\sqrt{d x^{2} + c} c^{4}} - \frac{24 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} b^{2} c^{2} d - 48 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} c^{3} d + 24 \, \sqrt{d x^{2} + c} b^{2} c^{4} d - 84 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} a b c d^{2} + 192 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b c^{2} d^{2} - 108 \, \sqrt{d x^{2} + c} a b c^{3} d^{2} + 57 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} a^{2} d^{3} - 136 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} c d^{3} + 87 \, \sqrt{d x^{2} + c} a^{2} c^{2} d^{3}}{48 \, c^{4} d^{3} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*x^7),x, algorithm="giac")
[Out]