Optimal. Leaf size=141 \[ -\frac{a^2}{5 c x^5 \sqrt{c+d x^2}}-\frac{2 d x \left (15 b^2 c^2-8 a d (5 b c-3 a d)\right )}{15 c^4 \sqrt{c+d x^2}}-\frac{15 b^2 c^2-8 a d (5 b c-3 a d)}{15 c^3 x \sqrt{c+d x^2}}-\frac{2 a (5 b c-3 a d)}{15 c^2 x^3 \sqrt{c+d x^2}} \]
[Out]
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Rubi [A] time = 0.30583, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{a^2}{5 c x^5 \sqrt{c+d x^2}}-\frac{15 b^2-\frac{8 a d (5 b c-3 a d)}{c^2}}{15 c x \sqrt{c+d x^2}}-\frac{2 d x \left (15 b^2 c^2-8 a d (5 b c-3 a d)\right )}{15 c^4 \sqrt{c+d x^2}}-\frac{2 a (5 b c-3 a d)}{15 c^2 x^3 \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^2/(x^6*(c + d*x^2)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 26.2176, size = 136, normalized size = 0.96 \[ - \frac{a^{2}}{5 c x^{5} \sqrt{c + d x^{2}}} + \frac{2 a \left (3 a d - 5 b c\right )}{15 c^{2} x^{3} \sqrt{c + d x^{2}}} - \frac{8 a d \left (3 a d - 5 b c\right ) + 15 b^{2} c^{2}}{15 c^{3} x \sqrt{c + d x^{2}}} - \frac{2 d x \left (8 a d \left (3 a d - 5 b c\right ) + 15 b^{2} c^{2}\right )}{15 c^{4} \sqrt{c + d x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2/x**6/(d*x**2+c)**(3/2),x)
[Out]
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Mathematica [A] time = 0.121269, size = 105, normalized size = 0.74 \[ \sqrt{c+d x^2} \left (\frac{-33 a^2 d^2+50 a b c d-15 b^2 c^2}{15 c^4 x}-\frac{a^2}{5 c^2 x^5}-\frac{d x (b c-a d)^2}{c^4 \left (c+d x^2\right )}+\frac{a (9 a d-10 b c)}{15 c^3 x^3}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^2/(x^6*(c + d*x^2)^(3/2)),x]
[Out]
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Maple [A] time = 0.011, size = 117, normalized size = 0.8 \[ -{\frac{48\,{a}^{2}{d}^{3}{x}^{6}-80\,abc{d}^{2}{x}^{6}+30\,{b}^{2}{c}^{2}d{x}^{6}+24\,{a}^{2}c{d}^{2}{x}^{4}-40\,ab{c}^{2}d{x}^{4}+15\,{b}^{2}{c}^{3}{x}^{4}-6\,{a}^{2}{c}^{2}d{x}^{2}+10\,ab{c}^{3}{x}^{2}+3\,{a}^{2}{c}^{3}}{15\,{x}^{5}{c}^{4}}{\frac{1}{\sqrt{d{x}^{2}+c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2/x^6/(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^6),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.286789, size = 163, normalized size = 1.16 \[ -\frac{{\left (2 \,{\left (15 \, b^{2} c^{2} d - 40 \, a b c d^{2} + 24 \, a^{2} d^{3}\right )} x^{6} + 3 \, a^{2} c^{3} +{\left (15 \, b^{2} c^{3} - 40 \, a b c^{2} d + 24 \, a^{2} c d^{2}\right )} x^{4} + 2 \,{\left (5 \, a b c^{3} - 3 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{15 \,{\left (c^{4} d x^{7} + c^{5} x^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*x^6),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2}\right )^{2}}{x^{6} \left (c + d x^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2/x**6/(d*x**2+c)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.245448, size = 610, normalized size = 4.33 \[ -\frac{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x}{\sqrt{d x^{2} + c} c^{4}} + \frac{2 \,{\left (15 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} b^{2} c^{2} \sqrt{d} - 30 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} a b c d^{\frac{3}{2}} + 15 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} a^{2} d^{\frac{5}{2}} - 60 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} b^{2} c^{3} \sqrt{d} + 180 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} a b c^{2} d^{\frac{3}{2}} - 90 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} a^{2} c d^{\frac{5}{2}} + 90 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b^{2} c^{4} \sqrt{d} - 320 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a b c^{3} d^{\frac{3}{2}} + 240 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a^{2} c^{2} d^{\frac{5}{2}} - 60 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b^{2} c^{5} \sqrt{d} + 220 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b c^{4} d^{\frac{3}{2}} - 150 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{2} c^{3} d^{\frac{5}{2}} + 15 \, b^{2} c^{6} \sqrt{d} - 50 \, a b c^{5} d^{\frac{3}{2}} + 33 \, a^{2} c^{4} d^{\frac{5}{2}}\right )}}{15 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{5} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*x^6),x, algorithm="giac")
[Out]