Optimal. Leaf size=73 \[ \frac{\left (a+b x^2\right ) (d x)^{m+1} \, _2F_1\left (5,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a^5 d (m+1) \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
[Out]
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Rubi [A] time = 0.0973308, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{\left (a+b x^2\right ) (d x)^{m+1} \, _2F_1\left (5,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a^5 d (m+1) \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^m/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 15.4337, size = 63, normalized size = 0.86 \[ \frac{\left (d x\right )^{m + 1} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}{{}_{2}F_{1}\left (\begin{matrix} 5, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a^{5} d \left (a + b x^{2}\right ) \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**m/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0691074, size = 60, normalized size = 0.82 \[ \frac{x \left (a+b x^2\right ) (d x)^m \, _2F_1\left (5,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a^5 (m+1) \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d*x)^m/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
[Out]
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Maple [F] time = 0.044, size = 0, normalized size = 0. \[ \int{ \left ( dx \right ) ^{m} \left ({b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2} \right ) ^{-{\frac{5}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^m/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d x\right )^{m}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^m/(b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (d x\right )^{m}}{{\left (b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}\right )} \sqrt{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^m/(b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d x\right )^{m}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)**m/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d x\right )^{m}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^m/(b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2),x, algorithm="giac")
[Out]