Optimal. Leaf size=121 \[ \frac{(c x)^{m+1} (A b-a C) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a b c (m+1)}+\frac{B (c x)^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-\frac{b x^2}{a}\right )}{a c^2 (m+2)}+\frac{C (c x)^{m+1}}{b c (m+1)} \]
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Rubi [A] time = 0.279132, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{(c x)^{m+1} (A b-a C) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a b c (m+1)}+\frac{B (c x)^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-\frac{b x^2}{a}\right )}{a c^2 (m+2)}+\frac{C (c x)^{m+1}}{b c (m+1)} \]
Antiderivative was successfully verified.
[In] Int[((c*x)^m*(A + B*x + C*x^2))/(a + b*x^2),x]
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Rubi in Sympy [A] time = 39.393, size = 90, normalized size = 0.74 \[ \frac{B \left (c x\right )^{m + 2}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a c^{2} \left (m + 2\right )} + \frac{C \left (c x\right )^{m + 1}}{b c \left (m + 1\right )} + \frac{\left (c x\right )^{m + 1} \left (A b - C a\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a b c \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x)**m*(C*x**2+B*x+A)/(b*x**2+a),x)
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Mathematica [A] time = 0.103921, size = 100, normalized size = 0.83 \[ \frac{x (c x)^m \left (b B (m+1) x \, _2F_1\left (1,\frac{m}{2}+1;\frac{m}{2}+2;-\frac{b x^2}{a}\right )-(m+2) \left ((a C-A b) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )-a C\right )\right )}{a b (m+1) (m+2)} \]
Antiderivative was successfully verified.
[In] Integrate[((c*x)^m*(A + B*x + C*x^2))/(a + b*x^2),x]
[Out]
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Maple [F] time = 0.047, size = 0, normalized size = 0. \[ \int{\frac{ \left ( cx \right ) ^{m} \left ( C{x}^{2}+Bx+A \right ) }{b{x}^{2}+a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x)^m*(C*x^2+B*x+A)/(b*x^2+a),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (C x^{2} + B x + A\right )} \left (c x\right )^{m}}{b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + B*x + A)*(c*x)^m/(b*x^2 + a),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (C x^{2} + B x + A\right )} \left (c x\right )^{m}}{b x^{2} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + B*x + A)*(c*x)^m/(b*x^2 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.67938, size = 298, normalized size = 2.46 \[ \frac{A c^{m} m x x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{A c^{m} x x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{B c^{m} m x^{2} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + 1\right ) \Gamma \left (\frac{m}{2} + 1\right )}{4 a \Gamma \left (\frac{m}{2} + 2\right )} + \frac{B c^{m} x^{2} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + 1\right ) \Gamma \left (\frac{m}{2} + 1\right )}{2 a \Gamma \left (\frac{m}{2} + 2\right )} + \frac{C c^{m} m x^{3} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} + \frac{3 C c^{m} x^{3} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)**m*(C*x**2+B*x+A)/(b*x**2+a),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (C x^{2} + B x + A\right )} \left (c x\right )^{m}}{b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + B*x + A)*(c*x)^m/(b*x^2 + a),x, algorithm="giac")
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