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 = 0.12, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1802, 808, 364} \[ \frac {(c x)^{m+1} (A b-a C) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{a b c (m+1)}+\frac {B (c x)^{m+2} \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};-\frac {b x^2}{a}\right )}{a c^2 (m+2)}+\frac {C (c x)^{m+1}}{b c (m+1)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 808
Rule 1802
Rubi steps
\begin {align*} \int \frac {(c x)^m \left (A+B x+C x^2\right )}{a+b x^2} \, dx &=\int \left (\frac {C (c x)^m}{b}+\frac {(c x)^m (A b-a C+b B x)}{b \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {C (c x)^{1+m}}{b c (1+m)}+\frac {\int \frac {(c x)^m (A b-a C+b B x)}{a+b x^2} \, dx}{b}\\ &=\frac {C (c x)^{1+m}}{b c (1+m)}+\frac {B \int \frac {(c x)^{1+m}}{a+b x^2} \, dx}{c}+\frac {(A b-a C) \int \frac {(c x)^m}{a+b x^2} \, dx}{b}\\ &=\frac {C (c x)^{1+m}}{b c (1+m)}+\frac {(A b-a C) (c x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{a b c (1+m)}+\frac {B (c x)^{2+m} \, _2F_1\left (1,\frac {2+m}{2};\frac {4+m}{2};-\frac {b x^2}{a}\right )}{a c^2 (2+m)}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 99, normalized size = 0.82 \[ \frac {x (c x)^m \left ((m+2) (A b-a C) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )+b B (m+1) x \, _2F_1\left (1,\frac {m}{2}+1;\frac {m}{2}+2;-\frac {b x^2}{a}\right )+a C (m+2)\right )}{a b (m+1) (m+2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C x^{2} + B x + A\right )} \left (c x\right )^{m}}{b x^{2} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C x^{2} + B x + A\right )} \left (c x\right )^{m}}{b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\left (C \,x^{2}+B x +A \right ) \left (c x \right )^{m}}{b \,x^{2}+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C x^{2} + B x + A\right )} \left (c x\right )^{m}}{b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c\,x\right )}^m\,\left (C\,x^2+B\,x+A\right )}{b\,x^2+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 7.60, 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.
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