Optimal. Leaf size=94 \[ \frac {x^{m+1} (b c-a d)^2 \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{a b^2 (m+1)}+\frac {d x^{m+1} (2 b c-a d)}{b^2 (m+1)}+\frac {d^2 x^{m+3}}{b (m+3)} \]
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Rubi [A] time = 0.06, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {461, 364} \[ \frac {x^{m+1} (b c-a d)^2 \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{a b^2 (m+1)}+\frac {d x^{m+1} (2 b c-a d)}{b^2 (m+1)}+\frac {d^2 x^{m+3}}{b (m+3)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 461
Rubi steps
\begin {align*} \int \frac {x^m \left (c+d x^2\right )^2}{a+b x^2} \, dx &=\int \left (\frac {d (2 b c-a d) x^m}{b^2}+\frac {d^2 x^{2+m}}{b}+\frac {\left (b^2 c^2-2 a b c d+a^2 d^2\right ) x^m}{b^2 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {d (2 b c-a d) x^{1+m}}{b^2 (1+m)}+\frac {d^2 x^{3+m}}{b (3+m)}+\frac {(b c-a d)^2 \int \frac {x^m}{a+b x^2} \, dx}{b^2}\\ &=\frac {d (2 b c-a d) x^{1+m}}{b^2 (1+m)}+\frac {d^2 x^{3+m}}{b (3+m)}+\frac {(b c-a d)^2 x^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{a b^2 (1+m)}\\ \end {align*}
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Mathematica [C] time = 0.44, size = 85, normalized size = 0.90 \[ \frac {x^{m+1} \left (c^2 \Phi \left (-\frac {b x^2}{a},1,\frac {m+1}{2}\right )+d x^2 \left (2 c \Phi \left (-\frac {b x^2}{a},1,\frac {m+3}{2}\right )+d x^2 \Phi \left (-\frac {b x^2}{a},1,\frac {m+5}{2}\right )\right )\right )}{2 a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d^{2} x^{4} + 2 \, c d x^{2} + c^{2}\right )} x^{m}}{b x^{2} + a}, x\right ) \]
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 \[ \int \frac {{\left (d x^{2} + c\right )}^{2} x^{m}}{b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \,x^{2}+c \right )^{2} x^{m}}{b \,x^{2}+a}\, dx \]
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 \[ \int \frac {{\left (d x^{2} + c\right )}^{2} x^{m}}{b x^{2} + a}\,{d x} \]
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 \[ \int \frac {x^m\,{\left (d\,x^2+c\right )}^2}{b\,x^2+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 6.57, size = 299, normalized size = 3.18 \[ \frac {c^{2} m x x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {c^{2} x x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {c d m x^{3} x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{2 a \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {3 c d x^{3} x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{2 a \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {d^{2} m x^{5} x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {5}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {7}{2}\right )} + \frac {5 d^{2} x^{5} x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {5}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {7}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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