Optimal. Leaf size=94 \[ \frac {x^{m+1} (b c-a d)^2 \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {d x^2}{c}\right )}{c d^2 (m+1)}-\frac {b x^{m+1} (b c-2 a d)}{d^2 (m+1)}+\frac {b^2 x^{m+3}}{d (m+3)} \]
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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} \[ \frac {x^{m+1} (b c-a d)^2 \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {d x^2}{c}\right )}{c d^2 (m+1)}-\frac {b x^{m+1} (b c-2 a d)}{d^2 (m+1)}+\frac {b^2 x^{m+3}}{d (m+3)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 461
Rubi steps
\begin {align*} \int \frac {x^m \left (a+b x^2\right )^2}{c+d x^2} \, dx &=\int \left (-\frac {b (b c-2 a d) x^m}{d^2}+\frac {b^2 x^{2+m}}{d}+\frac {\left (b^2 c^2-2 a b c d+a^2 d^2\right ) x^m}{d^2 \left (c+d x^2\right )}\right ) \, dx\\ &=-\frac {b (b c-2 a d) x^{1+m}}{d^2 (1+m)}+\frac {b^2 x^{3+m}}{d (3+m)}+\frac {(b c-a d)^2 \int \frac {x^m}{c+d x^2} \, dx}{d^2}\\ &=-\frac {b (b c-2 a d) x^{1+m}}{d^2 (1+m)}+\frac {b^2 x^{3+m}}{d (3+m)}+\frac {(b c-a d)^2 x^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )}{c d^2 (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 118, normalized size = 1.26 \[ \frac {x^{m+1} \left (\frac {a^2 \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {d x^2}{c}\right )}{m+1}+b x^2 \left (\frac {2 a \, _2F_1\left (1,\frac {m+3}{2};\frac {m+5}{2};-\frac {d x^2}{c}\right )}{m+3}+\frac {b x^2 \, _2F_1\left (1,\frac {m+5}{2};\frac {m+7}{2};-\frac {d x^2}{c}\right )}{m+5}\right )\right )}{c} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} x^{m}}{d x^{2} + c}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{2} x^{m}}{d x^{2} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{2}+a \right )^{2} x^{m}}{d \,x^{2}+c}\, dx \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{2} x^{m}}{d x^{2} + c}\,{d x} \]
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 \[ \int \frac {x^m\,{\left (b\,x^2+a\right )}^2}{d\,x^2+c} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 6.62, size = 299, normalized size = 3.18 \[ \frac {a^{2} m x x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {a^{2} x x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {a b m x^{3} x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{2 c \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {3 a b x^{3} x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{2 c \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {b^{2} m x^{5} x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {5}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {7}{2}\right )} + \frac {5 b^{2} x^{5} x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {5}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {7}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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