Optimal. Leaf size=133 \[ \frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{1+m}}{b^3 (1+m)}+\frac {d^2 (3 b c-a d) x^{3+m}}{b^2 (3+m)}+\frac {d^3 x^{5+m}}{b (5+m)}+\frac {(b c-a d)^3 x^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{a b^3 (1+m)} \]
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Rubi [A]
time = 0.06, antiderivative size = 133, 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 = {472, 371}
\begin {gather*} \frac {d x^{m+1} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3 (m+1)}+\frac {x^{m+1} (b c-a d)^3 \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{a b^3 (m+1)}+\frac {d^2 x^{m+3} (3 b c-a d)}{b^2 (m+3)}+\frac {d^3 x^{m+5}}{b (m+5)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 472
Rubi steps
\begin {align*} \int \frac {x^m \left (c+d x^2\right )^3}{a+b x^2} \, dx &=\int \left (\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^m}{b^3}+\frac {d^2 (3 b c-a d) x^{2+m}}{b^2}+\frac {d^3 x^{4+m}}{b}+\frac {\left (b^3 c^3-3 a b^2 c^2 d+3 a^2 b c d^2-a^3 d^3\right ) x^m}{b^3 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{1+m}}{b^3 (1+m)}+\frac {d^2 (3 b c-a d) x^{3+m}}{b^2 (3+m)}+\frac {d^3 x^{5+m}}{b (5+m)}+\frac {(b c-a d)^3 \int \frac {x^m}{a+b x^2} \, dx}{b^3}\\ &=\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{1+m}}{b^3 (1+m)}+\frac {d^2 (3 b c-a d) x^{3+m}}{b^2 (3+m)}+\frac {d^3 x^{5+m}}{b (5+m)}+\frac {(b c-a d)^3 x^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{a b^3 (1+m)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in
optimal.
time = 1.19, size = 114, normalized size = 0.86 \begin {gather*} \frac {x^{1+m} \left (c^3 \Phi \left (-\frac {b x^2}{a},1,\frac {1+m}{2}\right )+d x^2 \left (3 c^2 \Phi \left (-\frac {b x^2}{a},1,\frac {3+m}{2}\right )+d x^2 \left (3 c \Phi \left (-\frac {b x^2}{a},1,\frac {5+m}{2}\right )+d x^2 \Phi \left (-\frac {b x^2}{a},1,\frac {7+m}{2}\right )\right )\right )\right )}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{m} \left (d \,x^{2}+c \right )^{3}}{b \,x^{2}+a}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 4.82, size = 411, normalized size = 3.09 \begin {gather*} \frac {c^{3} 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^{3} 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 {3 c^{2} 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 )}{4 a \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {9 c^{2} 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 )}{4 a \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {3 c 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 {15 c 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 )} + \frac {d^{3} m x^{7} x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {7}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {7}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {9}{2}\right )} + \frac {7 d^{3} x^{7} x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {7}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {7}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {9}{2}\right )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^m\,{\left (d\,x^2+c\right )}^3}{b\,x^2+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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