Optimal. Leaf size=178 \[ \frac {\left (a^2 B d^2-a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) (e x)^{1+m}}{b^3 e (1+m)}+\frac {d (2 b B c+A b d-a B d) (e x)^{3+m}}{b^2 e^3 (3+m)}+\frac {B d^2 (e x)^{5+m}}{b e^5 (5+m)}+\frac {(A b-a B) (b c-a d)^2 (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{a b^3 e (1+m)} \]
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Rubi [A]
time = 0.11, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {584, 371}
\begin {gather*} \frac {(e x)^{m+1} \left (a^2 B d^2-a b d (A d+2 B c)+b^2 c (2 A d+B c)\right )}{b^3 e (m+1)}+\frac {(e x)^{m+1} (A b-a B) (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^3 e (m+1)}+\frac {d (e x)^{m+3} (-a B d+A b d+2 b B c)}{b^2 e^3 (m+3)}+\frac {B d^2 (e x)^{m+5}}{b e^5 (m+5)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 584
Rubi steps
\begin {align*} \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^2}{a+b x^2} \, dx &=\int \left (\frac {\left (a^2 B d^2-a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) (e x)^m}{b^3}+\frac {d (2 b B c+A b d-a B d) (e x)^{2+m}}{b^2 e^2}+\frac {B d^2 (e x)^{4+m}}{b e^4}+\frac {\left (A b^3 c^2-a b^2 B c^2-2 a A b^2 c d+2 a^2 b B c d+a^2 A b d^2-a^3 B d^2\right ) (e x)^m}{b^3 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {\left (a^2 B d^2-a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) (e x)^{1+m}}{b^3 e (1+m)}+\frac {d (2 b B c+A b d-a B d) (e x)^{3+m}}{b^2 e^3 (3+m)}+\frac {B d^2 (e x)^{5+m}}{b e^5 (5+m)}+\frac {\left ((A b-a B) (b c-a d)^2\right ) \int \frac {(e x)^m}{a+b x^2} \, dx}{b^3}\\ &=\frac {\left (a^2 B d^2-a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) (e x)^{1+m}}{b^3 e (1+m)}+\frac {d (2 b B c+A b d-a B d) (e x)^{3+m}}{b^2 e^3 (3+m)}+\frac {B d^2 (e x)^{5+m}}{b e^5 (5+m)}+\frac {(A b-a B) (b c-a d)^2 (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{a b^3 e (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 146, normalized size = 0.82 \begin {gather*} \frac {x (e x)^m \left (\frac {a^2 B d^2-a b d (2 B c+A d)+b^2 c (B c+2 A d)}{1+m}+\frac {b d (2 b B c+A b d-a B d) x^2}{3+m}+\frac {b^2 B d^2 x^4}{5+m}+\frac {(A b-a B) (b c-a d)^2 \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{a (1+m)}\right )}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m} \left (B \,x^{2}+A \right ) \left (d \,x^{2}+c \right )^{2}}{b \,x^{2}+a}\, dx\]
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 = 6.79, size = 666, normalized size = 3.74 \begin {gather*} \frac {A c^{2} e^{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^{2} e^{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 d e^{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 )}{2 a \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {3 A c d e^{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 {A d^{2} e^{m} 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 A d^{2} e^{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 {B c^{2} e^{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 B c^{2} e^{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 {B c d e^{m} 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 )}{2 a \Gamma \left (\frac {m}{2} + \frac {7}{2}\right )} + \frac {5 B c d e^{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 )}{2 a \Gamma \left (\frac {m}{2} + \frac {7}{2}\right )} + \frac {B d^{2} e^{m} 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 B d^{2} e^{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 )} \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 {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^m\,{\left (d\,x^2+c\right )}^2}{b\,x^2+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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