Optimal. Leaf size=180 \[ \frac {\left (a^2 B d^2+b^2 c (B c-A d)-2 a b d (B c-A d)\right ) (e x)^{1+m}}{d^3 e (1+m)}-\frac {b (b B c-A b d-2 a B d) (e x)^{3+m}}{d^2 e^3 (3+m)}+\frac {b^2 B (e x)^{5+m}}{d e^5 (5+m)}-\frac {(b c-a d)^2 (B c-A d) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )}{c d^3 e (1+m)} \]
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Rubi [A]
time = 0.12, antiderivative size = 180, 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^2-2 a b d (B c-A d)+b^2 c (B c-A d)\right )}{d^3 e (m+1)}-\frac {(e x)^{m+1} (b c-a d)^2 (B c-A d) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {d x^2}{c}\right )}{c d^3 e (m+1)}-\frac {b (e x)^{m+3} (-2 a B d-A b d+b B c)}{d^2 e^3 (m+3)}+\frac {b^2 B (e x)^{m+5}}{d 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 )^2 \left (A+B x^2\right )}{c+d x^2} \, dx &=\int \left (\frac {\left (a^2 B d^2+b^2 c (B c-A d)-2 a b d (B c-A d)\right ) (e x)^m}{d^3}-\frac {b (b B c-A b d-2 a B d) (e x)^{2+m}}{d^2 e^2}+\frac {b^2 B (e x)^{4+m}}{d e^4}+\frac {\left (-b^2 B c^3+A b^2 c^2 d+2 a b B c^2 d-2 a A b c d^2-a^2 B c d^2+a^2 A d^3\right ) (e x)^m}{d^3 \left (c+d x^2\right )}\right ) \, dx\\ &=\frac {\left (a^2 B d^2+b^2 c (B c-A d)-2 a b d (B c-A d)\right ) (e x)^{1+m}}{d^3 e (1+m)}-\frac {b (b B c-A b d-2 a B d) (e x)^{3+m}}{d^2 e^3 (3+m)}+\frac {b^2 B (e x)^{5+m}}{d e^5 (5+m)}-\frac {\left ((b c-a d)^2 (B c-A d)\right ) \int \frac {(e x)^m}{c+d x^2} \, dx}{d^3}\\ &=\frac {\left (a^2 B d^2+b^2 c (B c-A d)-2 a b d (B c-A d)\right ) (e x)^{1+m}}{d^3 e (1+m)}-\frac {b (b B c-A b d-2 a B d) (e x)^{3+m}}{d^2 e^3 (3+m)}+\frac {b^2 B (e x)^{5+m}}{d e^5 (5+m)}-\frac {(b c-a d)^2 (B c-A d) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )}{c d^3 e (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 147, normalized size = 0.82 \begin {gather*} \frac {x (e x)^m \left (\frac {a^2 B d^2+b^2 c (B c-A d)+2 a b d (-B c+A d)}{1+m}+\frac {b d (-b B c+A b d+2 a B d) x^2}{3+m}+\frac {b^2 B d^2 x^4}{5+m}-\frac {(b c-a d)^2 (B c-A d) \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )}{c (1+m)}\right )}{d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m} \left (b \,x^{2}+a \right )^{2} \left (B \,x^{2}+A \right )}{d \,x^{2}+c}\, dx\]
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 \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 = 6.99, size = 666, normalized size = 3.70 \begin {gather*} \frac {A a^{2} e^{m} 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 a^{2} e^{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 a b e^{m} 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 a b e^{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 {A b^{2} e^{m} 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 A b^{2} e^{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 {B a^{2} e^{m} 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 )}{4 c \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {3 B a^{2} e^{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 )}{4 c \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {B a b e^{m} 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 )}{2 c \Gamma \left (\frac {m}{2} + \frac {7}{2}\right )} + \frac {5 B a b e^{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 )}{2 c \Gamma \left (\frac {m}{2} + \frac {7}{2}\right )} + \frac {B b^{2} e^{m} m x^{7} x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {7}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {7}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {9}{2}\right )} + \frac {7 B b^{2} e^{m} x^{7} x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {7}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {7}{2}\right )}{4 c \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 (b\,x^2+a\right )}^2}{d\,x^2+c} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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