Optimal. Leaf size=247 \[ -\frac {d (A b (2 b c (1+m)-a d (3+m))-a B (2 b c (3+m)-a d (5+m))) (e x)^{1+m}}{2 a b^3 e (1+m)}-\frac {d^2 (A b (3+m)-a B (5+m)) (e x)^{3+m}}{2 a b^2 e^3 (3+m)}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^2\right )^2}{2 a b e \left (a+b x^2\right )}+\frac {(b c-a d) (a B (b c (1+m)-a d (5+m))+A b (a d (3+m)+b (c-c m))) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{2 a^2 b^3 e (1+m)} \]
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Rubi [A]
time = 0.29, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {591, 584, 371}
\begin {gather*} \frac {(e x)^{m+1} (b c-a d) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right ) (A b (a d (m+3)+b (c-c m))+a B (b c (m+1)-a d (m+5)))}{2 a^2 b^3 e (m+1)}-\frac {d (e x)^{m+1} (A b (2 b c (m+1)-a d (m+3))-a B (2 b c (m+3)-a d (m+5)))}{2 a b^3 e (m+1)}-\frac {d^2 (e x)^{m+3} (A b (m+3)-a B (m+5))}{2 a b^2 e^3 (m+3)}+\frac {\left (c+d x^2\right )^2 (e x)^{m+1} (A b-a B)}{2 a b e \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 584
Rule 591
Rubi steps
\begin {align*} \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx &=\frac {(A b-a B) (e x)^{1+m} \left (c+d x^2\right )^2}{2 a b e \left (a+b x^2\right )}-\frac {\int \frac {(e x)^m \left (c+d x^2\right ) \left (-c (A b (1-m)+a B (1+m))+d (A b (3+m)-a B (5+m)) x^2\right )}{a+b x^2} \, dx}{2 a b}\\ &=\frac {(A b-a B) (e x)^{1+m} \left (c+d x^2\right )^2}{2 a b e \left (a+b x^2\right )}-\frac {\int \left (\frac {d (A b (2 b c (1+m)-a d (3+m))-a B (2 b c (3+m)-a d (5+m))) (e x)^m}{b^2}+\frac {d^2 (A b (3+m)-a B (5+m)) (e x)^{2+m}}{b e^2}+\frac {\left (-A b^3 c^2-a b^2 B c^2-2 a A b^2 c d+6 a^2 b B c d+3 a^2 A b d^2-5 a^3 B d^2+A b^3 c^2 m-a b^2 B c^2 m-2 a A b^2 c d m+2 a^2 b B c d m+a^2 A b d^2 m-a^3 B d^2 m\right ) (e x)^m}{b^2 \left (a+b x^2\right )}\right ) \, dx}{2 a b}\\ &=-\frac {d (A b (2 b c (1+m)-a d (3+m))-a B (2 b c (3+m)-a d (5+m))) (e x)^{1+m}}{2 a b^3 e (1+m)}-\frac {d^2 (A b (3+m)-a B (5+m)) (e x)^{3+m}}{2 a b^2 e^3 (3+m)}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^2\right )^2}{2 a b e \left (a+b x^2\right )}-\frac {\left (-A b^3 c^2-a b^2 B c^2-2 a A b^2 c d+6 a^2 b B c d+3 a^2 A b d^2-5 a^3 B d^2+A b^3 c^2 m-a b^2 B c^2 m-2 a A b^2 c d m+2 a^2 b B c d m+a^2 A b d^2 m-a^3 B d^2 m\right ) \int \frac {(e x)^m}{a+b x^2} \, dx}{2 a b^3}\\ &=-\frac {d (A b (2 b c (1+m)-a d (3+m))-a B (2 b c (3+m)-a d (5+m))) (e x)^{1+m}}{2 a b^3 e (1+m)}-\frac {d^2 (A b (3+m)-a B (5+m)) (e x)^{3+m}}{2 a b^2 e^3 (3+m)}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^2\right )^2}{2 a b e \left (a+b x^2\right )}+\frac {(b c-a d) (A b (b c (1-m)+a d (3+m))+a B (b c (1+m)-a d (5+m))) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{2 a^2 b^3 e (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.75, size = 163, normalized size = 0.66 \begin {gather*} \frac {x (e x)^m \left (a^2 d \left (-2 a B d (3+m)+b \left (2 B c (3+m)+A d (3+m)+B d (1+m) x^2\right )\right )+a (b c-a d) (b B c+2 A b d-3 a B d) (3+m) \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )+(A b-a B) (b c-a d)^2 (3+m) \, _2F_1\left (2,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )\right )}{a^2 b^3 (1+m) (3+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m} \left (B \,x^{2}+A \right ) \left (d \,x^{2}+c \right )^{2}}{\left (b \,x^{2}+a \right )^{2}}\, dx\]
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e x\right )^{m} \left (A + B x^{2}\right ) \left (c + d x^{2}\right )^{2}}{\left (a + b x^{2}\right )^{2}}\, dx \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 = -0.00 \begin {gather*} \int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^m\,{\left (d\,x^2+c\right )}^2}{{\left (b\,x^2+a\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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