Optimal. Leaf size=340 \[ -\frac {b \left (3 a^2 d^2 (A d (1+m)-B c (3+m))-3 a b c d (A d (3+m)-B c (5+m))+b^2 c^2 (A d (5+m)-B c (7+m))\right ) (e x)^{1+m}}{2 c d^4 e (1+m)}-\frac {b^2 (3 a d (A d (3+m)-B c (5+m))-b c (A d (5+m)-B c (7+m))) (e x)^{3+m}}{2 c d^3 e^3 (3+m)}-\frac {b^3 (A d (5+m)-B c (7+m)) (e x)^{5+m}}{2 c d^2 e^5 (5+m)}-\frac {(B c-A d) (e x)^{1+m} \left (a+b x^2\right )^3}{2 c d e \left (c+d x^2\right )}+\frac {(b c-a d)^2 (a d (A d (1-m)+B c (1+m))+b c (A d (5+m)-B c (7+m))) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )}{2 c^2 d^4 e (1+m)} \]
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Rubi [A]
time = 0.49, antiderivative size = 340, 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 {b (e x)^{m+1} \left (3 a^2 d^2 (A d (m+1)-B c (m+3))-3 a b c d (A d (m+3)-B c (m+5))+b^2 c^2 (A d (m+5)-B c (m+7))\right )}{2 c d^4 e (m+1)}-\frac {b^2 (e x)^{m+3} (3 a d (A d (m+3)-B c (m+5))-b c (A d (m+5)-B c (m+7)))}{2 c d^3 e^3 (m+3)}+\frac {(e 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 ) (a d (A d (1-m)+B c (m+1))+b c (A d (m+5)-B c (m+7)))}{2 c^2 d^4 e (m+1)}-\frac {\left (a+b x^2\right )^3 (e x)^{m+1} (B c-A d)}{2 c d e \left (c+d x^2\right )}-\frac {b^3 (e x)^{m+5} (A d (m+5)-B c (m+7))}{2 c d^2 e^5 (m+5)} \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 )^3 \left (A+B x^2\right )}{\left (c+d x^2\right )^2} \, dx &=-\frac {(B c-A d) (e x)^{1+m} \left (a+b x^2\right )^3}{2 c d e \left (c+d x^2\right )}-\frac {\int \frac {(e x)^m \left (a+b x^2\right )^2 \left (-a (A d (1-m)+B c (1+m))+b (A d (5+m)-B c (7+m)) x^2\right )}{c+d x^2} \, dx}{2 c d}\\ &=-\frac {(B c-A d) (e x)^{1+m} \left (a+b x^2\right )^3}{2 c d e \left (c+d x^2\right )}-\frac {\int \left (\frac {b \left (3 a^2 d^2 (A d (1+m)-B c (3+m))-3 a b c d (A d (3+m)-B c (5+m))+b^2 c^2 (A d (5+m)-B c (7+m))\right ) (e x)^m}{d^3}+\frac {b^2 (3 a d (A d (3+m)-B c (5+m))-b c (A d (5+m)-B c (7+m))) (e x)^{2+m}}{d^2 e^2}+\frac {b^3 (A d (5+m)-B c (7+m)) (e x)^{4+m}}{d e^4}+\frac {\left (7 b^3 B c^4-5 A b^3 c^3 d-15 a b^2 B c^3 d+9 a A b^2 c^2 d^2+9 a^2 b B c^2 d^2-3 a^2 A b c d^3-a^3 B c d^3-a^3 A d^4+b^3 B c^4 m-A b^3 c^3 d m-3 a b^2 B c^3 d m+3 a A b^2 c^2 d^2 m+3 a^2 b B c^2 d^2 m-3 a^2 A b c d^3 m-a^3 B c d^3 m+a^3 A d^4 m\right ) (e x)^m}{d^3 \left (c+d x^2\right )}\right ) \, dx}{2 c d}\\ &=-\frac {b \left (3 a^2 d^2 (A d (1+m)-B c (3+m))-3 a b c d (A d (3+m)-B c (5+m))+b^2 c^2 (A d (5+m)-B c (7+m))\right ) (e x)^{1+m}}{2 c d^4 e (1+m)}-\frac {b^2 (3 a d (A d (3+m)-B c (5+m))-b c (A d (5+m)-B c (7+m))) (e x)^{3+m}}{2 c d^3 e^3 (3+m)}-\frac {b^3 (A d (5+m)-B c (7+m)) (e x)^{5+m}}{2 c d^2 e^5 (5+m)}-\frac {(B c-A d) (e x)^{1+m} \left (a+b x^2\right )^3}{2 c d e \left (c+d x^2\right )}+\frac {\left ((b c-a d)^2 (a d (A d (1-m)+B c (1+m))+b c (A d (5+m)-B c (7+m)))\right ) \int \frac {(e x)^m}{c+d x^2} \, dx}{2 c d^4}\\ &=-\frac {b \left (3 a^2 d^2 (A d (1+m)-B c (3+m))-3 a b c d (A d (3+m)-B c (5+m))+b^2 c^2 (A d (5+m)-B c (7+m))\right ) (e x)^{1+m}}{2 c d^4 e (1+m)}-\frac {b^2 (3 a d (A d (3+m)-B c (5+m))-b c (A d (5+m)-B c (7+m))) (e x)^{3+m}}{2 c d^3 e^3 (3+m)}-\frac {b^3 (A d (5+m)-B c (7+m)) (e x)^{5+m}}{2 c d^2 e^5 (5+m)}-\frac {(B c-A d) (e x)^{1+m} \left (a+b x^2\right )^3}{2 c d e \left (c+d x^2\right )}+\frac {(b c-a d)^2 (a d (A d (1-m)+B c (1+m))+b c (A d (5+m)-B c (7+m))) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )}{2 c^2 d^4 e (1+m)}\\ \end {align*}
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Mathematica [A]
time = 1.20, size = 239, normalized size = 0.70 \begin {gather*} \frac {x (e x)^m \left (b \left (\frac {3 a^2 B d^2}{1+m}+3 a b d \left (-\frac {2 B c}{1+m}+\frac {A d}{1+m}+\frac {B d x^2}{3+m}\right )+b^2 \left (A d \left (-\frac {2 c}{1+m}+\frac {d x^2}{3+m}\right )+B \left (\frac {3 c^2}{1+m}-\frac {2 c d x^2}{3+m}+\frac {d^2 x^4}{5+m}\right )\right )\right )-\frac {(b c-a d)^2 (4 b B c-3 A b d-a B d) \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )}{c (1+m)}+\frac {(b c-a d)^3 (B c-A d) \, _2F_1\left (2,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )}{c^2 (1+m)}\right )}{d^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m} \left (b \,x^{2}+a \right )^{3} \left (B \,x^{2}+A \right )}{\left (d \,x^{2}+c \right )^{2}}\, dx\]
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 (a + b x^{2}\right )^{3}}{\left (c + d 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 (b\,x^2+a\right )}^3}{{\left (d\,x^2+c\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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