Optimal. Leaf size=91 \[ \frac {d (e x)^{m+5} (A d+2 B c)}{e^5 (m+5)}+\frac {c (e x)^{m+3} (2 A d+B c)}{e^3 (m+3)}+\frac {A c^2 (e x)^{m+1}}{e (m+1)}+\frac {B d^2 (e x)^{m+7}}{e^7 (m+7)} \]
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Rubi [A] time = 0.07, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {448} \[ \frac {c (e x)^{m+3} (2 A d+B c)}{e^3 (m+3)}+\frac {d (e x)^{m+5} (A d+2 B c)}{e^5 (m+5)}+\frac {A c^2 (e x)^{m+1}}{e (m+1)}+\frac {B d^2 (e x)^{m+7}}{e^7 (m+7)} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin {align*} \int (e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx &=\int \left (A c^2 (e x)^m+\frac {c (B c+2 A d) (e x)^{2+m}}{e^2}+\frac {d (2 B c+A d) (e x)^{4+m}}{e^4}+\frac {B d^2 (e x)^{6+m}}{e^6}\right ) \, dx\\ &=\frac {A c^2 (e x)^{1+m}}{e (1+m)}+\frac {c (B c+2 A d) (e x)^{3+m}}{e^3 (3+m)}+\frac {d (2 B c+A d) (e x)^{5+m}}{e^5 (5+m)}+\frac {B d^2 (e x)^{7+m}}{e^7 (7+m)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 67, normalized size = 0.74 \[ x (e x)^m \left (\frac {d x^4 (A d+2 B c)}{m+5}+\frac {c x^2 (2 A d+B c)}{m+3}+\frac {A c^2}{m+1}+\frac {B d^2 x^6}{m+7}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.11, size = 217, normalized size = 2.38 \[ \frac {{\left ({\left (B d^{2} m^{3} + 9 \, B d^{2} m^{2} + 23 \, B d^{2} m + 15 \, B d^{2}\right )} x^{7} + {\left ({\left (2 \, B c d + A d^{2}\right )} m^{3} + 42 \, B c d + 21 \, A d^{2} + 11 \, {\left (2 \, B c d + A d^{2}\right )} m^{2} + 31 \, {\left (2 \, B c d + A d^{2}\right )} m\right )} x^{5} + {\left ({\left (B c^{2} + 2 \, A c d\right )} m^{3} + 35 \, B c^{2} + 70 \, A c d + 13 \, {\left (B c^{2} + 2 \, A c d\right )} m^{2} + 47 \, {\left (B c^{2} + 2 \, A c d\right )} m\right )} x^{3} + {\left (A c^{2} m^{3} + 15 \, A c^{2} m^{2} + 71 \, A c^{2} m + 105 \, A c^{2}\right )} x\right )} \left (e x\right )^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.45, size = 380, normalized size = 4.18 \[ \frac {B d^{2} m^{3} x^{7} x^{m} e^{m} + 9 \, B d^{2} m^{2} x^{7} x^{m} e^{m} + 2 \, B c d m^{3} x^{5} x^{m} e^{m} + A d^{2} m^{3} x^{5} x^{m} e^{m} + 23 \, B d^{2} m x^{7} x^{m} e^{m} + 22 \, B c d m^{2} x^{5} x^{m} e^{m} + 11 \, A d^{2} m^{2} x^{5} x^{m} e^{m} + 15 \, B d^{2} x^{7} x^{m} e^{m} + B c^{2} m^{3} x^{3} x^{m} e^{m} + 2 \, A c d m^{3} x^{3} x^{m} e^{m} + 62 \, B c d m x^{5} x^{m} e^{m} + 31 \, A d^{2} m x^{5} x^{m} e^{m} + 13 \, B c^{2} m^{2} x^{3} x^{m} e^{m} + 26 \, A c d m^{2} x^{3} x^{m} e^{m} + 42 \, B c d x^{5} x^{m} e^{m} + 21 \, A d^{2} x^{5} x^{m} e^{m} + A c^{2} m^{3} x x^{m} e^{m} + 47 \, B c^{2} m x^{3} x^{m} e^{m} + 94 \, A c d m x^{3} x^{m} e^{m} + 15 \, A c^{2} m^{2} x x^{m} e^{m} + 35 \, B c^{2} x^{3} x^{m} e^{m} + 70 \, A c d x^{3} x^{m} e^{m} + 71 \, A c^{2} m x x^{m} e^{m} + 105 \, A c^{2} x x^{m} e^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 263, normalized size = 2.89 \[ \frac {\left (B \,d^{2} m^{3} x^{6}+9 B \,d^{2} m^{2} x^{6}+A \,d^{2} m^{3} x^{4}+2 B c d \,m^{3} x^{4}+23 B \,d^{2} m \,x^{6}+11 A \,d^{2} m^{2} x^{4}+22 B c d \,m^{2} x^{4}+15 B \,d^{2} x^{6}+2 A c d \,m^{3} x^{2}+31 A \,d^{2} m \,x^{4}+B \,c^{2} m^{3} x^{2}+62 B c d m \,x^{4}+26 A c d \,m^{2} x^{2}+21 A \,d^{2} x^{4}+13 B \,c^{2} m^{2} x^{2}+42 B c d \,x^{4}+A \,c^{2} m^{3}+94 A c d m \,x^{2}+47 B \,c^{2} m \,x^{2}+15 A \,c^{2} m^{2}+70 A c d \,x^{2}+35 B \,c^{2} x^{2}+71 A \,c^{2} m +105 A \,c^{2}\right ) x \left (e x \right )^{m}}{\left (m +7\right ) \left (m +5\right ) \left (m +3\right ) \left (m +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.51, size = 116, normalized size = 1.27 \[ \frac {B d^{2} e^{m} x^{7} x^{m}}{m + 7} + \frac {2 \, B c d e^{m} x^{5} x^{m}}{m + 5} + \frac {A d^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {B c^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {2 \, A c d e^{m} x^{3} x^{m}}{m + 3} + \frac {\left (e x\right )^{m + 1} A c^{2}}{e {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.05, size = 179, normalized size = 1.97 \[ {\left (e\,x\right )}^m\,\left (\frac {B\,d^2\,x^7\,\left (m^3+9\,m^2+23\,m+15\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {A\,c^2\,x\,\left (m^3+15\,m^2+71\,m+105\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {c\,x^3\,\left (2\,A\,d+B\,c\right )\,\left (m^3+13\,m^2+47\,m+35\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {d\,x^5\,\left (A\,d+2\,B\,c\right )\,\left (m^3+11\,m^2+31\,m+21\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.17, size = 1137, normalized size = 12.49 \[ \begin {cases} \frac {- \frac {A c^{2}}{6 x^{6}} - \frac {A c d}{2 x^{4}} - \frac {A d^{2}}{2 x^{2}} - \frac {B c^{2}}{4 x^{4}} - \frac {B c d}{x^{2}} + B d^{2} \log {\relax (x )}}{e^{7}} & \text {for}\: m = -7 \\\frac {- \frac {A c^{2}}{4 x^{4}} - \frac {A c d}{x^{2}} + A d^{2} \log {\relax (x )} - \frac {B c^{2}}{2 x^{2}} + 2 B c d \log {\relax (x )} + \frac {B d^{2} x^{2}}{2}}{e^{5}} & \text {for}\: m = -5 \\\frac {- \frac {A c^{2}}{2 x^{2}} + 2 A c d \log {\relax (x )} + \frac {A d^{2} x^{2}}{2} + B c^{2} \log {\relax (x )} + B c d x^{2} + \frac {B d^{2} x^{4}}{4}}{e^{3}} & \text {for}\: m = -3 \\\frac {A c^{2} \log {\relax (x )} + A c d x^{2} + \frac {A d^{2} x^{4}}{4} + \frac {B c^{2} x^{2}}{2} + \frac {B c d x^{4}}{2} + \frac {B d^{2} x^{6}}{6}}{e} & \text {for}\: m = -1 \\\frac {A c^{2} e^{m} m^{3} x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {15 A c^{2} e^{m} m^{2} x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {71 A c^{2} e^{m} m x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {105 A c^{2} e^{m} x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {2 A c d e^{m} m^{3} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {26 A c d e^{m} m^{2} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {94 A c d e^{m} m x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {70 A c d e^{m} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {A d^{2} e^{m} m^{3} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {11 A d^{2} e^{m} m^{2} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {31 A d^{2} e^{m} m x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {21 A d^{2} e^{m} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {B c^{2} e^{m} m^{3} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {13 B c^{2} e^{m} m^{2} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {47 B c^{2} e^{m} m x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {35 B c^{2} e^{m} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {2 B c d e^{m} m^{3} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {22 B c d e^{m} m^{2} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {62 B c d e^{m} m x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {42 B c d e^{m} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {B d^{2} e^{m} m^{3} x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {9 B d^{2} e^{m} m^{2} x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {23 B d^{2} e^{m} m x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {15 B d^{2} e^{m} x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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