Optimal. Leaf size=121 \[ \frac {A c^3 (e x)^{1+m}}{e (1+m)}+\frac {c^2 (B c+3 A d) (e x)^{3+m}}{e^3 (3+m)}+\frac {3 c d (B c+A d) (e x)^{5+m}}{e^5 (5+m)}+\frac {d^2 (3 B c+A d) (e x)^{7+m}}{e^7 (7+m)}+\frac {B d^3 (e x)^{9+m}}{e^9 (9+m)} \]
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Rubi [A]
time = 0.05, antiderivative size = 121, 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 = {459}
\begin {gather*} \frac {c^2 (e x)^{m+3} (3 A d+B c)}{e^3 (m+3)}+\frac {d^2 (e x)^{m+7} (A d+3 B c)}{e^7 (m+7)}+\frac {3 c d (e x)^{m+5} (A d+B c)}{e^5 (m+5)}+\frac {A c^3 (e x)^{m+1}}{e (m+1)}+\frac {B d^3 (e x)^{m+9}}{e^9 (m+9)} \end {gather*}
Antiderivative was successfully verified.
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Rule 459
Rubi steps
\begin {align*} \int (e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx &=\int \left (A c^3 (e x)^m+\frac {c^2 (B c+3 A d) (e x)^{2+m}}{e^2}+\frac {3 c d (B c+A d) (e x)^{4+m}}{e^4}+\frac {d^2 (3 B c+A d) (e x)^{6+m}}{e^6}+\frac {B d^3 (e x)^{8+m}}{e^8}\right ) \, dx\\ &=\frac {A c^3 (e x)^{1+m}}{e (1+m)}+\frac {c^2 (B c+3 A d) (e x)^{3+m}}{e^3 (3+m)}+\frac {3 c d (B c+A d) (e x)^{5+m}}{e^5 (5+m)}+\frac {d^2 (3 B c+A d) (e x)^{7+m}}{e^7 (7+m)}+\frac {B d^3 (e x)^{9+m}}{e^9 (9+m)}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 90, normalized size = 0.74 \begin {gather*} x (e x)^m \left (\frac {A c^3}{1+m}+\frac {c^2 (B c+3 A d) x^2}{3+m}+\frac {3 c d (B c+A d) x^4}{5+m}+\frac {d^2 (3 B c+A d) x^6}{7+m}+\frac {B d^3 x^8}{9+m}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(474\) vs.
\(2(121)=242\).
time = 0.10, size = 475, normalized size = 3.93
method | result | size |
gosper | \(\frac {x \left (B \,d^{3} m^{4} x^{8}+16 B \,d^{3} m^{3} x^{8}+A \,d^{3} m^{4} x^{6}+3 B c \,d^{2} m^{4} x^{6}+86 B \,d^{3} m^{2} x^{8}+18 A \,d^{3} m^{3} x^{6}+54 B c \,d^{2} m^{3} x^{6}+176 m \,x^{8} B \,d^{3}+3 A c \,d^{2} m^{4} x^{4}+104 A \,d^{3} m^{2} x^{6}+3 B \,c^{2} d \,m^{4} x^{4}+312 B c \,d^{2} m^{2} x^{6}+105 B \,d^{3} x^{8}+60 A c \,d^{2} m^{3} x^{4}+222 A \,d^{3} x^{6} m +60 B \,c^{2} d \,m^{3} x^{4}+666 B c \,d^{2} x^{6} m +3 A \,c^{2} d \,m^{4} x^{2}+390 A c \,d^{2} m^{2} x^{4}+135 A \,d^{3} x^{6}+B \,c^{3} m^{4} x^{2}+390 B \,c^{2} d \,m^{2} x^{4}+405 B c \,d^{2} x^{6}+66 A \,c^{2} d \,m^{3} x^{2}+900 A c \,d^{2} x^{4} m +22 B \,c^{3} m^{3} x^{2}+900 B \,c^{2} d \,x^{4} m +A \,c^{3} m^{4}+492 A \,c^{2} d \,m^{2} x^{2}+567 A c \,d^{2} x^{4}+164 B \,c^{3} m^{2} x^{2}+567 B \,c^{2} d \,x^{4}+24 A \,c^{3} m^{3}+1374 A \,c^{2} d \,x^{2} m +458 B \,c^{3} x^{2} m +206 A \,c^{3} m^{2}+945 A \,c^{2} d \,x^{2}+315 B \,c^{3} x^{2}+744 A \,c^{3} m +945 A \,c^{3}\right ) \left (e x \right )^{m}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(475\) |
risch | \(\frac {x \left (B \,d^{3} m^{4} x^{8}+16 B \,d^{3} m^{3} x^{8}+A \,d^{3} m^{4} x^{6}+3 B c \,d^{2} m^{4} x^{6}+86 B \,d^{3} m^{2} x^{8}+18 A \,d^{3} m^{3} x^{6}+54 B c \,d^{2} m^{3} x^{6}+176 m \,x^{8} B \,d^{3}+3 A c \,d^{2} m^{4} x^{4}+104 A \,d^{3} m^{2} x^{6}+3 B \,c^{2} d \,m^{4} x^{4}+312 B c \,d^{2} m^{2} x^{6}+105 B \,d^{3} x^{8}+60 A c \,d^{2} m^{3} x^{4}+222 A \,d^{3} x^{6} m +60 B \,c^{2} d \,m^{3} x^{4}+666 B c \,d^{2} x^{6} m +3 A \,c^{2} d \,m^{4} x^{2}+390 A c \,d^{2} m^{2} x^{4}+135 A \,d^{3} x^{6}+B \,c^{3} m^{4} x^{2}+390 B \,c^{2} d \,m^{2} x^{4}+405 B c \,d^{2} x^{6}+66 A \,c^{2} d \,m^{3} x^{2}+900 A c \,d^{2} x^{4} m +22 B \,c^{3} m^{3} x^{2}+900 B \,c^{2} d \,x^{4} m +A \,c^{3} m^{4}+492 A \,c^{2} d \,m^{2} x^{2}+567 A c \,d^{2} x^{4}+164 B \,c^{3} m^{2} x^{2}+567 B \,c^{2} d \,x^{4}+24 A \,c^{3} m^{3}+1374 A \,c^{2} d \,x^{2} m +458 B \,c^{3} x^{2} m +206 A \,c^{3} m^{2}+945 A \,c^{2} d \,x^{2}+315 B \,c^{3} x^{2}+744 A \,c^{3} m +945 A \,c^{3}\right ) \left (e x \right )^{m}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(475\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 169, normalized size = 1.40 \begin {gather*} \frac {B d^{3} x^{9} e^{\left (m \log \left (x\right ) + m\right )}}{m + 9} + \frac {3 \, B c d^{2} x^{7} e^{\left (m \log \left (x\right ) + m\right )}}{m + 7} + \frac {A d^{3} x^{7} e^{\left (m \log \left (x\right ) + m\right )}}{m + 7} + \frac {3 \, B c^{2} d x^{5} e^{\left (m \log \left (x\right ) + m\right )}}{m + 5} + \frac {3 \, A c d^{2} x^{5} e^{\left (m \log \left (x\right ) + m\right )}}{m + 5} + \frac {B c^{3} x^{3} e^{\left (m \log \left (x\right ) + m\right )}}{m + 3} + \frac {3 \, A c^{2} d x^{3} e^{\left (m \log \left (x\right ) + m\right )}}{m + 3} + \frac {\left (x e\right )^{m + 1} A c^{3} e^{\left (-1\right )}}{m + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 382 vs.
\(2 (121) = 242\).
time = 1.30, size = 382, normalized size = 3.16 \begin {gather*} \frac {{\left ({\left (B d^{3} m^{4} + 16 \, B d^{3} m^{3} + 86 \, B d^{3} m^{2} + 176 \, B d^{3} m + 105 \, B d^{3}\right )} x^{9} + {\left ({\left (3 \, B c d^{2} + A d^{3}\right )} m^{4} + 405 \, B c d^{2} + 135 \, A d^{3} + 18 \, {\left (3 \, B c d^{2} + A d^{3}\right )} m^{3} + 104 \, {\left (3 \, B c d^{2} + A d^{3}\right )} m^{2} + 222 \, {\left (3 \, B c d^{2} + A d^{3}\right )} m\right )} x^{7} + 3 \, {\left ({\left (B c^{2} d + A c d^{2}\right )} m^{4} + 189 \, B c^{2} d + 189 \, A c d^{2} + 20 \, {\left (B c^{2} d + A c d^{2}\right )} m^{3} + 130 \, {\left (B c^{2} d + A c d^{2}\right )} m^{2} + 300 \, {\left (B c^{2} d + A c d^{2}\right )} m\right )} x^{5} + {\left ({\left (B c^{3} + 3 \, A c^{2} d\right )} m^{4} + 315 \, B c^{3} + 945 \, A c^{2} d + 22 \, {\left (B c^{3} + 3 \, A c^{2} d\right )} m^{3} + 164 \, {\left (B c^{3} + 3 \, A c^{2} d\right )} m^{2} + 458 \, {\left (B c^{3} + 3 \, A c^{2} d\right )} m\right )} x^{3} + {\left (A c^{3} m^{4} + 24 \, A c^{3} m^{3} + 206 \, A c^{3} m^{2} + 744 \, A c^{3} m + 945 \, A c^{3}\right )} x\right )} \left (x e\right )^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2152 vs.
\(2 (110) = 220\).
time = 0.60, size = 2152, normalized size = 17.79 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 673 vs.
\(2 (121) = 242\).
time = 1.11, size = 673, normalized size = 5.56 \begin {gather*} \frac {B d^{3} m^{4} x^{9} x^{m} e^{m} + 16 \, B d^{3} m^{3} x^{9} x^{m} e^{m} + 3 \, B c d^{2} m^{4} x^{7} x^{m} e^{m} + A d^{3} m^{4} x^{7} x^{m} e^{m} + 86 \, B d^{3} m^{2} x^{9} x^{m} e^{m} + 54 \, B c d^{2} m^{3} x^{7} x^{m} e^{m} + 18 \, A d^{3} m^{3} x^{7} x^{m} e^{m} + 176 \, B d^{3} m x^{9} x^{m} e^{m} + 3 \, B c^{2} d m^{4} x^{5} x^{m} e^{m} + 3 \, A c d^{2} m^{4} x^{5} x^{m} e^{m} + 312 \, B c d^{2} m^{2} x^{7} x^{m} e^{m} + 104 \, A d^{3} m^{2} x^{7} x^{m} e^{m} + 105 \, B d^{3} x^{9} x^{m} e^{m} + 60 \, B c^{2} d m^{3} x^{5} x^{m} e^{m} + 60 \, A c d^{2} m^{3} x^{5} x^{m} e^{m} + 666 \, B c d^{2} m x^{7} x^{m} e^{m} + 222 \, A d^{3} m x^{7} x^{m} e^{m} + B c^{3} m^{4} x^{3} x^{m} e^{m} + 3 \, A c^{2} d m^{4} x^{3} x^{m} e^{m} + 390 \, B c^{2} d m^{2} x^{5} x^{m} e^{m} + 390 \, A c d^{2} m^{2} x^{5} x^{m} e^{m} + 405 \, B c d^{2} x^{7} x^{m} e^{m} + 135 \, A d^{3} x^{7} x^{m} e^{m} + 22 \, B c^{3} m^{3} x^{3} x^{m} e^{m} + 66 \, A c^{2} d m^{3} x^{3} x^{m} e^{m} + 900 \, B c^{2} d m x^{5} x^{m} e^{m} + 900 \, A c d^{2} m x^{5} x^{m} e^{m} + A c^{3} m^{4} x x^{m} e^{m} + 164 \, B c^{3} m^{2} x^{3} x^{m} e^{m} + 492 \, A c^{2} d m^{2} x^{3} x^{m} e^{m} + 567 \, B c^{2} d x^{5} x^{m} e^{m} + 567 \, A c d^{2} x^{5} x^{m} e^{m} + 24 \, A c^{3} m^{3} x x^{m} e^{m} + 458 \, B c^{3} m x^{3} x^{m} e^{m} + 1374 \, A c^{2} d m x^{3} x^{m} e^{m} + 206 \, A c^{3} m^{2} x x^{m} e^{m} + 315 \, B c^{3} x^{3} x^{m} e^{m} + 945 \, A c^{2} d x^{3} x^{m} e^{m} + 744 \, A c^{3} m x x^{m} e^{m} + 945 \, A c^{3} x x^{m} e^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.13, size = 280, normalized size = 2.31 \begin {gather*} {\left (e\,x\right )}^m\,\left (\frac {A\,c^3\,x\,\left (m^4+24\,m^3+206\,m^2+744\,m+945\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {B\,d^3\,x^9\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {c^2\,x^3\,\left (3\,A\,d+B\,c\right )\,\left (m^4+22\,m^3+164\,m^2+458\,m+315\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {d^2\,x^7\,\left (A\,d+3\,B\,c\right )\,\left (m^4+18\,m^3+104\,m^2+222\,m+135\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {3\,c\,d\,x^5\,\left (A\,d+B\,c\right )\,\left (m^4+20\,m^3+130\,m^2+300\,m+189\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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