Optimal. Leaf size=143 \[ \frac {a^5 d^2 (e x)^{1+m}}{e (1+m)}+\frac {a^4 b d^2 (e x)^{2+m}}{e^2 (2+m)}-\frac {2 a^3 b^2 d^2 (e x)^{3+m}}{e^3 (3+m)}-\frac {2 a^2 b^3 d^2 (e x)^{4+m}}{e^4 (4+m)}+\frac {a b^4 d^2 (e x)^{5+m}}{e^5 (5+m)}+\frac {b^5 d^2 (e x)^{6+m}}{e^6 (6+m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {90}
\begin {gather*} \frac {a^5 d^2 (e x)^{m+1}}{e (m+1)}+\frac {a^4 b d^2 (e x)^{m+2}}{e^2 (m+2)}-\frac {2 a^3 b^2 d^2 (e x)^{m+3}}{e^3 (m+3)}-\frac {2 a^2 b^3 d^2 (e x)^{m+4}}{e^4 (m+4)}+\frac {a b^4 d^2 (e x)^{m+5}}{e^5 (m+5)}+\frac {b^5 d^2 (e x)^{m+6}}{e^6 (m+6)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 90
Rubi steps
\begin {align*} \int (e x)^m (a+b x)^3 (a d-b d x)^2 \, dx &=\int \left (a^5 d^2 (e x)^m+\frac {a^4 b d^2 (e x)^{1+m}}{e}-\frac {2 a^3 b^2 d^2 (e x)^{2+m}}{e^2}-\frac {2 a^2 b^3 d^2 (e x)^{3+m}}{e^3}+\frac {a b^4 d^2 (e x)^{4+m}}{e^4}+\frac {b^5 d^2 (e x)^{5+m}}{e^5}\right ) \, dx\\ &=\frac {a^5 d^2 (e x)^{1+m}}{e (1+m)}+\frac {a^4 b d^2 (e x)^{2+m}}{e^2 (2+m)}-\frac {2 a^3 b^2 d^2 (e x)^{3+m}}{e^3 (3+m)}-\frac {2 a^2 b^3 d^2 (e x)^{4+m}}{e^4 (4+m)}+\frac {a b^4 d^2 (e x)^{5+m}}{e^5 (5+m)}+\frac {b^5 d^2 (e x)^{6+m}}{e^6 (6+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.08, size = 88, normalized size = 0.62 \begin {gather*} d^2 x (e x)^m \left (\frac {a^5}{1+m}+\frac {a^4 b x}{2+m}-\frac {2 a^3 b^2 x^2}{3+m}-\frac {2 a^2 b^3 x^3}{4+m}+\frac {a b^4 x^4}{5+m}+\frac {b^5 x^5}{6+m}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.06, size = 142, normalized size = 0.99
method | result | size |
norman | \(\frac {a^{5} d^{2} x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}+\frac {b^{5} d^{2} x^{6} {\mathrm e}^{m \ln \left (e x \right )}}{6+m}+\frac {a \,b^{4} d^{2} x^{5} {\mathrm e}^{m \ln \left (e x \right )}}{5+m}+\frac {a^{4} b \,d^{2} x^{2} {\mathrm e}^{m \ln \left (e x \right )}}{2+m}-\frac {2 a^{2} b^{3} d^{2} x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}-\frac {2 a^{3} b^{2} d^{2} x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}\) | \(142\) |
gosper | \(\frac {d^{2} \left (e x \right )^{m} \left (b^{5} m^{5} x^{5}+a \,b^{4} m^{5} x^{4}+15 b^{5} m^{4} x^{5}-2 a^{2} b^{3} m^{5} x^{3}+16 a \,b^{4} m^{4} x^{4}+85 b^{5} m^{3} x^{5}-2 a^{3} b^{2} m^{5} x^{2}-34 a^{2} b^{3} m^{4} x^{3}+95 a \,b^{4} m^{3} x^{4}+225 b^{5} m^{2} x^{5}+a^{4} b \,m^{5} x -36 a^{3} b^{2} m^{4} x^{2}-214 a^{2} b^{3} m^{3} x^{3}+260 a \,b^{4} m^{2} x^{4}+274 m \,x^{5} b^{5}+a^{5} m^{5}+19 a^{4} b \,m^{4} x -242 a^{3} b^{2} m^{3} x^{2}-614 a^{2} b^{3} m^{2} x^{3}+324 a \,b^{4} m \,x^{4}+120 b^{5} x^{5}+20 a^{5} m^{4}+137 a^{4} b \,m^{3} x -744 a^{3} b^{2} m^{2} x^{2}-792 a^{2} b^{3} m \,x^{3}+144 a \,b^{4} x^{4}+155 a^{5} m^{3}+461 a^{4} b \,m^{2} x -1016 a^{3} b^{2} m \,x^{2}-360 a^{2} b^{3} x^{3}+580 a^{5} m^{2}+702 a^{4} b m x -480 a^{3} b^{2} x^{2}+1044 a^{5} m +360 a^{4} b x +720 a^{5}\right ) x}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(422\) |
risch | \(\frac {d^{2} \left (e x \right )^{m} \left (b^{5} m^{5} x^{5}+a \,b^{4} m^{5} x^{4}+15 b^{5} m^{4} x^{5}-2 a^{2} b^{3} m^{5} x^{3}+16 a \,b^{4} m^{4} x^{4}+85 b^{5} m^{3} x^{5}-2 a^{3} b^{2} m^{5} x^{2}-34 a^{2} b^{3} m^{4} x^{3}+95 a \,b^{4} m^{3} x^{4}+225 b^{5} m^{2} x^{5}+a^{4} b \,m^{5} x -36 a^{3} b^{2} m^{4} x^{2}-214 a^{2} b^{3} m^{3} x^{3}+260 a \,b^{4} m^{2} x^{4}+274 m \,x^{5} b^{5}+a^{5} m^{5}+19 a^{4} b \,m^{4} x -242 a^{3} b^{2} m^{3} x^{2}-614 a^{2} b^{3} m^{2} x^{3}+324 a \,b^{4} m \,x^{4}+120 b^{5} x^{5}+20 a^{5} m^{4}+137 a^{4} b \,m^{3} x -744 a^{3} b^{2} m^{2} x^{2}-792 a^{2} b^{3} m \,x^{3}+144 a \,b^{4} x^{4}+155 a^{5} m^{3}+461 a^{4} b \,m^{2} x -1016 a^{3} b^{2} m \,x^{2}-360 a^{2} b^{3} x^{3}+580 a^{5} m^{2}+702 a^{4} b m x -480 a^{3} b^{2} x^{2}+1044 a^{5} m +360 a^{4} b x +720 a^{5}\right ) x}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(422\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 143, normalized size = 1.00 \begin {gather*} \frac {b^{5} d^{2} x^{6} e^{\left (m \log \left (x\right ) + m\right )}}{m + 6} + \frac {a b^{4} d^{2} x^{5} e^{\left (m \log \left (x\right ) + m\right )}}{m + 5} - \frac {2 \, a^{2} b^{3} d^{2} x^{4} e^{\left (m \log \left (x\right ) + m\right )}}{m + 4} - \frac {2 \, a^{3} b^{2} d^{2} x^{3} e^{\left (m \log \left (x\right ) + m\right )}}{m + 3} + \frac {a^{4} b d^{2} x^{2} e^{\left (m \log \left (x\right ) + m\right )}}{m + 2} + \frac {\left (x e\right )^{m + 1} a^{5} d^{2} e^{\left (-1\right )}}{m + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 476 vs.
\(2 (143) = 286\).
time = 1.13, size = 476, normalized size = 3.33 \begin {gather*} \frac {{\left ({\left (b^{5} d^{2} m^{5} + 15 \, b^{5} d^{2} m^{4} + 85 \, b^{5} d^{2} m^{3} + 225 \, b^{5} d^{2} m^{2} + 274 \, b^{5} d^{2} m + 120 \, b^{5} d^{2}\right )} x^{6} + {\left (a b^{4} d^{2} m^{5} + 16 \, a b^{4} d^{2} m^{4} + 95 \, a b^{4} d^{2} m^{3} + 260 \, a b^{4} d^{2} m^{2} + 324 \, a b^{4} d^{2} m + 144 \, a b^{4} d^{2}\right )} x^{5} - 2 \, {\left (a^{2} b^{3} d^{2} m^{5} + 17 \, a^{2} b^{3} d^{2} m^{4} + 107 \, a^{2} b^{3} d^{2} m^{3} + 307 \, a^{2} b^{3} d^{2} m^{2} + 396 \, a^{2} b^{3} d^{2} m + 180 \, a^{2} b^{3} d^{2}\right )} x^{4} - 2 \, {\left (a^{3} b^{2} d^{2} m^{5} + 18 \, a^{3} b^{2} d^{2} m^{4} + 121 \, a^{3} b^{2} d^{2} m^{3} + 372 \, a^{3} b^{2} d^{2} m^{2} + 508 \, a^{3} b^{2} d^{2} m + 240 \, a^{3} b^{2} d^{2}\right )} x^{3} + {\left (a^{4} b d^{2} m^{5} + 19 \, a^{4} b d^{2} m^{4} + 137 \, a^{4} b d^{2} m^{3} + 461 \, a^{4} b d^{2} m^{2} + 702 \, a^{4} b d^{2} m + 360 \, a^{4} b d^{2}\right )} x^{2} + {\left (a^{5} d^{2} m^{5} + 20 \, a^{5} d^{2} m^{4} + 155 \, a^{5} d^{2} m^{3} + 580 \, a^{5} d^{2} m^{2} + 1044 \, a^{5} d^{2} m + 720 \, a^{5} d^{2}\right )} x\right )} \left (x e\right )^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2259 vs.
\(2 (133) = 266\).
time = 0.52, size = 2259, normalized size = 15.80 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 718 vs.
\(2 (143) = 286\).
time = 1.88, size = 718, normalized size = 5.02 \begin {gather*} \frac {b^{5} d^{2} m^{5} x^{6} x^{m} e^{m} + a b^{4} d^{2} m^{5} x^{5} x^{m} e^{m} + 15 \, b^{5} d^{2} m^{4} x^{6} x^{m} e^{m} - 2 \, a^{2} b^{3} d^{2} m^{5} x^{4} x^{m} e^{m} + 16 \, a b^{4} d^{2} m^{4} x^{5} x^{m} e^{m} + 85 \, b^{5} d^{2} m^{3} x^{6} x^{m} e^{m} - 2 \, a^{3} b^{2} d^{2} m^{5} x^{3} x^{m} e^{m} - 34 \, a^{2} b^{3} d^{2} m^{4} x^{4} x^{m} e^{m} + 95 \, a b^{4} d^{2} m^{3} x^{5} x^{m} e^{m} + 225 \, b^{5} d^{2} m^{2} x^{6} x^{m} e^{m} + a^{4} b d^{2} m^{5} x^{2} x^{m} e^{m} - 36 \, a^{3} b^{2} d^{2} m^{4} x^{3} x^{m} e^{m} - 214 \, a^{2} b^{3} d^{2} m^{3} x^{4} x^{m} e^{m} + 260 \, a b^{4} d^{2} m^{2} x^{5} x^{m} e^{m} + 274 \, b^{5} d^{2} m x^{6} x^{m} e^{m} + a^{5} d^{2} m^{5} x x^{m} e^{m} + 19 \, a^{4} b d^{2} m^{4} x^{2} x^{m} e^{m} - 242 \, a^{3} b^{2} d^{2} m^{3} x^{3} x^{m} e^{m} - 614 \, a^{2} b^{3} d^{2} m^{2} x^{4} x^{m} e^{m} + 324 \, a b^{4} d^{2} m x^{5} x^{m} e^{m} + 120 \, b^{5} d^{2} x^{6} x^{m} e^{m} + 20 \, a^{5} d^{2} m^{4} x x^{m} e^{m} + 137 \, a^{4} b d^{2} m^{3} x^{2} x^{m} e^{m} - 744 \, a^{3} b^{2} d^{2} m^{2} x^{3} x^{m} e^{m} - 792 \, a^{2} b^{3} d^{2} m x^{4} x^{m} e^{m} + 144 \, a b^{4} d^{2} x^{5} x^{m} e^{m} + 155 \, a^{5} d^{2} m^{3} x x^{m} e^{m} + 461 \, a^{4} b d^{2} m^{2} x^{2} x^{m} e^{m} - 1016 \, a^{3} b^{2} d^{2} m x^{3} x^{m} e^{m} - 360 \, a^{2} b^{3} d^{2} x^{4} x^{m} e^{m} + 580 \, a^{5} d^{2} m^{2} x x^{m} e^{m} + 702 \, a^{4} b d^{2} m x^{2} x^{m} e^{m} - 480 \, a^{3} b^{2} d^{2} x^{3} x^{m} e^{m} + 1044 \, a^{5} d^{2} m x x^{m} e^{m} + 360 \, a^{4} b d^{2} x^{2} x^{m} e^{m} + 720 \, a^{5} d^{2} x x^{m} e^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.61, size = 393, normalized size = 2.75 \begin {gather*} {\left (e\,x\right )}^m\,\left (\frac {b^5\,d^2\,x^6\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a^5\,d^2\,x\,\left (m^5+20\,m^4+155\,m^3+580\,m^2+1044\,m+720\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a\,b^4\,d^2\,x^5\,\left (m^5+16\,m^4+95\,m^3+260\,m^2+324\,m+144\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a^4\,b\,d^2\,x^2\,\left (m^5+19\,m^4+137\,m^3+461\,m^2+702\,m+360\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}-\frac {2\,a^2\,b^3\,d^2\,x^4\,\left (m^5+17\,m^4+107\,m^3+307\,m^2+396\,m+180\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}-\frac {2\,a^3\,b^2\,d^2\,x^3\,\left (m^5+18\,m^4+121\,m^3+372\,m^2+508\,m+240\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________