Optimal. Leaf size=141 \[ -\frac {\left (b^2-4 a c\right )^3 (b d+2 c d x)^{1+m}}{128 c^4 d (1+m)}+\frac {3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3+m}}{128 c^4 d^3 (3+m)}-\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5+m}}{128 c^4 d^5 (5+m)}+\frac {(b d+2 c d x)^{7+m}}{128 c^4 d^7 (7+m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 141, 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 = {697}
\begin {gather*} -\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{m+5}}{128 c^4 d^5 (m+5)}+\frac {3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{m+3}}{128 c^4 d^3 (m+3)}-\frac {\left (b^2-4 a c\right )^3 (b d+2 c d x)^{m+1}}{128 c^4 d (m+1)}+\frac {(b d+2 c d x)^{m+7}}{128 c^4 d^7 (m+7)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 697
Rubi steps
\begin {align*} \int (b d+2 c d x)^m \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^3 (b d+2 c d x)^m}{64 c^3}+\frac {3 \left (-b^2+4 a c\right )^2 (b d+2 c d x)^{2+m}}{64 c^3 d^2}+\frac {3 \left (-b^2+4 a c\right ) (b d+2 c d x)^{4+m}}{64 c^3 d^4}+\frac {(b d+2 c d x)^{6+m}}{64 c^3 d^6}\right ) \, dx\\ &=-\frac {\left (b^2-4 a c\right )^3 (b d+2 c d x)^{1+m}}{128 c^4 d (1+m)}+\frac {3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3+m}}{128 c^4 d^3 (3+m)}-\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5+m}}{128 c^4 d^5 (5+m)}+\frac {(b d+2 c d x)^{7+m}}{128 c^4 d^7 (7+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.18, size = 103, normalized size = 0.73 \begin {gather*} \frac {(b+2 c x) (d (b+2 c x))^m \left (-\frac {\left (b^2-4 a c\right )^3}{1+m}+\frac {3 \left (b^2-4 a c\right )^2 (b+2 c x)^2}{3+m}-\frac {3 \left (b^2-4 a c\right ) (b+2 c x)^4}{5+m}+\frac {(b+2 c x)^6}{7+m}\right )}{128 c^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(625\) vs.
\(2(133)=266\).
time = 0.72, size = 626, normalized size = 4.44
method | result | size |
norman | \(\frac {c^{3} x^{7} {\mathrm e}^{m \ln \left (2 c d x +b d \right )}}{7+m}+\frac {7 b \,c^{2} x^{6} {\mathrm e}^{m \ln \left (2 c d x +b d \right )}}{2 \left (7+m \right )}+\frac {5 b \left (6 a c m +2 b^{2} m +42 a c +7 b^{2}\right ) x^{4} {\mathrm e}^{m \ln \left (2 c d x +b d \right )}}{4 \left (m^{2}+12 m +35\right )}+\frac {b \left (4 a^{3} c^{3} m^{3}+60 a^{3} c^{3} m^{2}-6 a^{2} b^{2} c^{2} m^{2}+284 a^{3} c^{3} m -72 a^{2} b^{2} c^{2} m +6 a \,b^{4} c m +420 a^{3} c^{3}-210 a^{2} b^{2} c^{2}+42 a \,b^{4} c -3 b^{6}\right ) {\mathrm e}^{m \ln \left (2 c d x +b d \right )}}{8 c^{4} \left (m^{4}+16 m^{3}+86 m^{2}+176 m +105\right )}+\frac {\left (6 a^{2} c^{2} m^{2}+12 a \,b^{2} c \,m^{2}+b^{4} m^{2}+72 a^{2} c^{2} m +114 a \,b^{2} c m +2 b^{4} m +210 a^{2} c^{2}+210 a c \,b^{2}\right ) x^{3} {\mathrm e}^{m \ln \left (2 c d x +b d \right )}}{2 c \left (m^{3}+15 m^{2}+71 m +105\right )}+\frac {3 c \left (2 a c m +3 b^{2} m +14 a c +14 b^{2}\right ) x^{5} {\mathrm e}^{m \ln \left (2 c d x +b d \right )}}{2 \left (m^{2}+12 m +35\right )}+\frac {\left (4 a^{3} c^{3} m^{3}+6 a^{2} b^{2} c^{2} m^{3}+60 a^{3} c^{3} m^{2}+72 a^{2} b^{2} c^{2} m^{2}-6 a \,b^{4} c \,m^{2}+284 a^{3} c^{3} m +210 a^{2} b^{2} c^{2} m -42 a \,b^{4} c m +3 b^{6} m +420 a^{3} c^{3}\right ) x \,{\mathrm e}^{m \ln \left (2 c d x +b d \right )}}{4 c^{3} \left (m^{4}+16 m^{3}+86 m^{2}+176 m +105\right )}+\frac {3 \left (6 a^{2} c^{2} m^{2}+2 a \,b^{2} c \,m^{2}+72 a^{2} c^{2} m +14 a \,b^{2} c m -b^{4} m +210 a^{2} c^{2}\right ) b \,x^{2} {\mathrm e}^{m \ln \left (2 c d x +b d \right )}}{4 c^{2} \left (m^{3}+15 m^{2}+71 m +105\right )}\) | \(626\) |
gosper | \(\frac {\left (2 c d x +b d \right )^{m} \left (4 c^{6} m^{3} x^{6}+12 b \,c^{5} m^{3} x^{5}+36 c^{6} m^{2} x^{6}+12 a \,c^{5} m^{3} x^{4}+12 b^{2} c^{4} m^{3} x^{4}+108 b \,c^{5} m^{2} x^{5}+92 c^{6} m \,x^{6}+24 a b \,c^{4} m^{3} x^{3}+132 a \,c^{5} m^{2} x^{4}+4 b^{3} c^{3} m^{3} x^{3}+102 b^{2} c^{4} m^{2} x^{4}+276 b \,c^{5} m \,x^{5}+60 c^{6} x^{6}+12 a^{2} c^{4} m^{3} x^{2}+12 a \,b^{2} c^{3} m^{3} x^{2}+264 a b \,c^{4} m^{2} x^{3}+372 a \,c^{5} m \,x^{4}+24 b^{3} c^{3} m^{2} x^{3}+252 b^{2} c^{4} m \,x^{4}+180 b \,c^{5} x^{5}+12 a^{2} b \,c^{3} m^{3} x +156 a^{2} c^{4} m^{2} x^{2}+120 a \,b^{2} c^{3} m^{2} x^{2}+744 a b \,c^{4} m \,x^{3}+252 a \,c^{5} x^{4}-6 b^{4} c^{2} m^{2} x^{2}+44 b^{3} c^{3} m \,x^{3}+162 b^{2} c^{4} x^{4}+4 a^{3} c^{3} m^{3}+156 a^{2} b \,c^{3} m^{2} x +564 a^{2} c^{4} m \,x^{2}-12 a \,b^{3} c^{2} m^{2} x +276 a \,b^{2} c^{3} m \,x^{2}+504 a b \,c^{4} x^{3}-18 b^{4} c^{2} m \,x^{2}+24 b^{3} x^{3} c^{3}+60 a^{3} c^{3} m^{2}-6 a^{2} b^{2} c^{2} m^{2}+564 a^{2} b \,c^{3} m x +420 a^{2} c^{4} x^{2}-96 a \,b^{3} c^{2} m x +168 a \,b^{2} c^{3} x^{2}+6 b^{5} c m x -12 b^{4} c^{2} x^{2}+284 a^{3} c^{3} m -72 a^{2} b^{2} c^{2} m +420 a^{2} b \,c^{3} x +6 a \,b^{4} c m -84 a \,b^{3} c^{2} x +6 b^{5} c x +420 a^{3} c^{3}-210 a^{2} b^{2} c^{2}+42 a \,b^{4} c -3 b^{6}\right ) \left (2 c x +b \right )}{8 c^{4} \left (m^{4}+16 m^{3}+86 m^{2}+176 m +105\right )}\) | \(653\) |
risch | \(\frac {\left (8 c^{7} m^{3} x^{7}+28 b \,c^{6} m^{3} x^{6}+72 c^{7} m^{2} x^{7}+24 a \,c^{6} m^{3} x^{5}+36 b^{2} c^{5} m^{3} x^{5}+252 b \,c^{6} m^{2} x^{6}+184 c^{7} m \,x^{7}+60 a b \,c^{5} m^{3} x^{4}+264 a \,c^{6} m^{2} x^{5}+20 b^{3} c^{4} m^{3} x^{4}+312 b^{2} c^{5} m^{2} x^{5}+644 b \,c^{6} m \,x^{6}+120 c^{7} x^{7}+24 a^{2} c^{5} m^{3} x^{3}+48 a \,b^{2} c^{4} m^{3} x^{3}+660 a b \,c^{5} m^{2} x^{4}+744 a \,c^{6} m \,x^{5}+4 b^{4} c^{3} m^{3} x^{3}+150 b^{3} c^{4} m^{2} x^{4}+780 b^{2} c^{5} m \,x^{5}+420 b \,c^{6} x^{6}+36 a^{2} b \,c^{4} m^{3} x^{2}+312 a^{2} c^{5} m^{2} x^{3}+12 a \,b^{3} c^{3} m^{3} x^{2}+504 a \,b^{2} c^{4} m^{2} x^{3}+1860 a b \,c^{5} m \,x^{4}+504 a \,c^{6} x^{5}+12 b^{4} c^{3} m^{2} x^{3}+340 b^{3} c^{4} m \,x^{4}+504 b^{2} c^{5} x^{5}+8 a^{3} c^{4} m^{3} x +12 a^{2} b^{2} c^{3} m^{3} x +468 a^{2} b \,c^{4} m^{2} x^{2}+1128 a^{2} c^{5} m \,x^{3}+96 a \,b^{3} c^{3} m^{2} x^{2}+1296 a \,b^{2} c^{4} m \,x^{3}+1260 a b \,c^{5} x^{4}-6 b^{5} c^{2} m^{2} x^{2}+8 b^{4} c^{3} m \,x^{3}+210 b^{3} c^{4} x^{4}+4 a^{3} b \,c^{3} m^{3}+120 a^{3} c^{4} m^{2} x +144 a^{2} b^{2} c^{3} m^{2} x +1692 a^{2} b \,c^{4} m \,x^{2}+840 a^{2} c^{5} x^{3}-12 a \,b^{4} c^{2} m^{2} x +84 a \,b^{3} c^{3} m \,x^{2}+840 a \,b^{2} c^{4} x^{3}-6 b^{5} c^{2} m \,x^{2}+60 a^{3} b \,c^{3} m^{2}+568 a^{3} c^{4} m x -6 a^{2} b^{3} c^{2} m^{2}+420 a^{2} b^{2} c^{3} m x +1260 a^{2} b \,c^{4} x^{2}-84 a \,b^{4} c^{2} m x +6 b^{6} c m x +284 a^{3} b \,c^{3} m +840 a^{3} c^{4} x -72 a^{2} b^{3} c^{2} m +6 a \,b^{5} c m +420 a^{3} b \,c^{3}-210 a^{2} b^{3} c^{2}+42 a \,b^{5} c -3 b^{7}\right ) \left (d \left (2 c x +b \right )\right )^{m}}{8 \left (5+m \right ) \left (7+m \right ) \left (3+m \right ) \left (1+m \right ) c^{4}}\) | \(799\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1299 vs.
\(2 (133) = 266\).
time = 0.35, size = 1299, normalized size = 9.21 \begin {gather*} \frac {3 \, {\left (4 \, c^{2} d^{m} {\left (m + 1\right )} x^{2} + 2 \, b c d^{m} m x - b^{2} d^{m}\right )} {\left (2 \, c x + b\right )}^{m} a^{2} b}{4 \, {\left (m^{2} + 3 \, m + 2\right )} c^{2}} + \frac {3 \, {\left (4 \, {\left (m^{2} + 3 \, m + 2\right )} c^{3} d^{m} x^{3} + 2 \, {\left (m^{2} + m\right )} b c^{2} d^{m} x^{2} - 2 \, b^{2} c d^{m} m x + b^{3} d^{m}\right )} {\left (2 \, c x + b\right )}^{m} a b^{2}}{4 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} c^{3}} + \frac {3 \, {\left (4 \, {\left (m^{2} + 3 \, m + 2\right )} c^{3} d^{m} x^{3} + 2 \, {\left (m^{2} + m\right )} b c^{2} d^{m} x^{2} - 2 \, b^{2} c d^{m} m x + b^{3} d^{m}\right )} {\left (2 \, c x + b\right )}^{m} a^{2}}{4 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} c^{2}} + \frac {{\left (2 \, c d x + b d\right )}^{m + 1} a^{3}}{2 \, c d {\left (m + 1\right )}} + \frac {{\left (8 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} c^{4} d^{m} x^{4} + 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} b c^{3} d^{m} x^{3} - 6 \, {\left (m^{2} + m\right )} b^{2} c^{2} d^{m} x^{2} + 6 \, b^{3} c d^{m} m x - 3 \, b^{4} d^{m}\right )} {\left (2 \, c x + b\right )}^{m} b^{3}}{8 \, {\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} c^{4}} + \frac {3 \, {\left (8 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} c^{4} d^{m} x^{4} + 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} b c^{3} d^{m} x^{3} - 6 \, {\left (m^{2} + m\right )} b^{2} c^{2} d^{m} x^{2} + 6 \, b^{3} c d^{m} m x - 3 \, b^{4} d^{m}\right )} {\left (2 \, c x + b\right )}^{m} a b}{4 \, {\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} c^{3}} + \frac {3 \, {\left (4 \, {\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} c^{5} d^{m} x^{5} + 2 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} b c^{4} d^{m} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} b^{2} c^{3} d^{m} x^{3} + 6 \, {\left (m^{2} + m\right )} b^{3} c^{2} d^{m} x^{2} - 6 \, b^{4} c d^{m} m x + 3 \, b^{5} d^{m}\right )} {\left (2 \, c x + b\right )}^{m} b^{2}}{4 \, {\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} c^{4}} + \frac {3 \, {\left (4 \, {\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} c^{5} d^{m} x^{5} + 2 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} b c^{4} d^{m} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} b^{2} c^{3} d^{m} x^{3} + 6 \, {\left (m^{2} + m\right )} b^{3} c^{2} d^{m} x^{2} - 6 \, b^{4} c d^{m} m x + 3 \, b^{5} d^{m}\right )} {\left (2 \, c x + b\right )}^{m} a}{4 \, {\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} c^{3}} + \frac {3 \, {\left (8 \, {\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} c^{6} d^{m} x^{6} + 4 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} b c^{5} d^{m} x^{5} - 10 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} b^{2} c^{4} d^{m} x^{4} + 20 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} b^{3} c^{3} d^{m} x^{3} - 30 \, {\left (m^{2} + m\right )} b^{4} c^{2} d^{m} x^{2} + 30 \, b^{5} c d^{m} m x - 15 \, b^{6} d^{m}\right )} {\left (2 \, c x + b\right )}^{m} b}{8 \, {\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} c^{4}} + \frac {{\left (8 \, {\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} c^{7} d^{m} x^{7} + 4 \, {\left (m^{6} + 15 \, m^{5} + 85 \, m^{4} + 225 \, m^{3} + 274 \, m^{2} + 120 \, m\right )} b c^{6} d^{m} x^{6} - 12 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} b^{2} c^{5} d^{m} x^{5} + 30 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} b^{3} c^{4} d^{m} x^{4} - 60 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} b^{4} c^{3} d^{m} x^{3} + 90 \, {\left (m^{2} + m\right )} b^{5} c^{2} d^{m} x^{2} - 90 \, b^{6} c d^{m} m x + 45 \, b^{7} d^{m}\right )} {\left (2 \, c x + b\right )}^{m}}{8 \, {\left (m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040\right )} c^{4}} \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. 691 vs.
\(2 (133) = 266\).
time = 1.74, size = 691, normalized size = 4.90 \begin {gather*} \frac {{\left (4 \, a^{3} b c^{3} m^{3} + 8 \, {\left (c^{7} m^{3} + 9 \, c^{7} m^{2} + 23 \, c^{7} m + 15 \, c^{7}\right )} x^{7} - 3 \, b^{7} + 42 \, a b^{5} c - 210 \, a^{2} b^{3} c^{2} + 420 \, a^{3} b c^{3} + 28 \, {\left (b c^{6} m^{3} + 9 \, b c^{6} m^{2} + 23 \, b c^{6} m + 15 \, b c^{6}\right )} x^{6} + 12 \, {\left (42 \, b^{2} c^{5} + 42 \, a c^{6} + {\left (3 \, b^{2} c^{5} + 2 \, a c^{6}\right )} m^{3} + 2 \, {\left (13 \, b^{2} c^{5} + 11 \, a c^{6}\right )} m^{2} + {\left (65 \, b^{2} c^{5} + 62 \, a c^{6}\right )} m\right )} x^{5} + 10 \, {\left (21 \, b^{3} c^{4} + 126 \, a b c^{5} + 2 \, {\left (b^{3} c^{4} + 3 \, a b c^{5}\right )} m^{3} + 3 \, {\left (5 \, b^{3} c^{4} + 22 \, a b c^{5}\right )} m^{2} + 2 \, {\left (17 \, b^{3} c^{4} + 93 \, a b c^{5}\right )} m\right )} x^{4} + 4 \, {\left (210 \, a b^{2} c^{4} + 210 \, a^{2} c^{5} + {\left (b^{4} c^{3} + 12 \, a b^{2} c^{4} + 6 \, a^{2} c^{5}\right )} m^{3} + 3 \, {\left (b^{4} c^{3} + 42 \, a b^{2} c^{4} + 26 \, a^{2} c^{5}\right )} m^{2} + 2 \, {\left (b^{4} c^{3} + 162 \, a b^{2} c^{4} + 141 \, a^{2} c^{5}\right )} m\right )} x^{3} - 6 \, {\left (a^{2} b^{3} c^{2} - 10 \, a^{3} b c^{3}\right )} m^{2} + 6 \, {\left (210 \, a^{2} b c^{4} + 2 \, {\left (a b^{3} c^{3} + 3 \, a^{2} b c^{4}\right )} m^{3} - {\left (b^{5} c^{2} - 16 \, a b^{3} c^{3} - 78 \, a^{2} b c^{4}\right )} m^{2} - {\left (b^{5} c^{2} - 14 \, a b^{3} c^{3} - 282 \, a^{2} b c^{4}\right )} m\right )} x^{2} + 2 \, {\left (3 \, a b^{5} c - 36 \, a^{2} b^{3} c^{2} + 142 \, a^{3} b c^{3}\right )} m + 2 \, {\left (420 \, a^{3} c^{4} + 2 \, {\left (3 \, a^{2} b^{2} c^{3} + 2 \, a^{3} c^{4}\right )} m^{3} - 6 \, {\left (a b^{4} c^{2} - 12 \, a^{2} b^{2} c^{3} - 10 \, a^{3} c^{4}\right )} m^{2} + {\left (3 \, b^{6} c - 42 \, a b^{4} c^{2} + 210 \, a^{2} b^{2} c^{3} + 284 \, a^{3} c^{4}\right )} m\right )} x\right )} {\left (2 \, c d x + b d\right )}^{m}}{8 \, {\left (c^{4} m^{4} + 16 \, c^{4} m^{3} + 86 \, c^{4} m^{2} + 176 \, c^{4} m + 105 \, c^{4}\right )}} \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. 9491 vs.
\(2 (133) = 266\).
time = 2.09, size = 9491, normalized size = 67.31 \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. 1506 vs.
\(2 (133) = 266\).
time = 1.36, size = 1506, normalized size = 10.68 \begin {gather*} \frac {8 \, {\left (2 \, c d x + b d\right )}^{m} c^{7} m^{3} x^{7} + 28 \, {\left (2 \, c d x + b d\right )}^{m} b c^{6} m^{3} x^{6} + 72 \, {\left (2 \, c d x + b d\right )}^{m} c^{7} m^{2} x^{7} + 36 \, {\left (2 \, c d x + b d\right )}^{m} b^{2} c^{5} m^{3} x^{5} + 24 \, {\left (2 \, c d x + b d\right )}^{m} a c^{6} m^{3} x^{5} + 252 \, {\left (2 \, c d x + b d\right )}^{m} b c^{6} m^{2} x^{6} + 184 \, {\left (2 \, c d x + b d\right )}^{m} c^{7} m x^{7} + 20 \, {\left (2 \, c d x + b d\right )}^{m} b^{3} c^{4} m^{3} x^{4} + 60 \, {\left (2 \, c d x + b d\right )}^{m} a b c^{5} m^{3} x^{4} + 312 \, {\left (2 \, c d x + b d\right )}^{m} b^{2} c^{5} m^{2} x^{5} + 264 \, {\left (2 \, c d x + b d\right )}^{m} a c^{6} m^{2} x^{5} + 644 \, {\left (2 \, c d x + b d\right )}^{m} b c^{6} m x^{6} + 120 \, {\left (2 \, c d x + b d\right )}^{m} c^{7} x^{7} + 4 \, {\left (2 \, c d x + b d\right )}^{m} b^{4} c^{3} m^{3} x^{3} + 48 \, {\left (2 \, c d x + b d\right )}^{m} a b^{2} c^{4} m^{3} x^{3} + 24 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} c^{5} m^{3} x^{3} + 150 \, {\left (2 \, c d x + b d\right )}^{m} b^{3} c^{4} m^{2} x^{4} + 660 \, {\left (2 \, c d x + b d\right )}^{m} a b c^{5} m^{2} x^{4} + 780 \, {\left (2 \, c d x + b d\right )}^{m} b^{2} c^{5} m x^{5} + 744 \, {\left (2 \, c d x + b d\right )}^{m} a c^{6} m x^{5} + 420 \, {\left (2 \, c d x + b d\right )}^{m} b c^{6} x^{6} + 12 \, {\left (2 \, c d x + b d\right )}^{m} a b^{3} c^{3} m^{3} x^{2} + 36 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} b c^{4} m^{3} x^{2} + 12 \, {\left (2 \, c d x + b d\right )}^{m} b^{4} c^{3} m^{2} x^{3} + 504 \, {\left (2 \, c d x + b d\right )}^{m} a b^{2} c^{4} m^{2} x^{3} + 312 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} c^{5} m^{2} x^{3} + 340 \, {\left (2 \, c d x + b d\right )}^{m} b^{3} c^{4} m x^{4} + 1860 \, {\left (2 \, c d x + b d\right )}^{m} a b c^{5} m x^{4} + 504 \, {\left (2 \, c d x + b d\right )}^{m} b^{2} c^{5} x^{5} + 504 \, {\left (2 \, c d x + b d\right )}^{m} a c^{6} x^{5} + 12 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} b^{2} c^{3} m^{3} x + 8 \, {\left (2 \, c d x + b d\right )}^{m} a^{3} c^{4} m^{3} x - 6 \, {\left (2 \, c d x + b d\right )}^{m} b^{5} c^{2} m^{2} x^{2} + 96 \, {\left (2 \, c d x + b d\right )}^{m} a b^{3} c^{3} m^{2} x^{2} + 468 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} b c^{4} m^{2} x^{2} + 8 \, {\left (2 \, c d x + b d\right )}^{m} b^{4} c^{3} m x^{3} + 1296 \, {\left (2 \, c d x + b d\right )}^{m} a b^{2} c^{4} m x^{3} + 1128 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} c^{5} m x^{3} + 210 \, {\left (2 \, c d x + b d\right )}^{m} b^{3} c^{4} x^{4} + 1260 \, {\left (2 \, c d x + b d\right )}^{m} a b c^{5} x^{4} + 4 \, {\left (2 \, c d x + b d\right )}^{m} a^{3} b c^{3} m^{3} - 12 \, {\left (2 \, c d x + b d\right )}^{m} a b^{4} c^{2} m^{2} x + 144 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} b^{2} c^{3} m^{2} x + 120 \, {\left (2 \, c d x + b d\right )}^{m} a^{3} c^{4} m^{2} x - 6 \, {\left (2 \, c d x + b d\right )}^{m} b^{5} c^{2} m x^{2} + 84 \, {\left (2 \, c d x + b d\right )}^{m} a b^{3} c^{3} m x^{2} + 1692 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} b c^{4} m x^{2} + 840 \, {\left (2 \, c d x + b d\right )}^{m} a b^{2} c^{4} x^{3} + 840 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} c^{5} x^{3} - 6 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} b^{3} c^{2} m^{2} + 60 \, {\left (2 \, c d x + b d\right )}^{m} a^{3} b c^{3} m^{2} + 6 \, {\left (2 \, c d x + b d\right )}^{m} b^{6} c m x - 84 \, {\left (2 \, c d x + b d\right )}^{m} a b^{4} c^{2} m x + 420 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} b^{2} c^{3} m x + 568 \, {\left (2 \, c d x + b d\right )}^{m} a^{3} c^{4} m x + 1260 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} b c^{4} x^{2} + 6 \, {\left (2 \, c d x + b d\right )}^{m} a b^{5} c m - 72 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} b^{3} c^{2} m + 284 \, {\left (2 \, c d x + b d\right )}^{m} a^{3} b c^{3} m + 840 \, {\left (2 \, c d x + b d\right )}^{m} a^{3} c^{4} x - 3 \, {\left (2 \, c d x + b d\right )}^{m} b^{7} + 42 \, {\left (2 \, c d x + b d\right )}^{m} a b^{5} c - 210 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} b^{3} c^{2} + 420 \, {\left (2 \, c d x + b d\right )}^{m} a^{3} b c^{3}}{8 \, {\left (c^{4} m^{4} + 16 \, c^{4} m^{3} + 86 \, c^{4} m^{2} + 176 \, c^{4} m + 105 \, c^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.11, size = 655, normalized size = 4.65 \begin {gather*} {\left (b\,d+2\,c\,d\,x\right )}^m\,\left (\frac {4\,a^3\,b\,c^3\,m^3+60\,a^3\,b\,c^3\,m^2+284\,a^3\,b\,c^3\,m+420\,a^3\,b\,c^3-6\,a^2\,b^3\,c^2\,m^2-72\,a^2\,b^3\,c^2\,m-210\,a^2\,b^3\,c^2+6\,a\,b^5\,c\,m+42\,a\,b^5\,c-3\,b^7}{8\,c^4\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}+\frac {c^3\,x^7\,\left (m^3+9\,m^2+23\,m+15\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {x\,\left (8\,a^3\,c^4\,m^3+120\,a^3\,c^4\,m^2+568\,a^3\,c^4\,m+840\,a^3\,c^4+12\,a^2\,b^2\,c^3\,m^3+144\,a^2\,b^2\,c^3\,m^2+420\,a^2\,b^2\,c^3\,m-12\,a\,b^4\,c^2\,m^2-84\,a\,b^4\,c^2\,m+6\,b^6\,c\,m\right )}{8\,c^4\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}+\frac {5\,b\,x^4\,\left (m^2+4\,m+3\right )\,\left (42\,a\,c+2\,b^2\,m+7\,b^2+6\,a\,c\,m\right )}{4\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}+\frac {3\,c\,x^5\,\left (m^2+4\,m+3\right )\,\left (14\,a\,c+3\,b^2\,m+14\,b^2+2\,a\,c\,m\right )}{2\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}+\frac {x^3\,\left (m+1\right )\,\left (6\,a^2\,c^2\,m^2+72\,a^2\,c^2\,m+210\,a^2\,c^2+12\,a\,b^2\,c\,m^2+114\,a\,b^2\,c\,m+210\,a\,b^2\,c+b^4\,m^2+2\,b^4\,m\right )}{2\,c\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}+\frac {7\,b\,c^2\,x^6\,\left (m^3+9\,m^2+23\,m+15\right )}{2\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}+\frac {3\,b\,x^2\,\left (m+1\right )\,\left (6\,a^2\,c^2\,m^2+72\,a^2\,c^2\,m+210\,a^2\,c^2+2\,a\,b^2\,c\,m^2+14\,a\,b^2\,c\,m-b^4\,m\right )}{4\,c^2\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________