Optimal. Leaf size=137 \[ \frac {d^2 (c d-b e)^2 (d+e x)^5}{5 e^5}-\frac {d (c d-b e) (2 c d-b e) (d+e x)^6}{3 e^5}+\frac {\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^7}{7 e^5}-\frac {c (2 c d-b e) (d+e x)^8}{4 e^5}+\frac {c^2 (d+e x)^9}{9 e^5} \]
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Rubi [A]
time = 0.10, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712}
\begin {gather*} \frac {(d+e x)^7 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{7 e^5}+\frac {d^2 (d+e x)^5 (c d-b e)^2}{5 e^5}-\frac {c (d+e x)^8 (2 c d-b e)}{4 e^5}-\frac {d (d+e x)^6 (c d-b e) (2 c d-b e)}{3 e^5}+\frac {c^2 (d+e x)^9}{9 e^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rubi steps
\begin {align*} \int (d+e x)^4 \left (b x+c x^2\right )^2 \, dx &=\int \left (\frac {d^2 (c d-b e)^2 (d+e x)^4}{e^4}+\frac {2 d (c d-b e) (-2 c d+b e) (d+e x)^5}{e^4}+\frac {\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^6}{e^4}-\frac {2 c (2 c d-b e) (d+e x)^7}{e^4}+\frac {c^2 (d+e x)^8}{e^4}\right ) \, dx\\ &=\frac {d^2 (c d-b e)^2 (d+e x)^5}{5 e^5}-\frac {d (c d-b e) (2 c d-b e) (d+e x)^6}{3 e^5}+\frac {\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^7}{7 e^5}-\frac {c (2 c d-b e) (d+e x)^8}{4 e^5}+\frac {c^2 (d+e x)^9}{9 e^5}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 159, normalized size = 1.16 \begin {gather*} \frac {1}{3} b^2 d^4 x^3+\frac {1}{2} b d^3 (c d+2 b e) x^4+\frac {1}{5} d^2 \left (c^2 d^2+8 b c d e+6 b^2 e^2\right ) x^5+\frac {2}{3} d e \left (c^2 d^2+3 b c d e+b^2 e^2\right ) x^6+\frac {1}{7} e^2 \left (6 c^2 d^2+8 b c d e+b^2 e^2\right ) x^7+\frac {1}{4} c e^3 (2 c d+b e) x^8+\frac {1}{9} c^2 e^4 x^9 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 166, normalized size = 1.21
method | result | size |
norman | \(\frac {e^{4} c^{2} x^{9}}{9}+\left (\frac {1}{4} e^{4} b c +\frac {1}{2} d \,e^{3} c^{2}\right ) x^{8}+\left (\frac {1}{7} e^{4} b^{2}+\frac {8}{7} d \,e^{3} b c +\frac {6}{7} d^{2} e^{2} c^{2}\right ) x^{7}+\left (\frac {2}{3} d \,e^{3} b^{2}+2 d^{2} e^{2} b c +\frac {2}{3} d^{3} e \,c^{2}\right ) x^{6}+\left (\frac {6}{5} d^{2} e^{2} b^{2}+\frac {8}{5} d^{3} e b c +\frac {1}{5} c^{2} d^{4}\right ) x^{5}+\left (d^{3} e \,b^{2}+\frac {1}{2} d^{4} b c \right ) x^{4}+\frac {b^{2} d^{4} x^{3}}{3}\) | \(162\) |
default | \(\frac {e^{4} c^{2} x^{9}}{9}+\frac {\left (2 e^{4} b c +4 d \,e^{3} c^{2}\right ) x^{8}}{8}+\frac {\left (e^{4} b^{2}+8 d \,e^{3} b c +6 d^{2} e^{2} c^{2}\right ) x^{7}}{7}+\frac {\left (4 d \,e^{3} b^{2}+12 d^{2} e^{2} b c +4 d^{3} e \,c^{2}\right ) x^{6}}{6}+\frac {\left (6 d^{2} e^{2} b^{2}+8 d^{3} e b c +c^{2} d^{4}\right ) x^{5}}{5}+\frac {\left (4 d^{3} e \,b^{2}+2 d^{4} b c \right ) x^{4}}{4}+\frac {b^{2} d^{4} x^{3}}{3}\) | \(166\) |
gosper | \(\frac {x^{3} \left (140 e^{4} c^{2} x^{6}+315 x^{5} e^{4} b c +630 x^{5} d \,e^{3} c^{2}+180 x^{4} e^{4} b^{2}+1440 x^{4} d \,e^{3} b c +1080 x^{4} d^{2} e^{2} c^{2}+840 x^{3} d \,e^{3} b^{2}+2520 x^{3} d^{2} e^{2} b c +840 x^{3} d^{3} e \,c^{2}+1512 x^{2} d^{2} e^{2} b^{2}+2016 x^{2} d^{3} e b c +252 x^{2} c^{2} d^{4}+1260 x \,d^{3} e \,b^{2}+630 x \,d^{4} b c +420 b^{2} d^{4}\right )}{1260}\) | \(175\) |
risch | \(\frac {1}{9} e^{4} c^{2} x^{9}+\frac {1}{4} x^{8} e^{4} b c +\frac {1}{2} x^{8} d \,e^{3} c^{2}+\frac {1}{7} x^{7} e^{4} b^{2}+\frac {8}{7} x^{7} d \,e^{3} b c +\frac {6}{7} x^{7} d^{2} e^{2} c^{2}+\frac {2}{3} x^{6} d \,e^{3} b^{2}+2 x^{6} d^{2} e^{2} b c +\frac {2}{3} x^{6} d^{3} e \,c^{2}+\frac {6}{5} x^{5} d^{2} e^{2} b^{2}+\frac {8}{5} x^{5} d^{3} e b c +\frac {1}{5} x^{5} c^{2} d^{4}+x^{4} d^{3} e \,b^{2}+\frac {1}{2} x^{4} d^{4} b c +\frac {1}{3} b^{2} d^{4} x^{3}\) | \(176\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 155, normalized size = 1.13 \begin {gather*} \frac {1}{9} \, c^{2} x^{9} e^{4} + \frac {1}{3} \, b^{2} d^{4} x^{3} + \frac {1}{4} \, {\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, c^{2} d^{2} e^{2} + 8 \, b c d e^{3} + b^{2} e^{4}\right )} x^{7} + \frac {2}{3} \, {\left (c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{4} + 8 \, b c d^{3} e + 6 \, b^{2} d^{2} e^{2}\right )} x^{5} + \frac {1}{2} \, {\left (b c d^{4} + 2 \, b^{2} d^{3} e\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.60, size = 166, normalized size = 1.21 \begin {gather*} \frac {1}{5} \, c^{2} d^{4} x^{5} + \frac {1}{2} \, b c d^{4} x^{4} + \frac {1}{3} \, b^{2} d^{4} x^{3} + \frac {1}{252} \, {\left (28 \, c^{2} x^{9} + 63 \, b c x^{8} + 36 \, b^{2} x^{7}\right )} e^{4} + \frac {1}{42} \, {\left (21 \, c^{2} d x^{8} + 48 \, b c d x^{7} + 28 \, b^{2} d x^{6}\right )} e^{3} + \frac {2}{35} \, {\left (15 \, c^{2} d^{2} x^{7} + 35 \, b c d^{2} x^{6} + 21 \, b^{2} d^{2} x^{5}\right )} e^{2} + \frac {1}{15} \, {\left (10 \, c^{2} d^{3} x^{6} + 24 \, b c d^{3} x^{5} + 15 \, b^{2} d^{3} x^{4}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 178, normalized size = 1.30 \begin {gather*} \frac {b^{2} d^{4} x^{3}}{3} + \frac {c^{2} e^{4} x^{9}}{9} + x^{8} \left (\frac {b c e^{4}}{4} + \frac {c^{2} d e^{3}}{2}\right ) + x^{7} \left (\frac {b^{2} e^{4}}{7} + \frac {8 b c d e^{3}}{7} + \frac {6 c^{2} d^{2} e^{2}}{7}\right ) + x^{6} \cdot \left (\frac {2 b^{2} d e^{3}}{3} + 2 b c d^{2} e^{2} + \frac {2 c^{2} d^{3} e}{3}\right ) + x^{5} \cdot \left (\frac {6 b^{2} d^{2} e^{2}}{5} + \frac {8 b c d^{3} e}{5} + \frac {c^{2} d^{4}}{5}\right ) + x^{4} \left (b^{2} d^{3} e + \frac {b c d^{4}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.01, size = 169, normalized size = 1.23 \begin {gather*} \frac {1}{9} \, c^{2} x^{9} e^{4} + \frac {1}{2} \, c^{2} d x^{8} e^{3} + \frac {6}{7} \, c^{2} d^{2} x^{7} e^{2} + \frac {2}{3} \, c^{2} d^{3} x^{6} e + \frac {1}{5} \, c^{2} d^{4} x^{5} + \frac {1}{4} \, b c x^{8} e^{4} + \frac {8}{7} \, b c d x^{7} e^{3} + 2 \, b c d^{2} x^{6} e^{2} + \frac {8}{5} \, b c d^{3} x^{5} e + \frac {1}{2} \, b c d^{4} x^{4} + \frac {1}{7} \, b^{2} x^{7} e^{4} + \frac {2}{3} \, b^{2} d x^{6} e^{3} + \frac {6}{5} \, b^{2} d^{2} x^{5} e^{2} + b^{2} d^{3} x^{4} e + \frac {1}{3} \, b^{2} d^{4} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 149, normalized size = 1.09 \begin {gather*} x^5\,\left (\frac {6\,b^2\,d^2\,e^2}{5}+\frac {8\,b\,c\,d^3\,e}{5}+\frac {c^2\,d^4}{5}\right )+x^7\,\left (\frac {b^2\,e^4}{7}+\frac {8\,b\,c\,d\,e^3}{7}+\frac {6\,c^2\,d^2\,e^2}{7}\right )+\frac {b^2\,d^4\,x^3}{3}+\frac {c^2\,e^4\,x^9}{9}+\frac {b\,d^3\,x^4\,\left (2\,b\,e+c\,d\right )}{2}+\frac {c\,e^3\,x^8\,\left (b\,e+2\,c\,d\right )}{4}+\frac {2\,d\,e\,x^6\,\left (b^2\,e^2+3\,b\,c\,d\,e+c^2\,d^2\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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