Optimal. Leaf size=119 \[ \frac {(b d-a e)^4 (d+e x)^7}{7 e^5}-\frac {b (b d-a e)^3 (d+e x)^8}{2 e^5}+\frac {2 b^2 (b d-a e)^2 (d+e x)^9}{3 e^5}-\frac {2 b^3 (b d-a e) (d+e x)^{10}}{5 e^5}+\frac {b^4 (d+e x)^{11}}{11 e^5} \]
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Rubi [A]
time = 0.19, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45}
\begin {gather*} -\frac {2 b^3 (d+e x)^{10} (b d-a e)}{5 e^5}+\frac {2 b^2 (d+e x)^9 (b d-a e)^2}{3 e^5}-\frac {b (d+e x)^8 (b d-a e)^3}{2 e^5}+\frac {(d+e x)^7 (b d-a e)^4}{7 e^5}+\frac {b^4 (d+e x)^{11}}{11 e^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 45
Rubi steps
\begin {align*} \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (d+e x)^6 \, dx\\ &=\int \left (\frac {(-b d+a e)^4 (d+e x)^6}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^7}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^8}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^9}{e^4}+\frac {b^4 (d+e x)^{10}}{e^4}\right ) \, dx\\ &=\frac {(b d-a e)^4 (d+e x)^7}{7 e^5}-\frac {b (b d-a e)^3 (d+e x)^8}{2 e^5}+\frac {2 b^2 (b d-a e)^2 (d+e x)^9}{3 e^5}-\frac {2 b^3 (b d-a e) (d+e x)^{10}}{5 e^5}+\frac {b^4 (d+e x)^{11}}{11 e^5}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(398\) vs. \(2(119)=238\).
time = 0.04, size = 398, normalized size = 3.34 \begin {gather*} a^4 d^6 x+a^3 d^5 (2 b d+3 a e) x^2+a^2 d^4 \left (2 b^2 d^2+8 a b d e+5 a^2 e^2\right ) x^3+a d^3 \left (b^3 d^3+9 a b^2 d^2 e+15 a^2 b d e^2+5 a^3 e^3\right ) x^4+\frac {1}{5} d^2 \left (b^4 d^4+24 a b^3 d^3 e+90 a^2 b^2 d^2 e^2+80 a^3 b d e^3+15 a^4 e^4\right ) x^5+d e \left (b^4 d^4+10 a b^3 d^3 e+20 a^2 b^2 d^2 e^2+10 a^3 b d e^3+a^4 e^4\right ) x^6+\frac {1}{7} e^2 \left (15 b^4 d^4+80 a b^3 d^3 e+90 a^2 b^2 d^2 e^2+24 a^3 b d e^3+a^4 e^4\right ) x^7+\frac {1}{2} b e^3 \left (5 b^3 d^3+15 a b^2 d^2 e+9 a^2 b d e^2+a^3 e^3\right ) x^8+\frac {1}{3} b^2 e^4 \left (5 b^2 d^2+8 a b d e+2 a^2 e^2\right ) x^9+\frac {1}{5} b^3 e^5 (3 b d+2 a e) x^{10}+\frac {1}{11} b^4 e^6 x^{11} \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. \(426\) vs.
\(2(109)=218\).
time = 0.68, size = 427, normalized size = 3.59
method | result | size |
norman | \(\frac {b^{4} e^{6} x^{11}}{11}+\left (\frac {2}{5} a \,b^{3} e^{6}+\frac {3}{5} b^{4} d \,e^{5}\right ) x^{10}+\left (\frac {2}{3} a^{2} b^{2} e^{6}+\frac {8}{3} a \,b^{3} d \,e^{5}+\frac {5}{3} b^{4} d^{2} e^{4}\right ) x^{9}+\left (\frac {1}{2} a^{3} b \,e^{6}+\frac {9}{2} a^{2} b^{2} d \,e^{5}+\frac {15}{2} a \,b^{3} d^{2} e^{4}+\frac {5}{2} b^{4} d^{3} e^{3}\right ) x^{8}+\left (\frac {1}{7} a^{4} e^{6}+\frac {24}{7} a^{3} b d \,e^{5}+\frac {90}{7} a^{2} b^{2} d^{2} e^{4}+\frac {80}{7} a \,b^{3} d^{3} e^{3}+\frac {15}{7} b^{4} d^{4} e^{2}\right ) x^{7}+\left (a^{4} d \,e^{5}+10 a^{3} b \,d^{2} e^{4}+20 a^{2} b^{2} d^{3} e^{3}+10 a \,b^{3} d^{4} e^{2}+b^{4} d^{5} e \right ) x^{6}+\left (3 a^{4} d^{2} e^{4}+16 a^{3} b \,d^{3} e^{3}+18 a^{2} b^{2} d^{4} e^{2}+\frac {24}{5} a \,b^{3} d^{5} e +\frac {1}{5} b^{4} d^{6}\right ) x^{5}+\left (5 a^{4} d^{3} e^{3}+15 a^{3} b \,d^{4} e^{2}+9 a^{2} b^{2} d^{5} e +a \,b^{3} d^{6}\right ) x^{4}+\left (5 a^{4} d^{4} e^{2}+8 a^{3} b \,d^{5} e +2 a^{2} b^{2} d^{6}\right ) x^{3}+\left (3 a^{4} d^{5} e +2 a^{3} b \,d^{6}\right ) x^{2}+d^{6} a^{4} x\) | \(417\) |
default | \(\frac {b^{4} e^{6} x^{11}}{11}+\frac {\left (4 a \,b^{3} e^{6}+6 b^{4} d \,e^{5}\right ) x^{10}}{10}+\frac {\left (6 a^{2} b^{2} e^{6}+24 a \,b^{3} d \,e^{5}+15 b^{4} d^{2} e^{4}\right ) x^{9}}{9}+\frac {\left (4 a^{3} b \,e^{6}+36 a^{2} b^{2} d \,e^{5}+60 a \,b^{3} d^{2} e^{4}+20 b^{4} d^{3} e^{3}\right ) x^{8}}{8}+\frac {\left (a^{4} e^{6}+24 a^{3} b d \,e^{5}+90 a^{2} b^{2} d^{2} e^{4}+80 a \,b^{3} d^{3} e^{3}+15 b^{4} d^{4} e^{2}\right ) x^{7}}{7}+\frac {\left (6 a^{4} d \,e^{5}+60 a^{3} b \,d^{2} e^{4}+120 a^{2} b^{2} d^{3} e^{3}+60 a \,b^{3} d^{4} e^{2}+6 b^{4} d^{5} e \right ) x^{6}}{6}+\frac {\left (15 a^{4} d^{2} e^{4}+80 a^{3} b \,d^{3} e^{3}+90 a^{2} b^{2} d^{4} e^{2}+24 a \,b^{3} d^{5} e +b^{4} d^{6}\right ) x^{5}}{5}+\frac {\left (20 a^{4} d^{3} e^{3}+60 a^{3} b \,d^{4} e^{2}+36 a^{2} b^{2} d^{5} e +4 a \,b^{3} d^{6}\right ) x^{4}}{4}+\frac {\left (15 a^{4} d^{4} e^{2}+24 a^{3} b \,d^{5} e +6 a^{2} b^{2} d^{6}\right ) x^{3}}{3}+\frac {\left (6 a^{4} d^{5} e +4 a^{3} b \,d^{6}\right ) x^{2}}{2}+d^{6} a^{4} x\) | \(427\) |
risch | \(d^{6} a^{4} x +\frac {1}{7} x^{7} a^{4} e^{6}+\frac {1}{5} x^{5} b^{4} d^{6}+10 a^{3} b \,d^{2} e^{4} x^{6}+20 a^{2} b^{2} d^{3} e^{3} x^{6}+10 a \,b^{3} d^{4} e^{2} x^{6}+15 a^{3} b \,d^{4} e^{2} x^{4}+9 a^{2} b^{2} d^{5} e \,x^{4}+8 a^{3} b \,d^{5} e \,x^{3}+\frac {1}{11} b^{4} e^{6} x^{11}+\frac {2}{5} x^{10} a \,b^{3} e^{6}+\frac {3}{5} x^{10} b^{4} d \,e^{5}+\frac {2}{3} x^{9} a^{2} b^{2} e^{6}+\frac {5}{3} x^{9} b^{4} d^{2} e^{4}+\frac {1}{2} x^{8} a^{3} b \,e^{6}+\frac {5}{2} x^{8} b^{4} d^{3} e^{3}+\frac {15}{7} x^{7} b^{4} d^{4} e^{2}+3 x^{5} a^{4} d^{2} e^{4}+a^{4} d \,e^{5} x^{6}+b^{4} d^{5} e \,x^{6}+3 d^{5} e \,a^{4} x^{2}+16 x^{5} a^{3} b \,d^{3} e^{3}+18 x^{5} a^{2} b^{2} d^{4} e^{2}+\frac {24}{5} x^{5} a \,b^{3} d^{5} e +5 a^{4} d^{3} e^{3} x^{4}+a \,b^{3} d^{6} x^{4}+5 a^{4} d^{4} e^{2} x^{3}+2 a^{2} b^{2} d^{6} x^{3}+2 a^{3} b \,d^{6} x^{2}+\frac {8}{3} x^{9} a \,b^{3} d \,e^{5}+\frac {9}{2} x^{8} a^{2} b^{2} d \,e^{5}+\frac {15}{2} x^{8} a \,b^{3} d^{2} e^{4}+\frac {24}{7} x^{7} a^{3} b d \,e^{5}+\frac {90}{7} x^{7} a^{2} b^{2} d^{2} e^{4}+\frac {80}{7} x^{7} a \,b^{3} d^{3} e^{3}\) | \(471\) |
gosper | \(\frac {x \left (210 b^{4} e^{6} x^{10}+924 x^{9} a \,b^{3} e^{6}+1386 x^{9} b^{4} d \,e^{5}+1540 x^{8} a^{2} b^{2} e^{6}+6160 x^{8} a \,b^{3} d \,e^{5}+3850 x^{8} b^{4} d^{2} e^{4}+1155 x^{7} a^{3} b \,e^{6}+10395 x^{7} a^{2} b^{2} d \,e^{5}+17325 x^{7} a \,b^{3} d^{2} e^{4}+5775 x^{7} b^{4} d^{3} e^{3}+330 x^{6} a^{4} e^{6}+7920 x^{6} a^{3} b d \,e^{5}+29700 x^{6} a^{2} b^{2} d^{2} e^{4}+26400 x^{6} a \,b^{3} d^{3} e^{3}+4950 x^{6} b^{4} d^{4} e^{2}+2310 a^{4} d \,e^{5} x^{5}+23100 a^{3} b \,d^{2} e^{4} x^{5}+46200 a^{2} b^{2} d^{3} e^{3} x^{5}+23100 a \,b^{3} d^{4} e^{2} x^{5}+2310 b^{4} d^{5} e \,x^{5}+6930 x^{4} a^{4} d^{2} e^{4}+36960 x^{4} a^{3} b \,d^{3} e^{3}+41580 x^{4} a^{2} b^{2} d^{4} e^{2}+11088 x^{4} a \,b^{3} d^{5} e +462 x^{4} b^{4} d^{6}+11550 a^{4} d^{3} e^{3} x^{3}+34650 a^{3} b \,d^{4} e^{2} x^{3}+20790 a^{2} b^{2} d^{5} e \,x^{3}+2310 a \,b^{3} d^{6} x^{3}+11550 a^{4} d^{4} e^{2} x^{2}+18480 a^{3} b \,d^{5} e \,x^{2}+4620 a^{2} b^{2} d^{6} x^{2}+6930 a^{4} d^{5} e x +4620 a^{3} b \,d^{6} x +2310 a^{4} d^{6}\right )}{2310}\) | \(473\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 398 vs.
\(2 (113) = 226\).
time = 0.27, size = 398, normalized size = 3.34 \begin {gather*} \frac {1}{11} \, b^{4} x^{11} e^{6} + a^{4} d^{6} x + \frac {1}{5} \, {\left (3 \, b^{4} d e^{5} + 2 \, a b^{3} e^{6}\right )} x^{10} + \frac {1}{3} \, {\left (5 \, b^{4} d^{2} e^{4} + 8 \, a b^{3} d e^{5} + 2 \, a^{2} b^{2} e^{6}\right )} x^{9} + \frac {1}{2} \, {\left (5 \, b^{4} d^{3} e^{3} + 15 \, a b^{3} d^{2} e^{4} + 9 \, a^{2} b^{2} d e^{5} + a^{3} b e^{6}\right )} x^{8} + \frac {1}{7} \, {\left (15 \, b^{4} d^{4} e^{2} + 80 \, a b^{3} d^{3} e^{3} + 90 \, a^{2} b^{2} d^{2} e^{4} + 24 \, a^{3} b d e^{5} + a^{4} e^{6}\right )} x^{7} + {\left (b^{4} d^{5} e + 10 \, a b^{3} d^{4} e^{2} + 20 \, a^{2} b^{2} d^{3} e^{3} + 10 \, a^{3} b d^{2} e^{4} + a^{4} d e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{6} + 24 \, a b^{3} d^{5} e + 90 \, a^{2} b^{2} d^{4} e^{2} + 80 \, a^{3} b d^{3} e^{3} + 15 \, a^{4} d^{2} e^{4}\right )} x^{5} + {\left (a b^{3} d^{6} + 9 \, a^{2} b^{2} d^{5} e + 15 \, a^{3} b d^{4} e^{2} + 5 \, a^{4} d^{3} e^{3}\right )} x^{4} + {\left (2 \, a^{2} b^{2} d^{6} + 8 \, a^{3} b d^{5} e + 5 \, a^{4} d^{4} e^{2}\right )} x^{3} + {\left (2 \, a^{3} b d^{6} + 3 \, a^{4} d^{5} e\right )} x^{2} \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. 422 vs.
\(2 (113) = 226\).
time = 2.81, size = 422, normalized size = 3.55 \begin {gather*} \frac {1}{5} \, b^{4} d^{6} x^{5} + a b^{3} d^{6} x^{4} + 2 \, a^{2} b^{2} d^{6} x^{3} + 2 \, a^{3} b d^{6} x^{2} + a^{4} d^{6} x + \frac {1}{2310} \, {\left (210 \, b^{4} x^{11} + 924 \, a b^{3} x^{10} + 1540 \, a^{2} b^{2} x^{9} + 1155 \, a^{3} b x^{8} + 330 \, a^{4} x^{7}\right )} e^{6} + \frac {1}{210} \, {\left (126 \, b^{4} d x^{10} + 560 \, a b^{3} d x^{9} + 945 \, a^{2} b^{2} d x^{8} + 720 \, a^{3} b d x^{7} + 210 \, a^{4} d x^{6}\right )} e^{5} + \frac {1}{42} \, {\left (70 \, b^{4} d^{2} x^{9} + 315 \, a b^{3} d^{2} x^{8} + 540 \, a^{2} b^{2} d^{2} x^{7} + 420 \, a^{3} b d^{2} x^{6} + 126 \, a^{4} d^{2} x^{5}\right )} e^{4} + \frac {1}{14} \, {\left (35 \, b^{4} d^{3} x^{8} + 160 \, a b^{3} d^{3} x^{7} + 280 \, a^{2} b^{2} d^{3} x^{6} + 224 \, a^{3} b d^{3} x^{5} + 70 \, a^{4} d^{3} x^{4}\right )} e^{3} + \frac {1}{7} \, {\left (15 \, b^{4} d^{4} x^{7} + 70 \, a b^{3} d^{4} x^{6} + 126 \, a^{2} b^{2} d^{4} x^{5} + 105 \, a^{3} b d^{4} x^{4} + 35 \, a^{4} d^{4} x^{3}\right )} e^{2} + \frac {1}{5} \, {\left (5 \, b^{4} d^{5} x^{6} + 24 \, a b^{3} d^{5} x^{5} + 45 \, a^{2} b^{2} d^{5} x^{4} + 40 \, a^{3} b d^{5} x^{3} + 15 \, a^{4} d^{5} x^{2}\right )} e \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. 462 vs.
\(2 (105) = 210\).
time = 0.04, size = 462, normalized size = 3.88 \begin {gather*} a^{4} d^{6} x + \frac {b^{4} e^{6} x^{11}}{11} + x^{10} \cdot \left (\frac {2 a b^{3} e^{6}}{5} + \frac {3 b^{4} d e^{5}}{5}\right ) + x^{9} \cdot \left (\frac {2 a^{2} b^{2} e^{6}}{3} + \frac {8 a b^{3} d e^{5}}{3} + \frac {5 b^{4} d^{2} e^{4}}{3}\right ) + x^{8} \left (\frac {a^{3} b e^{6}}{2} + \frac {9 a^{2} b^{2} d e^{5}}{2} + \frac {15 a b^{3} d^{2} e^{4}}{2} + \frac {5 b^{4} d^{3} e^{3}}{2}\right ) + x^{7} \left (\frac {a^{4} e^{6}}{7} + \frac {24 a^{3} b d e^{5}}{7} + \frac {90 a^{2} b^{2} d^{2} e^{4}}{7} + \frac {80 a b^{3} d^{3} e^{3}}{7} + \frac {15 b^{4} d^{4} e^{2}}{7}\right ) + x^{6} \left (a^{4} d e^{5} + 10 a^{3} b d^{2} e^{4} + 20 a^{2} b^{2} d^{3} e^{3} + 10 a b^{3} d^{4} e^{2} + b^{4} d^{5} e\right ) + x^{5} \cdot \left (3 a^{4} d^{2} e^{4} + 16 a^{3} b d^{3} e^{3} + 18 a^{2} b^{2} d^{4} e^{2} + \frac {24 a b^{3} d^{5} e}{5} + \frac {b^{4} d^{6}}{5}\right ) + x^{4} \cdot \left (5 a^{4} d^{3} e^{3} + 15 a^{3} b d^{4} e^{2} + 9 a^{2} b^{2} d^{5} e + a b^{3} d^{6}\right ) + x^{3} \cdot \left (5 a^{4} d^{4} e^{2} + 8 a^{3} b d^{5} e + 2 a^{2} b^{2} d^{6}\right ) + x^{2} \cdot \left (3 a^{4} d^{5} e + 2 a^{3} b d^{6}\right ) \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. 450 vs.
\(2 (113) = 226\).
time = 1.66, size = 450, normalized size = 3.78 \begin {gather*} \frac {1}{11} \, b^{4} x^{11} e^{6} + \frac {3}{5} \, b^{4} d x^{10} e^{5} + \frac {5}{3} \, b^{4} d^{2} x^{9} e^{4} + \frac {5}{2} \, b^{4} d^{3} x^{8} e^{3} + \frac {15}{7} \, b^{4} d^{4} x^{7} e^{2} + b^{4} d^{5} x^{6} e + \frac {1}{5} \, b^{4} d^{6} x^{5} + \frac {2}{5} \, a b^{3} x^{10} e^{6} + \frac {8}{3} \, a b^{3} d x^{9} e^{5} + \frac {15}{2} \, a b^{3} d^{2} x^{8} e^{4} + \frac {80}{7} \, a b^{3} d^{3} x^{7} e^{3} + 10 \, a b^{3} d^{4} x^{6} e^{2} + \frac {24}{5} \, a b^{3} d^{5} x^{5} e + a b^{3} d^{6} x^{4} + \frac {2}{3} \, a^{2} b^{2} x^{9} e^{6} + \frac {9}{2} \, a^{2} b^{2} d x^{8} e^{5} + \frac {90}{7} \, a^{2} b^{2} d^{2} x^{7} e^{4} + 20 \, a^{2} b^{2} d^{3} x^{6} e^{3} + 18 \, a^{2} b^{2} d^{4} x^{5} e^{2} + 9 \, a^{2} b^{2} d^{5} x^{4} e + 2 \, a^{2} b^{2} d^{6} x^{3} + \frac {1}{2} \, a^{3} b x^{8} e^{6} + \frac {24}{7} \, a^{3} b d x^{7} e^{5} + 10 \, a^{3} b d^{2} x^{6} e^{4} + 16 \, a^{3} b d^{3} x^{5} e^{3} + 15 \, a^{3} b d^{4} x^{4} e^{2} + 8 \, a^{3} b d^{5} x^{3} e + 2 \, a^{3} b d^{6} x^{2} + \frac {1}{7} \, a^{4} x^{7} e^{6} + a^{4} d x^{6} e^{5} + 3 \, a^{4} d^{2} x^{5} e^{4} + 5 \, a^{4} d^{3} x^{4} e^{3} + 5 \, a^{4} d^{4} x^{3} e^{2} + 3 \, a^{4} d^{5} x^{2} e + a^{4} d^{6} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.62, size = 402, normalized size = 3.38 \begin {gather*} x^5\,\left (3\,a^4\,d^2\,e^4+16\,a^3\,b\,d^3\,e^3+18\,a^2\,b^2\,d^4\,e^2+\frac {24\,a\,b^3\,d^5\,e}{5}+\frac {b^4\,d^6}{5}\right )+x^7\,\left (\frac {a^4\,e^6}{7}+\frac {24\,a^3\,b\,d\,e^5}{7}+\frac {90\,a^2\,b^2\,d^2\,e^4}{7}+\frac {80\,a\,b^3\,d^3\,e^3}{7}+\frac {15\,b^4\,d^4\,e^2}{7}\right )+x^4\,\left (5\,a^4\,d^3\,e^3+15\,a^3\,b\,d^4\,e^2+9\,a^2\,b^2\,d^5\,e+a\,b^3\,d^6\right )+x^8\,\left (\frac {a^3\,b\,e^6}{2}+\frac {9\,a^2\,b^2\,d\,e^5}{2}+\frac {15\,a\,b^3\,d^2\,e^4}{2}+\frac {5\,b^4\,d^3\,e^3}{2}\right )+x^6\,\left (a^4\,d\,e^5+10\,a^3\,b\,d^2\,e^4+20\,a^2\,b^2\,d^3\,e^3+10\,a\,b^3\,d^4\,e^2+b^4\,d^5\,e\right )+a^4\,d^6\,x+\frac {b^4\,e^6\,x^{11}}{11}+a^3\,d^5\,x^2\,\left (3\,a\,e+2\,b\,d\right )+\frac {b^3\,e^5\,x^{10}\,\left (2\,a\,e+3\,b\,d\right )}{5}+a^2\,d^4\,x^3\,\left (5\,a^2\,e^2+8\,a\,b\,d\,e+2\,b^2\,d^2\right )+\frac {b^2\,e^4\,x^9\,\left (2\,a^2\,e^2+8\,a\,b\,d\,e+5\,b^2\,d^2\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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