Optimal. Leaf size=65 \[ \frac {(b c-a d)^2 (c+d x)^{11}}{11 d^3}-\frac {b (b c-a d) (c+d x)^{12}}{6 d^3}+\frac {b^2 (c+d x)^{13}}{13 d^3} \]
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Rubi [A]
time = 0.18, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45}
\begin {gather*} -\frac {b (c+d x)^{12} (b c-a d)}{6 d^3}+\frac {(c+d x)^{11} (b c-a d)^2}{11 d^3}+\frac {b^2 (c+d x)^{13}}{13 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int (a+b x)^2 (c+d x)^{10} \, dx &=\int \left (\frac {(-b c+a d)^2 (c+d x)^{10}}{d^2}-\frac {2 b (b c-a d) (c+d x)^{11}}{d^2}+\frac {b^2 (c+d x)^{12}}{d^2}\right ) \, dx\\ &=\frac {(b c-a d)^2 (c+d x)^{11}}{11 d^3}-\frac {b (b c-a d) (c+d x)^{12}}{6 d^3}+\frac {b^2 (c+d x)^{13}}{13 d^3}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(358\) vs. \(2(65)=130\).
time = 0.03, size = 358, normalized size = 5.51 \begin {gather*} a^2 c^{10} x+a c^9 (b c+5 a d) x^2+\frac {1}{3} c^8 \left (b^2 c^2+20 a b c d+45 a^2 d^2\right ) x^3+\frac {5}{2} c^7 d \left (b^2 c^2+9 a b c d+12 a^2 d^2\right ) x^4+3 c^6 d^2 \left (3 b^2 c^2+16 a b c d+14 a^2 d^2\right ) x^5+2 c^5 d^3 \left (10 b^2 c^2+35 a b c d+21 a^2 d^2\right ) x^6+6 c^4 d^4 \left (5 b^2 c^2+12 a b c d+5 a^2 d^2\right ) x^7+\frac {3}{2} c^3 d^5 \left (21 b^2 c^2+35 a b c d+10 a^2 d^2\right ) x^8+\frac {5}{3} c^2 d^6 \left (14 b^2 c^2+16 a b c d+3 a^2 d^2\right ) x^9+c d^7 \left (12 b^2 c^2+9 a b c d+a^2 d^2\right ) x^{10}+\frac {1}{11} d^8 \left (45 b^2 c^2+20 a b c d+a^2 d^2\right ) x^{11}+\frac {1}{6} b d^9 (5 b c+a d) x^{12}+\frac {1}{13} b^2 d^{10} x^{13} \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. \(390\) vs.
\(2(59)=118\).
time = 0.14, size = 391, normalized size = 6.02
method | result | size |
norman | \(\frac {b^{2} d^{10} x^{13}}{13}+\left (\frac {1}{6} a b \,d^{10}+\frac {5}{6} b^{2} c \,d^{9}\right ) x^{12}+\left (\frac {1}{11} a^{2} d^{10}+\frac {20}{11} a b c \,d^{9}+\frac {45}{11} b^{2} c^{2} d^{8}\right ) x^{11}+\left (a^{2} c \,d^{9}+9 a b \,c^{2} d^{8}+12 b^{2} c^{3} d^{7}\right ) x^{10}+\left (5 a^{2} c^{2} d^{8}+\frac {80}{3} a b \,c^{3} d^{7}+\frac {70}{3} b^{2} c^{4} d^{6}\right ) x^{9}+\left (15 a^{2} c^{3} d^{7}+\frac {105}{2} a b \,c^{4} d^{6}+\frac {63}{2} b^{2} c^{5} d^{5}\right ) x^{8}+\left (30 a^{2} c^{4} d^{6}+72 a b \,c^{5} d^{5}+30 b^{2} c^{6} d^{4}\right ) x^{7}+\left (42 a^{2} c^{5} d^{5}+70 a b \,c^{6} d^{4}+20 b^{2} c^{7} d^{3}\right ) x^{6}+\left (42 a^{2} c^{6} d^{4}+48 a b \,c^{7} d^{3}+9 b^{2} c^{8} d^{2}\right ) x^{5}+\left (30 a^{2} c^{7} d^{3}+\frac {45}{2} a b \,c^{8} d^{2}+\frac {5}{2} b^{2} c^{9} d \right ) x^{4}+\left (15 a^{2} c^{8} d^{2}+\frac {20}{3} a b \,c^{9} d +\frac {1}{3} b^{2} c^{10}\right ) x^{3}+\left (5 a^{2} c^{9} d +a b \,c^{10}\right ) x^{2}+a^{2} c^{10} x\) | \(380\) |
default | \(\frac {b^{2} d^{10} x^{13}}{13}+\frac {\left (2 a b \,d^{10}+10 b^{2} c \,d^{9}\right ) x^{12}}{12}+\frac {\left (a^{2} d^{10}+20 a b c \,d^{9}+45 b^{2} c^{2} d^{8}\right ) x^{11}}{11}+\frac {\left (10 a^{2} c \,d^{9}+90 a b \,c^{2} d^{8}+120 b^{2} c^{3} d^{7}\right ) x^{10}}{10}+\frac {\left (45 a^{2} c^{2} d^{8}+240 a b \,c^{3} d^{7}+210 b^{2} c^{4} d^{6}\right ) x^{9}}{9}+\frac {\left (120 a^{2} c^{3} d^{7}+420 a b \,c^{4} d^{6}+252 b^{2} c^{5} d^{5}\right ) x^{8}}{8}+\frac {\left (210 a^{2} c^{4} d^{6}+504 a b \,c^{5} d^{5}+210 b^{2} c^{6} d^{4}\right ) x^{7}}{7}+\frac {\left (252 a^{2} c^{5} d^{5}+420 a b \,c^{6} d^{4}+120 b^{2} c^{7} d^{3}\right ) x^{6}}{6}+\frac {\left (210 a^{2} c^{6} d^{4}+240 a b \,c^{7} d^{3}+45 b^{2} c^{8} d^{2}\right ) x^{5}}{5}+\frac {\left (120 a^{2} c^{7} d^{3}+90 a b \,c^{8} d^{2}+10 b^{2} c^{9} d \right ) x^{4}}{4}+\frac {\left (45 a^{2} c^{8} d^{2}+20 a b \,c^{9} d +b^{2} c^{10}\right ) x^{3}}{3}+\frac {\left (10 a^{2} c^{9} d +2 a b \,c^{10}\right ) x^{2}}{2}+a^{2} c^{10} x\) | \(391\) |
gosper | \(\frac {1}{6} x^{12} a b \,d^{10}+\frac {5}{6} x^{12} b^{2} c \,d^{9}+\frac {45}{11} x^{11} b^{2} c^{2} d^{8}+5 x^{9} a^{2} c^{2} d^{8}+\frac {70}{3} x^{9} b^{2} c^{4} d^{6}+15 x^{8} a^{2} c^{3} d^{7}+\frac {63}{2} x^{8} b^{2} c^{5} d^{5}+30 x^{4} a^{2} c^{7} d^{3}+\frac {5}{2} x^{4} b^{2} c^{9} d +15 x^{3} a^{2} c^{8} d^{2}+a^{2} c \,d^{9} x^{10}+12 b^{2} c^{3} d^{7} x^{10}+30 a^{2} c^{4} d^{6} x^{7}+30 b^{2} c^{6} d^{4} x^{7}+42 a^{2} c^{5} d^{5} x^{6}+20 b^{2} c^{7} d^{3} x^{6}+42 a^{2} c^{6} d^{4} x^{5}+9 b^{2} c^{8} d^{2} x^{5}+5 a^{2} c^{9} d \,x^{2}+a b \,c^{10} x^{2}+\frac {1}{3} x^{3} b^{2} c^{10}+\frac {1}{13} b^{2} d^{10} x^{13}+a^{2} c^{10} x +\frac {1}{11} x^{11} a^{2} d^{10}+9 a b \,c^{2} d^{8} x^{10}+72 a b \,c^{5} d^{5} x^{7}+70 a b \,c^{6} d^{4} x^{6}+48 a b \,c^{7} d^{3} x^{5}+\frac {20}{3} x^{3} a b \,c^{9} d +\frac {20}{11} x^{11} a b c \,d^{9}+\frac {80}{3} x^{9} a b \,c^{3} d^{7}+\frac {105}{2} x^{8} a b \,c^{4} d^{6}+\frac {45}{2} x^{4} a b \,c^{8} d^{2}\) | \(418\) |
risch | \(\frac {1}{6} x^{12} a b \,d^{10}+\frac {5}{6} x^{12} b^{2} c \,d^{9}+\frac {45}{11} x^{11} b^{2} c^{2} d^{8}+5 x^{9} a^{2} c^{2} d^{8}+\frac {70}{3} x^{9} b^{2} c^{4} d^{6}+15 x^{8} a^{2} c^{3} d^{7}+\frac {63}{2} x^{8} b^{2} c^{5} d^{5}+30 x^{4} a^{2} c^{7} d^{3}+\frac {5}{2} x^{4} b^{2} c^{9} d +15 x^{3} a^{2} c^{8} d^{2}+a^{2} c \,d^{9} x^{10}+12 b^{2} c^{3} d^{7} x^{10}+30 a^{2} c^{4} d^{6} x^{7}+30 b^{2} c^{6} d^{4} x^{7}+42 a^{2} c^{5} d^{5} x^{6}+20 b^{2} c^{7} d^{3} x^{6}+42 a^{2} c^{6} d^{4} x^{5}+9 b^{2} c^{8} d^{2} x^{5}+5 a^{2} c^{9} d \,x^{2}+a b \,c^{10} x^{2}+\frac {1}{3} x^{3} b^{2} c^{10}+\frac {1}{13} b^{2} d^{10} x^{13}+a^{2} c^{10} x +\frac {1}{11} x^{11} a^{2} d^{10}+9 a b \,c^{2} d^{8} x^{10}+72 a b \,c^{5} d^{5} x^{7}+70 a b \,c^{6} d^{4} x^{6}+48 a b \,c^{7} d^{3} x^{5}+\frac {20}{3} x^{3} a b \,c^{9} d +\frac {20}{11} x^{11} a b c \,d^{9}+\frac {80}{3} x^{9} a b \,c^{3} d^{7}+\frac {105}{2} x^{8} a b \,c^{4} d^{6}+\frac {45}{2} x^{4} a b \,c^{8} d^{2}\) | \(418\) |
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. 384 vs.
\(2 (59) = 118\).
time = 0.28, size = 384, normalized size = 5.91 \begin {gather*} \frac {1}{13} \, b^{2} d^{10} x^{13} + a^{2} c^{10} x + \frac {1}{6} \, {\left (5 \, b^{2} c d^{9} + a b d^{10}\right )} x^{12} + \frac {1}{11} \, {\left (45 \, b^{2} c^{2} d^{8} + 20 \, a b c d^{9} + a^{2} d^{10}\right )} x^{11} + {\left (12 \, b^{2} c^{3} d^{7} + 9 \, a b c^{2} d^{8} + a^{2} c d^{9}\right )} x^{10} + \frac {5}{3} \, {\left (14 \, b^{2} c^{4} d^{6} + 16 \, a b c^{3} d^{7} + 3 \, a^{2} c^{2} d^{8}\right )} x^{9} + \frac {3}{2} \, {\left (21 \, b^{2} c^{5} d^{5} + 35 \, a b c^{4} d^{6} + 10 \, a^{2} c^{3} d^{7}\right )} x^{8} + 6 \, {\left (5 \, b^{2} c^{6} d^{4} + 12 \, a b c^{5} d^{5} + 5 \, a^{2} c^{4} d^{6}\right )} x^{7} + 2 \, {\left (10 \, b^{2} c^{7} d^{3} + 35 \, a b c^{6} d^{4} + 21 \, a^{2} c^{5} d^{5}\right )} x^{6} + 3 \, {\left (3 \, b^{2} c^{8} d^{2} + 16 \, a b c^{7} d^{3} + 14 \, a^{2} c^{6} d^{4}\right )} x^{5} + \frac {5}{2} \, {\left (b^{2} c^{9} d + 9 \, a b c^{8} d^{2} + 12 \, a^{2} c^{7} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} c^{10} + 20 \, a b c^{9} d + 45 \, a^{2} c^{8} d^{2}\right )} x^{3} + {\left (a b c^{10} + 5 \, a^{2} c^{9} d\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. 384 vs.
\(2 (59) = 118\).
time = 0.49, size = 384, normalized size = 5.91 \begin {gather*} \frac {1}{13} \, b^{2} d^{10} x^{13} + a^{2} c^{10} x + \frac {1}{6} \, {\left (5 \, b^{2} c d^{9} + a b d^{10}\right )} x^{12} + \frac {1}{11} \, {\left (45 \, b^{2} c^{2} d^{8} + 20 \, a b c d^{9} + a^{2} d^{10}\right )} x^{11} + {\left (12 \, b^{2} c^{3} d^{7} + 9 \, a b c^{2} d^{8} + a^{2} c d^{9}\right )} x^{10} + \frac {5}{3} \, {\left (14 \, b^{2} c^{4} d^{6} + 16 \, a b c^{3} d^{7} + 3 \, a^{2} c^{2} d^{8}\right )} x^{9} + \frac {3}{2} \, {\left (21 \, b^{2} c^{5} d^{5} + 35 \, a b c^{4} d^{6} + 10 \, a^{2} c^{3} d^{7}\right )} x^{8} + 6 \, {\left (5 \, b^{2} c^{6} d^{4} + 12 \, a b c^{5} d^{5} + 5 \, a^{2} c^{4} d^{6}\right )} x^{7} + 2 \, {\left (10 \, b^{2} c^{7} d^{3} + 35 \, a b c^{6} d^{4} + 21 \, a^{2} c^{5} d^{5}\right )} x^{6} + 3 \, {\left (3 \, b^{2} c^{8} d^{2} + 16 \, a b c^{7} d^{3} + 14 \, a^{2} c^{6} d^{4}\right )} x^{5} + \frac {5}{2} \, {\left (b^{2} c^{9} d + 9 \, a b c^{8} d^{2} + 12 \, a^{2} c^{7} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} c^{10} + 20 \, a b c^{9} d + 45 \, a^{2} c^{8} d^{2}\right )} x^{3} + {\left (a b c^{10} + 5 \, a^{2} c^{9} d\right )} x^{2} \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. 415 vs.
\(2 (54) = 108\).
time = 0.04, size = 415, normalized size = 6.38 \begin {gather*} a^{2} c^{10} x + \frac {b^{2} d^{10} x^{13}}{13} + x^{12} \left (\frac {a b d^{10}}{6} + \frac {5 b^{2} c d^{9}}{6}\right ) + x^{11} \left (\frac {a^{2} d^{10}}{11} + \frac {20 a b c d^{9}}{11} + \frac {45 b^{2} c^{2} d^{8}}{11}\right ) + x^{10} \left (a^{2} c d^{9} + 9 a b c^{2} d^{8} + 12 b^{2} c^{3} d^{7}\right ) + x^{9} \cdot \left (5 a^{2} c^{2} d^{8} + \frac {80 a b c^{3} d^{7}}{3} + \frac {70 b^{2} c^{4} d^{6}}{3}\right ) + x^{8} \cdot \left (15 a^{2} c^{3} d^{7} + \frac {105 a b c^{4} d^{6}}{2} + \frac {63 b^{2} c^{5} d^{5}}{2}\right ) + x^{7} \cdot \left (30 a^{2} c^{4} d^{6} + 72 a b c^{5} d^{5} + 30 b^{2} c^{6} d^{4}\right ) + x^{6} \cdot \left (42 a^{2} c^{5} d^{5} + 70 a b c^{6} d^{4} + 20 b^{2} c^{7} d^{3}\right ) + x^{5} \cdot \left (42 a^{2} c^{6} d^{4} + 48 a b c^{7} d^{3} + 9 b^{2} c^{8} d^{2}\right ) + x^{4} \cdot \left (30 a^{2} c^{7} d^{3} + \frac {45 a b c^{8} d^{2}}{2} + \frac {5 b^{2} c^{9} d}{2}\right ) + x^{3} \cdot \left (15 a^{2} c^{8} d^{2} + \frac {20 a b c^{9} d}{3} + \frac {b^{2} c^{10}}{3}\right ) + x^{2} \cdot \left (5 a^{2} c^{9} d + a b c^{10}\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. 417 vs.
\(2 (59) = 118\).
time = 0.68, size = 417, normalized size = 6.42 \begin {gather*} \frac {1}{13} \, b^{2} d^{10} x^{13} + \frac {5}{6} \, b^{2} c d^{9} x^{12} + \frac {1}{6} \, a b d^{10} x^{12} + \frac {45}{11} \, b^{2} c^{2} d^{8} x^{11} + \frac {20}{11} \, a b c d^{9} x^{11} + \frac {1}{11} \, a^{2} d^{10} x^{11} + 12 \, b^{2} c^{3} d^{7} x^{10} + 9 \, a b c^{2} d^{8} x^{10} + a^{2} c d^{9} x^{10} + \frac {70}{3} \, b^{2} c^{4} d^{6} x^{9} + \frac {80}{3} \, a b c^{3} d^{7} x^{9} + 5 \, a^{2} c^{2} d^{8} x^{9} + \frac {63}{2} \, b^{2} c^{5} d^{5} x^{8} + \frac {105}{2} \, a b c^{4} d^{6} x^{8} + 15 \, a^{2} c^{3} d^{7} x^{8} + 30 \, b^{2} c^{6} d^{4} x^{7} + 72 \, a b c^{5} d^{5} x^{7} + 30 \, a^{2} c^{4} d^{6} x^{7} + 20 \, b^{2} c^{7} d^{3} x^{6} + 70 \, a b c^{6} d^{4} x^{6} + 42 \, a^{2} c^{5} d^{5} x^{6} + 9 \, b^{2} c^{8} d^{2} x^{5} + 48 \, a b c^{7} d^{3} x^{5} + 42 \, a^{2} c^{6} d^{4} x^{5} + \frac {5}{2} \, b^{2} c^{9} d x^{4} + \frac {45}{2} \, a b c^{8} d^{2} x^{4} + 30 \, a^{2} c^{7} d^{3} x^{4} + \frac {1}{3} \, b^{2} c^{10} x^{3} + \frac {20}{3} \, a b c^{9} d x^{3} + 15 \, a^{2} c^{8} d^{2} x^{3} + a b c^{10} x^{2} + 5 \, a^{2} c^{9} d x^{2} + a^{2} c^{10} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.32, size = 348, normalized size = 5.35 \begin {gather*} x^3\,\left (15\,a^2\,c^8\,d^2+\frac {20\,a\,b\,c^9\,d}{3}+\frac {b^2\,c^{10}}{3}\right )+x^{11}\,\left (\frac {a^2\,d^{10}}{11}+\frac {20\,a\,b\,c\,d^9}{11}+\frac {45\,b^2\,c^2\,d^8}{11}\right )+a^2\,c^{10}\,x+\frac {b^2\,d^{10}\,x^{13}}{13}+a\,c^9\,x^2\,\left (5\,a\,d+b\,c\right )+\frac {b\,d^9\,x^{12}\,\left (a\,d+5\,b\,c\right )}{6}+\frac {5\,c^7\,d\,x^4\,\left (12\,a^2\,d^2+9\,a\,b\,c\,d+b^2\,c^2\right )}{2}+c\,d^7\,x^{10}\,\left (a^2\,d^2+9\,a\,b\,c\,d+12\,b^2\,c^2\right )+6\,c^4\,d^4\,x^7\,\left (5\,a^2\,d^2+12\,a\,b\,c\,d+5\,b^2\,c^2\right )+3\,c^6\,d^2\,x^5\,\left (14\,a^2\,d^2+16\,a\,b\,c\,d+3\,b^2\,c^2\right )+\frac {5\,c^2\,d^6\,x^9\,\left (3\,a^2\,d^2+16\,a\,b\,c\,d+14\,b^2\,c^2\right )}{3}+2\,c^5\,d^3\,x^6\,\left (21\,a^2\,d^2+35\,a\,b\,c\,d+10\,b^2\,c^2\right )+\frac {3\,c^3\,d^5\,x^8\,\left (10\,a^2\,d^2+35\,a\,b\,c\,d+21\,b^2\,c^2\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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