Optimal. Leaf size=65 \[ \frac {(b c-a d)^2 (c+d x)^8}{8 d^3}-\frac {2 b (b c-a d) (c+d x)^9}{9 d^3}+\frac {b^2 (c+d x)^{10}}{10 d^3} \]
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Rubi [A]
time = 0.11, 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 {2 b (c+d x)^9 (b c-a d)}{9 d^3}+\frac {(c+d x)^8 (b c-a d)^2}{8 d^3}+\frac {b^2 (c+d x)^{10}}{10 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)^7 \, dx &=\int \left (\frac {(-b c+a d)^2 (c+d x)^7}{d^2}-\frac {2 b (b c-a d) (c+d x)^8}{d^2}+\frac {b^2 (c+d x)^9}{d^2}\right ) \, dx\\ &=\frac {(b c-a d)^2 (c+d x)^8}{8 d^3}-\frac {2 b (b c-a d) (c+d x)^9}{9 d^3}+\frac {b^2 (c+d x)^{10}}{10 d^3}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(261\) vs. \(2(65)=130\).
time = 0.02, size = 261, normalized size = 4.02 \begin {gather*} a^2 c^7 x+\frac {1}{2} a c^6 (2 b c+7 a d) x^2+\frac {1}{3} c^5 \left (b^2 c^2+14 a b c d+21 a^2 d^2\right ) x^3+\frac {7}{4} c^4 d \left (b^2 c^2+6 a b c d+5 a^2 d^2\right ) x^4+\frac {7}{5} c^3 d^2 \left (3 b^2 c^2+10 a b c d+5 a^2 d^2\right ) x^5+\frac {7}{6} c^2 d^3 \left (5 b^2 c^2+10 a b c d+3 a^2 d^2\right ) x^6+c d^4 \left (5 b^2 c^2+6 a b c d+a^2 d^2\right ) x^7+\frac {1}{8} d^5 \left (21 b^2 c^2+14 a b c d+a^2 d^2\right ) x^8+\frac {1}{9} b d^6 (7 b c+2 a d) x^9+\frac {1}{10} b^2 d^7 x^{10} \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. \(276\) vs.
\(2(59)=118\).
time = 0.13, size = 277, normalized size = 4.26
method | result | size |
norman | \(\frac {b^{2} d^{7} x^{10}}{10}+\left (\frac {2}{9} a b \,d^{7}+\frac {7}{9} b^{2} c \,d^{6}\right ) x^{9}+\left (\frac {1}{8} a^{2} d^{7}+\frac {7}{4} a b c \,d^{6}+\frac {21}{8} b^{2} c^{2} d^{5}\right ) x^{8}+\left (a^{2} c \,d^{6}+6 a b \,c^{2} d^{5}+5 b^{2} c^{3} d^{4}\right ) x^{7}+\left (\frac {7}{2} a^{2} c^{2} d^{5}+\frac {35}{3} a b \,c^{3} d^{4}+\frac {35}{6} b^{2} c^{4} d^{3}\right ) x^{6}+\left (7 a^{2} c^{3} d^{4}+14 a b \,c^{4} d^{3}+\frac {21}{5} b^{2} c^{5} d^{2}\right ) x^{5}+\left (\frac {35}{4} a^{2} c^{4} d^{3}+\frac {21}{2} a b \,c^{5} d^{2}+\frac {7}{4} b^{2} c^{6} d \right ) x^{4}+\left (7 a^{2} c^{5} d^{2}+\frac {14}{3} a b \,c^{6} d +\frac {1}{3} b^{2} c^{7}\right ) x^{3}+\left (\frac {7}{2} a^{2} c^{6} d +a b \,c^{7}\right ) x^{2}+a^{2} c^{7} x\) | \(269\) |
default | \(\frac {b^{2} d^{7} x^{10}}{10}+\frac {\left (2 a b \,d^{7}+7 b^{2} c \,d^{6}\right ) x^{9}}{9}+\frac {\left (a^{2} d^{7}+14 a b c \,d^{6}+21 b^{2} c^{2} d^{5}\right ) x^{8}}{8}+\frac {\left (7 a^{2} c \,d^{6}+42 a b \,c^{2} d^{5}+35 b^{2} c^{3} d^{4}\right ) x^{7}}{7}+\frac {\left (21 a^{2} c^{2} d^{5}+70 a b \,c^{3} d^{4}+35 b^{2} c^{4} d^{3}\right ) x^{6}}{6}+\frac {\left (35 a^{2} c^{3} d^{4}+70 a b \,c^{4} d^{3}+21 b^{2} c^{5} d^{2}\right ) x^{5}}{5}+\frac {\left (35 a^{2} c^{4} d^{3}+42 a b \,c^{5} d^{2}+7 b^{2} c^{6} d \right ) x^{4}}{4}+\frac {\left (21 a^{2} c^{5} d^{2}+14 a b \,c^{6} d +b^{2} c^{7}\right ) x^{3}}{3}+\frac {\left (7 a^{2} c^{6} d +2 a b \,c^{7}\right ) x^{2}}{2}+a^{2} c^{7} x\) | \(277\) |
gosper | \(\frac {1}{10} b^{2} d^{7} x^{10}+\frac {2}{9} x^{9} a b \,d^{7}+\frac {7}{9} x^{9} b^{2} c \,d^{6}+\frac {1}{8} x^{8} a^{2} d^{7}+\frac {7}{4} x^{8} a b c \,d^{6}+\frac {21}{8} x^{8} b^{2} c^{2} d^{5}+a^{2} c \,d^{6} x^{7}+6 a b \,c^{2} d^{5} x^{7}+5 b^{2} c^{3} d^{4} x^{7}+\frac {7}{2} x^{6} a^{2} c^{2} d^{5}+\frac {35}{3} x^{6} a b \,c^{3} d^{4}+\frac {35}{6} x^{6} b^{2} c^{4} d^{3}+7 x^{5} a^{2} c^{3} d^{4}+14 x^{5} a b \,c^{4} d^{3}+\frac {21}{5} x^{5} b^{2} c^{5} d^{2}+\frac {35}{4} x^{4} a^{2} c^{4} d^{3}+\frac {21}{2} x^{4} a b \,c^{5} d^{2}+\frac {7}{4} x^{4} b^{2} c^{6} d +7 x^{3} a^{2} c^{5} d^{2}+\frac {14}{3} x^{3} a b \,c^{6} d +\frac {1}{3} x^{3} b^{2} c^{7}+\frac {7}{2} x^{2} a^{2} c^{6} d +x^{2} a b \,c^{7}+a^{2} c^{7} x\) | \(295\) |
risch | \(\frac {1}{10} b^{2} d^{7} x^{10}+\frac {2}{9} x^{9} a b \,d^{7}+\frac {7}{9} x^{9} b^{2} c \,d^{6}+\frac {1}{8} x^{8} a^{2} d^{7}+\frac {7}{4} x^{8} a b c \,d^{6}+\frac {21}{8} x^{8} b^{2} c^{2} d^{5}+a^{2} c \,d^{6} x^{7}+6 a b \,c^{2} d^{5} x^{7}+5 b^{2} c^{3} d^{4} x^{7}+\frac {7}{2} x^{6} a^{2} c^{2} d^{5}+\frac {35}{3} x^{6} a b \,c^{3} d^{4}+\frac {35}{6} x^{6} b^{2} c^{4} d^{3}+7 x^{5} a^{2} c^{3} d^{4}+14 x^{5} a b \,c^{4} d^{3}+\frac {21}{5} x^{5} b^{2} c^{5} d^{2}+\frac {35}{4} x^{4} a^{2} c^{4} d^{3}+\frac {21}{2} x^{4} a b \,c^{5} d^{2}+\frac {7}{4} x^{4} b^{2} c^{6} d +7 x^{3} a^{2} c^{5} d^{2}+\frac {14}{3} x^{3} a b \,c^{6} d +\frac {1}{3} x^{3} b^{2} c^{7}+\frac {7}{2} x^{2} a^{2} c^{6} d +x^{2} a b \,c^{7}+a^{2} c^{7} x\) | \(295\) |
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. 273 vs.
\(2 (59) = 118\).
time = 0.27, size = 273, normalized size = 4.20 \begin {gather*} \frac {1}{10} \, b^{2} d^{7} x^{10} + a^{2} c^{7} x + \frac {1}{9} \, {\left (7 \, b^{2} c d^{6} + 2 \, a b d^{7}\right )} x^{9} + \frac {1}{8} \, {\left (21 \, b^{2} c^{2} d^{5} + 14 \, a b c d^{6} + a^{2} d^{7}\right )} x^{8} + {\left (5 \, b^{2} c^{3} d^{4} + 6 \, a b c^{2} d^{5} + a^{2} c d^{6}\right )} x^{7} + \frac {7}{6} \, {\left (5 \, b^{2} c^{4} d^{3} + 10 \, a b c^{3} d^{4} + 3 \, a^{2} c^{2} d^{5}\right )} x^{6} + \frac {7}{5} \, {\left (3 \, b^{2} c^{5} d^{2} + 10 \, a b c^{4} d^{3} + 5 \, a^{2} c^{3} d^{4}\right )} x^{5} + \frac {7}{4} \, {\left (b^{2} c^{6} d + 6 \, a b c^{5} d^{2} + 5 \, a^{2} c^{4} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} c^{7} + 14 \, a b c^{6} d + 21 \, a^{2} c^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b c^{7} + 7 \, a^{2} c^{6} 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. 273 vs.
\(2 (59) = 118\).
time = 0.99, size = 273, normalized size = 4.20 \begin {gather*} \frac {1}{10} \, b^{2} d^{7} x^{10} + a^{2} c^{7} x + \frac {1}{9} \, {\left (7 \, b^{2} c d^{6} + 2 \, a b d^{7}\right )} x^{9} + \frac {1}{8} \, {\left (21 \, b^{2} c^{2} d^{5} + 14 \, a b c d^{6} + a^{2} d^{7}\right )} x^{8} + {\left (5 \, b^{2} c^{3} d^{4} + 6 \, a b c^{2} d^{5} + a^{2} c d^{6}\right )} x^{7} + \frac {7}{6} \, {\left (5 \, b^{2} c^{4} d^{3} + 10 \, a b c^{3} d^{4} + 3 \, a^{2} c^{2} d^{5}\right )} x^{6} + \frac {7}{5} \, {\left (3 \, b^{2} c^{5} d^{2} + 10 \, a b c^{4} d^{3} + 5 \, a^{2} c^{3} d^{4}\right )} x^{5} + \frac {7}{4} \, {\left (b^{2} c^{6} d + 6 \, a b c^{5} d^{2} + 5 \, a^{2} c^{4} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} c^{7} + 14 \, a b c^{6} d + 21 \, a^{2} c^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b c^{7} + 7 \, a^{2} c^{6} 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. 303 vs.
\(2 (56) = 112\).
time = 0.03, size = 303, normalized size = 4.66 \begin {gather*} a^{2} c^{7} x + \frac {b^{2} d^{7} x^{10}}{10} + x^{9} \cdot \left (\frac {2 a b d^{7}}{9} + \frac {7 b^{2} c d^{6}}{9}\right ) + x^{8} \left (\frac {a^{2} d^{7}}{8} + \frac {7 a b c d^{6}}{4} + \frac {21 b^{2} c^{2} d^{5}}{8}\right ) + x^{7} \left (a^{2} c d^{6} + 6 a b c^{2} d^{5} + 5 b^{2} c^{3} d^{4}\right ) + x^{6} \cdot \left (\frac {7 a^{2} c^{2} d^{5}}{2} + \frac {35 a b c^{3} d^{4}}{3} + \frac {35 b^{2} c^{4} d^{3}}{6}\right ) + x^{5} \cdot \left (7 a^{2} c^{3} d^{4} + 14 a b c^{4} d^{3} + \frac {21 b^{2} c^{5} d^{2}}{5}\right ) + x^{4} \cdot \left (\frac {35 a^{2} c^{4} d^{3}}{4} + \frac {21 a b c^{5} d^{2}}{2} + \frac {7 b^{2} c^{6} d}{4}\right ) + x^{3} \cdot \left (7 a^{2} c^{5} d^{2} + \frac {14 a b c^{6} d}{3} + \frac {b^{2} c^{7}}{3}\right ) + x^{2} \cdot \left (\frac {7 a^{2} c^{6} d}{2} + a b c^{7}\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. 294 vs.
\(2 (59) = 118\).
time = 0.73, size = 294, normalized size = 4.52 \begin {gather*} \frac {1}{10} \, b^{2} d^{7} x^{10} + \frac {7}{9} \, b^{2} c d^{6} x^{9} + \frac {2}{9} \, a b d^{7} x^{9} + \frac {21}{8} \, b^{2} c^{2} d^{5} x^{8} + \frac {7}{4} \, a b c d^{6} x^{8} + \frac {1}{8} \, a^{2} d^{7} x^{8} + 5 \, b^{2} c^{3} d^{4} x^{7} + 6 \, a b c^{2} d^{5} x^{7} + a^{2} c d^{6} x^{7} + \frac {35}{6} \, b^{2} c^{4} d^{3} x^{6} + \frac {35}{3} \, a b c^{3} d^{4} x^{6} + \frac {7}{2} \, a^{2} c^{2} d^{5} x^{6} + \frac {21}{5} \, b^{2} c^{5} d^{2} x^{5} + 14 \, a b c^{4} d^{3} x^{5} + 7 \, a^{2} c^{3} d^{4} x^{5} + \frac {7}{4} \, b^{2} c^{6} d x^{4} + \frac {21}{2} \, a b c^{5} d^{2} x^{4} + \frac {35}{4} \, a^{2} c^{4} d^{3} x^{4} + \frac {1}{3} \, b^{2} c^{7} x^{3} + \frac {14}{3} \, a b c^{6} d x^{3} + 7 \, a^{2} c^{5} d^{2} x^{3} + a b c^{7} x^{2} + \frac {7}{2} \, a^{2} c^{6} d x^{2} + a^{2} c^{7} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 249, normalized size = 3.83 \begin {gather*} x^3\,\left (7\,a^2\,c^5\,d^2+\frac {14\,a\,b\,c^6\,d}{3}+\frac {b^2\,c^7}{3}\right )+x^8\,\left (\frac {a^2\,d^7}{8}+\frac {7\,a\,b\,c\,d^6}{4}+\frac {21\,b^2\,c^2\,d^5}{8}\right )+a^2\,c^7\,x+\frac {b^2\,d^7\,x^{10}}{10}+\frac {a\,c^6\,x^2\,\left (7\,a\,d+2\,b\,c\right )}{2}+\frac {b\,d^6\,x^9\,\left (2\,a\,d+7\,b\,c\right )}{9}+\frac {7\,c^4\,d\,x^4\,\left (5\,a^2\,d^2+6\,a\,b\,c\,d+b^2\,c^2\right )}{4}+c\,d^4\,x^7\,\left (a^2\,d^2+6\,a\,b\,c\,d+5\,b^2\,c^2\right )+\frac {7\,c^3\,d^2\,x^5\,\left (5\,a^2\,d^2+10\,a\,b\,c\,d+3\,b^2\,c^2\right )}{5}+\frac {7\,c^2\,d^3\,x^6\,\left (3\,a^2\,d^2+10\,a\,b\,c\,d+5\,b^2\,c^2\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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