Optimal. Leaf size=92 \[ -\frac {(b c-a d)^3 (c+d x)^8}{8 d^4}+\frac {b (b c-a d)^2 (c+d x)^9}{3 d^4}-\frac {3 b^2 (b c-a d) (c+d x)^{10}}{10 d^4}+\frac {b^3 (c+d x)^{11}}{11 d^4} \]
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Rubi [A]
time = 0.15, antiderivative size = 92, 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 {3 b^2 (c+d x)^{10} (b c-a d)}{10 d^4}+\frac {b (c+d x)^9 (b c-a d)^2}{3 d^4}-\frac {(c+d x)^8 (b c-a d)^3}{8 d^4}+\frac {b^3 (c+d x)^{11}}{11 d^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int (a+b x)^3 (c+d x)^7 \, dx &=\int \left (\frac {(-b c+a d)^3 (c+d x)^7}{d^3}+\frac {3 b (b c-a d)^2 (c+d x)^8}{d^3}-\frac {3 b^2 (b c-a d) (c+d x)^9}{d^3}+\frac {b^3 (c+d x)^{10}}{d^3}\right ) \, dx\\ &=-\frac {(b c-a d)^3 (c+d x)^8}{8 d^4}+\frac {b (b c-a d)^2 (c+d x)^9}{3 d^4}-\frac {3 b^2 (b c-a d) (c+d x)^{10}}{10 d^4}+\frac {b^3 (c+d x)^{11}}{11 d^4}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(360\) vs. \(2(92)=184\).
time = 0.03, size = 360, normalized size = 3.91 \begin {gather*} a^3 c^7 x+\frac {1}{2} a^2 c^6 (3 b c+7 a d) x^2+a c^5 \left (b^2 c^2+7 a b c d+7 a^2 d^2\right ) x^3+\frac {1}{4} c^4 \left (b^3 c^3+21 a b^2 c^2 d+63 a^2 b c d^2+35 a^3 d^3\right ) x^4+\frac {7}{5} c^3 d \left (b^3 c^3+9 a b^2 c^2 d+15 a^2 b c d^2+5 a^3 d^3\right ) x^5+\frac {7}{2} c^2 d^2 \left (b^3 c^3+5 a b^2 c^2 d+5 a^2 b c d^2+a^3 d^3\right ) x^6+c d^3 \left (5 b^3 c^3+15 a b^2 c^2 d+9 a^2 b c d^2+a^3 d^3\right ) x^7+\frac {1}{8} d^4 \left (35 b^3 c^3+63 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) x^8+\frac {1}{3} b d^5 \left (7 b^2 c^2+7 a b c d+a^2 d^2\right ) x^9+\frac {1}{10} b^2 d^6 (7 b c+3 a d) x^{10}+\frac {1}{11} b^3 d^7 x^{11} \end {gather*}
Antiderivative was successfully verified.
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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(359\) vs. \(2(92)=184\).
time = 4.35, size = 343, normalized size = 3.73 \begin {gather*} x \left (a^3 c^7+\frac {a^2 c^6 x \left (7 a d+3 b c\right )}{2}+a c^5 x^2 \left (7 a^2 d^2+7 a b c d+b^2 c^2\right )+\frac {b d^5 x^8 \left (a^2 d^2+7 a b c d+7 b^2 c^2\right )}{3}+\frac {b^2 d^6 x^9 \left (3 a d+7 b c\right )}{10}+\frac {c^4 x^3 \left (35 a^3 d^3+63 a^2 b c d^2+21 a b^2 c^2 d+b^3 c^3\right )}{4}+\frac {d^4 x^7 \left (a^3 d^3+21 a^2 b c d^2+63 a b^2 c^2 d+35 b^3 c^3\right )}{8}+\frac {b^3 d^7 x^{10}}{11}+\frac {7 c^3 d x^4 \left (5 a^3 d^3+15 a^2 b c d^2+9 a b^2 c^2 d+b^3 c^3\right )}{5}+\frac {7 c^2 d^2 x^5 \left (a^3 d^3+5 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right )}{2}+c d^3 x^6 \left (a^3 d^3+9 a^2 b c d^2+15 a b^2 c^2 d+5 b^3 c^3\right )\right ) \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. \(384\) vs.
\(2(84)=168\).
time = 0.14, size = 385, normalized size = 4.18
method | result | size |
norman | \(\frac {b^{3} d^{7} x^{11}}{11}+\left (\frac {3}{10} a \,b^{2} d^{7}+\frac {7}{10} b^{3} c \,d^{6}\right ) x^{10}+\left (\frac {1}{3} a^{2} b \,d^{7}+\frac {7}{3} a \,b^{2} c \,d^{6}+\frac {7}{3} b^{3} c^{2} d^{5}\right ) x^{9}+\left (\frac {1}{8} a^{3} d^{7}+\frac {21}{8} a^{2} b c \,d^{6}+\frac {63}{8} a \,b^{2} c^{2} d^{5}+\frac {35}{8} b^{3} c^{3} d^{4}\right ) x^{8}+\left (a^{3} c \,d^{6}+9 a^{2} b \,c^{2} d^{5}+15 a \,b^{2} c^{3} d^{4}+5 b^{3} c^{4} d^{3}\right ) x^{7}+\left (\frac {7}{2} a^{3} c^{2} d^{5}+\frac {35}{2} a^{2} b \,c^{3} d^{4}+\frac {35}{2} a \,b^{2} c^{4} d^{3}+\frac {7}{2} b^{3} c^{5} d^{2}\right ) x^{6}+\left (7 a^{3} c^{3} d^{4}+21 a^{2} b \,c^{4} d^{3}+\frac {63}{5} a \,b^{2} c^{5} d^{2}+\frac {7}{5} b^{3} c^{6} d \right ) x^{5}+\left (\frac {35}{4} a^{3} c^{4} d^{3}+\frac {63}{4} a^{2} b \,c^{5} d^{2}+\frac {21}{4} a \,b^{2} c^{6} d +\frac {1}{4} b^{3} c^{7}\right ) x^{4}+\left (7 a^{3} c^{5} d^{2}+7 a^{2} b \,c^{6} d +a \,b^{2} c^{7}\right ) x^{3}+\left (\frac {7}{2} a^{3} c^{6} d +\frac {3}{2} a^{2} b \,c^{7}\right ) x^{2}+a^{3} c^{7} x\) | \(376\) |
default | \(\frac {b^{3} d^{7} x^{11}}{11}+\frac {\left (3 a \,b^{2} d^{7}+7 b^{3} c \,d^{6}\right ) x^{10}}{10}+\frac {\left (3 a^{2} b \,d^{7}+21 a \,b^{2} c \,d^{6}+21 b^{3} c^{2} d^{5}\right ) x^{9}}{9}+\frac {\left (a^{3} d^{7}+21 a^{2} b c \,d^{6}+63 a \,b^{2} c^{2} d^{5}+35 b^{3} c^{3} d^{4}\right ) x^{8}}{8}+\frac {\left (7 a^{3} c \,d^{6}+63 a^{2} b \,c^{2} d^{5}+105 a \,b^{2} c^{3} d^{4}+35 b^{3} c^{4} d^{3}\right ) x^{7}}{7}+\frac {\left (21 a^{3} c^{2} d^{5}+105 a^{2} b \,c^{3} d^{4}+105 a \,b^{2} c^{4} d^{3}+21 b^{3} c^{5} d^{2}\right ) x^{6}}{6}+\frac {\left (35 a^{3} c^{3} d^{4}+105 a^{2} b \,c^{4} d^{3}+63 a \,b^{2} c^{5} d^{2}+7 b^{3} c^{6} d \right ) x^{5}}{5}+\frac {\left (35 a^{3} c^{4} d^{3}+63 a^{2} b \,c^{5} d^{2}+21 a \,b^{2} c^{6} d +b^{3} c^{7}\right ) x^{4}}{4}+\frac {\left (21 a^{3} c^{5} d^{2}+21 a^{2} b \,c^{6} d +3 a \,b^{2} c^{7}\right ) x^{3}}{3}+\frac {\left (7 a^{3} c^{6} d +3 a^{2} b \,c^{7}\right ) x^{2}}{2}+a^{3} c^{7} x\) | \(385\) |
gosper | \(5 b^{3} c^{4} d^{3} x^{7}+7 a^{3} c^{5} d^{2} x^{3}+a \,b^{2} c^{7} x^{3}+\frac {3}{10} x^{10} a \,b^{2} d^{7}+\frac {7}{10} x^{10} b^{3} c \,d^{6}+\frac {1}{3} x^{9} a^{2} b \,d^{7}+\frac {7}{3} x^{9} b^{3} c^{2} d^{5}+\frac {35}{8} x^{8} b^{3} c^{3} d^{4}+\frac {7}{2} x^{6} a^{3} c^{2} d^{5}+\frac {7}{2} x^{2} a^{3} c^{6} d +\frac {3}{2} x^{2} a^{2} b \,c^{7}+a^{3} c \,d^{6} x^{7}+\frac {35}{4} x^{4} a^{3} c^{4} d^{3}+\frac {7}{2} x^{6} b^{3} c^{5} d^{2}+7 x^{5} a^{3} c^{3} d^{4}+\frac {7}{5} x^{5} b^{3} c^{6} d +\frac {7}{3} x^{9} a \,b^{2} c \,d^{6}+\frac {21}{8} x^{8} a^{2} b c \,d^{6}+\frac {63}{8} x^{8} a \,b^{2} c^{2} d^{5}+\frac {35}{2} x^{6} a^{2} b \,c^{3} d^{4}+\frac {35}{2} x^{6} a \,b^{2} c^{4} d^{3}+21 x^{5} a^{2} b \,c^{4} d^{3}+\frac {63}{5} x^{5} a \,b^{2} c^{5} d^{2}+\frac {63}{4} x^{4} a^{2} b \,c^{5} d^{2}+\frac {1}{11} b^{3} d^{7} x^{11}+a^{3} c^{7} x +\frac {1}{8} x^{8} a^{3} d^{7}+\frac {1}{4} x^{4} b^{3} c^{7}+\frac {21}{4} x^{4} a \,b^{2} c^{6} d +9 a^{2} b \,c^{2} d^{5} x^{7}+15 a \,b^{2} c^{3} d^{4} x^{7}+7 a^{2} b \,c^{6} d \,x^{3}\) | \(421\) |
risch | \(5 b^{3} c^{4} d^{3} x^{7}+7 a^{3} c^{5} d^{2} x^{3}+a \,b^{2} c^{7} x^{3}+\frac {3}{10} x^{10} a \,b^{2} d^{7}+\frac {7}{10} x^{10} b^{3} c \,d^{6}+\frac {1}{3} x^{9} a^{2} b \,d^{7}+\frac {7}{3} x^{9} b^{3} c^{2} d^{5}+\frac {35}{8} x^{8} b^{3} c^{3} d^{4}+\frac {7}{2} x^{6} a^{3} c^{2} d^{5}+\frac {7}{2} x^{2} a^{3} c^{6} d +\frac {3}{2} x^{2} a^{2} b \,c^{7}+a^{3} c \,d^{6} x^{7}+\frac {35}{4} x^{4} a^{3} c^{4} d^{3}+\frac {7}{2} x^{6} b^{3} c^{5} d^{2}+7 x^{5} a^{3} c^{3} d^{4}+\frac {7}{5} x^{5} b^{3} c^{6} d +\frac {7}{3} x^{9} a \,b^{2} c \,d^{6}+\frac {21}{8} x^{8} a^{2} b c \,d^{6}+\frac {63}{8} x^{8} a \,b^{2} c^{2} d^{5}+\frac {35}{2} x^{6} a^{2} b \,c^{3} d^{4}+\frac {35}{2} x^{6} a \,b^{2} c^{4} d^{3}+21 x^{5} a^{2} b \,c^{4} d^{3}+\frac {63}{5} x^{5} a \,b^{2} c^{5} d^{2}+\frac {63}{4} x^{4} a^{2} b \,c^{5} d^{2}+\frac {1}{11} b^{3} d^{7} x^{11}+a^{3} c^{7} x +\frac {1}{8} x^{8} a^{3} d^{7}+\frac {1}{4} x^{4} b^{3} c^{7}+\frac {21}{4} x^{4} a \,b^{2} c^{6} d +9 a^{2} b \,c^{2} d^{5} x^{7}+15 a \,b^{2} c^{3} d^{4} x^{7}+7 a^{2} b \,c^{6} d \,x^{3}\) | \(421\) |
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. 376 vs.
\(2 (84) = 168\).
time = 0.27, size = 376, normalized size = 4.09 \begin {gather*} \frac {1}{11} \, b^{3} d^{7} x^{11} + a^{3} c^{7} x + \frac {1}{10} \, {\left (7 \, b^{3} c d^{6} + 3 \, a b^{2} d^{7}\right )} x^{10} + \frac {1}{3} \, {\left (7 \, b^{3} c^{2} d^{5} + 7 \, a b^{2} c d^{6} + a^{2} b d^{7}\right )} x^{9} + \frac {1}{8} \, {\left (35 \, b^{3} c^{3} d^{4} + 63 \, a b^{2} c^{2} d^{5} + 21 \, a^{2} b c d^{6} + a^{3} d^{7}\right )} x^{8} + {\left (5 \, b^{3} c^{4} d^{3} + 15 \, a b^{2} c^{3} d^{4} + 9 \, a^{2} b c^{2} d^{5} + a^{3} c d^{6}\right )} x^{7} + \frac {7}{2} \, {\left (b^{3} c^{5} d^{2} + 5 \, a b^{2} c^{4} d^{3} + 5 \, a^{2} b c^{3} d^{4} + a^{3} c^{2} d^{5}\right )} x^{6} + \frac {7}{5} \, {\left (b^{3} c^{6} d + 9 \, a b^{2} c^{5} d^{2} + 15 \, a^{2} b c^{4} d^{3} + 5 \, a^{3} c^{3} d^{4}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} c^{7} + 21 \, a b^{2} c^{6} d + 63 \, a^{2} b c^{5} d^{2} + 35 \, a^{3} c^{4} d^{3}\right )} x^{4} + {\left (a b^{2} c^{7} + 7 \, a^{2} b c^{6} d + 7 \, a^{3} c^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b c^{7} + 7 \, a^{3} 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. 376 vs.
\(2 (84) = 168\).
time = 0.29, size = 376, normalized size = 4.09 \begin {gather*} \frac {1}{11} \, b^{3} d^{7} x^{11} + a^{3} c^{7} x + \frac {1}{10} \, {\left (7 \, b^{3} c d^{6} + 3 \, a b^{2} d^{7}\right )} x^{10} + \frac {1}{3} \, {\left (7 \, b^{3} c^{2} d^{5} + 7 \, a b^{2} c d^{6} + a^{2} b d^{7}\right )} x^{9} + \frac {1}{8} \, {\left (35 \, b^{3} c^{3} d^{4} + 63 \, a b^{2} c^{2} d^{5} + 21 \, a^{2} b c d^{6} + a^{3} d^{7}\right )} x^{8} + {\left (5 \, b^{3} c^{4} d^{3} + 15 \, a b^{2} c^{3} d^{4} + 9 \, a^{2} b c^{2} d^{5} + a^{3} c d^{6}\right )} x^{7} + \frac {7}{2} \, {\left (b^{3} c^{5} d^{2} + 5 \, a b^{2} c^{4} d^{3} + 5 \, a^{2} b c^{3} d^{4} + a^{3} c^{2} d^{5}\right )} x^{6} + \frac {7}{5} \, {\left (b^{3} c^{6} d + 9 \, a b^{2} c^{5} d^{2} + 15 \, a^{2} b c^{4} d^{3} + 5 \, a^{3} c^{3} d^{4}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} c^{7} + 21 \, a b^{2} c^{6} d + 63 \, a^{2} b c^{5} d^{2} + 35 \, a^{3} c^{4} d^{3}\right )} x^{4} + {\left (a b^{2} c^{7} + 7 \, a^{2} b c^{6} d + 7 \, a^{3} c^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b c^{7} + 7 \, a^{3} 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. 427 vs.
\(2 (80) = 160\).
time = 0.06, size = 427, normalized size = 4.64 \begin {gather*} a^{3} c^{7} x + \frac {b^{3} d^{7} x^{11}}{11} + x^{10} \cdot \left (\frac {3 a b^{2} d^{7}}{10} + \frac {7 b^{3} c d^{6}}{10}\right ) + x^{9} \left (\frac {a^{2} b d^{7}}{3} + \frac {7 a b^{2} c d^{6}}{3} + \frac {7 b^{3} c^{2} d^{5}}{3}\right ) + x^{8} \left (\frac {a^{3} d^{7}}{8} + \frac {21 a^{2} b c d^{6}}{8} + \frac {63 a b^{2} c^{2} d^{5}}{8} + \frac {35 b^{3} c^{3} d^{4}}{8}\right ) + x^{7} \left (a^{3} c d^{6} + 9 a^{2} b c^{2} d^{5} + 15 a b^{2} c^{3} d^{4} + 5 b^{3} c^{4} d^{3}\right ) + x^{6} \cdot \left (\frac {7 a^{3} c^{2} d^{5}}{2} + \frac {35 a^{2} b c^{3} d^{4}}{2} + \frac {35 a b^{2} c^{4} d^{3}}{2} + \frac {7 b^{3} c^{5} d^{2}}{2}\right ) + x^{5} \cdot \left (7 a^{3} c^{3} d^{4} + 21 a^{2} b c^{4} d^{3} + \frac {63 a b^{2} c^{5} d^{2}}{5} + \frac {7 b^{3} c^{6} d}{5}\right ) + x^{4} \cdot \left (\frac {35 a^{3} c^{4} d^{3}}{4} + \frac {63 a^{2} b c^{5} d^{2}}{4} + \frac {21 a b^{2} c^{6} d}{4} + \frac {b^{3} c^{7}}{4}\right ) + x^{3} \cdot \left (7 a^{3} c^{5} d^{2} + 7 a^{2} b c^{6} d + a b^{2} c^{7}\right ) + x^{2} \cdot \left (\frac {7 a^{3} c^{6} d}{2} + \frac {3 a^{2} b c^{7}}{2}\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. 420 vs.
\(2 (84) = 168\).
time = 0.00, size = 464, normalized size = 5.04 \begin {gather*} \frac {1}{11} x^{11} b^{3} d^{7}+\frac {7}{10} x^{10} b^{3} d^{6} c+\frac {3}{10} x^{10} b^{2} a d^{7}+\frac {7}{3} x^{9} b^{3} d^{5} c^{2}+\frac {7}{3} x^{9} b^{2} a d^{6} c+\frac {1}{3} x^{9} b a^{2} d^{7}+\frac {35}{8} x^{8} b^{3} d^{4} c^{3}+\frac {63}{8} x^{8} b^{2} a d^{5} c^{2}+\frac {21}{8} x^{8} b a^{2} d^{6} c+\frac {1}{8} x^{8} a^{3} d^{7}+5 x^{7} b^{3} d^{3} c^{4}+15 x^{7} b^{2} a d^{4} c^{3}+9 x^{7} b a^{2} d^{5} c^{2}+x^{7} a^{3} d^{6} c+\frac {7}{2} x^{6} b^{3} d^{2} c^{5}+\frac {35}{2} x^{6} b^{2} a d^{3} c^{4}+\frac {35}{2} x^{6} b a^{2} d^{4} c^{3}+\frac {7}{2} x^{6} a^{3} d^{5} c^{2}+\frac {7}{5} x^{5} b^{3} d c^{6}+\frac {63}{5} x^{5} b^{2} a d^{2} c^{5}+21 x^{5} b a^{2} d^{3} c^{4}+7 x^{5} a^{3} d^{4} c^{3}+\frac {1}{4} x^{4} b^{3} c^{7}+\frac {21}{4} x^{4} b^{2} a d c^{6}+\frac {63}{4} x^{4} b a^{2} d^{2} c^{5}+\frac {35}{4} x^{4} a^{3} d^{3} c^{4}+x^{3} b^{2} a c^{7}+7 x^{3} b a^{2} d c^{6}+7 x^{3} a^{3} d^{2} c^{5}+\frac {3}{2} x^{2} b a^{2} c^{7}+\frac {7}{2} x^{2} a^{3} d c^{6}+x a^{3} c^{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.27, size = 356, normalized size = 3.87 \begin {gather*} x^7\,\left (a^3\,c\,d^6+9\,a^2\,b\,c^2\,d^5+15\,a\,b^2\,c^3\,d^4+5\,b^3\,c^4\,d^3\right )+x^5\,\left (7\,a^3\,c^3\,d^4+21\,a^2\,b\,c^4\,d^3+\frac {63\,a\,b^2\,c^5\,d^2}{5}+\frac {7\,b^3\,c^6\,d}{5}\right )+x^4\,\left (\frac {35\,a^3\,c^4\,d^3}{4}+\frac {63\,a^2\,b\,c^5\,d^2}{4}+\frac {21\,a\,b^2\,c^6\,d}{4}+\frac {b^3\,c^7}{4}\right )+x^8\,\left (\frac {a^3\,d^7}{8}+\frac {21\,a^2\,b\,c\,d^6}{8}+\frac {63\,a\,b^2\,c^2\,d^5}{8}+\frac {35\,b^3\,c^3\,d^4}{8}\right )+a^3\,c^7\,x+\frac {b^3\,d^7\,x^{11}}{11}+\frac {7\,c^2\,d^2\,x^6\,\left (a^3\,d^3+5\,a^2\,b\,c\,d^2+5\,a\,b^2\,c^2\,d+b^3\,c^3\right )}{2}+\frac {a^2\,c^6\,x^2\,\left (7\,a\,d+3\,b\,c\right )}{2}+\frac {b^2\,d^6\,x^{10}\,\left (3\,a\,d+7\,b\,c\right )}{10}+a\,c^5\,x^3\,\left (7\,a^2\,d^2+7\,a\,b\,c\,d+b^2\,c^2\right )+\frac {b\,d^5\,x^9\,\left (a^2\,d^2+7\,a\,b\,c\,d+7\,b^2\,c^2\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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