Optimal. Leaf size=119 \[ \frac {(b c-a d)^4 (c+d x)^8}{8 d^5}-\frac {4 b (b c-a d)^3 (c+d x)^9}{9 d^5}+\frac {3 b^2 (b c-a d)^2 (c+d x)^{10}}{5 d^5}-\frac {4 b^3 (b c-a d) (c+d x)^{11}}{11 d^5}+\frac {b^4 (c+d x)^{12}}{12 d^5} \]
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Rubi [A]
time = 0.19, antiderivative size = 119, 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 {4 b^3 (c+d x)^{11} (b c-a d)}{11 d^5}+\frac {3 b^2 (c+d x)^{10} (b c-a d)^2}{5 d^5}-\frac {4 b (c+d x)^9 (b c-a d)^3}{9 d^5}+\frac {(c+d x)^8 (b c-a d)^4}{8 d^5}+\frac {b^4 (c+d x)^{12}}{12 d^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int (a+b x)^4 (c+d x)^7 \, dx &=\int \left (\frac {(-b c+a d)^4 (c+d x)^7}{d^4}-\frac {4 b (b c-a d)^3 (c+d x)^8}{d^4}+\frac {6 b^2 (b c-a d)^2 (c+d x)^9}{d^4}-\frac {4 b^3 (b c-a d) (c+d x)^{10}}{d^4}+\frac {b^4 (c+d x)^{11}}{d^4}\right ) \, dx\\ &=\frac {(b c-a d)^4 (c+d x)^8}{8 d^5}-\frac {4 b (b c-a d)^3 (c+d x)^9}{9 d^5}+\frac {3 b^2 (b c-a d)^2 (c+d x)^{10}}{5 d^5}-\frac {4 b^3 (b c-a d) (c+d x)^{11}}{11 d^5}+\frac {b^4 (c+d x)^{12}}{12 d^5}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(473\) vs. \(2(119)=238\).
time = 0.03, size = 473, normalized size = 3.97 \begin {gather*} a^4 c^7 x+\frac {1}{2} a^3 c^6 (4 b c+7 a d) x^2+\frac {1}{3} a^2 c^5 \left (6 b^2 c^2+28 a b c d+21 a^2 d^2\right ) x^3+\frac {1}{4} a c^4 \left (4 b^3 c^3+42 a b^2 c^2 d+84 a^2 b c d^2+35 a^3 d^3\right ) x^4+\frac {1}{5} c^3 \left (b^4 c^4+28 a b^3 c^3 d+126 a^2 b^2 c^2 d^2+140 a^3 b c d^3+35 a^4 d^4\right ) x^5+\frac {7}{6} c^2 d \left (b^4 c^4+12 a b^3 c^3 d+30 a^2 b^2 c^2 d^2+20 a^3 b c d^3+3 a^4 d^4\right ) x^6+c d^2 \left (3 b^4 c^4+20 a b^3 c^3 d+30 a^2 b^2 c^2 d^2+12 a^3 b c d^3+a^4 d^4\right ) x^7+\frac {1}{8} d^3 \left (35 b^4 c^4+140 a b^3 c^3 d+126 a^2 b^2 c^2 d^2+28 a^3 b c d^3+a^4 d^4\right ) x^8+\frac {1}{9} b d^4 \left (35 b^3 c^3+84 a b^2 c^2 d+42 a^2 b c d^2+4 a^3 d^3\right ) x^9+\frac {1}{10} b^2 d^5 \left (21 b^2 c^2+28 a b c d+6 a^2 d^2\right ) x^{10}+\frac {1}{11} b^3 d^6 (7 b c+4 a d) x^{11}+\frac {1}{12} b^4 d^7 x^{12} \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. \(492\) vs.
\(2(109)=218\).
time = 0.14, size = 493, normalized size = 4.14 Too large to display
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. 489 vs.
\(2 (109) = 218\).
time = 0.28, size = 489, normalized size = 4.11 \begin {gather*} \frac {1}{12} \, b^{4} d^{7} x^{12} + a^{4} c^{7} x + \frac {1}{11} \, {\left (7 \, b^{4} c d^{6} + 4 \, a b^{3} d^{7}\right )} x^{11} + \frac {1}{10} \, {\left (21 \, b^{4} c^{2} d^{5} + 28 \, a b^{3} c d^{6} + 6 \, a^{2} b^{2} d^{7}\right )} x^{10} + \frac {1}{9} \, {\left (35 \, b^{4} c^{3} d^{4} + 84 \, a b^{3} c^{2} d^{5} + 42 \, a^{2} b^{2} c d^{6} + 4 \, a^{3} b d^{7}\right )} x^{9} + \frac {1}{8} \, {\left (35 \, b^{4} c^{4} d^{3} + 140 \, a b^{3} c^{3} d^{4} + 126 \, a^{2} b^{2} c^{2} d^{5} + 28 \, a^{3} b c d^{6} + a^{4} d^{7}\right )} x^{8} + {\left (3 \, b^{4} c^{5} d^{2} + 20 \, a b^{3} c^{4} d^{3} + 30 \, a^{2} b^{2} c^{3} d^{4} + 12 \, a^{3} b c^{2} d^{5} + a^{4} c d^{6}\right )} x^{7} + \frac {7}{6} \, {\left (b^{4} c^{6} d + 12 \, a b^{3} c^{5} d^{2} + 30 \, a^{2} b^{2} c^{4} d^{3} + 20 \, a^{3} b c^{3} d^{4} + 3 \, a^{4} c^{2} d^{5}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} c^{7} + 28 \, a b^{3} c^{6} d + 126 \, a^{2} b^{2} c^{5} d^{2} + 140 \, a^{3} b c^{4} d^{3} + 35 \, a^{4} c^{3} d^{4}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, a b^{3} c^{7} + 42 \, a^{2} b^{2} c^{6} d + 84 \, a^{3} b c^{5} d^{2} + 35 \, a^{4} c^{4} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} c^{7} + 28 \, a^{3} b c^{6} d + 21 \, a^{4} c^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b c^{7} + 7 \, a^{4} 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. 489 vs.
\(2 (109) = 218\).
time = 0.61, size = 489, normalized size = 4.11 \begin {gather*} \frac {1}{12} \, b^{4} d^{7} x^{12} + a^{4} c^{7} x + \frac {1}{11} \, {\left (7 \, b^{4} c d^{6} + 4 \, a b^{3} d^{7}\right )} x^{11} + \frac {1}{10} \, {\left (21 \, b^{4} c^{2} d^{5} + 28 \, a b^{3} c d^{6} + 6 \, a^{2} b^{2} d^{7}\right )} x^{10} + \frac {1}{9} \, {\left (35 \, b^{4} c^{3} d^{4} + 84 \, a b^{3} c^{2} d^{5} + 42 \, a^{2} b^{2} c d^{6} + 4 \, a^{3} b d^{7}\right )} x^{9} + \frac {1}{8} \, {\left (35 \, b^{4} c^{4} d^{3} + 140 \, a b^{3} c^{3} d^{4} + 126 \, a^{2} b^{2} c^{2} d^{5} + 28 \, a^{3} b c d^{6} + a^{4} d^{7}\right )} x^{8} + {\left (3 \, b^{4} c^{5} d^{2} + 20 \, a b^{3} c^{4} d^{3} + 30 \, a^{2} b^{2} c^{3} d^{4} + 12 \, a^{3} b c^{2} d^{5} + a^{4} c d^{6}\right )} x^{7} + \frac {7}{6} \, {\left (b^{4} c^{6} d + 12 \, a b^{3} c^{5} d^{2} + 30 \, a^{2} b^{2} c^{4} d^{3} + 20 \, a^{3} b c^{3} d^{4} + 3 \, a^{4} c^{2} d^{5}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} c^{7} + 28 \, a b^{3} c^{6} d + 126 \, a^{2} b^{2} c^{5} d^{2} + 140 \, a^{3} b c^{4} d^{3} + 35 \, a^{4} c^{3} d^{4}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, a b^{3} c^{7} + 42 \, a^{2} b^{2} c^{6} d + 84 \, a^{3} b c^{5} d^{2} + 35 \, a^{4} c^{4} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} c^{7} + 28 \, a^{3} b c^{6} d + 21 \, a^{4} c^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b c^{7} + 7 \, a^{4} 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. 549 vs.
\(2 (107) = 214\).
time = 0.04, size = 549, normalized size = 4.61 \begin {gather*} a^{4} c^{7} x + \frac {b^{4} d^{7} x^{12}}{12} + x^{11} \cdot \left (\frac {4 a b^{3} d^{7}}{11} + \frac {7 b^{4} c d^{6}}{11}\right ) + x^{10} \cdot \left (\frac {3 a^{2} b^{2} d^{7}}{5} + \frac {14 a b^{3} c d^{6}}{5} + \frac {21 b^{4} c^{2} d^{5}}{10}\right ) + x^{9} \cdot \left (\frac {4 a^{3} b d^{7}}{9} + \frac {14 a^{2} b^{2} c d^{6}}{3} + \frac {28 a b^{3} c^{2} d^{5}}{3} + \frac {35 b^{4} c^{3} d^{4}}{9}\right ) + x^{8} \left (\frac {a^{4} d^{7}}{8} + \frac {7 a^{3} b c d^{6}}{2} + \frac {63 a^{2} b^{2} c^{2} d^{5}}{4} + \frac {35 a b^{3} c^{3} d^{4}}{2} + \frac {35 b^{4} c^{4} d^{3}}{8}\right ) + x^{7} \left (a^{4} c d^{6} + 12 a^{3} b c^{2} d^{5} + 30 a^{2} b^{2} c^{3} d^{4} + 20 a b^{3} c^{4} d^{3} + 3 b^{4} c^{5} d^{2}\right ) + x^{6} \cdot \left (\frac {7 a^{4} c^{2} d^{5}}{2} + \frac {70 a^{3} b c^{3} d^{4}}{3} + 35 a^{2} b^{2} c^{4} d^{3} + 14 a b^{3} c^{5} d^{2} + \frac {7 b^{4} c^{6} d}{6}\right ) + x^{5} \cdot \left (7 a^{4} c^{3} d^{4} + 28 a^{3} b c^{4} d^{3} + \frac {126 a^{2} b^{2} c^{5} d^{2}}{5} + \frac {28 a b^{3} c^{6} d}{5} + \frac {b^{4} c^{7}}{5}\right ) + x^{4} \cdot \left (\frac {35 a^{4} c^{4} d^{3}}{4} + 21 a^{3} b c^{5} d^{2} + \frac {21 a^{2} b^{2} c^{6} d}{2} + a b^{3} c^{7}\right ) + x^{3} \cdot \left (7 a^{4} c^{5} d^{2} + \frac {28 a^{3} b c^{6} d}{3} + 2 a^{2} b^{2} c^{7}\right ) + x^{2} \cdot \left (\frac {7 a^{4} c^{6} d}{2} + 2 a^{3} 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. 546 vs.
\(2 (109) = 218\).
time = 0.54, size = 546, normalized size = 4.59 \begin {gather*} \frac {1}{12} \, b^{4} d^{7} x^{12} + \frac {7}{11} \, b^{4} c d^{6} x^{11} + \frac {4}{11} \, a b^{3} d^{7} x^{11} + \frac {21}{10} \, b^{4} c^{2} d^{5} x^{10} + \frac {14}{5} \, a b^{3} c d^{6} x^{10} + \frac {3}{5} \, a^{2} b^{2} d^{7} x^{10} + \frac {35}{9} \, b^{4} c^{3} d^{4} x^{9} + \frac {28}{3} \, a b^{3} c^{2} d^{5} x^{9} + \frac {14}{3} \, a^{2} b^{2} c d^{6} x^{9} + \frac {4}{9} \, a^{3} b d^{7} x^{9} + \frac {35}{8} \, b^{4} c^{4} d^{3} x^{8} + \frac {35}{2} \, a b^{3} c^{3} d^{4} x^{8} + \frac {63}{4} \, a^{2} b^{2} c^{2} d^{5} x^{8} + \frac {7}{2} \, a^{3} b c d^{6} x^{8} + \frac {1}{8} \, a^{4} d^{7} x^{8} + 3 \, b^{4} c^{5} d^{2} x^{7} + 20 \, a b^{3} c^{4} d^{3} x^{7} + 30 \, a^{2} b^{2} c^{3} d^{4} x^{7} + 12 \, a^{3} b c^{2} d^{5} x^{7} + a^{4} c d^{6} x^{7} + \frac {7}{6} \, b^{4} c^{6} d x^{6} + 14 \, a b^{3} c^{5} d^{2} x^{6} + 35 \, a^{2} b^{2} c^{4} d^{3} x^{6} + \frac {70}{3} \, a^{3} b c^{3} d^{4} x^{6} + \frac {7}{2} \, a^{4} c^{2} d^{5} x^{6} + \frac {1}{5} \, b^{4} c^{7} x^{5} + \frac {28}{5} \, a b^{3} c^{6} d x^{5} + \frac {126}{5} \, a^{2} b^{2} c^{5} d^{2} x^{5} + 28 \, a^{3} b c^{4} d^{3} x^{5} + 7 \, a^{4} c^{3} d^{4} x^{5} + a b^{3} c^{7} x^{4} + \frac {21}{2} \, a^{2} b^{2} c^{6} d x^{4} + 21 \, a^{3} b c^{5} d^{2} x^{4} + \frac {35}{4} \, a^{4} c^{4} d^{3} x^{4} + 2 \, a^{2} b^{2} c^{7} x^{3} + \frac {28}{3} \, a^{3} b c^{6} d x^{3} + 7 \, a^{4} c^{5} d^{2} x^{3} + 2 \, a^{3} b c^{7} x^{2} + \frac {7}{2} \, a^{4} c^{6} d x^{2} + a^{4} c^{7} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.31, size = 470, normalized size = 3.95 \begin {gather*} x^5\,\left (7\,a^4\,c^3\,d^4+28\,a^3\,b\,c^4\,d^3+\frac {126\,a^2\,b^2\,c^5\,d^2}{5}+\frac {28\,a\,b^3\,c^6\,d}{5}+\frac {b^4\,c^7}{5}\right )+x^8\,\left (\frac {a^4\,d^7}{8}+\frac {7\,a^3\,b\,c\,d^6}{2}+\frac {63\,a^2\,b^2\,c^2\,d^5}{4}+\frac {35\,a\,b^3\,c^3\,d^4}{2}+\frac {35\,b^4\,c^4\,d^3}{8}\right )+x^4\,\left (\frac {35\,a^4\,c^4\,d^3}{4}+21\,a^3\,b\,c^5\,d^2+\frac {21\,a^2\,b^2\,c^6\,d}{2}+a\,b^3\,c^7\right )+x^9\,\left (\frac {4\,a^3\,b\,d^7}{9}+\frac {14\,a^2\,b^2\,c\,d^6}{3}+\frac {28\,a\,b^3\,c^2\,d^5}{3}+\frac {35\,b^4\,c^3\,d^4}{9}\right )+x^7\,\left (a^4\,c\,d^6+12\,a^3\,b\,c^2\,d^5+30\,a^2\,b^2\,c^3\,d^4+20\,a\,b^3\,c^4\,d^3+3\,b^4\,c^5\,d^2\right )+x^6\,\left (\frac {7\,a^4\,c^2\,d^5}{2}+\frac {70\,a^3\,b\,c^3\,d^4}{3}+35\,a^2\,b^2\,c^4\,d^3+14\,a\,b^3\,c^5\,d^2+\frac {7\,b^4\,c^6\,d}{6}\right )+a^4\,c^7\,x+\frac {b^4\,d^7\,x^{12}}{12}+\frac {a^3\,c^6\,x^2\,\left (7\,a\,d+4\,b\,c\right )}{2}+\frac {b^3\,d^6\,x^{11}\,\left (4\,a\,d+7\,b\,c\right )}{11}+\frac {a^2\,c^5\,x^3\,\left (21\,a^2\,d^2+28\,a\,b\,c\,d+6\,b^2\,c^2\right )}{3}+\frac {b^2\,d^5\,x^{10}\,\left (6\,a^2\,d^2+28\,a\,b\,c\,d+21\,b^2\,c^2\right )}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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