Optimal. Leaf size=119 \[ \frac {(b c-a d)^4 (c+d x)^{11}}{11 d^5}-\frac {b (b c-a d)^3 (c+d x)^{12}}{3 d^5}+\frac {6 b^2 (b c-a d)^2 (c+d x)^{13}}{13 d^5}-\frac {2 b^3 (b c-a d) (c+d x)^{14}}{7 d^5}+\frac {b^4 (c+d x)^{15}}{15 d^5} \]
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Rubi [A]
time = 0.31, 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 {2 b^3 (c+d x)^{14} (b c-a d)}{7 d^5}+\frac {6 b^2 (c+d x)^{13} (b c-a d)^2}{13 d^5}-\frac {b (c+d x)^{12} (b c-a d)^3}{3 d^5}+\frac {(c+d x)^{11} (b c-a d)^4}{11 d^5}+\frac {b^4 (c+d x)^{15}}{15 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)^{10} \, dx &=\int \left (\frac {(-b c+a d)^4 (c+d x)^{10}}{d^4}-\frac {4 b (b c-a d)^3 (c+d x)^{11}}{d^4}+\frac {6 b^2 (b c-a d)^2 (c+d x)^{12}}{d^4}-\frac {4 b^3 (b c-a d) (c+d x)^{13}}{d^4}+\frac {b^4 (c+d x)^{14}}{d^4}\right ) \, dx\\ &=\frac {(b c-a d)^4 (c+d x)^{11}}{11 d^5}-\frac {b (b c-a d)^3 (c+d x)^{12}}{3 d^5}+\frac {6 b^2 (b c-a d)^2 (c+d x)^{13}}{13 d^5}-\frac {2 b^3 (b c-a d) (c+d x)^{14}}{7 d^5}+\frac {b^4 (c+d x)^{15}}{15 d^5}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(660\) vs. \(2(119)=238\).
time = 0.05, size = 660, normalized size = 5.55 \begin {gather*} a^4 c^{10} x+a^3 c^9 (2 b c+5 a d) x^2+\frac {1}{3} a^2 c^8 \left (6 b^2 c^2+40 a b c d+45 a^2 d^2\right ) x^3+a c^7 \left (b^3 c^3+15 a b^2 c^2 d+45 a^2 b c d^2+30 a^3 d^3\right ) x^4+\frac {1}{5} c^6 \left (b^4 c^4+40 a b^3 c^3 d+270 a^2 b^2 c^2 d^2+480 a^3 b c d^3+210 a^4 d^4\right ) x^5+\frac {1}{3} c^5 d \left (5 b^4 c^4+90 a b^3 c^3 d+360 a^2 b^2 c^2 d^2+420 a^3 b c d^3+126 a^4 d^4\right ) x^6+\frac {3}{7} c^4 d^2 \left (15 b^4 c^4+160 a b^3 c^3 d+420 a^2 b^2 c^2 d^2+336 a^3 b c d^3+70 a^4 d^4\right ) x^7+3 c^3 d^3 \left (5 b^4 c^4+35 a b^3 c^3 d+63 a^2 b^2 c^2 d^2+35 a^3 b c d^3+5 a^4 d^4\right ) x^8+\frac {1}{3} c^2 d^4 \left (70 b^4 c^4+336 a b^3 c^3 d+420 a^2 b^2 c^2 d^2+160 a^3 b c d^3+15 a^4 d^4\right ) x^9+\frac {1}{5} c d^5 \left (126 b^4 c^4+420 a b^3 c^3 d+360 a^2 b^2 c^2 d^2+90 a^3 b c d^3+5 a^4 d^4\right ) x^{10}+\frac {1}{11} d^6 \left (210 b^4 c^4+480 a b^3 c^3 d+270 a^2 b^2 c^2 d^2+40 a^3 b c d^3+a^4 d^4\right ) x^{11}+\frac {1}{3} b d^7 \left (30 b^3 c^3+45 a b^2 c^2 d+15 a^2 b c d^2+a^3 d^3\right ) x^{12}+\frac {1}{13} b^2 d^8 \left (45 b^2 c^2+40 a b c d+6 a^2 d^2\right ) x^{13}+\frac {1}{7} b^3 d^9 (5 b c+2 a d) x^{14}+\frac {1}{15} b^4 d^{10} x^{15} \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. \(690\) vs.
\(2(109)=218\).
time = 0.13, size = 691, normalized size = 5.81 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. 686 vs.
\(2 (109) = 218\).
time = 0.30, size = 686, normalized size = 5.76 \begin {gather*} \frac {1}{15} \, b^{4} d^{10} x^{15} + a^{4} c^{10} x + \frac {1}{7} \, {\left (5 \, b^{4} c d^{9} + 2 \, a b^{3} d^{10}\right )} x^{14} + \frac {1}{13} \, {\left (45 \, b^{4} c^{2} d^{8} + 40 \, a b^{3} c d^{9} + 6 \, a^{2} b^{2} d^{10}\right )} x^{13} + \frac {1}{3} \, {\left (30 \, b^{4} c^{3} d^{7} + 45 \, a b^{3} c^{2} d^{8} + 15 \, a^{2} b^{2} c d^{9} + a^{3} b d^{10}\right )} x^{12} + \frac {1}{11} \, {\left (210 \, b^{4} c^{4} d^{6} + 480 \, a b^{3} c^{3} d^{7} + 270 \, a^{2} b^{2} c^{2} d^{8} + 40 \, a^{3} b c d^{9} + a^{4} d^{10}\right )} x^{11} + \frac {1}{5} \, {\left (126 \, b^{4} c^{5} d^{5} + 420 \, a b^{3} c^{4} d^{6} + 360 \, a^{2} b^{2} c^{3} d^{7} + 90 \, a^{3} b c^{2} d^{8} + 5 \, a^{4} c d^{9}\right )} x^{10} + \frac {1}{3} \, {\left (70 \, b^{4} c^{6} d^{4} + 336 \, a b^{3} c^{5} d^{5} + 420 \, a^{2} b^{2} c^{4} d^{6} + 160 \, a^{3} b c^{3} d^{7} + 15 \, a^{4} c^{2} d^{8}\right )} x^{9} + 3 \, {\left (5 \, b^{4} c^{7} d^{3} + 35 \, a b^{3} c^{6} d^{4} + 63 \, a^{2} b^{2} c^{5} d^{5} + 35 \, a^{3} b c^{4} d^{6} + 5 \, a^{4} c^{3} d^{7}\right )} x^{8} + \frac {3}{7} \, {\left (15 \, b^{4} c^{8} d^{2} + 160 \, a b^{3} c^{7} d^{3} + 420 \, a^{2} b^{2} c^{6} d^{4} + 336 \, a^{3} b c^{5} d^{5} + 70 \, a^{4} c^{4} d^{6}\right )} x^{7} + \frac {1}{3} \, {\left (5 \, b^{4} c^{9} d + 90 \, a b^{3} c^{8} d^{2} + 360 \, a^{2} b^{2} c^{7} d^{3} + 420 \, a^{3} b c^{6} d^{4} + 126 \, a^{4} c^{5} d^{5}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} c^{10} + 40 \, a b^{3} c^{9} d + 270 \, a^{2} b^{2} c^{8} d^{2} + 480 \, a^{3} b c^{7} d^{3} + 210 \, a^{4} c^{6} d^{4}\right )} x^{5} + {\left (a b^{3} c^{10} + 15 \, a^{2} b^{2} c^{9} d + 45 \, a^{3} b c^{8} d^{2} + 30 \, a^{4} c^{7} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} c^{10} + 40 \, a^{3} b c^{9} d + 45 \, a^{4} c^{8} d^{2}\right )} x^{3} + {\left (2 \, a^{3} b c^{10} + 5 \, a^{4} 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. 686 vs.
\(2 (109) = 218\).
time = 0.60, size = 686, normalized size = 5.76 \begin {gather*} \frac {1}{15} \, b^{4} d^{10} x^{15} + a^{4} c^{10} x + \frac {1}{7} \, {\left (5 \, b^{4} c d^{9} + 2 \, a b^{3} d^{10}\right )} x^{14} + \frac {1}{13} \, {\left (45 \, b^{4} c^{2} d^{8} + 40 \, a b^{3} c d^{9} + 6 \, a^{2} b^{2} d^{10}\right )} x^{13} + \frac {1}{3} \, {\left (30 \, b^{4} c^{3} d^{7} + 45 \, a b^{3} c^{2} d^{8} + 15 \, a^{2} b^{2} c d^{9} + a^{3} b d^{10}\right )} x^{12} + \frac {1}{11} \, {\left (210 \, b^{4} c^{4} d^{6} + 480 \, a b^{3} c^{3} d^{7} + 270 \, a^{2} b^{2} c^{2} d^{8} + 40 \, a^{3} b c d^{9} + a^{4} d^{10}\right )} x^{11} + \frac {1}{5} \, {\left (126 \, b^{4} c^{5} d^{5} + 420 \, a b^{3} c^{4} d^{6} + 360 \, a^{2} b^{2} c^{3} d^{7} + 90 \, a^{3} b c^{2} d^{8} + 5 \, a^{4} c d^{9}\right )} x^{10} + \frac {1}{3} \, {\left (70 \, b^{4} c^{6} d^{4} + 336 \, a b^{3} c^{5} d^{5} + 420 \, a^{2} b^{2} c^{4} d^{6} + 160 \, a^{3} b c^{3} d^{7} + 15 \, a^{4} c^{2} d^{8}\right )} x^{9} + 3 \, {\left (5 \, b^{4} c^{7} d^{3} + 35 \, a b^{3} c^{6} d^{4} + 63 \, a^{2} b^{2} c^{5} d^{5} + 35 \, a^{3} b c^{4} d^{6} + 5 \, a^{4} c^{3} d^{7}\right )} x^{8} + \frac {3}{7} \, {\left (15 \, b^{4} c^{8} d^{2} + 160 \, a b^{3} c^{7} d^{3} + 420 \, a^{2} b^{2} c^{6} d^{4} + 336 \, a^{3} b c^{5} d^{5} + 70 \, a^{4} c^{4} d^{6}\right )} x^{7} + \frac {1}{3} \, {\left (5 \, b^{4} c^{9} d + 90 \, a b^{3} c^{8} d^{2} + 360 \, a^{2} b^{2} c^{7} d^{3} + 420 \, a^{3} b c^{6} d^{4} + 126 \, a^{4} c^{5} d^{5}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} c^{10} + 40 \, a b^{3} c^{9} d + 270 \, a^{2} b^{2} c^{8} d^{2} + 480 \, a^{3} b c^{7} d^{3} + 210 \, a^{4} c^{6} d^{4}\right )} x^{5} + {\left (a b^{3} c^{10} + 15 \, a^{2} b^{2} c^{9} d + 45 \, a^{3} b c^{8} d^{2} + 30 \, a^{4} c^{7} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} c^{10} + 40 \, a^{3} b c^{9} d + 45 \, a^{4} c^{8} d^{2}\right )} x^{3} + {\left (2 \, a^{3} b c^{10} + 5 \, a^{4} 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. 748 vs.
\(2 (105) = 210\).
time = 0.06, size = 748, normalized size = 6.29 \begin {gather*} a^{4} c^{10} x + \frac {b^{4} d^{10} x^{15}}{15} + x^{14} \cdot \left (\frac {2 a b^{3} d^{10}}{7} + \frac {5 b^{4} c d^{9}}{7}\right ) + x^{13} \cdot \left (\frac {6 a^{2} b^{2} d^{10}}{13} + \frac {40 a b^{3} c d^{9}}{13} + \frac {45 b^{4} c^{2} d^{8}}{13}\right ) + x^{12} \left (\frac {a^{3} b d^{10}}{3} + 5 a^{2} b^{2} c d^{9} + 15 a b^{3} c^{2} d^{8} + 10 b^{4} c^{3} d^{7}\right ) + x^{11} \left (\frac {a^{4} d^{10}}{11} + \frac {40 a^{3} b c d^{9}}{11} + \frac {270 a^{2} b^{2} c^{2} d^{8}}{11} + \frac {480 a b^{3} c^{3} d^{7}}{11} + \frac {210 b^{4} c^{4} d^{6}}{11}\right ) + x^{10} \left (a^{4} c d^{9} + 18 a^{3} b c^{2} d^{8} + 72 a^{2} b^{2} c^{3} d^{7} + 84 a b^{3} c^{4} d^{6} + \frac {126 b^{4} c^{5} d^{5}}{5}\right ) + x^{9} \cdot \left (5 a^{4} c^{2} d^{8} + \frac {160 a^{3} b c^{3} d^{7}}{3} + 140 a^{2} b^{2} c^{4} d^{6} + 112 a b^{3} c^{5} d^{5} + \frac {70 b^{4} c^{6} d^{4}}{3}\right ) + x^{8} \cdot \left (15 a^{4} c^{3} d^{7} + 105 a^{3} b c^{4} d^{6} + 189 a^{2} b^{2} c^{5} d^{5} + 105 a b^{3} c^{6} d^{4} + 15 b^{4} c^{7} d^{3}\right ) + x^{7} \cdot \left (30 a^{4} c^{4} d^{6} + 144 a^{3} b c^{5} d^{5} + 180 a^{2} b^{2} c^{6} d^{4} + \frac {480 a b^{3} c^{7} d^{3}}{7} + \frac {45 b^{4} c^{8} d^{2}}{7}\right ) + x^{6} \cdot \left (42 a^{4} c^{5} d^{5} + 140 a^{3} b c^{6} d^{4} + 120 a^{2} b^{2} c^{7} d^{3} + 30 a b^{3} c^{8} d^{2} + \frac {5 b^{4} c^{9} d}{3}\right ) + x^{5} \cdot \left (42 a^{4} c^{6} d^{4} + 96 a^{3} b c^{7} d^{3} + 54 a^{2} b^{2} c^{8} d^{2} + 8 a b^{3} c^{9} d + \frac {b^{4} c^{10}}{5}\right ) + x^{4} \cdot \left (30 a^{4} c^{7} d^{3} + 45 a^{3} b c^{8} d^{2} + 15 a^{2} b^{2} c^{9} d + a b^{3} c^{10}\right ) + x^{3} \cdot \left (15 a^{4} c^{8} d^{2} + \frac {40 a^{3} b c^{9} d}{3} + 2 a^{2} b^{2} c^{10}\right ) + x^{2} \cdot \left (5 a^{4} c^{9} d + 2 a^{3} 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. 771 vs.
\(2 (109) = 218\).
time = 0.55, size = 771, normalized size = 6.48 \begin {gather*} \frac {1}{15} \, b^{4} d^{10} x^{15} + \frac {5}{7} \, b^{4} c d^{9} x^{14} + \frac {2}{7} \, a b^{3} d^{10} x^{14} + \frac {45}{13} \, b^{4} c^{2} d^{8} x^{13} + \frac {40}{13} \, a b^{3} c d^{9} x^{13} + \frac {6}{13} \, a^{2} b^{2} d^{10} x^{13} + 10 \, b^{4} c^{3} d^{7} x^{12} + 15 \, a b^{3} c^{2} d^{8} x^{12} + 5 \, a^{2} b^{2} c d^{9} x^{12} + \frac {1}{3} \, a^{3} b d^{10} x^{12} + \frac {210}{11} \, b^{4} c^{4} d^{6} x^{11} + \frac {480}{11} \, a b^{3} c^{3} d^{7} x^{11} + \frac {270}{11} \, a^{2} b^{2} c^{2} d^{8} x^{11} + \frac {40}{11} \, a^{3} b c d^{9} x^{11} + \frac {1}{11} \, a^{4} d^{10} x^{11} + \frac {126}{5} \, b^{4} c^{5} d^{5} x^{10} + 84 \, a b^{3} c^{4} d^{6} x^{10} + 72 \, a^{2} b^{2} c^{3} d^{7} x^{10} + 18 \, a^{3} b c^{2} d^{8} x^{10} + a^{4} c d^{9} x^{10} + \frac {70}{3} \, b^{4} c^{6} d^{4} x^{9} + 112 \, a b^{3} c^{5} d^{5} x^{9} + 140 \, a^{2} b^{2} c^{4} d^{6} x^{9} + \frac {160}{3} \, a^{3} b c^{3} d^{7} x^{9} + 5 \, a^{4} c^{2} d^{8} x^{9} + 15 \, b^{4} c^{7} d^{3} x^{8} + 105 \, a b^{3} c^{6} d^{4} x^{8} + 189 \, a^{2} b^{2} c^{5} d^{5} x^{8} + 105 \, a^{3} b c^{4} d^{6} x^{8} + 15 \, a^{4} c^{3} d^{7} x^{8} + \frac {45}{7} \, b^{4} c^{8} d^{2} x^{7} + \frac {480}{7} \, a b^{3} c^{7} d^{3} x^{7} + 180 \, a^{2} b^{2} c^{6} d^{4} x^{7} + 144 \, a^{3} b c^{5} d^{5} x^{7} + 30 \, a^{4} c^{4} d^{6} x^{7} + \frac {5}{3} \, b^{4} c^{9} d x^{6} + 30 \, a b^{3} c^{8} d^{2} x^{6} + 120 \, a^{2} b^{2} c^{7} d^{3} x^{6} + 140 \, a^{3} b c^{6} d^{4} x^{6} + 42 \, a^{4} c^{5} d^{5} x^{6} + \frac {1}{5} \, b^{4} c^{10} x^{5} + 8 \, a b^{3} c^{9} d x^{5} + 54 \, a^{2} b^{2} c^{8} d^{2} x^{5} + 96 \, a^{3} b c^{7} d^{3} x^{5} + 42 \, a^{4} c^{6} d^{4} x^{5} + a b^{3} c^{10} x^{4} + 15 \, a^{2} b^{2} c^{9} d x^{4} + 45 \, a^{3} b c^{8} d^{2} x^{4} + 30 \, a^{4} c^{7} d^{3} x^{4} + 2 \, a^{2} b^{2} c^{10} x^{3} + \frac {40}{3} \, a^{3} b c^{9} d x^{3} + 15 \, a^{4} c^{8} d^{2} x^{3} + 2 \, a^{3} b c^{10} x^{2} + 5 \, a^{4} c^{9} d x^{2} + a^{4} c^{10} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.43, size = 664, normalized size = 5.58 \begin {gather*} x^5\,\left (42\,a^4\,c^6\,d^4+96\,a^3\,b\,c^7\,d^3+54\,a^2\,b^2\,c^8\,d^2+8\,a\,b^3\,c^9\,d+\frac {b^4\,c^{10}}{5}\right )+x^{11}\,\left (\frac {a^4\,d^{10}}{11}+\frac {40\,a^3\,b\,c\,d^9}{11}+\frac {270\,a^2\,b^2\,c^2\,d^8}{11}+\frac {480\,a\,b^3\,c^3\,d^7}{11}+\frac {210\,b^4\,c^4\,d^6}{11}\right )+x^8\,\left (15\,a^4\,c^3\,d^7+105\,a^3\,b\,c^4\,d^6+189\,a^2\,b^2\,c^5\,d^5+105\,a\,b^3\,c^6\,d^4+15\,b^4\,c^7\,d^3\right )+x^9\,\left (5\,a^4\,c^2\,d^8+\frac {160\,a^3\,b\,c^3\,d^7}{3}+140\,a^2\,b^2\,c^4\,d^6+112\,a\,b^3\,c^5\,d^5+\frac {70\,b^4\,c^6\,d^4}{3}\right )+x^7\,\left (30\,a^4\,c^4\,d^6+144\,a^3\,b\,c^5\,d^5+180\,a^2\,b^2\,c^6\,d^4+\frac {480\,a\,b^3\,c^7\,d^3}{7}+\frac {45\,b^4\,c^8\,d^2}{7}\right )+x^4\,\left (30\,a^4\,c^7\,d^3+45\,a^3\,b\,c^8\,d^2+15\,a^2\,b^2\,c^9\,d+a\,b^3\,c^{10}\right )+x^{12}\,\left (\frac {a^3\,b\,d^{10}}{3}+5\,a^2\,b^2\,c\,d^9+15\,a\,b^3\,c^2\,d^8+10\,b^4\,c^3\,d^7\right )+x^{10}\,\left (a^4\,c\,d^9+18\,a^3\,b\,c^2\,d^8+72\,a^2\,b^2\,c^3\,d^7+84\,a\,b^3\,c^4\,d^6+\frac {126\,b^4\,c^5\,d^5}{5}\right )+x^6\,\left (42\,a^4\,c^5\,d^5+140\,a^3\,b\,c^6\,d^4+120\,a^2\,b^2\,c^7\,d^3+30\,a\,b^3\,c^8\,d^2+\frac {5\,b^4\,c^9\,d}{3}\right )+a^4\,c^{10}\,x+\frac {b^4\,d^{10}\,x^{15}}{15}+a^3\,c^9\,x^2\,\left (5\,a\,d+2\,b\,c\right )+\frac {b^3\,d^9\,x^{14}\,\left (2\,a\,d+5\,b\,c\right )}{7}+\frac {a^2\,c^8\,x^3\,\left (45\,a^2\,d^2+40\,a\,b\,c\,d+6\,b^2\,c^2\right )}{3}+\frac {b^2\,d^8\,x^{13}\,\left (6\,a^2\,d^2+40\,a\,b\,c\,d+45\,b^2\,c^2\right )}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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