Optimal. Leaf size=133 \[ \frac {a^7 x^{1+m}}{1+m}+\frac {7 a^6 b x^{2+m}}{2+m}+\frac {21 a^5 b^2 x^{3+m}}{3+m}+\frac {35 a^4 b^3 x^{4+m}}{4+m}+\frac {35 a^3 b^4 x^{5+m}}{5+m}+\frac {21 a^2 b^5 x^{6+m}}{6+m}+\frac {7 a b^6 x^{7+m}}{7+m}+\frac {b^7 x^{8+m}}{8+m} \]
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Rubi [A]
time = 0.04, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45}
\begin {gather*} \frac {a^7 x^{m+1}}{m+1}+\frac {7 a^6 b x^{m+2}}{m+2}+\frac {21 a^5 b^2 x^{m+3}}{m+3}+\frac {35 a^4 b^3 x^{m+4}}{m+4}+\frac {35 a^3 b^4 x^{m+5}}{m+5}+\frac {21 a^2 b^5 x^{m+6}}{m+6}+\frac {7 a b^6 x^{m+7}}{m+7}+\frac {b^7 x^{m+8}}{m+8} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int x^m (a+b x)^7 \, dx &=\int \left (a^7 x^m+7 a^6 b x^{1+m}+21 a^5 b^2 x^{2+m}+35 a^4 b^3 x^{3+m}+35 a^3 b^4 x^{4+m}+21 a^2 b^5 x^{5+m}+7 a b^6 x^{6+m}+b^7 x^{7+m}\right ) \, dx\\ &=\frac {a^7 x^{1+m}}{1+m}+\frac {7 a^6 b x^{2+m}}{2+m}+\frac {21 a^5 b^2 x^{3+m}}{3+m}+\frac {35 a^4 b^3 x^{4+m}}{4+m}+\frac {35 a^3 b^4 x^{5+m}}{5+m}+\frac {21 a^2 b^5 x^{6+m}}{6+m}+\frac {7 a b^6 x^{7+m}}{7+m}+\frac {b^7 x^{8+m}}{8+m}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 118, normalized size = 0.89 \begin {gather*} x^{1+m} \left (\frac {a^7}{1+m}+\frac {7 a^6 b x}{2+m}+\frac {21 a^5 b^2 x^2}{3+m}+\frac {35 a^4 b^3 x^3}{4+m}+\frac {35 a^3 b^4 x^4}{5+m}+\frac {21 a^2 b^5 x^5}{6+m}+\frac {7 a b^6 x^6}{7+m}+\frac {b^7 x^7}{8+m}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 156, normalized size = 1.17
method | result | size |
norman | \(\frac {a^{7} x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {b^{7} x^{8} {\mathrm e}^{m \ln \left (x \right )}}{8+m}+\frac {7 a \,b^{6} x^{7} {\mathrm e}^{m \ln \left (x \right )}}{7+m}+\frac {21 a^{2} b^{5} x^{6} {\mathrm e}^{m \ln \left (x \right )}}{6+m}+\frac {35 a^{3} b^{4} x^{5} {\mathrm e}^{m \ln \left (x \right )}}{5+m}+\frac {35 a^{4} b^{3} x^{4} {\mathrm e}^{m \ln \left (x \right )}}{4+m}+\frac {21 a^{5} b^{2} x^{3} {\mathrm e}^{m \ln \left (x \right )}}{3+m}+\frac {7 a^{6} b \,x^{2} {\mathrm e}^{m \ln \left (x \right )}}{2+m}\) | \(156\) |
risch | \(\frac {x \left (b^{7} m^{7} x^{7}+7 a \,b^{6} m^{7} x^{6}+28 b^{7} m^{6} x^{7}+21 a^{2} b^{5} m^{7} x^{5}+203 a \,b^{6} m^{6} x^{6}+322 b^{7} m^{5} x^{7}+35 a^{3} b^{4} m^{7} x^{4}+630 a^{2} b^{5} m^{6} x^{5}+2401 a \,b^{6} m^{5} x^{6}+1960 b^{7} m^{4} x^{7}+35 a^{4} b^{3} m^{7} x^{3}+1085 a^{3} b^{4} m^{6} x^{4}+7686 a^{2} b^{5} m^{5} x^{5}+14945 a \,b^{6} m^{4} x^{6}+6769 b^{7} m^{3} x^{7}+21 a^{5} b^{2} m^{7} x^{2}+1120 a^{4} b^{3} m^{6} x^{3}+13685 a^{3} b^{4} m^{5} x^{4}+49140 a^{2} b^{5} m^{4} x^{5}+52528 a \,b^{6} m^{3} x^{6}+13132 b^{7} m^{2} x^{7}+7 a^{6} b \,m^{7} x +693 a^{5} b^{2} m^{6} x^{2}+14630 a^{4} b^{3} m^{5} x^{3}+90335 a^{3} b^{4} m^{4} x^{4}+176589 a^{2} b^{5} m^{3} x^{5}+103292 a \,b^{6} m^{2} x^{6}+13068 m \,x^{7} b^{7}+a^{7} m^{7}+238 a^{6} b \,m^{6} x +9387 a^{5} b^{2} m^{5} x^{2}+100240 a^{4} b^{3} m^{4} x^{3}+334040 a^{3} b^{4} m^{3} x^{4}+353430 a^{2} b^{5} m^{2} x^{5}+103824 m \,x^{6} a \,b^{6}+5040 b^{7} x^{7}+35 a^{7} m^{6}+3346 a^{6} b \,m^{5} x +67095 a^{5} b^{2} m^{4} x^{2}+384755 a^{4} b^{3} m^{3} x^{3}+684740 a^{3} b^{4} m^{2} x^{4}+360024 m \,x^{5} a^{2} b^{5}+40320 a \,b^{6} x^{6}+511 a^{7} m^{5}+25060 a^{6} b \,m^{4} x +270144 a^{5} b^{2} m^{3} x^{2}+815920 a^{4} b^{3} m^{2} x^{3}+710640 m \,x^{4} a^{3} b^{4}+141120 a^{2} b^{5} x^{5}+4025 a^{7} m^{4}+107023 a^{6} b \,m^{3} x +602532 a^{5} b^{2} m^{2} x^{2}+870660 m \,x^{3} a^{4} b^{3}+282240 a^{3} b^{4} x^{4}+18424 a^{7} m^{3}+256942 a^{6} b \,m^{2} x +673008 m \,x^{2} a^{5} b^{2}+352800 a^{4} b^{3} x^{3}+48860 a^{7} m^{2}+312984 m x \,a^{6} b +282240 a^{5} b^{2} x^{2}+69264 m \,a^{7}+141120 a^{6} b x +40320 a^{7}\right ) x^{m}}{\left (8+m \right ) \left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(781\) |
gosper | \(\frac {x^{1+m} \left (b^{7} m^{7} x^{7}+7 a \,b^{6} m^{7} x^{6}+28 b^{7} m^{6} x^{7}+21 a^{2} b^{5} m^{7} x^{5}+203 a \,b^{6} m^{6} x^{6}+322 b^{7} m^{5} x^{7}+35 a^{3} b^{4} m^{7} x^{4}+630 a^{2} b^{5} m^{6} x^{5}+2401 a \,b^{6} m^{5} x^{6}+1960 b^{7} m^{4} x^{7}+35 a^{4} b^{3} m^{7} x^{3}+1085 a^{3} b^{4} m^{6} x^{4}+7686 a^{2} b^{5} m^{5} x^{5}+14945 a \,b^{6} m^{4} x^{6}+6769 b^{7} m^{3} x^{7}+21 a^{5} b^{2} m^{7} x^{2}+1120 a^{4} b^{3} m^{6} x^{3}+13685 a^{3} b^{4} m^{5} x^{4}+49140 a^{2} b^{5} m^{4} x^{5}+52528 a \,b^{6} m^{3} x^{6}+13132 b^{7} m^{2} x^{7}+7 a^{6} b \,m^{7} x +693 a^{5} b^{2} m^{6} x^{2}+14630 a^{4} b^{3} m^{5} x^{3}+90335 a^{3} b^{4} m^{4} x^{4}+176589 a^{2} b^{5} m^{3} x^{5}+103292 a \,b^{6} m^{2} x^{6}+13068 m \,x^{7} b^{7}+a^{7} m^{7}+238 a^{6} b \,m^{6} x +9387 a^{5} b^{2} m^{5} x^{2}+100240 a^{4} b^{3} m^{4} x^{3}+334040 a^{3} b^{4} m^{3} x^{4}+353430 a^{2} b^{5} m^{2} x^{5}+103824 m \,x^{6} a \,b^{6}+5040 b^{7} x^{7}+35 a^{7} m^{6}+3346 a^{6} b \,m^{5} x +67095 a^{5} b^{2} m^{4} x^{2}+384755 a^{4} b^{3} m^{3} x^{3}+684740 a^{3} b^{4} m^{2} x^{4}+360024 m \,x^{5} a^{2} b^{5}+40320 a \,b^{6} x^{6}+511 a^{7} m^{5}+25060 a^{6} b \,m^{4} x +270144 a^{5} b^{2} m^{3} x^{2}+815920 a^{4} b^{3} m^{2} x^{3}+710640 m \,x^{4} a^{3} b^{4}+141120 a^{2} b^{5} x^{5}+4025 a^{7} m^{4}+107023 a^{6} b \,m^{3} x +602532 a^{5} b^{2} m^{2} x^{2}+870660 m \,x^{3} a^{4} b^{3}+282240 a^{3} b^{4} x^{4}+18424 a^{7} m^{3}+256942 a^{6} b \,m^{2} x +673008 m \,x^{2} a^{5} b^{2}+352800 a^{4} b^{3} x^{3}+48860 a^{7} m^{2}+312984 m x \,a^{6} b +282240 a^{5} b^{2} x^{2}+69264 m \,a^{7}+141120 a^{6} b x +40320 a^{7}\right )}{\left (8+m \right ) \left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(782\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 133, normalized size = 1.00 \begin {gather*} \frac {b^{7} x^{m + 8}}{m + 8} + \frac {7 \, a b^{6} x^{m + 7}}{m + 7} + \frac {21 \, a^{2} b^{5} x^{m + 6}}{m + 6} + \frac {35 \, a^{3} b^{4} x^{m + 5}}{m + 5} + \frac {35 \, a^{4} b^{3} x^{m + 4}}{m + 4} + \frac {21 \, a^{5} b^{2} x^{m + 3}}{m + 3} + \frac {7 \, a^{6} b x^{m + 2}}{m + 2} + \frac {a^{7} x^{m + 1}}{m + 1} \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. 665 vs.
\(2 (133) = 266\).
time = 0.84, size = 665, normalized size = 5.00 \begin {gather*} \frac {{\left ({\left (b^{7} m^{7} + 28 \, b^{7} m^{6} + 322 \, b^{7} m^{5} + 1960 \, b^{7} m^{4} + 6769 \, b^{7} m^{3} + 13132 \, b^{7} m^{2} + 13068 \, b^{7} m + 5040 \, b^{7}\right )} x^{8} + 7 \, {\left (a b^{6} m^{7} + 29 \, a b^{6} m^{6} + 343 \, a b^{6} m^{5} + 2135 \, a b^{6} m^{4} + 7504 \, a b^{6} m^{3} + 14756 \, a b^{6} m^{2} + 14832 \, a b^{6} m + 5760 \, a b^{6}\right )} x^{7} + 21 \, {\left (a^{2} b^{5} m^{7} + 30 \, a^{2} b^{5} m^{6} + 366 \, a^{2} b^{5} m^{5} + 2340 \, a^{2} b^{5} m^{4} + 8409 \, a^{2} b^{5} m^{3} + 16830 \, a^{2} b^{5} m^{2} + 17144 \, a^{2} b^{5} m + 6720 \, a^{2} b^{5}\right )} x^{6} + 35 \, {\left (a^{3} b^{4} m^{7} + 31 \, a^{3} b^{4} m^{6} + 391 \, a^{3} b^{4} m^{5} + 2581 \, a^{3} b^{4} m^{4} + 9544 \, a^{3} b^{4} m^{3} + 19564 \, a^{3} b^{4} m^{2} + 20304 \, a^{3} b^{4} m + 8064 \, a^{3} b^{4}\right )} x^{5} + 35 \, {\left (a^{4} b^{3} m^{7} + 32 \, a^{4} b^{3} m^{6} + 418 \, a^{4} b^{3} m^{5} + 2864 \, a^{4} b^{3} m^{4} + 10993 \, a^{4} b^{3} m^{3} + 23312 \, a^{4} b^{3} m^{2} + 24876 \, a^{4} b^{3} m + 10080 \, a^{4} b^{3}\right )} x^{4} + 21 \, {\left (a^{5} b^{2} m^{7} + 33 \, a^{5} b^{2} m^{6} + 447 \, a^{5} b^{2} m^{5} + 3195 \, a^{5} b^{2} m^{4} + 12864 \, a^{5} b^{2} m^{3} + 28692 \, a^{5} b^{2} m^{2} + 32048 \, a^{5} b^{2} m + 13440 \, a^{5} b^{2}\right )} x^{3} + 7 \, {\left (a^{6} b m^{7} + 34 \, a^{6} b m^{6} + 478 \, a^{6} b m^{5} + 3580 \, a^{6} b m^{4} + 15289 \, a^{6} b m^{3} + 36706 \, a^{6} b m^{2} + 44712 \, a^{6} b m + 20160 \, a^{6} b\right )} x^{2} + {\left (a^{7} m^{7} + 35 \, a^{7} m^{6} + 511 \, a^{7} m^{5} + 4025 \, a^{7} m^{4} + 18424 \, a^{7} m^{3} + 48860 \, a^{7} m^{2} + 69264 \, a^{7} m + 40320 \, a^{7}\right )} x\right )} x^{m}}{m^{8} + 36 \, m^{7} + 546 \, m^{6} + 4536 \, m^{5} + 22449 \, m^{4} + 67284 \, m^{3} + 118124 \, m^{2} + 109584 \, m + 40320} \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. 4257 vs.
\(2 (121) = 242\).
time = 0.64, size = 4257, normalized size = 32.01 \begin {gather*} \text {Too large to display} \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. 992 vs.
\(2 (133) = 266\).
time = 1.47, size = 992, normalized size = 7.46 \begin {gather*} \frac {b^{7} m^{7} x^{8} x^{m} + 7 \, a b^{6} m^{7} x^{7} x^{m} + 28 \, b^{7} m^{6} x^{8} x^{m} + 21 \, a^{2} b^{5} m^{7} x^{6} x^{m} + 203 \, a b^{6} m^{6} x^{7} x^{m} + 322 \, b^{7} m^{5} x^{8} x^{m} + 35 \, a^{3} b^{4} m^{7} x^{5} x^{m} + 630 \, a^{2} b^{5} m^{6} x^{6} x^{m} + 2401 \, a b^{6} m^{5} x^{7} x^{m} + 1960 \, b^{7} m^{4} x^{8} x^{m} + 35 \, a^{4} b^{3} m^{7} x^{4} x^{m} + 1085 \, a^{3} b^{4} m^{6} x^{5} x^{m} + 7686 \, a^{2} b^{5} m^{5} x^{6} x^{m} + 14945 \, a b^{6} m^{4} x^{7} x^{m} + 6769 \, b^{7} m^{3} x^{8} x^{m} + 21 \, a^{5} b^{2} m^{7} x^{3} x^{m} + 1120 \, a^{4} b^{3} m^{6} x^{4} x^{m} + 13685 \, a^{3} b^{4} m^{5} x^{5} x^{m} + 49140 \, a^{2} b^{5} m^{4} x^{6} x^{m} + 52528 \, a b^{6} m^{3} x^{7} x^{m} + 13132 \, b^{7} m^{2} x^{8} x^{m} + 7 \, a^{6} b m^{7} x^{2} x^{m} + 693 \, a^{5} b^{2} m^{6} x^{3} x^{m} + 14630 \, a^{4} b^{3} m^{5} x^{4} x^{m} + 90335 \, a^{3} b^{4} m^{4} x^{5} x^{m} + 176589 \, a^{2} b^{5} m^{3} x^{6} x^{m} + 103292 \, a b^{6} m^{2} x^{7} x^{m} + 13068 \, b^{7} m x^{8} x^{m} + a^{7} m^{7} x x^{m} + 238 \, a^{6} b m^{6} x^{2} x^{m} + 9387 \, a^{5} b^{2} m^{5} x^{3} x^{m} + 100240 \, a^{4} b^{3} m^{4} x^{4} x^{m} + 334040 \, a^{3} b^{4} m^{3} x^{5} x^{m} + 353430 \, a^{2} b^{5} m^{2} x^{6} x^{m} + 103824 \, a b^{6} m x^{7} x^{m} + 5040 \, b^{7} x^{8} x^{m} + 35 \, a^{7} m^{6} x x^{m} + 3346 \, a^{6} b m^{5} x^{2} x^{m} + 67095 \, a^{5} b^{2} m^{4} x^{3} x^{m} + 384755 \, a^{4} b^{3} m^{3} x^{4} x^{m} + 684740 \, a^{3} b^{4} m^{2} x^{5} x^{m} + 360024 \, a^{2} b^{5} m x^{6} x^{m} + 40320 \, a b^{6} x^{7} x^{m} + 511 \, a^{7} m^{5} x x^{m} + 25060 \, a^{6} b m^{4} x^{2} x^{m} + 270144 \, a^{5} b^{2} m^{3} x^{3} x^{m} + 815920 \, a^{4} b^{3} m^{2} x^{4} x^{m} + 710640 \, a^{3} b^{4} m x^{5} x^{m} + 141120 \, a^{2} b^{5} x^{6} x^{m} + 4025 \, a^{7} m^{4} x x^{m} + 107023 \, a^{6} b m^{3} x^{2} x^{m} + 602532 \, a^{5} b^{2} m^{2} x^{3} x^{m} + 870660 \, a^{4} b^{3} m x^{4} x^{m} + 282240 \, a^{3} b^{4} x^{5} x^{m} + 18424 \, a^{7} m^{3} x x^{m} + 256942 \, a^{6} b m^{2} x^{2} x^{m} + 673008 \, a^{5} b^{2} m x^{3} x^{m} + 352800 \, a^{4} b^{3} x^{4} x^{m} + 48860 \, a^{7} m^{2} x x^{m} + 312984 \, a^{6} b m x^{2} x^{m} + 282240 \, a^{5} b^{2} x^{3} x^{m} + 69264 \, a^{7} m x x^{m} + 141120 \, a^{6} b x^{2} x^{m} + 40320 \, a^{7} x x^{m}}{m^{8} + 36 \, m^{7} + 546 \, m^{6} + 4536 \, m^{5} + 22449 \, m^{4} + 67284 \, m^{3} + 118124 \, m^{2} + 109584 \, m + 40320} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.78, size = 683, normalized size = 5.14 \begin {gather*} \frac {a^7\,x\,x^m\,\left (m^7+35\,m^6+511\,m^5+4025\,m^4+18424\,m^3+48860\,m^2+69264\,m+40320\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {b^7\,x^m\,x^8\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {21\,a^2\,b^5\,x^m\,x^6\,\left (m^7+30\,m^6+366\,m^5+2340\,m^4+8409\,m^3+16830\,m^2+17144\,m+6720\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {35\,a^3\,b^4\,x^m\,x^5\,\left (m^7+31\,m^6+391\,m^5+2581\,m^4+9544\,m^3+19564\,m^2+20304\,m+8064\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {35\,a^4\,b^3\,x^m\,x^4\,\left (m^7+32\,m^6+418\,m^5+2864\,m^4+10993\,m^3+23312\,m^2+24876\,m+10080\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {21\,a^5\,b^2\,x^m\,x^3\,\left (m^7+33\,m^6+447\,m^5+3195\,m^4+12864\,m^3+28692\,m^2+32048\,m+13440\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {7\,a\,b^6\,x^m\,x^7\,\left (m^7+29\,m^6+343\,m^5+2135\,m^4+7504\,m^3+14756\,m^2+14832\,m+5760\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {7\,a^6\,b\,x^m\,x^2\,\left (m^7+34\,m^6+478\,m^5+3580\,m^4+15289\,m^3+36706\,m^2+44712\,m+20160\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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