Optimal. Leaf size=150 \[ \frac {a^6 (d x)^{1+m}}{d (1+m)}+\frac {6 a^5 b (d x)^{3+m}}{d^3 (3+m)}+\frac {15 a^4 b^2 (d x)^{5+m}}{d^5 (5+m)}+\frac {20 a^3 b^3 (d x)^{7+m}}{d^7 (7+m)}+\frac {15 a^2 b^4 (d x)^{9+m}}{d^9 (9+m)}+\frac {6 a b^5 (d x)^{11+m}}{d^{11} (11+m)}+\frac {b^6 (d x)^{13+m}}{d^{13} (13+m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {28, 276}
\begin {gather*} \frac {a^6 (d x)^{m+1}}{d (m+1)}+\frac {6 a^5 b (d x)^{m+3}}{d^3 (m+3)}+\frac {15 a^4 b^2 (d x)^{m+5}}{d^5 (m+5)}+\frac {20 a^3 b^3 (d x)^{m+7}}{d^7 (m+7)}+\frac {15 a^2 b^4 (d x)^{m+9}}{d^9 (m+9)}+\frac {6 a b^5 (d x)^{m+11}}{d^{11} (m+11)}+\frac {b^6 (d x)^{m+13}}{d^{13} (m+13)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 28
Rule 276
Rubi steps
\begin {align*} \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx &=\frac {\int (d x)^m \left (a b+b^2 x^2\right )^6 \, dx}{b^6}\\ &=\frac {\int \left (a^6 b^6 (d x)^m+\frac {6 a^5 b^7 (d x)^{2+m}}{d^2}+\frac {15 a^4 b^8 (d x)^{4+m}}{d^4}+\frac {20 a^3 b^9 (d x)^{6+m}}{d^6}+\frac {15 a^2 b^{10} (d x)^{8+m}}{d^8}+\frac {6 a b^{11} (d x)^{10+m}}{d^{10}}+\frac {b^{12} (d x)^{12+m}}{d^{12}}\right ) \, dx}{b^6}\\ &=\frac {a^6 (d x)^{1+m}}{d (1+m)}+\frac {6 a^5 b (d x)^{3+m}}{d^3 (3+m)}+\frac {15 a^4 b^2 (d x)^{5+m}}{d^5 (5+m)}+\frac {20 a^3 b^3 (d x)^{7+m}}{d^7 (7+m)}+\frac {15 a^2 b^4 (d x)^{9+m}}{d^9 (9+m)}+\frac {6 a b^5 (d x)^{11+m}}{d^{11} (11+m)}+\frac {b^6 (d x)^{13+m}}{d^{13} (13+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.18, size = 105, normalized size = 0.70 \begin {gather*} x (d x)^m \left (\frac {a^6}{1+m}+\frac {6 a^5 b x^2}{3+m}+\frac {15 a^4 b^2 x^4}{5+m}+\frac {20 a^3 b^3 x^6}{7+m}+\frac {15 a^2 b^4 x^8}{9+m}+\frac {6 a b^5 x^{10}}{11+m}+\frac {b^6 x^{12}}{13+m}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(601\) vs.
\(2(150)=300\).
time = 0.02, size = 602, normalized size = 4.01
method | result | size |
gosper | \(\frac {\left (d x \right )^{m} \left (b^{6} m^{6} x^{12}+36 b^{6} m^{5} x^{12}+6 a \,b^{5} m^{6} x^{10}+505 b^{6} m^{4} x^{12}+228 a \,b^{5} m^{5} x^{10}+3480 b^{6} m^{3} x^{12}+15 a^{2} b^{4} m^{6} x^{8}+3330 a \,b^{5} m^{4} x^{10}+12139 b^{6} m^{2} x^{12}+600 a^{2} b^{4} m^{5} x^{8}+23640 a \,b^{5} m^{3} x^{10}+19524 m \,x^{12} b^{6}+20 a^{3} b^{3} m^{6} x^{6}+9195 a^{2} b^{4} m^{4} x^{8}+84234 a \,b^{5} m^{2} x^{10}+10395 b^{6} x^{12}+840 a^{3} b^{3} m^{5} x^{6}+67920 a^{2} b^{4} m^{3} x^{8}+137412 m \,x^{10} a \,b^{5}+15 a^{4} b^{2} m^{6} x^{4}+13580 a^{3} b^{3} m^{4} x^{6}+249405 a^{2} b^{4} m^{2} x^{8}+73710 a \,b^{5} x^{10}+660 a^{4} b^{2} m^{5} x^{4}+105840 a^{3} b^{3} m^{3} x^{6}+415320 m \,x^{8} a^{2} b^{4}+6 a^{5} b \,m^{6} x^{2}+11295 a^{4} b^{2} m^{4} x^{4}+406700 a^{3} b^{3} m^{2} x^{6}+225225 a^{2} b^{4} x^{8}+276 a^{5} b \,m^{5} x^{2}+94200 a^{4} b^{2} m^{3} x^{4}+699720 m \,x^{6} a^{3} b^{3}+a^{6} m^{6}+5010 a^{5} b \,m^{4} x^{2}+389685 a^{4} b^{2} m^{2} x^{4}+386100 a^{3} b^{3} x^{6}+48 a^{6} m^{5}+45240 a^{5} b \,m^{3} x^{2}+711540 m \,x^{4} a^{4} b^{2}+925 a^{6} m^{4}+208554 a^{5} b \,m^{2} x^{2}+405405 a^{4} b^{2} x^{4}+9120 a^{6} m^{3}+438324 m \,x^{2} a^{5} b +48259 a^{6} m^{2}+270270 a^{5} b \,x^{2}+129072 m \,a^{6}+135135 a^{6}\right ) x}{\left (13+m \right ) \left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(602\) |
risch | \(\frac {\left (d x \right )^{m} \left (b^{6} m^{6} x^{12}+36 b^{6} m^{5} x^{12}+6 a \,b^{5} m^{6} x^{10}+505 b^{6} m^{4} x^{12}+228 a \,b^{5} m^{5} x^{10}+3480 b^{6} m^{3} x^{12}+15 a^{2} b^{4} m^{6} x^{8}+3330 a \,b^{5} m^{4} x^{10}+12139 b^{6} m^{2} x^{12}+600 a^{2} b^{4} m^{5} x^{8}+23640 a \,b^{5} m^{3} x^{10}+19524 m \,x^{12} b^{6}+20 a^{3} b^{3} m^{6} x^{6}+9195 a^{2} b^{4} m^{4} x^{8}+84234 a \,b^{5} m^{2} x^{10}+10395 b^{6} x^{12}+840 a^{3} b^{3} m^{5} x^{6}+67920 a^{2} b^{4} m^{3} x^{8}+137412 m \,x^{10} a \,b^{5}+15 a^{4} b^{2} m^{6} x^{4}+13580 a^{3} b^{3} m^{4} x^{6}+249405 a^{2} b^{4} m^{2} x^{8}+73710 a \,b^{5} x^{10}+660 a^{4} b^{2} m^{5} x^{4}+105840 a^{3} b^{3} m^{3} x^{6}+415320 m \,x^{8} a^{2} b^{4}+6 a^{5} b \,m^{6} x^{2}+11295 a^{4} b^{2} m^{4} x^{4}+406700 a^{3} b^{3} m^{2} x^{6}+225225 a^{2} b^{4} x^{8}+276 a^{5} b \,m^{5} x^{2}+94200 a^{4} b^{2} m^{3} x^{4}+699720 m \,x^{6} a^{3} b^{3}+a^{6} m^{6}+5010 a^{5} b \,m^{4} x^{2}+389685 a^{4} b^{2} m^{2} x^{4}+386100 a^{3} b^{3} x^{6}+48 a^{6} m^{5}+45240 a^{5} b \,m^{3} x^{2}+711540 m \,x^{4} a^{4} b^{2}+925 a^{6} m^{4}+208554 a^{5} b \,m^{2} x^{2}+405405 a^{4} b^{2} x^{4}+9120 a^{6} m^{3}+438324 m \,x^{2} a^{5} b +48259 a^{6} m^{2}+270270 a^{5} b \,x^{2}+129072 m \,a^{6}+135135 a^{6}\right ) x}{\left (13+m \right ) \left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(602\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 144, normalized size = 0.96 \begin {gather*} \frac {b^{6} d^{m} x^{13} x^{m}}{m + 13} + \frac {6 \, a b^{5} d^{m} x^{11} x^{m}}{m + 11} + \frac {15 \, a^{2} b^{4} d^{m} x^{9} x^{m}}{m + 9} + \frac {20 \, a^{3} b^{3} d^{m} x^{7} x^{m}}{m + 7} + \frac {15 \, a^{4} b^{2} d^{m} x^{5} x^{m}}{m + 5} + \frac {6 \, a^{5} b d^{m} x^{3} x^{m}}{m + 3} + \frac {\left (d x\right )^{m + 1} a^{6}}{d {\left (m + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 507 vs.
\(2 (150) = 300\).
time = 0.34, size = 507, normalized size = 3.38 \begin {gather*} \frac {{\left ({\left (b^{6} m^{6} + 36 \, b^{6} m^{5} + 505 \, b^{6} m^{4} + 3480 \, b^{6} m^{3} + 12139 \, b^{6} m^{2} + 19524 \, b^{6} m + 10395 \, b^{6}\right )} x^{13} + 6 \, {\left (a b^{5} m^{6} + 38 \, a b^{5} m^{5} + 555 \, a b^{5} m^{4} + 3940 \, a b^{5} m^{3} + 14039 \, a b^{5} m^{2} + 22902 \, a b^{5} m + 12285 \, a b^{5}\right )} x^{11} + 15 \, {\left (a^{2} b^{4} m^{6} + 40 \, a^{2} b^{4} m^{5} + 613 \, a^{2} b^{4} m^{4} + 4528 \, a^{2} b^{4} m^{3} + 16627 \, a^{2} b^{4} m^{2} + 27688 \, a^{2} b^{4} m + 15015 \, a^{2} b^{4}\right )} x^{9} + 20 \, {\left (a^{3} b^{3} m^{6} + 42 \, a^{3} b^{3} m^{5} + 679 \, a^{3} b^{3} m^{4} + 5292 \, a^{3} b^{3} m^{3} + 20335 \, a^{3} b^{3} m^{2} + 34986 \, a^{3} b^{3} m + 19305 \, a^{3} b^{3}\right )} x^{7} + 15 \, {\left (a^{4} b^{2} m^{6} + 44 \, a^{4} b^{2} m^{5} + 753 \, a^{4} b^{2} m^{4} + 6280 \, a^{4} b^{2} m^{3} + 25979 \, a^{4} b^{2} m^{2} + 47436 \, a^{4} b^{2} m + 27027 \, a^{4} b^{2}\right )} x^{5} + 6 \, {\left (a^{5} b m^{6} + 46 \, a^{5} b m^{5} + 835 \, a^{5} b m^{4} + 7540 \, a^{5} b m^{3} + 34759 \, a^{5} b m^{2} + 73054 \, a^{5} b m + 45045 \, a^{5} b\right )} x^{3} + {\left (a^{6} m^{6} + 48 \, a^{6} m^{5} + 925 \, a^{6} m^{4} + 9120 \, a^{6} m^{3} + 48259 \, a^{6} m^{2} + 129072 \, a^{6} m + 135135 \, a^{6}\right )} x\right )} \left (d x\right )^{m}}{m^{7} + 49 \, m^{6} + 973 \, m^{5} + 10045 \, m^{4} + 57379 \, m^{3} + 177331 \, m^{2} + 264207 \, m + 135135} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 3104 vs.
\(2 (138) = 276\).
time = 0.95, size = 3104, normalized size = 20.69 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 847 vs.
\(2 (150) = 300\).
time = 4.43, size = 847, normalized size = 5.65 \begin {gather*} \frac {\left (d x\right )^{m} b^{6} m^{6} x^{13} + 36 \, \left (d x\right )^{m} b^{6} m^{5} x^{13} + 6 \, \left (d x\right )^{m} a b^{5} m^{6} x^{11} + 505 \, \left (d x\right )^{m} b^{6} m^{4} x^{13} + 228 \, \left (d x\right )^{m} a b^{5} m^{5} x^{11} + 3480 \, \left (d x\right )^{m} b^{6} m^{3} x^{13} + 15 \, \left (d x\right )^{m} a^{2} b^{4} m^{6} x^{9} + 3330 \, \left (d x\right )^{m} a b^{5} m^{4} x^{11} + 12139 \, \left (d x\right )^{m} b^{6} m^{2} x^{13} + 600 \, \left (d x\right )^{m} a^{2} b^{4} m^{5} x^{9} + 23640 \, \left (d x\right )^{m} a b^{5} m^{3} x^{11} + 19524 \, \left (d x\right )^{m} b^{6} m x^{13} + 20 \, \left (d x\right )^{m} a^{3} b^{3} m^{6} x^{7} + 9195 \, \left (d x\right )^{m} a^{2} b^{4} m^{4} x^{9} + 84234 \, \left (d x\right )^{m} a b^{5} m^{2} x^{11} + 10395 \, \left (d x\right )^{m} b^{6} x^{13} + 840 \, \left (d x\right )^{m} a^{3} b^{3} m^{5} x^{7} + 67920 \, \left (d x\right )^{m} a^{2} b^{4} m^{3} x^{9} + 137412 \, \left (d x\right )^{m} a b^{5} m x^{11} + 15 \, \left (d x\right )^{m} a^{4} b^{2} m^{6} x^{5} + 13580 \, \left (d x\right )^{m} a^{3} b^{3} m^{4} x^{7} + 249405 \, \left (d x\right )^{m} a^{2} b^{4} m^{2} x^{9} + 73710 \, \left (d x\right )^{m} a b^{5} x^{11} + 660 \, \left (d x\right )^{m} a^{4} b^{2} m^{5} x^{5} + 105840 \, \left (d x\right )^{m} a^{3} b^{3} m^{3} x^{7} + 415320 \, \left (d x\right )^{m} a^{2} b^{4} m x^{9} + 6 \, \left (d x\right )^{m} a^{5} b m^{6} x^{3} + 11295 \, \left (d x\right )^{m} a^{4} b^{2} m^{4} x^{5} + 406700 \, \left (d x\right )^{m} a^{3} b^{3} m^{2} x^{7} + 225225 \, \left (d x\right )^{m} a^{2} b^{4} x^{9} + 276 \, \left (d x\right )^{m} a^{5} b m^{5} x^{3} + 94200 \, \left (d x\right )^{m} a^{4} b^{2} m^{3} x^{5} + 699720 \, \left (d x\right )^{m} a^{3} b^{3} m x^{7} + \left (d x\right )^{m} a^{6} m^{6} x + 5010 \, \left (d x\right )^{m} a^{5} b m^{4} x^{3} + 389685 \, \left (d x\right )^{m} a^{4} b^{2} m^{2} x^{5} + 386100 \, \left (d x\right )^{m} a^{3} b^{3} x^{7} + 48 \, \left (d x\right )^{m} a^{6} m^{5} x + 45240 \, \left (d x\right )^{m} a^{5} b m^{3} x^{3} + 711540 \, \left (d x\right )^{m} a^{4} b^{2} m x^{5} + 925 \, \left (d x\right )^{m} a^{6} m^{4} x + 208554 \, \left (d x\right )^{m} a^{5} b m^{2} x^{3} + 405405 \, \left (d x\right )^{m} a^{4} b^{2} x^{5} + 9120 \, \left (d x\right )^{m} a^{6} m^{3} x + 438324 \, \left (d x\right )^{m} a^{5} b m x^{3} + 48259 \, \left (d x\right )^{m} a^{6} m^{2} x + 270270 \, \left (d x\right )^{m} a^{5} b x^{3} + 129072 \, \left (d x\right )^{m} a^{6} m x + 135135 \, \left (d x\right )^{m} a^{6} x}{m^{7} + 49 \, m^{6} + 973 \, m^{5} + 10045 \, m^{4} + 57379 \, m^{3} + 177331 \, m^{2} + 264207 \, m + 135135} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.58, size = 540, normalized size = 3.60 \begin {gather*} \frac {a^6\,x\,{\left (d\,x\right )}^m\,\left (m^6+48\,m^5+925\,m^4+9120\,m^3+48259\,m^2+129072\,m+135135\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {b^6\,x^{13}\,{\left (d\,x\right )}^m\,\left (m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {6\,a\,b^5\,x^{11}\,{\left (d\,x\right )}^m\,\left (m^6+38\,m^5+555\,m^4+3940\,m^3+14039\,m^2+22902\,m+12285\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {6\,a^5\,b\,x^3\,{\left (d\,x\right )}^m\,\left (m^6+46\,m^5+835\,m^4+7540\,m^3+34759\,m^2+73054\,m+45045\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {15\,a^2\,b^4\,x^9\,{\left (d\,x\right )}^m\,\left (m^6+40\,m^5+613\,m^4+4528\,m^3+16627\,m^2+27688\,m+15015\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {20\,a^3\,b^3\,x^7\,{\left (d\,x\right )}^m\,\left (m^6+42\,m^5+679\,m^4+5292\,m^3+20335\,m^2+34986\,m+19305\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {15\,a^4\,b^2\,x^5\,{\left (d\,x\right )}^m\,\left (m^6+44\,m^5+753\,m^4+6280\,m^3+25979\,m^2+47436\,m+27027\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________