Optimal. Leaf size=97 \[ \frac {a^5 x^{1+m}}{1+m}+\frac {5 a^4 b x^{4+m}}{4+m}+\frac {10 a^3 b^2 x^{7+m}}{7+m}+\frac {10 a^2 b^3 x^{10+m}}{10+m}+\frac {5 a b^4 x^{13+m}}{13+m}+\frac {b^5 x^{16+m}}{16+m} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {276}
\begin {gather*} \frac {a^5 x^{m+1}}{m+1}+\frac {5 a^4 b x^{m+4}}{m+4}+\frac {10 a^3 b^2 x^{m+7}}{m+7}+\frac {10 a^2 b^3 x^{m+10}}{m+10}+\frac {5 a b^4 x^{m+13}}{m+13}+\frac {b^5 x^{m+16}}{m+16} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 276
Rubi steps
\begin {align*} \int x^m \left (a+b x^3\right )^5 \, dx &=\int \left (a^5 x^m+5 a^4 b x^{3+m}+10 a^3 b^2 x^{6+m}+10 a^2 b^3 x^{9+m}+5 a b^4 x^{12+m}+b^5 x^{15+m}\right ) \, dx\\ &=\frac {a^5 x^{1+m}}{1+m}+\frac {5 a^4 b x^{4+m}}{4+m}+\frac {10 a^3 b^2 x^{7+m}}{7+m}+\frac {10 a^2 b^3 x^{10+m}}{10+m}+\frac {5 a b^4 x^{13+m}}{13+m}+\frac {b^5 x^{16+m}}{16+m}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.18, size = 88, normalized size = 0.91 \begin {gather*} x^{1+m} \left (\frac {a^5}{1+m}+\frac {5 a^4 b x^3}{4+m}+\frac {10 a^3 b^2 x^6}{7+m}+\frac {10 a^2 b^3 x^9}{10+m}+\frac {5 a b^4 x^{12}}{13+m}+\frac {b^5 x^{15}}{16+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. \(430\) vs.
\(2(97)=194\).
time = 0.13, size = 431, normalized size = 4.44
method | result | size |
risch | \(\frac {x \left (b^{5} m^{5} x^{15}+35 b^{5} m^{4} x^{15}+445 b^{5} m^{3} x^{15}+5 a \,b^{4} m^{5} x^{12}+2485 b^{5} m^{2} x^{15}+190 a \,b^{4} m^{4} x^{12}+5714 m \,x^{15} b^{5}+2555 a \,b^{4} m^{3} x^{12}+3640 b^{5} x^{15}+10 a^{2} b^{3} m^{5} x^{9}+14810 a \,b^{4} m^{2} x^{12}+410 a^{2} b^{3} m^{4} x^{9}+34840 m \,x^{12} a \,b^{4}+5950 a^{2} b^{3} m^{3} x^{9}+22400 a \,b^{4} x^{12}+10 a^{3} b^{2} m^{5} x^{6}+36550 a^{2} b^{3} m^{2} x^{9}+440 a^{3} b^{2} m^{4} x^{6}+89240 m \,x^{9} a^{2} b^{3}+6970 a^{3} b^{2} m^{3} x^{6}+58240 a^{2} b^{3} x^{9}+5 a^{4} b \,m^{5} x^{3}+47260 a^{3} b^{2} m^{2} x^{6}+235 a^{4} b \,m^{4} x^{3}+123920 m \,x^{6} a^{3} b^{2}+4085 a^{4} b \,m^{3} x^{3}+83200 a^{3} b^{2} x^{6}+a^{5} m^{5}+31685 a^{4} b \,m^{2} x^{3}+50 a^{5} m^{4}+100630 m \,x^{3} a^{4} b +955 a^{5} m^{3}+72800 a^{4} b \,x^{3}+8650 a^{5} m^{2}+36824 m \,a^{5}+58240 a^{5}\right ) x^{m}}{\left (1+m \right ) \left (4+m \right ) \left (7+m \right ) \left (10+m \right ) \left (13+m \right ) \left (16+m \right )}\) | \(431\) |
gosper | \(\frac {x^{1+m} \left (b^{5} m^{5} x^{15}+35 b^{5} m^{4} x^{15}+445 b^{5} m^{3} x^{15}+5 a \,b^{4} m^{5} x^{12}+2485 b^{5} m^{2} x^{15}+190 a \,b^{4} m^{4} x^{12}+5714 m \,x^{15} b^{5}+2555 a \,b^{4} m^{3} x^{12}+3640 b^{5} x^{15}+10 a^{2} b^{3} m^{5} x^{9}+14810 a \,b^{4} m^{2} x^{12}+410 a^{2} b^{3} m^{4} x^{9}+34840 m \,x^{12} a \,b^{4}+5950 a^{2} b^{3} m^{3} x^{9}+22400 a \,b^{4} x^{12}+10 a^{3} b^{2} m^{5} x^{6}+36550 a^{2} b^{3} m^{2} x^{9}+440 a^{3} b^{2} m^{4} x^{6}+89240 m \,x^{9} a^{2} b^{3}+6970 a^{3} b^{2} m^{3} x^{6}+58240 a^{2} b^{3} x^{9}+5 a^{4} b \,m^{5} x^{3}+47260 a^{3} b^{2} m^{2} x^{6}+235 a^{4} b \,m^{4} x^{3}+123920 m \,x^{6} a^{3} b^{2}+4085 a^{4} b \,m^{3} x^{3}+83200 a^{3} b^{2} x^{6}+a^{5} m^{5}+31685 a^{4} b \,m^{2} x^{3}+50 a^{5} m^{4}+100630 m \,x^{3} a^{4} b +955 a^{5} m^{3}+72800 a^{4} b \,x^{3}+8650 a^{5} m^{2}+36824 m \,a^{5}+58240 a^{5}\right )}{\left (1+m \right ) \left (4+m \right ) \left (7+m \right ) \left (10+m \right ) \left (13+m \right ) \left (16+m \right )}\) | \(432\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 97, normalized size = 1.00 \begin {gather*} \frac {b^{5} x^{m + 16}}{m + 16} + \frac {5 \, a b^{4} x^{m + 13}}{m + 13} + \frac {10 \, a^{2} b^{3} x^{m + 10}}{m + 10} + \frac {10 \, a^{3} b^{2} x^{m + 7}}{m + 7} + \frac {5 \, a^{4} b x^{m + 4}}{m + 4} + \frac {a^{5} x^{m + 1}}{m + 1} \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. 367 vs.
\(2 (97) = 194\).
time = 0.37, size = 367, normalized size = 3.78 \begin {gather*} \frac {{\left ({\left (b^{5} m^{5} + 35 \, b^{5} m^{4} + 445 \, b^{5} m^{3} + 2485 \, b^{5} m^{2} + 5714 \, b^{5} m + 3640 \, b^{5}\right )} x^{16} + 5 \, {\left (a b^{4} m^{5} + 38 \, a b^{4} m^{4} + 511 \, a b^{4} m^{3} + 2962 \, a b^{4} m^{2} + 6968 \, a b^{4} m + 4480 \, a b^{4}\right )} x^{13} + 10 \, {\left (a^{2} b^{3} m^{5} + 41 \, a^{2} b^{3} m^{4} + 595 \, a^{2} b^{3} m^{3} + 3655 \, a^{2} b^{3} m^{2} + 8924 \, a^{2} b^{3} m + 5824 \, a^{2} b^{3}\right )} x^{10} + 10 \, {\left (a^{3} b^{2} m^{5} + 44 \, a^{3} b^{2} m^{4} + 697 \, a^{3} b^{2} m^{3} + 4726 \, a^{3} b^{2} m^{2} + 12392 \, a^{3} b^{2} m + 8320 \, a^{3} b^{2}\right )} x^{7} + 5 \, {\left (a^{4} b m^{5} + 47 \, a^{4} b m^{4} + 817 \, a^{4} b m^{3} + 6337 \, a^{4} b m^{2} + 20126 \, a^{4} b m + 14560 \, a^{4} b\right )} x^{4} + {\left (a^{5} m^{5} + 50 \, a^{5} m^{4} + 955 \, a^{5} m^{3} + 8650 \, a^{5} m^{2} + 36824 \, a^{5} m + 58240 \, a^{5}\right )} x\right )} x^{m}}{m^{6} + 51 \, m^{5} + 1005 \, m^{4} + 9605 \, m^{3} + 45474 \, m^{2} + 95064 \, m + 58240} \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. 2006 vs.
\(2 (87) = 174\).
time = 1.12, size = 2006, normalized size = 20.68 \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. 540 vs.
\(2 (97) = 194\).
time = 1.46, size = 540, normalized size = 5.57 \begin {gather*} \frac {b^{5} m^{5} x^{16} x^{m} + 35 \, b^{5} m^{4} x^{16} x^{m} + 445 \, b^{5} m^{3} x^{16} x^{m} + 5 \, a b^{4} m^{5} x^{13} x^{m} + 2485 \, b^{5} m^{2} x^{16} x^{m} + 190 \, a b^{4} m^{4} x^{13} x^{m} + 5714 \, b^{5} m x^{16} x^{m} + 2555 \, a b^{4} m^{3} x^{13} x^{m} + 3640 \, b^{5} x^{16} x^{m} + 10 \, a^{2} b^{3} m^{5} x^{10} x^{m} + 14810 \, a b^{4} m^{2} x^{13} x^{m} + 410 \, a^{2} b^{3} m^{4} x^{10} x^{m} + 34840 \, a b^{4} m x^{13} x^{m} + 5950 \, a^{2} b^{3} m^{3} x^{10} x^{m} + 22400 \, a b^{4} x^{13} x^{m} + 10 \, a^{3} b^{2} m^{5} x^{7} x^{m} + 36550 \, a^{2} b^{3} m^{2} x^{10} x^{m} + 440 \, a^{3} b^{2} m^{4} x^{7} x^{m} + 89240 \, a^{2} b^{3} m x^{10} x^{m} + 6970 \, a^{3} b^{2} m^{3} x^{7} x^{m} + 58240 \, a^{2} b^{3} x^{10} x^{m} + 5 \, a^{4} b m^{5} x^{4} x^{m} + 47260 \, a^{3} b^{2} m^{2} x^{7} x^{m} + 235 \, a^{4} b m^{4} x^{4} x^{m} + 123920 \, a^{3} b^{2} m x^{7} x^{m} + 4085 \, a^{4} b m^{3} x^{4} x^{m} + 83200 \, a^{3} b^{2} x^{7} x^{m} + a^{5} m^{5} x x^{m} + 31685 \, a^{4} b m^{2} x^{4} x^{m} + 50 \, a^{5} m^{4} x x^{m} + 100630 \, a^{4} b m x^{4} x^{m} + 955 \, a^{5} m^{3} x x^{m} + 72800 \, a^{4} b x^{4} x^{m} + 8650 \, a^{5} m^{2} x x^{m} + 36824 \, a^{5} m x x^{m} + 58240 \, a^{5} x x^{m}}{m^{6} + 51 \, m^{5} + 1005 \, m^{4} + 9605 \, m^{3} + 45474 \, m^{2} + 95064 \, m + 58240} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.55, size = 389, normalized size = 4.01 \begin {gather*} \frac {a^5\,x\,x^m\,\left (m^5+50\,m^4+955\,m^3+8650\,m^2+36824\,m+58240\right )}{m^6+51\,m^5+1005\,m^4+9605\,m^3+45474\,m^2+95064\,m+58240}+\frac {b^5\,x^m\,x^{16}\,\left (m^5+35\,m^4+445\,m^3+2485\,m^2+5714\,m+3640\right )}{m^6+51\,m^5+1005\,m^4+9605\,m^3+45474\,m^2+95064\,m+58240}+\frac {5\,a\,b^4\,x^m\,x^{13}\,\left (m^5+38\,m^4+511\,m^3+2962\,m^2+6968\,m+4480\right )}{m^6+51\,m^5+1005\,m^4+9605\,m^3+45474\,m^2+95064\,m+58240}+\frac {5\,a^4\,b\,x^m\,x^4\,\left (m^5+47\,m^4+817\,m^3+6337\,m^2+20126\,m+14560\right )}{m^6+51\,m^5+1005\,m^4+9605\,m^3+45474\,m^2+95064\,m+58240}+\frac {10\,a^2\,b^3\,x^m\,x^{10}\,\left (m^5+41\,m^4+595\,m^3+3655\,m^2+8924\,m+5824\right )}{m^6+51\,m^5+1005\,m^4+9605\,m^3+45474\,m^2+95064\,m+58240}+\frac {10\,a^3\,b^2\,x^m\,x^7\,\left (m^5+44\,m^4+697\,m^3+4726\,m^2+12392\,m+8320\right )}{m^6+51\,m^5+1005\,m^4+9605\,m^3+45474\,m^2+95064\,m+58240} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________