Optimal. Leaf size=183 \[ \frac {3 a^8 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^9}-\frac {24 a^7 \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^9}+\frac {14 a^6 \left (a+b \sqrt [3]{x}\right )^{18}}{3 b^9}-\frac {168 a^5 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^9}+\frac {21 a^4 \left (a+b \sqrt [3]{x}\right )^{20}}{2 b^9}-\frac {8 a^3 \left (a+b \sqrt [3]{x}\right )^{21}}{b^9}+\frac {42 a^2 \left (a+b \sqrt [3]{x}\right )^{22}}{11 b^9}+\frac {\left (a+b \sqrt [3]{x}\right )^{24}}{8 b^9}-\frac {24 a \left (a+b \sqrt [3]{x}\right )^{23}}{23 b^9} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac {42 a^2 \left (a+b \sqrt [3]{x}\right )^{22}}{11 b^9}-\frac {8 a^3 \left (a+b \sqrt [3]{x}\right )^{21}}{b^9}+\frac {21 a^4 \left (a+b \sqrt [3]{x}\right )^{20}}{2 b^9}-\frac {168 a^5 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^9}+\frac {14 a^6 \left (a+b \sqrt [3]{x}\right )^{18}}{3 b^9}-\frac {24 a^7 \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^9}+\frac {3 a^8 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^9}+\frac {\left (a+b \sqrt [3]{x}\right )^{24}}{8 b^9}-\frac {24 a \left (a+b \sqrt [3]{x}\right )^{23}}{23 b^9} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \left (a+b \sqrt [3]{x}\right )^{15} x^2 \, dx &=3 \operatorname {Subst}\left (\int x^8 (a+b x)^{15} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {a^8 (a+b x)^{15}}{b^8}-\frac {8 a^7 (a+b x)^{16}}{b^8}+\frac {28 a^6 (a+b x)^{17}}{b^8}-\frac {56 a^5 (a+b x)^{18}}{b^8}+\frac {70 a^4 (a+b x)^{19}}{b^8}-\frac {56 a^3 (a+b x)^{20}}{b^8}+\frac {28 a^2 (a+b x)^{21}}{b^8}-\frac {8 a (a+b x)^{22}}{b^8}+\frac {(a+b x)^{23}}{b^8}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3 a^8 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^9}-\frac {24 a^7 \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^9}+\frac {14 a^6 \left (a+b \sqrt [3]{x}\right )^{18}}{3 b^9}-\frac {168 a^5 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^9}+\frac {21 a^4 \left (a+b \sqrt [3]{x}\right )^{20}}{2 b^9}-\frac {8 a^3 \left (a+b \sqrt [3]{x}\right )^{21}}{b^9}+\frac {42 a^2 \left (a+b \sqrt [3]{x}\right )^{22}}{11 b^9}-\frac {24 a \left (a+b \sqrt [3]{x}\right )^{23}}{23 b^9}+\frac {\left (a+b \sqrt [3]{x}\right )^{24}}{8 b^9}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 113, normalized size = 0.62 \[ \frac {\left (a+b \sqrt [3]{x}\right )^{16} \left (a^8-16 a^7 b \sqrt [3]{x}+136 a^6 b^2 x^{2/3}-816 a^5 b^3 x+3876 a^4 b^4 x^{4/3}-15504 a^3 b^5 x^{5/3}+54264 a^2 b^6 x^2-170544 a b^7 x^{7/3}+490314 b^8 x^{8/3}\right )}{3922512 b^9} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.54, size = 179, normalized size = 0.98 \[ \frac {1}{8} \, b^{15} x^{8} + 65 \, a^{3} b^{12} x^{7} + \frac {5005}{6} \, a^{6} b^{9} x^{6} + 1001 \, a^{9} b^{6} x^{5} + \frac {455}{4} \, a^{12} b^{3} x^{4} + \frac {1}{3} \, a^{15} x^{3} + \frac {9}{17204} \, {\left (3740 \, a b^{14} x^{7} + 391391 \, a^{4} b^{11} x^{6} + 2170740 \, a^{7} b^{8} x^{5} + 1230086 \, a^{10} b^{5} x^{4} + 54740 \, a^{13} b^{2} x^{3}\right )} x^{\frac {2}{3}} + \frac {9}{3344} \, {\left (5320 \, a^{2} b^{13} x^{7} + 176176 \, a^{5} b^{10} x^{6} + 448305 \, a^{8} b^{7} x^{5} + 117040 \, a^{11} b^{4} x^{4} + 1672 \, a^{14} b x^{3}\right )} x^{\frac {1}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 167, normalized size = 0.91 \[ \frac {1}{8} \, b^{15} x^{8} + \frac {45}{23} \, a b^{14} x^{\frac {23}{3}} + \frac {315}{22} \, a^{2} b^{13} x^{\frac {22}{3}} + 65 \, a^{3} b^{12} x^{7} + \frac {819}{4} \, a^{4} b^{11} x^{\frac {20}{3}} + \frac {9009}{19} \, a^{5} b^{10} x^{\frac {19}{3}} + \frac {5005}{6} \, a^{6} b^{9} x^{6} + \frac {19305}{17} \, a^{7} b^{8} x^{\frac {17}{3}} + \frac {19305}{16} \, a^{8} b^{7} x^{\frac {16}{3}} + 1001 \, a^{9} b^{6} x^{5} + \frac {1287}{2} \, a^{10} b^{5} x^{\frac {14}{3}} + 315 \, a^{11} b^{4} x^{\frac {13}{3}} + \frac {455}{4} \, a^{12} b^{3} x^{4} + \frac {315}{11} \, a^{13} b^{2} x^{\frac {11}{3}} + \frac {9}{2} \, a^{14} b x^{\frac {10}{3}} + \frac {1}{3} \, a^{15} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 168, normalized size = 0.92 \[ \frac {b^{15} x^{8}}{8}+\frac {45 a \,b^{14} x^{\frac {23}{3}}}{23}+\frac {315 a^{2} b^{13} x^{\frac {22}{3}}}{22}+65 a^{3} b^{12} x^{7}+\frac {819 a^{4} b^{11} x^{\frac {20}{3}}}{4}+\frac {9009 a^{5} b^{10} x^{\frac {19}{3}}}{19}+\frac {5005 a^{6} b^{9} x^{6}}{6}+\frac {19305 a^{7} b^{8} x^{\frac {17}{3}}}{17}+\frac {19305 a^{8} b^{7} x^{\frac {16}{3}}}{16}+1001 a^{9} b^{6} x^{5}+\frac {1287 a^{10} b^{5} x^{\frac {14}{3}}}{2}+315 a^{11} b^{4} x^{\frac {13}{3}}+\frac {455 a^{12} b^{3} x^{4}}{4}+\frac {315 a^{13} b^{2} x^{\frac {11}{3}}}{11}+\frac {9 a^{14} b \,x^{\frac {10}{3}}}{2}+\frac {a^{15} x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.92, size = 149, normalized size = 0.81 \[ \frac {{\left (b x^{\frac {1}{3}} + a\right )}^{24}}{8 \, b^{9}} - \frac {24 \, {\left (b x^{\frac {1}{3}} + a\right )}^{23} a}{23 \, b^{9}} + \frac {42 \, {\left (b x^{\frac {1}{3}} + a\right )}^{22} a^{2}}{11 \, b^{9}} - \frac {8 \, {\left (b x^{\frac {1}{3}} + a\right )}^{21} a^{3}}{b^{9}} + \frac {21 \, {\left (b x^{\frac {1}{3}} + a\right )}^{20} a^{4}}{2 \, b^{9}} - \frac {168 \, {\left (b x^{\frac {1}{3}} + a\right )}^{19} a^{5}}{19 \, b^{9}} + \frac {14 \, {\left (b x^{\frac {1}{3}} + a\right )}^{18} a^{6}}{3 \, b^{9}} - \frac {24 \, {\left (b x^{\frac {1}{3}} + a\right )}^{17} a^{7}}{17 \, b^{9}} + \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{16} a^{8}}{16 \, b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.16, size = 167, normalized size = 0.91 \[ \frac {a^{15}\,x^3}{3}+\frac {b^{15}\,x^8}{8}+\frac {9\,a^{14}\,b\,x^{10/3}}{2}+\frac {45\,a\,b^{14}\,x^{23/3}}{23}+\frac {455\,a^{12}\,b^3\,x^4}{4}+1001\,a^9\,b^6\,x^5+\frac {5005\,a^6\,b^9\,x^6}{6}+65\,a^3\,b^{12}\,x^7+\frac {315\,a^{13}\,b^2\,x^{11/3}}{11}+315\,a^{11}\,b^4\,x^{13/3}+\frac {1287\,a^{10}\,b^5\,x^{14/3}}{2}+\frac {19305\,a^8\,b^7\,x^{16/3}}{16}+\frac {19305\,a^7\,b^8\,x^{17/3}}{17}+\frac {9009\,a^5\,b^{10}\,x^{19/3}}{19}+\frac {819\,a^4\,b^{11}\,x^{20/3}}{4}+\frac {315\,a^2\,b^{13}\,x^{22/3}}{22} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 5.63, size = 214, normalized size = 1.17 \[ \frac {a^{15} x^{3}}{3} + \frac {9 a^{14} b x^{\frac {10}{3}}}{2} + \frac {315 a^{13} b^{2} x^{\frac {11}{3}}}{11} + \frac {455 a^{12} b^{3} x^{4}}{4} + 315 a^{11} b^{4} x^{\frac {13}{3}} + \frac {1287 a^{10} b^{5} x^{\frac {14}{3}}}{2} + 1001 a^{9} b^{6} x^{5} + \frac {19305 a^{8} b^{7} x^{\frac {16}{3}}}{16} + \frac {19305 a^{7} b^{8} x^{\frac {17}{3}}}{17} + \frac {5005 a^{6} b^{9} x^{6}}{6} + \frac {9009 a^{5} b^{10} x^{\frac {19}{3}}}{19} + \frac {819 a^{4} b^{11} x^{\frac {20}{3}}}{4} + 65 a^{3} b^{12} x^{7} + \frac {315 a^{2} b^{13} x^{\frac {22}{3}}}{22} + \frac {45 a b^{14} x^{\frac {23}{3}}}{23} + \frac {b^{15} x^{8}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________