Optimal. Leaf size=162 \[ -\frac {2 a^7 \left (a+b \sqrt {x}\right )^{11}}{11 b^8}+\frac {7 a^6 \left (a+b \sqrt {x}\right )^{12}}{6 b^8}-\frac {42 a^5 \left (a+b \sqrt {x}\right )^{13}}{13 b^8}+\frac {5 a^4 \left (a+b \sqrt {x}\right )^{14}}{b^8}-\frac {14 a^3 \left (a+b \sqrt {x}\right )^{15}}{3 b^8}+\frac {21 a^2 \left (a+b \sqrt {x}\right )^{16}}{8 b^8}+\frac {\left (a+b \sqrt {x}\right )^{18}}{9 b^8}-\frac {14 a \left (a+b \sqrt {x}\right )^{17}}{17 b^8} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 162, 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} \[ \frac {21 a^2 \left (a+b \sqrt {x}\right )^{16}}{8 b^8}-\frac {14 a^3 \left (a+b \sqrt {x}\right )^{15}}{3 b^8}+\frac {5 a^4 \left (a+b \sqrt {x}\right )^{14}}{b^8}-\frac {42 a^5 \left (a+b \sqrt {x}\right )^{13}}{13 b^8}+\frac {7 a^6 \left (a+b \sqrt {x}\right )^{12}}{6 b^8}-\frac {2 a^7 \left (a+b \sqrt {x}\right )^{11}}{11 b^8}+\frac {\left (a+b \sqrt {x}\right )^{18}}{9 b^8}-\frac {14 a \left (a+b \sqrt {x}\right )^{17}}{17 b^8} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \left (a+b \sqrt {x}\right )^{10} x^3 \, dx &=2 \operatorname {Subst}\left (\int x^7 (a+b x)^{10} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\frac {a^7 (a+b x)^{10}}{b^7}+\frac {7 a^6 (a+b x)^{11}}{b^7}-\frac {21 a^5 (a+b x)^{12}}{b^7}+\frac {35 a^4 (a+b x)^{13}}{b^7}-\frac {35 a^3 (a+b x)^{14}}{b^7}+\frac {21 a^2 (a+b x)^{15}}{b^7}-\frac {7 a (a+b x)^{16}}{b^7}+\frac {(a+b x)^{17}}{b^7}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 a^7 \left (a+b \sqrt {x}\right )^{11}}{11 b^8}+\frac {7 a^6 \left (a+b \sqrt {x}\right )^{12}}{6 b^8}-\frac {42 a^5 \left (a+b \sqrt {x}\right )^{13}}{13 b^8}+\frac {5 a^4 \left (a+b \sqrt {x}\right )^{14}}{b^8}-\frac {14 a^3 \left (a+b \sqrt {x}\right )^{15}}{3 b^8}+\frac {21 a^2 \left (a+b \sqrt {x}\right )^{16}}{8 b^8}-\frac {14 a \left (a+b \sqrt {x}\right )^{17}}{17 b^8}+\frac {\left (a+b \sqrt {x}\right )^{18}}{9 b^8}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 136, normalized size = 0.84 \[ \frac {a^{10} x^4}{4}+\frac {20}{9} a^9 b x^{9/2}+9 a^8 b^2 x^5+\frac {240}{11} a^7 b^3 x^{11/2}+35 a^6 b^4 x^6+\frac {504}{13} a^5 b^5 x^{13/2}+30 a^4 b^6 x^7+16 a^3 b^7 x^{15/2}+\frac {45}{8} a^2 b^8 x^8+\frac {20}{17} a b^9 x^{17/2}+\frac {b^{10} x^9}{9} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.96, size = 118, normalized size = 0.73 \[ \frac {1}{9} \, b^{10} x^{9} + \frac {45}{8} \, a^{2} b^{8} x^{8} + 30 \, a^{4} b^{6} x^{7} + 35 \, a^{6} b^{4} x^{6} + 9 \, a^{8} b^{2} x^{5} + \frac {1}{4} \, a^{10} x^{4} + \frac {4}{21879} \, {\left (6435 \, a b^{9} x^{8} + 87516 \, a^{3} b^{7} x^{7} + 212058 \, a^{5} b^{5} x^{6} + 119340 \, a^{7} b^{3} x^{5} + 12155 \, a^{9} b x^{4}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 112, normalized size = 0.69 \[ \frac {1}{9} \, b^{10} x^{9} + \frac {20}{17} \, a b^{9} x^{\frac {17}{2}} + \frac {45}{8} \, a^{2} b^{8} x^{8} + 16 \, a^{3} b^{7} x^{\frac {15}{2}} + 30 \, a^{4} b^{6} x^{7} + \frac {504}{13} \, a^{5} b^{5} x^{\frac {13}{2}} + 35 \, a^{6} b^{4} x^{6} + \frac {240}{11} \, a^{7} b^{3} x^{\frac {11}{2}} + 9 \, a^{8} b^{2} x^{5} + \frac {20}{9} \, a^{9} b x^{\frac {9}{2}} + \frac {1}{4} \, a^{10} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 113, normalized size = 0.70 \[ \frac {b^{10} x^{9}}{9}+\frac {20 a \,b^{9} x^{\frac {17}{2}}}{17}+\frac {45 a^{2} b^{8} x^{8}}{8}+16 a^{3} b^{7} x^{\frac {15}{2}}+30 a^{4} b^{6} x^{7}+\frac {504 a^{5} b^{5} x^{\frac {13}{2}}}{13}+35 a^{6} b^{4} x^{6}+\frac {240 a^{7} b^{3} x^{\frac {11}{2}}}{11}+9 a^{8} b^{2} x^{5}+\frac {20 a^{9} b \,x^{\frac {9}{2}}}{9}+\frac {a^{10} x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.87, size = 132, normalized size = 0.81 \[ \frac {{\left (b \sqrt {x} + a\right )}^{18}}{9 \, b^{8}} - \frac {14 \, {\left (b \sqrt {x} + a\right )}^{17} a}{17 \, b^{8}} + \frac {21 \, {\left (b \sqrt {x} + a\right )}^{16} a^{2}}{8 \, b^{8}} - \frac {14 \, {\left (b \sqrt {x} + a\right )}^{15} a^{3}}{3 \, b^{8}} + \frac {5 \, {\left (b \sqrt {x} + a\right )}^{14} a^{4}}{b^{8}} - \frac {42 \, {\left (b \sqrt {x} + a\right )}^{13} a^{5}}{13 \, b^{8}} + \frac {7 \, {\left (b \sqrt {x} + a\right )}^{12} a^{6}}{6 \, b^{8}} - \frac {2 \, {\left (b \sqrt {x} + a\right )}^{11} a^{7}}{11 \, b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.06, size = 112, normalized size = 0.69 \[ \frac {a^{10}\,x^4}{4}+\frac {b^{10}\,x^9}{9}+\frac {20\,a^9\,b\,x^{9/2}}{9}+\frac {20\,a\,b^9\,x^{17/2}}{17}+9\,a^8\,b^2\,x^5+35\,a^6\,b^4\,x^6+30\,a^4\,b^6\,x^7+\frac {45\,a^2\,b^8\,x^8}{8}+\frac {240\,a^7\,b^3\,x^{11/2}}{11}+\frac {504\,a^5\,b^5\,x^{13/2}}{13}+16\,a^3\,b^7\,x^{15/2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 6.05, size = 136, normalized size = 0.84 \[ \frac {a^{10} x^{4}}{4} + \frac {20 a^{9} b x^{\frac {9}{2}}}{9} + 9 a^{8} b^{2} x^{5} + \frac {240 a^{7} b^{3} x^{\frac {11}{2}}}{11} + 35 a^{6} b^{4} x^{6} + \frac {504 a^{5} b^{5} x^{\frac {13}{2}}}{13} + 30 a^{4} b^{6} x^{7} + 16 a^{3} b^{7} x^{\frac {15}{2}} + \frac {45 a^{2} b^{8} x^{8}}{8} + \frac {20 a b^{9} x^{\frac {17}{2}}}{17} + \frac {b^{10} x^{9}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________