Optimal. Leaf size=110 \[ -\frac {2 a^5 (a+b x)^{11/2}}{11 b^6}+\frac {10 a^4 (a+b x)^{13/2}}{13 b^6}-\frac {4 a^3 (a+b x)^{15/2}}{3 b^6}+\frac {20 a^2 (a+b x)^{17/2}}{17 b^6}-\frac {10 a (a+b x)^{19/2}}{19 b^6}+\frac {2 (a+b x)^{21/2}}{21 b^6} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 110, 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 = {45}
\begin {gather*} -\frac {2 a^5 (a+b x)^{11/2}}{11 b^6}+\frac {10 a^4 (a+b x)^{13/2}}{13 b^6}-\frac {4 a^3 (a+b x)^{15/2}}{3 b^6}+\frac {20 a^2 (a+b x)^{17/2}}{17 b^6}+\frac {2 (a+b x)^{21/2}}{21 b^6}-\frac {10 a (a+b x)^{19/2}}{19 b^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rubi steps
\begin {align*} \int x^5 (a+b x)^{9/2} \, dx &=\int \left (-\frac {a^5 (a+b x)^{9/2}}{b^5}+\frac {5 a^4 (a+b x)^{11/2}}{b^5}-\frac {10 a^3 (a+b x)^{13/2}}{b^5}+\frac {10 a^2 (a+b x)^{15/2}}{b^5}-\frac {5 a (a+b x)^{17/2}}{b^5}+\frac {(a+b x)^{19/2}}{b^5}\right ) \, dx\\ &=-\frac {2 a^5 (a+b x)^{11/2}}{11 b^6}+\frac {10 a^4 (a+b x)^{13/2}}{13 b^6}-\frac {4 a^3 (a+b x)^{15/2}}{3 b^6}+\frac {20 a^2 (a+b x)^{17/2}}{17 b^6}-\frac {10 a (a+b x)^{19/2}}{19 b^6}+\frac {2 (a+b x)^{21/2}}{21 b^6}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 68, normalized size = 0.62 \begin {gather*} \frac {2 (a+b x)^{11/2} \left (-256 a^5+1408 a^4 b x-4576 a^3 b^2 x^2+11440 a^2 b^3 x^3-24310 a b^4 x^4+46189 b^5 x^5\right )}{969969 b^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in
optimal.
time = 3.78, size = 132, normalized size = 1.20 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \left (-256 a^{10}+128 a^9 b x-96 a^8 b^2 x^2+80 a^7 b^3 x^3-70 a^6 b^4 x^4+63 a^5 b^5 x^5+11 b^6 x^6 \left (7343 a^4+24674 a^3 b x+31980 a^2 b^2 x^2+18785 a b^3 x^3+4199 b^4 x^4\right )\right ) \sqrt {a+b x}}{969969 b^6},b\text {!=}0\right \}\right \},\frac {a^{\frac {9}{2}} x^6}{6}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 74, normalized size = 0.67
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {11}{2}} \left (-46189 b^{5} x^{5}+24310 a \,b^{4} x^{4}-11440 a^{2} b^{3} x^{3}+4576 a^{3} b^{2} x^{2}-1408 a^{4} b x +256 a^{5}\right )}{969969 b^{6}}\) | \(65\) |
derivativedivides | \(\frac {\frac {2 \left (b x +a \right )^{\frac {21}{2}}}{21}-\frac {10 a \left (b x +a \right )^{\frac {19}{2}}}{19}+\frac {20 a^{2} \left (b x +a \right )^{\frac {17}{2}}}{17}-\frac {4 a^{3} \left (b x +a \right )^{\frac {15}{2}}}{3}+\frac {10 a^{4} \left (b x +a \right )^{\frac {13}{2}}}{13}-\frac {2 a^{5} \left (b x +a \right )^{\frac {11}{2}}}{11}}{b^{6}}\) | \(74\) |
default | \(\frac {\frac {2 \left (b x +a \right )^{\frac {21}{2}}}{21}-\frac {10 a \left (b x +a \right )^{\frac {19}{2}}}{19}+\frac {20 a^{2} \left (b x +a \right )^{\frac {17}{2}}}{17}-\frac {4 a^{3} \left (b x +a \right )^{\frac {15}{2}}}{3}+\frac {10 a^{4} \left (b x +a \right )^{\frac {13}{2}}}{13}-\frac {2 a^{5} \left (b x +a \right )^{\frac {11}{2}}}{11}}{b^{6}}\) | \(74\) |
trager | \(-\frac {2 \left (-46189 b^{10} x^{10}-206635 a \,b^{9} x^{9}-351780 a^{2} b^{8} x^{8}-271414 a^{3} b^{7} x^{7}-80773 a^{4} b^{6} x^{6}-63 a^{5} b^{5} x^{5}+70 a^{6} b^{4} x^{4}-80 a^{7} b^{3} x^{3}+96 a^{8} b^{2} x^{2}-128 a^{9} b x +256 a^{10}\right ) \sqrt {b x +a}}{969969 b^{6}}\) | \(120\) |
risch | \(-\frac {2 \left (-46189 b^{10} x^{10}-206635 a \,b^{9} x^{9}-351780 a^{2} b^{8} x^{8}-271414 a^{3} b^{7} x^{7}-80773 a^{4} b^{6} x^{6}-63 a^{5} b^{5} x^{5}+70 a^{6} b^{4} x^{4}-80 a^{7} b^{3} x^{3}+96 a^{8} b^{2} x^{2}-128 a^{9} b x +256 a^{10}\right ) \sqrt {b x +a}}{969969 b^{6}}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 86, normalized size = 0.78 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {21}{2}}}{21 \, b^{6}} - \frac {10 \, {\left (b x + a\right )}^{\frac {19}{2}} a}{19 \, b^{6}} + \frac {20 \, {\left (b x + a\right )}^{\frac {17}{2}} a^{2}}{17 \, b^{6}} - \frac {4 \, {\left (b x + a\right )}^{\frac {15}{2}} a^{3}}{3 \, b^{6}} + \frac {10 \, {\left (b x + a\right )}^{\frac {13}{2}} a^{4}}{13 \, b^{6}} - \frac {2 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{5}}{11 \, b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.31, size = 119, normalized size = 1.08 \begin {gather*} \frac {2 \, {\left (46189 \, b^{10} x^{10} + 206635 \, a b^{9} x^{9} + 351780 \, a^{2} b^{8} x^{8} + 271414 \, a^{3} b^{7} x^{7} + 80773 \, a^{4} b^{6} x^{6} + 63 \, a^{5} b^{5} x^{5} - 70 \, a^{6} b^{4} x^{4} + 80 \, a^{7} b^{3} x^{3} - 96 \, a^{8} b^{2} x^{2} + 128 \, a^{9} b x - 256 \, a^{10}\right )} \sqrt {b x + a}}{969969 \, b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 1.34, size = 235, normalized size = 2.14 \begin {gather*} \begin {cases} - \frac {512 a^{10} \sqrt {a + b x}}{969969 b^{6}} + \frac {256 a^{9} x \sqrt {a + b x}}{969969 b^{5}} - \frac {64 a^{8} x^{2} \sqrt {a + b x}}{323323 b^{4}} + \frac {160 a^{7} x^{3} \sqrt {a + b x}}{969969 b^{3}} - \frac {20 a^{6} x^{4} \sqrt {a + b x}}{138567 b^{2}} + \frac {6 a^{5} x^{5} \sqrt {a + b x}}{46189 b} + \frac {2098 a^{4} x^{6} \sqrt {a + b x}}{12597} + \frac {3796 a^{3} b x^{7} \sqrt {a + b x}}{6783} + \frac {1640 a^{2} b^{2} x^{8} \sqrt {a + b x}}{2261} + \frac {170 a b^{3} x^{9} \sqrt {a + b x}}{399} + \frac {2 b^{4} x^{10} \sqrt {a + b x}}{21} & \text {for}\: b \neq 0 \\\frac {a^{\frac {9}{2}} x^{6}}{6} & \text {otherwise} \end {cases} \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. 637 vs.
\(2 (86) = 172\).
time = 0.00, size = 1141, normalized size = 10.37 \begin {gather*} \frac {\frac {2 b^{5} \left (\frac {1}{21} \sqrt {a+b x} \left (a+b x\right )^{10}-\frac {10}{19} \sqrt {a+b x} \left (a+b x\right )^{9} a+\frac {45}{17} \sqrt {a+b x} \left (a+b x\right )^{8} a^{2}-8 \sqrt {a+b x} \left (a+b x\right )^{7} a^{3}+\frac {210}{13} \sqrt {a+b x} \left (a+b x\right )^{6} a^{4}-\frac {252}{11} \sqrt {a+b x} \left (a+b x\right )^{5} a^{5}+\frac {70}{3} \sqrt {a+b x} \left (a+b x\right )^{4} a^{6}-\frac {120}{7} \sqrt {a+b x} \left (a+b x\right )^{3} a^{7}+9 \sqrt {a+b x} \left (a+b x\right )^{2} a^{8}-\frac {10}{3} \sqrt {a+b x} \left (a+b x\right ) a^{9}+\sqrt {a+b x} a^{10}\right )}{b^{10}}+\frac {10 a b^{4} \left (\frac {1}{19} \sqrt {a+b x} \left (a+b x\right )^{9}-\frac {9}{17} \sqrt {a+b x} \left (a+b x\right )^{8} a+\frac {12}{5} \sqrt {a+b x} \left (a+b x\right )^{7} a^{2}-\frac {84}{13} \sqrt {a+b x} \left (a+b x\right )^{6} a^{3}+\frac {126}{11} \sqrt {a+b x} \left (a+b x\right )^{5} a^{4}-14 \sqrt {a+b x} \left (a+b x\right )^{4} a^{5}+12 \sqrt {a+b x} \left (a+b x\right )^{3} a^{6}-\frac {36}{5} \sqrt {a+b x} \left (a+b x\right )^{2} a^{7}+3 \sqrt {a+b x} \left (a+b x\right ) a^{8}-\sqrt {a+b x} a^{9}\right )}{b^{9}}+\frac {20 a^{2} b^{3} \left (\frac {1}{17} \sqrt {a+b x} \left (a+b x\right )^{8}-\frac {8}{15} \sqrt {a+b x} \left (a+b x\right )^{7} a+\frac {28}{13} \sqrt {a+b x} \left (a+b x\right )^{6} a^{2}-\frac {56}{11} \sqrt {a+b x} \left (a+b x\right )^{5} a^{3}+\frac {70}{9} \sqrt {a+b x} \left (a+b x\right )^{4} a^{4}-8 \sqrt {a+b x} \left (a+b x\right )^{3} a^{5}+\frac {28}{5} \sqrt {a+b x} \left (a+b x\right )^{2} a^{6}-\frac {8}{3} \sqrt {a+b x} \left (a+b x\right ) a^{7}+\sqrt {a+b x} a^{8}\right )}{b^{8}}+\frac {20 a^{3} b^{2} \left (\frac {1}{15} \sqrt {a+b x} \left (a+b x\right )^{7}-\frac {7}{13} \sqrt {a+b x} \left (a+b x\right )^{6} a+\frac {21}{11} \sqrt {a+b x} \left (a+b x\right )^{5} a^{2}-\frac {35}{9} \sqrt {a+b x} \left (a+b x\right )^{4} a^{3}+5 \sqrt {a+b x} \left (a+b x\right )^{3} a^{4}-\frac {21}{5} \sqrt {a+b x} \left (a+b x\right )^{2} a^{5}+\frac {7}{3} \sqrt {a+b x} \left (a+b x\right ) a^{6}-\sqrt {a+b x} a^{7}\right )}{b^{7}}+\frac {10 a^{4} b \left (\frac {1}{13} \sqrt {a+b x} \left (a+b x\right )^{6}-\frac {6}{11} \sqrt {a+b x} \left (a+b x\right )^{5} a+\frac {5}{3} \sqrt {a+b x} \left (a+b x\right )^{4} a^{2}-\frac {20}{7} \sqrt {a+b x} \left (a+b x\right )^{3} a^{3}+3 \sqrt {a+b x} \left (a+b x\right )^{2} a^{4}-2 \sqrt {a+b x} \left (a+b x\right ) a^{5}+\sqrt {a+b x} a^{6}\right )}{b^{6}}+\frac {2 a^{5} \left (\frac {1}{11} \sqrt {a+b x} \left (a+b x\right )^{5}-\frac {5}{9} \sqrt {a+b x} \left (a+b x\right )^{4} a+\frac {10}{7} \sqrt {a+b x} \left (a+b x\right )^{3} a^{2}-2 \sqrt {a+b x} \left (a+b x\right )^{2} a^{3}+\frac {5}{3} \sqrt {a+b x} \left (a+b x\right ) a^{4}-\sqrt {a+b x} a^{5}\right )}{b^{5}}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.03, size = 86, normalized size = 0.78 \begin {gather*} \frac {2\,{\left (a+b\,x\right )}^{21/2}}{21\,b^6}-\frac {2\,a^5\,{\left (a+b\,x\right )}^{11/2}}{11\,b^6}+\frac {10\,a^4\,{\left (a+b\,x\right )}^{13/2}}{13\,b^6}-\frac {4\,a^3\,{\left (a+b\,x\right )}^{15/2}}{3\,b^6}+\frac {20\,a^2\,{\left (a+b\,x\right )}^{17/2}}{17\,b^6}-\frac {10\,a\,{\left (a+b\,x\right )}^{19/2}}{19\,b^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________