Optimal. Leaf size=80 \[ -\frac {2 a^3 \left (a+b x^3\right )^{5/2}}{15 b^4}+\frac {2 a^2 \left (a+b x^3\right )^{7/2}}{7 b^4}-\frac {2 a \left (a+b x^3\right )^{9/2}}{9 b^4}+\frac {2 \left (a+b x^3\right )^{11/2}}{33 b^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 80, 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 = {272, 45}
\begin {gather*} -\frac {2 a^3 \left (a+b x^3\right )^{5/2}}{15 b^4}+\frac {2 a^2 \left (a+b x^3\right )^{7/2}}{7 b^4}+\frac {2 \left (a+b x^3\right )^{11/2}}{33 b^4}-\frac {2 a \left (a+b x^3\right )^{9/2}}{9 b^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^{11} \left (a+b x^3\right )^{3/2} \, dx &=\frac {1}{3} \text {Subst}\left (\int x^3 (a+b x)^{3/2} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (-\frac {a^3 (a+b x)^{3/2}}{b^3}+\frac {3 a^2 (a+b x)^{5/2}}{b^3}-\frac {3 a (a+b x)^{7/2}}{b^3}+\frac {(a+b x)^{9/2}}{b^3}\right ) \, dx,x,x^3\right )\\ &=-\frac {2 a^3 \left (a+b x^3\right )^{5/2}}{15 b^4}+\frac {2 a^2 \left (a+b x^3\right )^{7/2}}{7 b^4}-\frac {2 a \left (a+b x^3\right )^{9/2}}{9 b^4}+\frac {2 \left (a+b x^3\right )^{11/2}}{33 b^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 50, normalized size = 0.62 \begin {gather*} \frac {2 \left (a+b x^3\right )^{5/2} \left (-16 a^3+40 a^2 b x^3-70 a b^2 x^6+105 b^3 x^9\right )}{3465 b^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.14, size = 109, normalized size = 1.36
method | result | size |
gosper | \(-\frac {2 \left (b \,x^{3}+a \right )^{\frac {5}{2}} \left (-105 b^{3} x^{9}+70 a \,b^{2} x^{6}-40 a^{2} b \,x^{3}+16 a^{3}\right )}{3465 b^{4}}\) | \(47\) |
trager | \(-\frac {2 \left (-105 b^{5} x^{15}-140 a \,b^{4} x^{12}-5 a^{2} b^{3} x^{9}+6 a^{3} b^{2} x^{6}-8 a^{4} b \,x^{3}+16 a^{5}\right ) \sqrt {b \,x^{3}+a}}{3465 b^{4}}\) | \(69\) |
risch | \(-\frac {2 \left (-105 b^{5} x^{15}-140 a \,b^{4} x^{12}-5 a^{2} b^{3} x^{9}+6 a^{3} b^{2} x^{6}-8 a^{4} b \,x^{3}+16 a^{5}\right ) \sqrt {b \,x^{3}+a}}{3465 b^{4}}\) | \(69\) |
default | \(\frac {2 b \,x^{15} \sqrt {b \,x^{3}+a}}{33}+\frac {8 a \,x^{12} \sqrt {b \,x^{3}+a}}{99}+\frac {2 a^{2} x^{9} \sqrt {b \,x^{3}+a}}{693 b}-\frac {4 a^{3} x^{6} \sqrt {b \,x^{3}+a}}{1155 b^{2}}+\frac {16 a^{4} x^{3} \sqrt {b \,x^{3}+a}}{3465 b^{3}}-\frac {32 a^{5} \sqrt {b \,x^{3}+a}}{3465 b^{4}}\) | \(109\) |
elliptic | \(\frac {2 b \,x^{15} \sqrt {b \,x^{3}+a}}{33}+\frac {8 a \,x^{12} \sqrt {b \,x^{3}+a}}{99}+\frac {2 a^{2} x^{9} \sqrt {b \,x^{3}+a}}{693 b}-\frac {4 a^{3} x^{6} \sqrt {b \,x^{3}+a}}{1155 b^{2}}+\frac {16 a^{4} x^{3} \sqrt {b \,x^{3}+a}}{3465 b^{3}}-\frac {32 a^{5} \sqrt {b \,x^{3}+a}}{3465 b^{4}}\) | \(109\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 64, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {11}{2}}}{33 \, b^{4}} - \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {9}{2}} a}{9 \, b^{4}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} a^{2}}{7 \, b^{4}} - \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a^{3}}{15 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.34, size = 68, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (105 \, b^{5} x^{15} + 140 \, a b^{4} x^{12} + 5 \, a^{2} b^{3} x^{9} - 6 \, a^{3} b^{2} x^{6} + 8 \, a^{4} b x^{3} - 16 \, a^{5}\right )} \sqrt {b x^{3} + a}}{3465 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.58, size = 136, normalized size = 1.70 \begin {gather*} \begin {cases} - \frac {32 a^{5} \sqrt {a + b x^{3}}}{3465 b^{4}} + \frac {16 a^{4} x^{3} \sqrt {a + b x^{3}}}{3465 b^{3}} - \frac {4 a^{3} x^{6} \sqrt {a + b x^{3}}}{1155 b^{2}} + \frac {2 a^{2} x^{9} \sqrt {a + b x^{3}}}{693 b} + \frac {8 a x^{12} \sqrt {a + b x^{3}}}{99} + \frac {2 b x^{15} \sqrt {a + b x^{3}}}{33} & \text {for}\: b \neq 0 \\\frac {a^{\frac {3}{2}} x^{12}}{12} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.30, size = 57, normalized size = 0.71 \begin {gather*} \frac {2 \, {\left (105 \, {\left (b x^{3} + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x^{3} + a\right )}^{\frac {9}{2}} a + 495 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} a^{2} - 231 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a^{3}\right )}}{3465 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.12, size = 108, normalized size = 1.35 \begin {gather*} \frac {8\,a\,x^{12}\,\sqrt {b\,x^3+a}}{99}+\frac {2\,b\,x^{15}\,\sqrt {b\,x^3+a}}{33}-\frac {32\,a^5\,\sqrt {b\,x^3+a}}{3465\,b^4}+\frac {16\,a^4\,x^3\,\sqrt {b\,x^3+a}}{3465\,b^3}-\frac {4\,a^3\,x^6\,\sqrt {b\,x^3+a}}{1155\,b^2}+\frac {2\,a^2\,x^9\,\sqrt {b\,x^3+a}}{693\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________