Optimal. Leaf size=69 \[ \frac {a^5 x^{14}}{14}+\frac {5}{16} a^4 b x^{16}+\frac {5}{9} a^3 b^2 x^{18}+\frac {1}{2} a^2 b^3 x^{20}+\frac {5}{22} a b^4 x^{22}+\frac {b^5 x^{24}}{24} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \begin {gather*} \frac {1}{2} a^2 b^3 x^{20}+\frac {5}{9} a^3 b^2 x^{18}+\frac {5}{16} a^4 b x^{16}+\frac {a^5 x^{14}}{14}+\frac {5}{22} a b^4 x^{22}+\frac {b^5 x^{24}}{24} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^{13} \left (a+b x^2\right )^5 \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^6 (a+b x)^5 \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (a^5 x^6+5 a^4 b x^7+10 a^3 b^2 x^8+10 a^2 b^3 x^9+5 a b^4 x^{10}+b^5 x^{11}\right ) \, dx,x,x^2\right )\\ &=\frac {a^5 x^{14}}{14}+\frac {5}{16} a^4 b x^{16}+\frac {5}{9} a^3 b^2 x^{18}+\frac {1}{2} a^2 b^3 x^{20}+\frac {5}{22} a b^4 x^{22}+\frac {b^5 x^{24}}{24}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 69, normalized size = 1.00 \begin {gather*} \frac {a^5 x^{14}}{14}+\frac {5}{16} a^4 b x^{16}+\frac {5}{9} a^3 b^2 x^{18}+\frac {1}{2} a^2 b^3 x^{20}+\frac {5}{22} a b^4 x^{22}+\frac {b^5 x^{24}}{24} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{13} \left (a+b x^2\right )^5 \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 57, normalized size = 0.83 \begin {gather*} \frac {1}{24} x^{24} b^{5} + \frac {5}{22} x^{22} b^{4} a + \frac {1}{2} x^{20} b^{3} a^{2} + \frac {5}{9} x^{18} b^{2} a^{3} + \frac {5}{16} x^{16} b a^{4} + \frac {1}{14} x^{14} a^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.87, size = 57, normalized size = 0.83 \begin {gather*} \frac {1}{24} \, b^{5} x^{24} + \frac {5}{22} \, a b^{4} x^{22} + \frac {1}{2} \, a^{2} b^{3} x^{20} + \frac {5}{9} \, a^{3} b^{2} x^{18} + \frac {5}{16} \, a^{4} b x^{16} + \frac {1}{14} \, a^{5} x^{14} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 58, normalized size = 0.84 \begin {gather*} \frac {1}{24} b^{5} x^{24}+\frac {5}{22} a \,b^{4} x^{22}+\frac {1}{2} a^{2} b^{3} x^{20}+\frac {5}{9} a^{3} b^{2} x^{18}+\frac {5}{16} a^{4} b \,x^{16}+\frac {1}{14} a^{5} x^{14} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.43, size = 57, normalized size = 0.83 \begin {gather*} \frac {1}{24} \, b^{5} x^{24} + \frac {5}{22} \, a b^{4} x^{22} + \frac {1}{2} \, a^{2} b^{3} x^{20} + \frac {5}{9} \, a^{3} b^{2} x^{18} + \frac {5}{16} \, a^{4} b x^{16} + \frac {1}{14} \, a^{5} x^{14} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 57, normalized size = 0.83 \begin {gather*} \frac {a^5\,x^{14}}{14}+\frac {5\,a^4\,b\,x^{16}}{16}+\frac {5\,a^3\,b^2\,x^{18}}{9}+\frac {a^2\,b^3\,x^{20}}{2}+\frac {5\,a\,b^4\,x^{22}}{22}+\frac {b^5\,x^{24}}{24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.08, size = 65, normalized size = 0.94 \begin {gather*} \frac {a^{5} x^{14}}{14} + \frac {5 a^{4} b x^{16}}{16} + \frac {5 a^{3} b^{2} x^{18}}{9} + \frac {a^{2} b^{3} x^{20}}{2} + \frac {5 a b^{4} x^{22}}{22} + \frac {b^{5} x^{24}}{24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________