Optimal. Leaf size=82 \[ \frac {a^6 x^3}{3}+\frac {6}{5} a^5 b x^5+\frac {15}{7} a^4 b^2 x^7+\frac {20}{9} a^3 b^3 x^9+\frac {15}{11} a^2 b^4 x^{11}+\frac {6}{13} a b^5 x^{13}+\frac {b^6 x^{15}}{15} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {28, 270} \begin {gather*} \frac {15}{11} a^2 b^4 x^{11}+\frac {20}{9} a^3 b^3 x^9+\frac {15}{7} a^4 b^2 x^7+\frac {6}{5} a^5 b x^5+\frac {a^6 x^3}{3}+\frac {6}{13} a b^5 x^{13}+\frac {b^6 x^{15}}{15} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 28
Rule 270
Rubi steps
\begin {align*} \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx &=\frac {\int x^2 \left (a b+b^2 x^2\right )^6 \, dx}{b^6}\\ &=\frac {\int \left (a^6 b^6 x^2+6 a^5 b^7 x^4+15 a^4 b^8 x^6+20 a^3 b^9 x^8+15 a^2 b^{10} x^{10}+6 a b^{11} x^{12}+b^{12} x^{14}\right ) \, dx}{b^6}\\ &=\frac {a^6 x^3}{3}+\frac {6}{5} a^5 b x^5+\frac {15}{7} a^4 b^2 x^7+\frac {20}{9} a^3 b^3 x^9+\frac {15}{11} a^2 b^4 x^{11}+\frac {6}{13} a b^5 x^{13}+\frac {b^6 x^{15}}{15}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 82, normalized size = 1.00 \begin {gather*} \frac {a^6 x^3}{3}+\frac {6}{5} a^5 b x^5+\frac {15}{7} a^4 b^2 x^7+\frac {20}{9} a^3 b^3 x^9+\frac {15}{11} a^2 b^4 x^{11}+\frac {6}{13} a b^5 x^{13}+\frac {b^6 x^{15}}{15} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.76, size = 68, normalized size = 0.83 \begin {gather*} \frac {1}{15} x^{15} b^{6} + \frac {6}{13} x^{13} b^{5} a + \frac {15}{11} x^{11} b^{4} a^{2} + \frac {20}{9} x^{9} b^{3} a^{3} + \frac {15}{7} x^{7} b^{2} a^{4} + \frac {6}{5} x^{5} b a^{5} + \frac {1}{3} x^{3} a^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 68, normalized size = 0.83 \begin {gather*} \frac {1}{15} \, b^{6} x^{15} + \frac {6}{13} \, a b^{5} x^{13} + \frac {15}{11} \, a^{2} b^{4} x^{11} + \frac {20}{9} \, a^{3} b^{3} x^{9} + \frac {15}{7} \, a^{4} b^{2} x^{7} + \frac {6}{5} \, a^{5} b x^{5} + \frac {1}{3} \, a^{6} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 69, normalized size = 0.84 \begin {gather*} \frac {1}{15} b^{6} x^{15}+\frac {6}{13} a \,b^{5} x^{13}+\frac {15}{11} a^{2} b^{4} x^{11}+\frac {20}{9} a^{3} b^{3} x^{9}+\frac {15}{7} a^{4} b^{2} x^{7}+\frac {6}{5} a^{5} b \,x^{5}+\frac {1}{3} a^{6} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.36, size = 68, normalized size = 0.83 \begin {gather*} \frac {1}{15} \, b^{6} x^{15} + \frac {6}{13} \, a b^{5} x^{13} + \frac {15}{11} \, a^{2} b^{4} x^{11} + \frac {20}{9} \, a^{3} b^{3} x^{9} + \frac {15}{7} \, a^{4} b^{2} x^{7} + \frac {6}{5} \, a^{5} b x^{5} + \frac {1}{3} \, a^{6} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 68, normalized size = 0.83 \begin {gather*} \frac {a^6\,x^3}{3}+\frac {6\,a^5\,b\,x^5}{5}+\frac {15\,a^4\,b^2\,x^7}{7}+\frac {20\,a^3\,b^3\,x^9}{9}+\frac {15\,a^2\,b^4\,x^{11}}{11}+\frac {6\,a\,b^5\,x^{13}}{13}+\frac {b^6\,x^{15}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.09, size = 80, normalized size = 0.98 \begin {gather*} \frac {a^{6} x^{3}}{3} + \frac {6 a^{5} b x^{5}}{5} + \frac {15 a^{4} b^{2} x^{7}}{7} + \frac {20 a^{3} b^{3} x^{9}}{9} + \frac {15 a^{2} b^{4} x^{11}}{11} + \frac {6 a b^{5} x^{13}}{13} + \frac {b^{6} x^{15}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________