Optimal. Leaf size=62 \[ -\frac {b^2 \left (a+b x^2\right )^7}{504 a^3 x^{14}}+\frac {b \left (a+b x^2\right )^7}{72 a^2 x^{16}}-\frac {\left (a+b x^2\right )^7}{18 a x^{18}} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {28, 266, 45, 37} \begin {gather*} -\frac {b^2 \left (a+b x^2\right )^7}{504 a^3 x^{14}}+\frac {b \left (a+b x^2\right )^7}{72 a^2 x^{16}}-\frac {\left (a+b x^2\right )^7}{18 a x^{18}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 28
Rule 37
Rule 45
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{19}} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^6}{x^{19}} \, dx}{b^6}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^6}{x^{10}} \, dx,x,x^2\right )}{2 b^6}\\ &=-\frac {\left (a+b x^2\right )^7}{18 a x^{18}}-\frac {\operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^6}{x^9} \, dx,x,x^2\right )}{9 a b^5}\\ &=-\frac {\left (a+b x^2\right )^7}{18 a x^{18}}+\frac {b \left (a+b x^2\right )^7}{72 a^2 x^{16}}+\frac {\operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^6}{x^8} \, dx,x,x^2\right )}{72 a^2 b^4}\\ &=-\frac {\left (a+b x^2\right )^7}{18 a x^{18}}+\frac {b \left (a+b x^2\right )^7}{72 a^2 x^{16}}-\frac {b^2 \left (a+b x^2\right )^7}{504 a^3 x^{14}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 82, normalized size = 1.32 \begin {gather*} -\frac {a^6}{18 x^{18}}-\frac {3 a^5 b}{8 x^{16}}-\frac {15 a^4 b^2}{14 x^{14}}-\frac {5 a^3 b^3}{3 x^{12}}-\frac {3 a^2 b^4}{2 x^{10}}-\frac {3 a b^5}{4 x^8}-\frac {b^6}{6 x^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{19}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.64, size = 70, normalized size = 1.13 \begin {gather*} -\frac {84 \, b^{6} x^{12} + 378 \, a b^{5} x^{10} + 756 \, a^{2} b^{4} x^{8} + 840 \, a^{3} b^{3} x^{6} + 540 \, a^{4} b^{2} x^{4} + 189 \, a^{5} b x^{2} + 28 \, a^{6}}{504 \, x^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 70, normalized size = 1.13 \begin {gather*} -\frac {84 \, b^{6} x^{12} + 378 \, a b^{5} x^{10} + 756 \, a^{2} b^{4} x^{8} + 840 \, a^{3} b^{3} x^{6} + 540 \, a^{4} b^{2} x^{4} + 189 \, a^{5} b x^{2} + 28 \, a^{6}}{504 \, x^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 69, normalized size = 1.11 \begin {gather*} -\frac {b^{6}}{6 x^{6}}-\frac {3 a \,b^{5}}{4 x^{8}}-\frac {3 a^{2} b^{4}}{2 x^{10}}-\frac {5 a^{3} b^{3}}{3 x^{12}}-\frac {15 a^{4} b^{2}}{14 x^{14}}-\frac {3 a^{5} b}{8 x^{16}}-\frac {a^{6}}{18 x^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.34, size = 70, normalized size = 1.13 \begin {gather*} -\frac {84 \, b^{6} x^{12} + 378 \, a b^{5} x^{10} + 756 \, a^{2} b^{4} x^{8} + 840 \, a^{3} b^{3} x^{6} + 540 \, a^{4} b^{2} x^{4} + 189 \, a^{5} b x^{2} + 28 \, a^{6}}{504 \, x^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.31, size = 70, normalized size = 1.13 \begin {gather*} -\frac {\frac {a^6}{18}+\frac {3\,a^5\,b\,x^2}{8}+\frac {15\,a^4\,b^2\,x^4}{14}+\frac {5\,a^3\,b^3\,x^6}{3}+\frac {3\,a^2\,b^4\,x^8}{2}+\frac {3\,a\,b^5\,x^{10}}{4}+\frac {b^6\,x^{12}}{6}}{x^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.74, size = 75, normalized size = 1.21 \begin {gather*} \frac {- 28 a^{6} - 189 a^{5} b x^{2} - 540 a^{4} b^{2} x^{4} - 840 a^{3} b^{3} x^{6} - 756 a^{2} b^{4} x^{8} - 378 a b^{5} x^{10} - 84 b^{6} x^{12}}{504 x^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________