Optimal. Leaf size=79 \[ -\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )}-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1355, 14} \[ -\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )}-\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 1355
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^{10}} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {a b+b^2 x^3}{x^{10}} \, dx}{a b+b^2 x^3}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (\frac {a b}{x^{10}}+\frac {b^2}{x^7}\right ) \, dx}{a b+b^2 x^3}\\ &=-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )}-\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 39, normalized size = 0.49 \[ -\frac {\sqrt {\left (a+b x^3\right )^2} \left (2 a+3 b x^3\right )}{18 x^9 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.87, size = 15, normalized size = 0.19 \[ -\frac {3 \, b x^{3} + 2 \, a}{18 \, x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.36, size = 31, normalized size = 0.39 \[ -\frac {3 \, b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 2 \, a \mathrm {sgn}\left (b x^{3} + a\right )}{18 \, x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 36, normalized size = 0.46 \[ -\frac {\left (3 b \,x^{3}+2 a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{18 \left (b \,x^{3}+a \right ) x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.47, size = 117, normalized size = 1.48 \[ -\frac {\sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} b^{3}}{6 \, a^{3}} - \frac {\sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} b^{2}}{6 \, a^{2} x^{3}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} b}{6 \, a^{3} x^{6}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}}}{9 \, a^{2} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.15, size = 35, normalized size = 0.44 \[ -\frac {\left (3\,b\,x^3+2\,a\right )\,\sqrt {{\left (b\,x^3+a\right )}^2}}{18\,x^9\,\left (b\,x^3+a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.21, size = 15, normalized size = 0.19 \[ \frac {- 2 a - 3 b x^{3}}{18 x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________