Optimal. Leaf size=41 \[ \frac {2 \sqrt {a x^3-b} \left (3 a^2 x^6+4 a b x^3+8 b^2\right )}{45 a^3} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.59, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {266, 43} \begin {gather*} \frac {2 b^2 \sqrt {a x^3-b}}{3 a^3}+\frac {2 \left (a x^3-b\right )^{5/2}}{15 a^3}+\frac {4 b \left (a x^3-b\right )^{3/2}}{9 a^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^8}{\sqrt {-b+a x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-b+a x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {b^2}{a^2 \sqrt {-b+a x}}+\frac {2 b \sqrt {-b+a x}}{a^2}+\frac {(-b+a x)^{3/2}}{a^2}\right ) \, dx,x,x^3\right )\\ &=\frac {2 b^2 \sqrt {-b+a x^3}}{3 a^3}+\frac {4 b \left (-b+a x^3\right )^{3/2}}{9 a^3}+\frac {2 \left (-b+a x^3\right )^{5/2}}{15 a^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 41, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {a x^3-b} \left (3 a^2 x^6+4 a b x^3+8 b^2\right )}{45 a^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.03, size = 41, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {a x^3-b} \left (3 a^2 x^6+4 a b x^3+8 b^2\right )}{45 a^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 37, normalized size = 0.90 \begin {gather*} \frac {2 \, {\left (3 \, a^{2} x^{6} + 4 \, a b x^{3} + 8 \, b^{2}\right )} \sqrt {a x^{3} - b}}{45 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.30, size = 53, normalized size = 1.29 \begin {gather*} \frac {2 \, \sqrt {a x^{3} - b} b^{2}}{3 \, a^{3}} + \frac {2 \, {\left (3 \, {\left (a x^{3} - b\right )}^{\frac {5}{2}} + 10 \, {\left (a x^{3} - b\right )}^{\frac {3}{2}} b\right )}}{45 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 38, normalized size = 0.93 \begin {gather*} \frac {2 \sqrt {a \,x^{3}-b}\, \left (3 a^{2} x^{6}+4 a b \,x^{3}+8 b^{2}\right )}{45 a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.46, size = 53, normalized size = 1.29 \begin {gather*} \frac {2 \, {\left (a x^{3} - b\right )}^{\frac {5}{2}}}{15 \, a^{3}} + \frac {4 \, {\left (a x^{3} - b\right )}^{\frac {3}{2}} b}{9 \, a^{3}} + \frac {2 \, \sqrt {a x^{3} - b} b^{2}}{3 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.62, size = 37, normalized size = 0.90 \begin {gather*} \frac {2\,\sqrt {a\,x^3-b}\,\left (3\,a^2\,x^6+4\,a\,b\,x^3+8\,b^2\right )}{45\,a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.25, size = 71, normalized size = 1.73 \begin {gather*} \begin {cases} \frac {2 x^{6} \sqrt {a x^{3} - b}}{15 a} + \frac {8 b x^{3} \sqrt {a x^{3} - b}}{45 a^{2}} + \frac {16 b^{2} \sqrt {a x^{3} - b}}{45 a^{3}} & \text {for}\: a \neq 0 \\\frac {x^{9}}{9 \sqrt {- b}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________