Optimal. Leaf size=56 \[ \frac {4 \left (a+b \left (c x^3\right )^{3/2}\right )^{5/2}}{45 b^2 c^3}-\frac {4 a \left (a+b \left (c x^3\right )^{3/2}\right )^{3/2}}{27 b^2 c^3} \]
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Rubi [A] time = 0.04, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {369, 266, 43} \[ \frac {4 \left (a+b \left (c x^3\right )^{3/2}\right )^{5/2}}{45 b^2 c^3}-\frac {4 a \left (a+b \left (c x^3\right )^{3/2}\right )^{3/2}}{27 b^2 c^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 369
Rubi steps
\begin {align*} \int x^8 \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx &=\operatorname {Subst}\left (\int x^8 \sqrt {a+b c^{3/2} x^{9/2}} \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\operatorname {Subst}\left (\frac {2}{9} \operatorname {Subst}\left (\int x \sqrt {a+b c^{3/2} x} \, dx,x,x^{9/2}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\operatorname {Subst}\left (\frac {2}{9} \operatorname {Subst}\left (\int \left (-\frac {a \sqrt {a+b c^{3/2} x}}{b c^{3/2}}+\frac {\left (a+b c^{3/2} x\right )^{3/2}}{b c^{3/2}}\right ) \, dx,x,x^{9/2}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\frac {4 a \left (a+b \left (c x^3\right )^{3/2}\right )^{3/2}}{27 b^2 c^3}+\frac {4 \left (a+b \left (c x^3\right )^{3/2}\right )^{5/2}}{45 b^2 c^3}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 43, normalized size = 0.77 \[ \frac {4 \left (a+b \left (c x^3\right )^{3/2}\right )^{3/2} \left (3 b \left (c x^3\right )^{3/2}-2 a\right )}{135 b^2 c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.86, size = 56, normalized size = 1.00 \[ \frac {4 \, {\left (3 \, b^{2} c^{3} x^{9} + \sqrt {c x^{3}} a b c x^{3} - 2 \, a^{2}\right )} \sqrt {\sqrt {c x^{3}} b c x^{3} + a}}{135 \, b^{2} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 63, normalized size = 1.12 \[ -\frac {4 \, {\left (5 \, {\left (\sqrt {c x} b c^{5} x^{4} + a c^{4}\right )}^{\frac {3}{2}} a c^{4} - 3 \, {\left (\sqrt {c x} b c^{5} x^{4} + a c^{4}\right )}^{\frac {5}{2}}\right )} {\left | c \right |}^{2}}{135 \, b^{2} c^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.26, size = 0, normalized size = 0.00 \[ \int \sqrt {a +\left (c \,x^{3}\right )^{\frac {3}{2}} b}\, x^{8}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 43, normalized size = 0.77 \[ \frac {4 \, {\left (\frac {3 \, {\left (\left (c x^{3}\right )^{\frac {3}{2}} b + a\right )}^{\frac {5}{2}}}{b^{2}} - \frac {5 \, {\left (\left (c x^{3}\right )^{\frac {3}{2}} b + a\right )}^{\frac {3}{2}} a}{b^{2}}\right )}}{135 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^8\,\sqrt {a+b\,{\left (c\,x^3\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{8} \sqrt {a + b \left (c x^{3}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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