Optimal. Leaf size=113 \[ \frac {2 a^2 x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b^3 \left (c x^2\right )^{9/2}}+\frac {2 x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{7/2}}{21 b^3 \left (c x^2\right )^{9/2}}-\frac {4 a x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{5/2}}{15 b^3 \left (c x^2\right )^{9/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 113, 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 = {368, 266, 43} \[ \frac {2 a^2 x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b^3 \left (c x^2\right )^{9/2}}+\frac {2 x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{7/2}}{21 b^3 \left (c x^2\right )^{9/2}}-\frac {4 a x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{5/2}}{15 b^3 \left (c x^2\right )^{9/2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 368
Rubi steps
\begin {align*} \int x^8 \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx &=\frac {x^9 \operatorname {Subst}\left (\int x^8 \sqrt {a+b x^3} \, dx,x,\sqrt {c x^2}\right )}{\left (c x^2\right )^{9/2}}\\ &=\frac {x^9 \operatorname {Subst}\left (\int x^2 \sqrt {a+b x} \, dx,x,\left (c x^2\right )^{3/2}\right )}{3 \left (c x^2\right )^{9/2}}\\ &=\frac {x^9 \operatorname {Subst}\left (\int \left (\frac {a^2 \sqrt {a+b x}}{b^2}-\frac {2 a (a+b x)^{3/2}}{b^2}+\frac {(a+b x)^{5/2}}{b^2}\right ) \, dx,x,\left (c x^2\right )^{3/2}\right )}{3 \left (c x^2\right )^{9/2}}\\ &=\frac {2 a^2 x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b^3 \left (c x^2\right )^{9/2}}-\frac {4 a x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{5/2}}{15 b^3 \left (c x^2\right )^{9/2}}+\frac {2 x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{7/2}}{21 b^3 \left (c x^2\right )^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 67, normalized size = 0.59 \[ \frac {2 x \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2} \left (8 a^2-12 a b \left (c x^2\right )^{3/2}+15 b^2 c^3 x^6\right )}{315 b^3 c^4 \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 78, normalized size = 0.69 \[ \frac {2 \, {\left (15 \, b^{3} c^{5} x^{10} - 4 \, a^{2} b c^{2} x^{4} + {\left (3 \, a b^{2} c^{3} x^{6} + 8 \, a^{3}\right )} \sqrt {c x^{2}}\right )} \sqrt {\sqrt {c x^{2}} b c x^{2} + a}}{315 \, b^{3} c^{5} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 64, normalized size = 0.57 \[ \frac {2 \, {\left (15 \, {\left (b c^{\frac {3}{2}} x^{3} + a\right )}^{\frac {7}{2}} \sqrt {c} - 42 \, {\left (b c^{\frac {3}{2}} x^{3} + a\right )}^{\frac {5}{2}} a \sqrt {c} + 35 \, {\left (b c^{\frac {3}{2}} x^{3} + a\right )}^{\frac {3}{2}} a^{2} \sqrt {c}\right )}}{315 \, b^{3} c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \sqrt {a +\left (c \,x^{2}\right )^{\frac {3}{2}} b}\, x^{8}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^8\,\sqrt {a+b\,{\left (c\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int x^{8} \sqrt {a + b \left (c x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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