Optimal. Leaf size=101 \[ \frac{b^2 c^3 \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{\sqrt{a}}\right )}{18 a^{3/2}}-\frac{b c^3 \sqrt{a+b \left (c x^3\right )^{3/2}}}{18 a \left (c x^3\right )^{3/2}}-\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{9 x^9} \]
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Rubi [A] time = 0.065108, antiderivative size = 104, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {369, 266, 47, 51, 63, 208} \[ \frac{b^2 c^3 \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{\sqrt{a}}\right )}{18 a^{3/2}}-\frac{b c^6 x^9 \sqrt{a+b \left (c x^3\right )^{3/2}}}{18 a \left (c x^3\right )^{9/2}}-\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{9 x^9} \]
Antiderivative was successfully verified.
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Rule 369
Rule 266
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{x^{10}} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{a+b c^{3/2} x^{9/2}}}{x^{10}} \, dx,\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\operatorname{Subst}\left (\frac{2}{9} \operatorname{Subst}\left (\int \frac{\sqrt{a+b c^{3/2} x}}{x^3} \, dx,x,x^{9/2}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{9 x^9}+\operatorname{Subst}\left (\frac{1}{18} \left (b c^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b c^{3/2} x}} \, dx,x,x^{9/2}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{9 x^9}-\frac{b c^6 x^9 \sqrt{a+b \left (c x^3\right )^{3/2}}}{18 a \left (c x^3\right )^{9/2}}-\operatorname{Subst}\left (\frac{\left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b c^{3/2} x}} \, dx,x,x^{9/2}\right )}{36 a},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{9 x^9}-\frac{b c^6 x^9 \sqrt{a+b \left (c x^3\right )^{3/2}}}{18 a \left (c x^3\right )^{9/2}}-\operatorname{Subst}\left (\frac{\left (b c^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b c^{3/2}}+\frac{x^2}{b c^{3/2}}} \, dx,x,\sqrt{a+b c^{3/2} x^{9/2}}\right )}{18 a},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{9 x^9}-\frac{b c^6 x^9 \sqrt{a+b \left (c x^3\right )^{3/2}}}{18 a \left (c x^3\right )^{9/2}}+\frac{b^2 c^3 \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{\sqrt{a}}\right )}{18 a^{3/2}}\\ \end{align*}
Mathematica [F] time = 0.0495551, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{x^{10}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{10}}\sqrt{a+b \left ( c{x}^{3} \right ) ^{{\frac{3}{2}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b \left (c x^{3}\right )^{\frac{3}{2}}}}{x^{10}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.41312, size = 158, normalized size = 1.56 \begin{align*} -\frac{1}{18} \, b^{2} c^{\frac{13}{2}}{\left (\frac{\arctan \left (\frac{\sqrt{\sqrt{c x} b c^{4} x^{4} + a c^{3}}}{\sqrt{-a c} c}\right )}{\sqrt{-a c} a c^{4}} + \frac{\sqrt{\sqrt{c x} b c^{4} x^{4} + a c^{3}} a c^{3} +{\left (\sqrt{c x} b c^{4} x^{4} + a c^{3}\right )}^{\frac{3}{2}}}{a b^{2} c^{12} x^{9}}\right )}{\left | c \right |} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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