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 = 0.07, 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 47
Rule 51
Rule 63
Rule 208
Rule 266
Rule 369
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*}
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Mathematica [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^{10}} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 130, normalized size = 1.29 \[ -\frac {{\left (\frac {b^{3} c^{6} \arctan \left (\frac {\sqrt {\sqrt {c x} b c^{5} x^{4} + a c^{4}}}{\sqrt {-a} c^{2}}\right )}{\sqrt {-a} a} + \frac {\sqrt {\sqrt {c x} b c^{5} x^{4} + a c^{4}} a b^{3} c^{12} + {\left (\sqrt {c x} b c^{5} x^{4} + a c^{4}\right )}^{\frac {3}{2}} b^{3} c^{8}}{a b^{2} c^{11} x^{9}}\right )} {\left | c \right |}^{2}}{18 \, b c^{5}} \]
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 \[ \int \frac {\sqrt {a +\left (c \,x^{3}\right )^{\frac {3}{2}} b}}{x^{10}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 128, normalized size = 1.27 \[ -\frac {1}{36} \, {\left (\frac {b^{2} \log \left (\frac {\sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a} - \sqrt {a}}{\sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2 \, {\left ({\left (\left (c x^{3}\right )^{\frac {3}{2}} b + a\right )}^{\frac {3}{2}} b^{2} + \sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a} a b^{2}\right )}}{{\left (\left (c x^{3}\right )^{\frac {3}{2}} b + a\right )}^{2} a - 2 \, {\left (\left (c x^{3}\right )^{\frac {3}{2}} b + a\right )} a^{2} + a^{3}}\right )} c^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {a+b\,{\left (c\,x^3\right )}^{3/2}}}{x^{10}} \,d x \]
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 \[ \int \frac {\sqrt {a + b \left (c x^{3}\right )^{\frac {3}{2}}}}{x^{10}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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