Optimal. Leaf size=139 \[ -\frac {45 b^2 c^3 x \sqrt {\frac {b \left (c x^3\right )^{3/2}}{a}+1} \, _2F_1\left (\frac {2}{9},\frac {1}{2};\frac {11}{9};-\frac {b \left (c x^3\right )^{3/2}}{a}\right )}{448 a \sqrt {a+b \left (c x^3\right )^{3/2}}}-\frac {9 b c^3 x \sqrt {a+b \left (c x^3\right )^{3/2}}}{112 a \left (c x^3\right )^{3/2}}-\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{8 x^8} \]
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Rubi [A] time = 0.09, antiderivative size = 141, normalized size of antiderivative = 1.01, 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, 341, 277, 325, 365, 364} \[ -\frac {45 b^2 c^3 x \sqrt {\frac {b \left (c x^3\right )^{3/2}}{a}+1} \, _2F_1\left (\frac {2}{9},\frac {1}{2};\frac {11}{9};-\frac {b \left (c x^3\right )^{3/2}}{a}\right )}{448 a \sqrt {a+b \left (c x^3\right )^{3/2}}}-\frac {9 b c^5 x^7 \sqrt {a+b \left (c x^3\right )^{3/2}}}{112 a \left (c x^3\right )^{7/2}}-\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{8 x^8} \]
Antiderivative was successfully verified.
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Rule 277
Rule 325
Rule 341
Rule 364
Rule 365
Rule 369
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^9} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt {a+b c^{3/2} x^{9/2}}}{x^9} \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\operatorname {Subst}\left (2 \operatorname {Subst}\left (\int \frac {\sqrt {a+b c^{3/2} x^9}}{x^{17}} \, dx,x,\sqrt {x}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{8 x^8}+\operatorname {Subst}\left (\frac {1}{16} \left (9 b c^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{x^8 \sqrt {a+b c^{3/2} x^9}} \, dx,x,\sqrt {x}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{8 x^8}-\frac {9 b c^5 x^7 \sqrt {a+b \left (c x^3\right )^{3/2}}}{112 a \left (c x^3\right )^{7/2}}-\operatorname {Subst}\left (\frac {\left (45 b^2 c^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b c^{3/2} x^9}} \, dx,x,\sqrt {x}\right )}{224 a},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{8 x^8}-\frac {9 b c^5 x^7 \sqrt {a+b \left (c x^3\right )^{3/2}}}{112 a \left (c x^3\right )^{7/2}}-\operatorname {Subst}\left (\frac {\left (45 b^2 c^3 \sqrt {1+\frac {b c^{3/2} x^{9/2}}{a}}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+\frac {b c^{3/2} x^9}{a}}} \, dx,x,\sqrt {x}\right )}{224 a \sqrt {a+b c^{3/2} x^{9/2}}},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{8 x^8}-\frac {9 b c^5 x^7 \sqrt {a+b \left (c x^3\right )^{3/2}}}{112 a \left (c x^3\right )^{7/2}}-\frac {45 b^2 c^3 x \sqrt {1+\frac {b \left (c x^3\right )^{3/2}}{a}} \, _2F_1\left (\frac {2}{9},\frac {1}{2};\frac {11}{9};-\frac {b \left (c x^3\right )^{3/2}}{a}\right )}{448 a \sqrt {a+b \left (c x^3\right )^{3/2}}}\\ \end {align*}
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Mathematica [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^9} \, dx \]
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 [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a}}{x^{9}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a +\left (c \,x^{3}\right )^{\frac {3}{2}} b}}{x^{9}}\, dx \]
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 \[ \int \frac {\sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a}}{x^{9}}\,{d x} \]
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 \[ \int \frac {\sqrt {a+b\,{\left (c\,x^3\right )}^{3/2}}}{x^9} \,d x \]
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 \[ \int \frac {\sqrt {a + b \left (c x^{3}\right )^{\frac {3}{2}}}}{x^{9}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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