Optimal. Leaf size=116 \[ \frac {4 a^3 \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2}}{3 b^4 c^2}-\frac {12 a^2 \left (a+b \sqrt {\frac {c}{x}}\right )^{5/2}}{5 b^4 c^2}-\frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{9/2}}{9 b^4 c^2}+\frac {12 a \left (a+b \sqrt {\frac {c}{x}}\right )^{7/2}}{7 b^4 c^2} \]
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Rubi [A] time = 0.07, antiderivative size = 116, 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} \[ -\frac {12 a^2 \left (a+b \sqrt {\frac {c}{x}}\right )^{5/2}}{5 b^4 c^2}+\frac {4 a^3 \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2}}{3 b^4 c^2}-\frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{9/2}}{9 b^4 c^2}+\frac {12 a \left (a+b \sqrt {\frac {c}{x}}\right )^{7/2}}{7 b^4 c^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 369
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{x^3} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{x^3} \, dx,\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=-\operatorname {Subst}\left (2 \operatorname {Subst}\left (\int x^3 \sqrt {a+b \sqrt {c} x} \, dx,x,\frac {1}{\sqrt {x}}\right ),\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=-\operatorname {Subst}\left (2 \operatorname {Subst}\left (\int \left (-\frac {a^3 \sqrt {a+b \sqrt {c} x}}{b^3 c^{3/2}}+\frac {3 a^2 \left (a+b \sqrt {c} x\right )^{3/2}}{b^3 c^{3/2}}-\frac {3 a \left (a+b \sqrt {c} x\right )^{5/2}}{b^3 c^{3/2}}+\frac {\left (a+b \sqrt {c} x\right )^{7/2}}{b^3 c^{3/2}}\right ) \, dx,x,\frac {1}{\sqrt {x}}\right ),\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=\frac {4 a^3 \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2}}{3 b^4 c^2}-\frac {12 a^2 \left (a+b \sqrt {\frac {c}{x}}\right )^{5/2}}{5 b^4 c^2}+\frac {12 a \left (a+b \sqrt {\frac {c}{x}}\right )^{7/2}}{7 b^4 c^2}-\frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{9/2}}{9 b^4 c^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 75, normalized size = 0.65 \[ \frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2} \left (16 a^3 x-24 a^2 b x \sqrt {\frac {c}{x}}+30 a b^2 c-35 b^3 c \sqrt {\frac {c}{x}}\right )}{315 b^4 c^2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 77, normalized size = 0.66 \[ -\frac {4 \, {\left (35 \, b^{4} c^{2} - 6 \, a^{2} b^{2} c x - 16 \, a^{4} x^{2} + {\left (5 \, a b^{3} c x + 8 \, a^{3} b x^{2}\right )} \sqrt {\frac {c}{x}}\right )} \sqrt {b \sqrt {\frac {c}{x}} + a}}{315 \, b^{4} c^{2} x^{2}} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 97, normalized size = 0.84 \[ -\frac {4 \sqrt {a +\sqrt {\frac {c}{x}}\, b}\, \left (a x +\sqrt {\frac {c}{x}}\, b x \right )^{\frac {3}{2}} \left (-16 a^{3} x +24 \sqrt {\frac {c}{x}}\, a^{2} b x -30 a \,b^{2} c +35 \left (\frac {c}{x}\right )^{\frac {3}{2}} b^{3} x \right )}{315 \sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, b^{4} c^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 85, normalized size = 0.73 \[ -\frac {4 \, {\left (\frac {35 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {9}{2}}}{b^{4}} - \frac {135 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {7}{2}} a}{b^{4}} + \frac {189 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {5}{2}} a^{2}}{b^{4}} - \frac {105 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {3}{2}} a^{3}}{b^{4}}\right )}}{315 \, c^{2}} \]
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 \[ \int \frac {\sqrt {a+b\,\sqrt {\frac {c}{x}}}}{x^3} \,d x \]
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 \[ \int \frac {\sqrt {a + b \sqrt {\frac {c}{x}}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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