Optimal. Leaf size=355 \[ \frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{2/3} \sqrt [3]{c} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} F\left (\sin ^{-1}\left (\frac {\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \sqrt {a+b \sqrt {c x^3}}}-\frac {\sqrt {a+b \sqrt {c x^3}}}{x} \]
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Rubi [A] time = 0.15, antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {369, 341, 277, 218} \[ \frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{2/3} \sqrt [3]{c} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} F\left (\sin ^{-1}\left (\frac {\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \sqrt {a+b \sqrt {c x^3}}}-\frac {\sqrt {a+b \sqrt {c x^3}}}{x} \]
Antiderivative was successfully verified.
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Rule 218
Rule 277
Rule 341
Rule 369
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^2} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt {a+b \sqrt {c} x^{3/2}}}{x^2} \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\operatorname {Subst}\left (2 \operatorname {Subst}\left (\int \frac {\sqrt {a+b \sqrt {c} x^3}}{x^3} \, dx,x,\sqrt {x}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\frac {\sqrt {a+b \sqrt {c x^3}}}{x}+\operatorname {Subst}\left (\frac {1}{2} \left (3 b \sqrt {c}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \sqrt {c} x^3}} \, dx,x,\sqrt {x}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\frac {\sqrt {a+b \sqrt {c x^3}}}{x}+\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{2/3} \sqrt [3]{c} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} \sqrt [3]{c} x-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \sqrt {a+b \sqrt {c x^3}}}\\ \end {align*}
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Mathematica [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 6.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\sqrt {c x^{3}} b + a}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 306, normalized size = 0.86 \[ -\frac {i \sqrt {3}\, \left (-a \,b^{2} c \right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {-\frac {i \left (-2 \sqrt {c \,x^{3}}\, b +i \sqrt {3}\, \left (-a \,b^{2} c \right )^{\frac {1}{3}} x -\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}\, \sqrt {\frac {\sqrt {c \,x^{3}}\, b -\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} \left (i \sqrt {3}-3\right ) x}}\, \sqrt {-\frac {i \left (2 \sqrt {c \,x^{3}}\, b +i \sqrt {3}\, \left (-a \,b^{2} c \right )^{\frac {1}{3}} x +\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}\, x \EllipticF \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {-\frac {i \left (-2 \sqrt {c \,x^{3}}\, b +i \sqrt {3}\, \left (-a \,b^{2} c \right )^{\frac {1}{3}} x -\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{i \sqrt {3}-3}}\right )+2 a +2 \sqrt {c \,x^{3}}\, b}{2 \sqrt {a +\sqrt {c \,x^{3}}\, b}\, x} \]
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 \[ \int \frac {\sqrt {\sqrt {c x^{3}} b + a}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {a+b\,\sqrt {c\,x^3}}}{x^2} \,d x \]
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 \[ \int \frac {\sqrt {a + b \sqrt {c x^{3}}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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