Optimal. Leaf size=810 \[ -\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} c^{2/3} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right ) \sqrt{\frac{-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x+a^{2/3}}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} E\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right ) b^{4/3}}{8 a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right )}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}+\frac{3^{3/4} c^{2/3} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right ) \sqrt{\frac{-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x+a^{2/3}}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt{3}\right ) b^{4/3}}{2 \sqrt{2} a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right )}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}+\frac{3 c^{2/3} \sqrt{a+b \sqrt{c x^3}} b^{4/3}}{4 a \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )}-\frac{3 c x \sqrt{a+b \sqrt{c x^3}} b}{4 a \sqrt{c x^3}}-\frac{\sqrt{a+b \sqrt{c x^3}}}{2 x^2} \]
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Rubi [A] time = 0.458766, antiderivative size = 810, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {369, 341, 277, 325, 303, 218, 1877} \[ -\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} c^{2/3} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right ) \sqrt{\frac{-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x+a^{2/3}}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} E\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right ) b^{4/3}}{8 a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right )}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}+\frac{3^{3/4} c^{2/3} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right ) \sqrt{\frac{-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x+a^{2/3}}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} F\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right ) b^{4/3}}{2 \sqrt{2} a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right )}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}+\frac{3 c^{2/3} \sqrt{a+b \sqrt{c x^3}} b^{4/3}}{4 a \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )}-\frac{3 c x \sqrt{a+b \sqrt{c x^3}} b}{4 a \sqrt{c x^3}}-\frac{\sqrt{a+b \sqrt{c x^3}}}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 369
Rule 341
Rule 277
Rule 325
Rule 303
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b \sqrt{c x^3}}}{x^3} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{a+b \sqrt{c} x^{3/2}}}{x^3} \, 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^5} \, dx,x,\sqrt{x}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{\sqrt{a+b \sqrt{c x^3}}}{2 x^2}+\operatorname{Subst}\left (\frac{1}{4} \left (3 b \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \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}}}{2 x^2}-\frac{3 b c x \sqrt{a+b \sqrt{c x^3}}}{4 a \sqrt{c x^3}}+\operatorname{Subst}\left (\frac{\left (3 b^2 c\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \sqrt{c} x^3}} \, dx,x,\sqrt{x}\right )}{8 a},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{\sqrt{a+b \sqrt{c x^3}}}{2 x^2}-\frac{3 b c x \sqrt{a+b \sqrt{c x^3}}}{4 a \sqrt{c x^3}}+\operatorname{Subst}\left (\frac{\left (3 b^{5/3} c^{5/6}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt [6]{c} x}{\sqrt{a+b \sqrt{c} x^3}} \, dx,x,\sqrt{x}\right )}{8 a},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )+\operatorname{Subst}\left (\frac{\left (3 \sqrt{\frac{1}{2} \left (2-\sqrt{3}\right )} b^{5/3} c^{5/6}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \sqrt{c} x^3}} \, dx,x,\sqrt{x}\right )}{4 a^{2/3}},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{\sqrt{a+b \sqrt{c x^3}}}{2 x^2}-\frac{3 b c x \sqrt{a+b \sqrt{c x^3}}}{4 a \sqrt{c x^3}}+\frac{3 b^{4/3} c^{2/3} \sqrt{a+b \sqrt{c x^3}}}{4 a \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{4/3} c^{2/3} \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}} E\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 )}{8 a^{2/3} \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{3^{3/4} b^{4/3} c^{2/3} \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 )}{2 \sqrt{2} a^{2/3} \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*}
Mathematica [F] time = 0.0401099, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sqrt{c x^3}}}{x^3} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.298, size = 869, normalized size = 1.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\sqrt{c x^{3}} b + a}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{\sqrt{c x^{3}} b + a}}{x^{3}}, x\right ) \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 \sqrt{c x^{3}}}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\sqrt{c x^{3}} b + a}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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