Integrand size = 21, antiderivative size = 298 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^3} \, dx=-\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{2 x^2}+\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{2/3} c \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} c x^2-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}} \]
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Time = 0.14 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {375, 283, 224} \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^3} \, dx=\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{2/3} c \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} \sqrt {c x^2}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt {c x^2}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}}-\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{2 x^2} \]
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Rule 224
Rule 283
Rule 375
Rubi steps \begin{align*} \text {integral}& = c \text {Subst}\left (\int \frac {\sqrt {a+b x^3}}{x^3} \, dx,x,\sqrt {c x^2}\right ) \\ & = -\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{2 x^2}+\frac {1}{4} (3 b c) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^3}} \, dx,x,\sqrt {c x^2}\right ) \\ & = -\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{2 x^2}+\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{2/3} c \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} c x^2-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.23 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^3} \, dx=-\frac {\sqrt {a+b \left (c x^2\right )^{3/2}} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{2},\frac {1}{3},-\frac {b \left (c x^2\right )^{3/2}}{a}\right )}{2 x^2 \sqrt {1+\frac {b \left (c x^2\right )^{3/2}}{a}}} \]
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\[\int \frac {\sqrt {a +b \left (c \,x^{2}\right )^{\frac {3}{2}}}}{x^{3}}d x\]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.22 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^3} \, dx=\frac {3 \, \sqrt {\frac {\sqrt {c x^{2}} b c}{x}} x^{2} {\rm weierstrassPInverse}\left (0, -\frac {4 \, \sqrt {c x^{2}} a}{b c^{2} x}, x\right ) - \sqrt {\sqrt {c x^{2}} b c x^{2} + a}}{2 \, x^{2}} \]
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\[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^3} \, dx=\int \frac {\sqrt {a + b \left (c x^{2}\right )^{\frac {3}{2}}}}{x^{3}}\, dx \]
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\[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^3} \, dx=\int { \frac {\sqrt {\left (c x^{2}\right )^{\frac {3}{2}} b + a}}{x^{3}} \,d x } \]
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\[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^3} \, dx=\int { \frac {\sqrt {\left (c x^{2}\right )^{\frac {3}{2}} b + a}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^3} \, dx=\int \frac {\sqrt {a+b\,{\left (c\,x^2\right )}^{3/2}}}{x^3} \,d x \]
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