Optimal. Leaf size=97 \[ \frac{b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{b c \sqrt{a+b \sqrt{c x^2}}}{4 a \sqrt{c x^2}}-\frac{\sqrt{a+b \sqrt{c x^2}}}{2 x^2} \]
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Rubi [A] time = 0.0427247, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {368, 47, 51, 63, 208} \[ \frac{b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{b c \sqrt{a+b \sqrt{c x^2}}}{4 a \sqrt{c x^2}}-\frac{\sqrt{a+b \sqrt{c x^2}}}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 368
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b \sqrt{c x^2}}}{x^3} \, dx &=c \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^3} \, dx,x,\sqrt{c x^2}\right )\\ &=-\frac{\sqrt{a+b \sqrt{c x^2}}}{2 x^2}+\frac{1}{4} (b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\sqrt{c x^2}\right )\\ &=-\frac{\sqrt{a+b \sqrt{c x^2}}}{2 x^2}-\frac{b c \sqrt{a+b \sqrt{c x^2}}}{4 a \sqrt{c x^2}}-\frac{\left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sqrt{c x^2}\right )}{8 a}\\ &=-\frac{\sqrt{a+b \sqrt{c x^2}}}{2 x^2}-\frac{b c \sqrt{a+b \sqrt{c x^2}}}{4 a \sqrt{c x^2}}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sqrt{c x^2}}\right )}{4 a}\\ &=-\frac{\sqrt{a+b \sqrt{c x^2}}}{2 x^2}-\frac{b c \sqrt{a+b \sqrt{c x^2}}}{4 a \sqrt{c x^2}}+\frac{b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{4 a^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0115763, size = 52, normalized size = 0.54 \[ -\frac{2 b^2 c \left (a+b \sqrt{c x^2}\right )^{3/2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{\sqrt{c x^2} b}{a}+1\right )}{3 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 72, normalized size = 0.7 \begin{align*} -{\frac{1}{4\,{x}^{2}} \left ( -{\it Artanh} \left ({\sqrt{a+b\sqrt{c{x}^{2}}}{\frac{1}{\sqrt{a}}}} \right ) c{x}^{2}a{b}^{2}+ \left ( a+b\sqrt{c{x}^{2}} \right ) ^{{\frac{3}{2}}}{a}^{{\frac{3}{2}}}+\sqrt{a+b\sqrt{c{x}^{2}}}{a}^{{\frac{5}{2}}} \right ){a}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31517, size = 419, normalized size = 4.32 \begin{align*} \left [\frac{\sqrt{a} b^{2} c x^{2} \log \left (\frac{b c x^{2} + 2 \, \sqrt{c x^{2}} \sqrt{\sqrt{c x^{2}} b + a} \sqrt{a} + 2 \, \sqrt{c x^{2}} a}{x^{2}}\right ) - 2 \,{\left (\sqrt{c x^{2}} a b + 2 \, a^{2}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{8 \, a^{2} x^{2}}, -\frac{\sqrt{-a} b^{2} c x^{2} \arctan \left (\frac{\sqrt{\sqrt{c x^{2}} b + a} \sqrt{-a}}{a}\right ) +{\left (\sqrt{c x^{2}} a b + 2 \, a^{2}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{4 \, a^{2} x^{2}}\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^{2}}}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22736, size = 122, normalized size = 1.26 \begin{align*} -\frac{\frac{b^{3} c^{\frac{3}{2}} \arctan \left (\frac{\sqrt{b \sqrt{c} x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{{\left (b \sqrt{c} x + a\right )}^{\frac{3}{2}} b^{3} c^{\frac{3}{2}} + \sqrt{b \sqrt{c} x + a} a b^{3} c^{\frac{3}{2}}}{a b^{2} c x^{2}}}{4 \, b \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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