Optimal. Leaf size=144 \[ \frac{b^2 c \sqrt{a+b \sqrt{c x^2}}}{8 a^2 x}-\frac{b^3 \left (c x^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{8 a^{5/2} x^3}-\frac{b \left (c x^2\right )^{3/2} \sqrt{a+b \sqrt{c x^2}}}{12 a c x^5}-\frac{\sqrt{a+b \sqrt{c x^2}}}{3 x^3} \]
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Rubi [A] time = 0.0592248, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{b^2 c \sqrt{a+b \sqrt{c x^2}}}{8 a^2 x}-\frac{b^3 \left (c x^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{8 a^{5/2} x^3}-\frac{b \left (c x^2\right )^{3/2} \sqrt{a+b \sqrt{c x^2}}}{12 a c x^5}-\frac{\sqrt{a+b \sqrt{c x^2}}}{3 x^3} \]
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^4} \, dx &=\frac{\left (c x^2\right )^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^4} \, dx,x,\sqrt{c x^2}\right )}{x^3}\\ &=-\frac{\sqrt{a+b \sqrt{c x^2}}}{3 x^3}+\frac{\left (b \left (c x^2\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,\sqrt{c x^2}\right )}{6 x^3}\\ &=-\frac{\sqrt{a+b \sqrt{c x^2}}}{3 x^3}-\frac{b \left (c x^2\right )^{3/2} \sqrt{a+b \sqrt{c x^2}}}{12 a c x^5}-\frac{\left (b^2 \left (c x^2\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\sqrt{c x^2}\right )}{8 a x^3}\\ &=-\frac{\sqrt{a+b \sqrt{c x^2}}}{3 x^3}+\frac{b^2 c \sqrt{a+b \sqrt{c x^2}}}{8 a^2 x}-\frac{b \left (c x^2\right )^{3/2} \sqrt{a+b \sqrt{c x^2}}}{12 a c x^5}+\frac{\left (b^3 \left (c x^2\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sqrt{c x^2}\right )}{16 a^2 x^3}\\ &=-\frac{\sqrt{a+b \sqrt{c x^2}}}{3 x^3}+\frac{b^2 c \sqrt{a+b \sqrt{c x^2}}}{8 a^2 x}-\frac{b \left (c x^2\right )^{3/2} \sqrt{a+b \sqrt{c x^2}}}{12 a c x^5}+\frac{\left (b^2 \left (c x^2\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sqrt{c x^2}}\right )}{8 a^2 x^3}\\ &=-\frac{\sqrt{a+b \sqrt{c x^2}}}{3 x^3}+\frac{b^2 c \sqrt{a+b \sqrt{c x^2}}}{8 a^2 x}-\frac{b \left (c x^2\right )^{3/2} \sqrt{a+b \sqrt{c x^2}}}{12 a c x^5}-\frac{b^3 \left (c x^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{8 a^{5/2} x^3}\\ \end{align*}
Mathematica [C] time = 0.0137765, size = 63, normalized size = 0.44 \[ \frac{2 b^3 \left (c x^2\right )^{3/2} \left (a+b \sqrt{c x^2}\right )^{3/2} \, _2F_1\left (\frac{3}{2},4;\frac{5}{2};\frac{\sqrt{c x^2} b}{a}+1\right )}{3 a^4 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 97, normalized size = 0.7 \begin{align*} -{\frac{1}{24\,{x}^{3}} \left ( -3\, \left ( a+b\sqrt{c{x}^{2}} \right ) ^{5/2}{a}^{5/2}+3\,{\it Artanh} \left ({\frac{\sqrt{a+b\sqrt{c{x}^{2}}}}{\sqrt{a}}} \right ){a}^{2}{b}^{3} \left ( c{x}^{2} \right ) ^{3/2}+8\, \left ( a+b\sqrt{c{x}^{2}} \right ) ^{3/2}{a}^{7/2}+3\,\sqrt{a+b\sqrt{c{x}^{2}}}{a}^{9/2} \right ){a}^{-{\frac{9}{2}}}} \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^{2}} b + a}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53949, size = 556, normalized size = 3.86 \begin{align*} \left [\frac{3 \, b^{3} c x^{3} \sqrt{\frac{c}{a}} \log \left (\frac{b c x^{2} - 2 \, \sqrt{\sqrt{c x^{2}} b + a} a x \sqrt{\frac{c}{a}} + 2 \, \sqrt{c x^{2}} a}{x^{2}}\right ) + 2 \,{\left (3 \, b^{2} c x^{2} - 2 \, \sqrt{c x^{2}} a b - 8 \, a^{2}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{48 \, a^{2} x^{3}}, -\frac{3 \, b^{3} c x^{3} \sqrt{-\frac{c}{a}} \arctan \left (-\frac{{\left (a b c x^{2} \sqrt{-\frac{c}{a}} - \sqrt{c x^{2}} a^{2} \sqrt{-\frac{c}{a}}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{b^{2} c^{2} x^{3} - a^{2} c x}\right ) -{\left (3 \, b^{2} c x^{2} - 2 \, \sqrt{c x^{2}} a b - 8 \, a^{2}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{24 \, a^{2} x^{3}}\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^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19208, size = 154, normalized size = 1.07 \begin{align*} \frac{\frac{3 \, b^{4} c^{2} \arctan \left (\frac{\sqrt{b \sqrt{c} x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{3 \,{\left (b \sqrt{c} x + a\right )}^{\frac{5}{2}} b^{4} c^{2} - 8 \,{\left (b \sqrt{c} x + a\right )}^{\frac{3}{2}} a b^{4} c^{2} - 3 \, \sqrt{b \sqrt{c} x + a} a^{2} b^{4} c^{2}}{a^{2} b^{3} c^{\frac{3}{2}} x^{3}}}{24 \, b \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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