Optimal. Leaf size=219 \[ \frac{7 b^4 c^2 \sqrt{a+b \sqrt{c x^2}}}{128 a^4 x}-\frac{7 b^3 \left (c x^2\right )^{5/2} \sqrt{a+b \sqrt{c x^2}}}{192 a^3 c x^7}+\frac{7 b^2 c \sqrt{a+b \sqrt{c x^2}}}{240 a^2 x^3}-\frac{7 b^5 \left (c x^2\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{128 a^{9/2} x^5}-\frac{b \left (c x^2\right )^{5/2} \sqrt{a+b \sqrt{c x^2}}}{40 a c^2 x^9}-\frac{\sqrt{a+b \sqrt{c x^2}}}{5 x^5} \]
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Rubi [A] time = 0.0939607, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{7 b^4 c^2 \sqrt{a+b \sqrt{c x^2}}}{128 a^4 x}-\frac{7 b^3 \left (c x^2\right )^{5/2} \sqrt{a+b \sqrt{c x^2}}}{192 a^3 c x^7}+\frac{7 b^2 c \sqrt{a+b \sqrt{c x^2}}}{240 a^2 x^3}-\frac{7 b^5 \left (c x^2\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{128 a^{9/2} x^5}-\frac{b \left (c x^2\right )^{5/2} \sqrt{a+b \sqrt{c x^2}}}{40 a c^2 x^9}-\frac{\sqrt{a+b \sqrt{c x^2}}}{5 x^5} \]
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^6} \, dx &=\frac{\left (c x^2\right )^{5/2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^6} \, dx,x,\sqrt{c x^2}\right )}{x^5}\\ &=-\frac{\sqrt{a+b \sqrt{c x^2}}}{5 x^5}+\frac{\left (b \left (c x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^5 \sqrt{a+b x}} \, dx,x,\sqrt{c x^2}\right )}{10 x^5}\\ &=-\frac{\sqrt{a+b \sqrt{c x^2}}}{5 x^5}-\frac{b \left (c x^2\right )^{5/2} \sqrt{a+b \sqrt{c x^2}}}{40 a c^2 x^9}-\frac{\left (7 b^2 \left (c x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{a+b x}} \, dx,x,\sqrt{c x^2}\right )}{80 a x^5}\\ &=-\frac{\sqrt{a+b \sqrt{c x^2}}}{5 x^5}+\frac{7 b^2 c \sqrt{a+b \sqrt{c x^2}}}{240 a^2 x^3}-\frac{b \left (c x^2\right )^{5/2} \sqrt{a+b \sqrt{c x^2}}}{40 a c^2 x^9}+\frac{\left (7 b^3 \left (c x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,\sqrt{c x^2}\right )}{96 a^2 x^5}\\ &=-\frac{\sqrt{a+b \sqrt{c x^2}}}{5 x^5}+\frac{7 b^2 c \sqrt{a+b \sqrt{c x^2}}}{240 a^2 x^3}-\frac{b \left (c x^2\right )^{5/2} \sqrt{a+b \sqrt{c x^2}}}{40 a c^2 x^9}-\frac{7 b^3 \left (c x^2\right )^{5/2} \sqrt{a+b \sqrt{c x^2}}}{192 a^3 c x^7}-\frac{\left (7 b^4 \left (c x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\sqrt{c x^2}\right )}{128 a^3 x^5}\\ &=-\frac{\sqrt{a+b \sqrt{c x^2}}}{5 x^5}+\frac{7 b^2 c \sqrt{a+b \sqrt{c x^2}}}{240 a^2 x^3}+\frac{7 b^4 c^2 \sqrt{a+b \sqrt{c x^2}}}{128 a^4 x}-\frac{b \left (c x^2\right )^{5/2} \sqrt{a+b \sqrt{c x^2}}}{40 a c^2 x^9}-\frac{7 b^3 \left (c x^2\right )^{5/2} \sqrt{a+b \sqrt{c x^2}}}{192 a^3 c x^7}+\frac{\left (7 b^5 \left (c x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sqrt{c x^2}\right )}{256 a^4 x^5}\\ &=-\frac{\sqrt{a+b \sqrt{c x^2}}}{5 x^5}+\frac{7 b^2 c \sqrt{a+b \sqrt{c x^2}}}{240 a^2 x^3}+\frac{7 b^4 c^2 \sqrt{a+b \sqrt{c x^2}}}{128 a^4 x}-\frac{b \left (c x^2\right )^{5/2} \sqrt{a+b \sqrt{c x^2}}}{40 a c^2 x^9}-\frac{7 b^3 \left (c x^2\right )^{5/2} \sqrt{a+b \sqrt{c x^2}}}{192 a^3 c x^7}+\frac{\left (7 b^4 \left (c x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sqrt{c x^2}}\right )}{128 a^4 x^5}\\ &=-\frac{\sqrt{a+b \sqrt{c x^2}}}{5 x^5}+\frac{7 b^2 c \sqrt{a+b \sqrt{c x^2}}}{240 a^2 x^3}+\frac{7 b^4 c^2 \sqrt{a+b \sqrt{c x^2}}}{128 a^4 x}-\frac{b \left (c x^2\right )^{5/2} \sqrt{a+b \sqrt{c x^2}}}{40 a c^2 x^9}-\frac{7 b^3 \left (c x^2\right )^{5/2} \sqrt{a+b \sqrt{c x^2}}}{192 a^3 c x^7}-\frac{7 b^5 \left (c x^2\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{128 a^{9/2} x^5}\\ \end{align*}
Mathematica [C] time = 0.0134367, size = 63, normalized size = 0.29 \[ \frac{2 b^5 \left (c x^2\right )^{5/2} \left (a+b \sqrt{c x^2}\right )^{3/2} \, _2F_1\left (\frac{3}{2},6;\frac{5}{2};\frac{\sqrt{c x^2} b}{a}+1\right )}{3 a^6 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 133, normalized size = 0.6 \begin{align*} -{\frac{1}{1920\,{x}^{5}} \left ( -105\, \left ( a+b\sqrt{c{x}^{2}} \right ) ^{9/2}{a}^{9/2}+490\, \left ( a+b\sqrt{c{x}^{2}} \right ) ^{7/2}{a}^{11/2}-896\, \left ( a+b\sqrt{c{x}^{2}} \right ) ^{5/2}{a}^{13/2}+105\,{\it Artanh} \left ({\frac{\sqrt{a+b\sqrt{c{x}^{2}}}}{\sqrt{a}}} \right ){a}^{4}{b}^{5} \left ( c{x}^{2} \right ) ^{5/2}+790\, \left ( a+b\sqrt{c{x}^{2}} \right ) ^{3/2}{a}^{15/2}+105\,\sqrt{a+b\sqrt{c{x}^{2}}}{a}^{17/2} \right ){a}^{-{\frac{17}{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^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43592, size = 705, normalized size = 3.22 \begin{align*} \left [\frac{105 \, b^{5} c^{2} x^{5} \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 (105 \, b^{4} c^{2} x^{4} + 56 \, a^{2} b^{2} c x^{2} - 384 \, a^{4} - 2 \,{\left (35 \, a b^{3} c x^{2} + 24 \, a^{3} b\right )} \sqrt{c x^{2}}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{3840 \, a^{4} x^{5}}, -\frac{105 \, b^{5} c^{2} x^{5} \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 (105 \, b^{4} c^{2} x^{4} + 56 \, a^{2} b^{2} c x^{2} - 384 \, a^{4} - 2 \,{\left (35 \, a b^{3} c x^{2} + 24 \, a^{3} b\right )} \sqrt{c x^{2}}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{1920 \, a^{4} x^{5}}\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^{6}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20528, size = 211, normalized size = 0.96 \begin{align*} \frac{\frac{105 \, b^{6} c^{3} \arctan \left (\frac{\sqrt{b \sqrt{c} x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} + \frac{105 \,{\left (b \sqrt{c} x + a\right )}^{\frac{9}{2}} b^{6} c^{3} - 490 \,{\left (b \sqrt{c} x + a\right )}^{\frac{7}{2}} a b^{6} c^{3} + 896 \,{\left (b \sqrt{c} x + a\right )}^{\frac{5}{2}} a^{2} b^{6} c^{3} - 790 \,{\left (b \sqrt{c} x + a\right )}^{\frac{3}{2}} a^{3} b^{6} c^{3} - 105 \, \sqrt{b \sqrt{c} x + a} a^{4} b^{6} c^{3}}{a^{4} b^{5} c^{\frac{5}{2}} x^{5}}}{1920 \, b \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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