Optimal. Leaf size=172 \[ -\frac{40 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^6 c^3}+\frac{8 a^3 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{b^6 c^3}-\frac{20 a^4 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^6 c^3}+\frac{4 a^5 \sqrt{a+b \sqrt{\frac{c}{x}}}}{b^6 c^3}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{11/2}}{11 b^6 c^3}+\frac{20 a \left (a+b \sqrt{\frac{c}{x}}\right )^{9/2}}{9 b^6 c^3} \]
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Rubi [A] time = 0.103033, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {369, 266, 43} \[ -\frac{40 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^6 c^3}+\frac{8 a^3 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{b^6 c^3}-\frac{20 a^4 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^6 c^3}+\frac{4 a^5 \sqrt{a+b \sqrt{\frac{c}{x}}}}{b^6 c^3}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{11/2}}{11 b^6 c^3}+\frac{20 a \left (a+b \sqrt{\frac{c}{x}}\right )^{9/2}}{9 b^6 c^3} \]
Antiderivative was successfully verified.
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Rule 369
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b \sqrt{\frac{c}{x}}} x^4} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b \sqrt{c}}{\sqrt{x}}} x^4} \, dx,\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\operatorname{Subst}\left (2 \operatorname{Subst}\left (\int \frac{x^5}{\sqrt{a+b \sqrt{c} x}} \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\operatorname{Subst}\left (2 \operatorname{Subst}\left (\int \left (-\frac{a^5}{b^5 c^{5/2} \sqrt{a+b \sqrt{c} x}}+\frac{5 a^4 \sqrt{a+b \sqrt{c} x}}{b^5 c^{5/2}}-\frac{10 a^3 \left (a+b \sqrt{c} x\right )^{3/2}}{b^5 c^{5/2}}+\frac{10 a^2 \left (a+b \sqrt{c} x\right )^{5/2}}{b^5 c^{5/2}}-\frac{5 a \left (a+b \sqrt{c} x\right )^{7/2}}{b^5 c^{5/2}}+\frac{\left (a+b \sqrt{c} x\right )^{9/2}}{b^5 c^{5/2}}\right ) \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\frac{4 a^5 \sqrt{a+b \sqrt{\frac{c}{x}}}}{b^6 c^3}-\frac{20 a^4 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^6 c^3}+\frac{8 a^3 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{b^6 c^3}-\frac{40 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^6 c^3}+\frac{20 a \left (a+b \sqrt{\frac{c}{x}}\right )^{9/2}}{9 b^6 c^3}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{11/2}}{11 b^6 c^3}\\ \end{align*}
Mathematica [A] time = 0.0873034, size = 111, normalized size = 0.65 \[ \frac{4 \sqrt{a+b \sqrt{\frac{c}{x}}} \left (96 a^3 b^2 c x-80 a^2 b^3 c x \sqrt{\frac{c}{x}}-128 a^4 b x^2 \sqrt{\frac{c}{x}}+256 a^5 x^2+70 a b^4 c^2-63 b^5 c x \left (\frac{c}{x}\right )^{3/2}\right )}{693 b^6 c^3 x^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.051, size = 400, normalized size = 2.3 \begin{align*} -{\frac{1}{693\,{b}^{7}}\sqrt{a+b\sqrt{{\frac{c}{x}}}} \left ( 1386\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}{a}^{13/2}{x}^{7/2}+1386\,\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }{a}^{13/2}{x}^{7/2}+252\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2} \left ({\frac{c}{x}} \right ) ^{5/2}\sqrt{a}{x}^{5/2}{b}^{5}+852\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2} \left ({\frac{c}{x}} \right ) ^{3/2}{a}^{5/2}{x}^{5/2}{b}^{3}+1748\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}\sqrt{{\frac{c}{x}}}{a}^{9/2}{x}^{5/2}b-2772\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{a}^{11/2}{x}^{5/2}-1236\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{a}^{7/2}{x}^{3/2}{b}^{2}c-532\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{a}^{3/2}\sqrt{x}{b}^{4}{c}^{2}+693\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ) \sqrt{{\frac{c}{x}}}{x}^{4}{a}^{6}b-693\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }\sqrt{a}+2\,a\sqrt{x} \right ) } \right ) \sqrt{{\frac{c}{x}}}{x}^{4}{a}^{6}b \right ){x}^{-{\frac{13}{2}}}{\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}} \left ({\frac{c}{x}} \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt{a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01123, size = 171, normalized size = 0.99 \begin{align*} -\frac{4 \,{\left (\frac{63 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{11}{2}}}{b^{6}} - \frac{385 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{9}{2}} a}{b^{6}} + \frac{990 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{7}{2}} a^{2}}{b^{6}} - \frac{1386 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{5}{2}} a^{3}}{b^{6}} + \frac{1155 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{3}{2}} a^{4}}{b^{6}} - \frac{693 \, \sqrt{b \sqrt{\frac{c}{x}} + a} a^{5}}{b^{6}}\right )}}{693 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57795, size = 198, normalized size = 1.15 \begin{align*} \frac{4 \,{\left (70 \, a b^{4} c^{2} + 96 \, a^{3} b^{2} c x + 256 \, a^{5} x^{2} -{\left (63 \, b^{5} c^{2} + 80 \, a^{2} b^{3} c x + 128 \, a^{4} b x^{2}\right )} \sqrt{\frac{c}{x}}\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{693 \, b^{6} c^{3} x^{2}} \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{1}{x^{4} \sqrt{a + b \sqrt{\frac{c}{x}}}}\, 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{1}{\sqrt{b \sqrt{\frac{c}{x}} + a} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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