Optimal. Leaf size=165 \[ \frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} \left (16 a c-15 b^2 d+10 b c \sqrt{\frac{d}{x}}\right )}{12 c^3}-\frac{b \sqrt{d} \left (12 a c-5 b^2 d\right ) \tanh ^{-1}\left (\frac{b d+2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{8 c^{7/2}}-\frac{2 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{3 c x} \]
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Rubi [A] time = 0.22797, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1970, 1357, 742, 779, 621, 206} \[ \frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} \left (16 a c-15 b^2 d+10 b c \sqrt{\frac{d}{x}}\right )}{12 c^3}-\frac{b \sqrt{d} \left (12 a c-5 b^2 d\right ) \tanh ^{-1}\left (\frac{b d+2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{8 c^{7/2}}-\frac{2 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{3 c x} \]
Antiderivative was successfully verified.
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Rule 1970
Rule 1357
Rule 742
Rule 779
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^3} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \sqrt{x}+\frac{c x}{d}}} \, dx,x,\frac{d}{x}\right )}{d^2}\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{a+b x+\frac{c x^2}{d}}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{d^2}\\ &=-\frac{2 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{3 c x}-\frac{2 \operatorname{Subst}\left (\int \frac{x \left (-2 a-\frac{5 b x}{2}\right )}{\sqrt{a+b x+\frac{c x^2}{d}}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{3 c d}\\ &=\frac{\left (16 a c-5 b \left (3 b d-2 c \sqrt{\frac{d}{x}}\right )\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{12 c^3}-\frac{2 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{3 c x}-\frac{\left (b \left (12 a c-5 b^2 d\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+\frac{c x^2}{d}}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{8 c^3}\\ &=\frac{\left (16 a c-5 b \left (3 b d-2 c \sqrt{\frac{d}{x}}\right )\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{12 c^3}-\frac{2 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{3 c x}-\frac{\left (b \left (12 a c-5 b^2 d\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{4 c}{d}-x^2} \, dx,x,\frac{b+\frac{2 c \sqrt{\frac{d}{x}}}{d}}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{4 c^3}\\ &=\frac{\left (16 a c-5 b \left (3 b d-2 c \sqrt{\frac{d}{x}}\right )\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{12 c^3}-\frac{2 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{3 c x}-\frac{b \sqrt{d} \left (12 a c-5 b^2 d\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \left (b+\frac{2 c \sqrt{\frac{d}{x}}}{d}\right )}{2 \sqrt{c} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{8 c^{7/2}}\\ \end{align*}
Mathematica [F] time = 0.190929, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^3} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.138, size = 267, normalized size = 1.6 \begin{align*}{\frac{1}{24\,x}\sqrt{{\frac{1}{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) }} \left ( 15\,\ln \left ({\frac{1}{\sqrt{x}} \left ( 2\,c+b\sqrt{{\frac{d}{x}}}x+2\,\sqrt{c}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \right ) } \right ) \left ({\frac{d}{x}} \right ) ^{3/2}{x}^{3}{b}^{3}c+20\,{c}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}xb-36\,\ln \left ({\frac{1}{\sqrt{x}} \left ( 2\,c+b\sqrt{{\frac{d}{x}}}x+2\,\sqrt{c}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \right ) } \right ) \sqrt{{\frac{d}{x}}}{x}^{2}ab{c}^{2}-16\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{c}^{7/2}+32\,{c}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}xa-30\,{c}^{3/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}dx{b}^{2} \right ){\frac{1}{\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}}}{c}^{-{\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{1}{\sqrt{b \sqrt{\frac{d}{x}} + a + \frac{c}{x}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{3} \sqrt{a + b \sqrt{\frac{d}{x}} + \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{d}{x}} + a + \frac{c}{x}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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