Optimal. Leaf size=230 \[ \frac{x^{m+1} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} F_1\left (-2 (m+1);-\frac{1}{2},-\frac{1}{2};-2 m-1;-\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d}-\sqrt{b^2 d-4 a c}\right )},-\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (\sqrt{d} b+\sqrt{b^2 d-4 a c}\right )}\right )}{(m+1) \sqrt{\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d}-\sqrt{b^2 d-4 a c}\right )}+1} \sqrt{\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (\sqrt{b^2 d-4 a c}+b \sqrt{d}\right )}+1}} \]
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Rubi [A] time = 0.719473, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {1971, 1379, 759, 133} \[ \frac{x^{m+1} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} F_1\left (-2 (m+1);-\frac{1}{2},-\frac{1}{2};-2 m-1;-\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d}-\sqrt{b^2 d-4 a c}\right )},-\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (\sqrt{d} b+\sqrt{b^2 d-4 a c}\right )}\right )}{(m+1) \sqrt{\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d}-\sqrt{b^2 d-4 a c}\right )}+1} \sqrt{\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (\sqrt{b^2 d-4 a c}+b \sqrt{d}\right )}+1}} \]
Antiderivative was successfully verified.
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Rule 1971
Rule 1379
Rule 759
Rule 133
Rubi steps
\begin{align*} \int \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^m \, dx &=-\left (\left (d \left (\frac{d}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int x^{-2-m} \sqrt{a+b \sqrt{x}+\frac{c x}{d}} \, dx,x,\frac{d}{x}\right )\right )\\ &=-\left (\left (2 d \left (\frac{d}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int x^{-1+2 (-1-m)} \sqrt{a+b x+\frac{c x^2}{d}} \, dx,x,\sqrt{\frac{d}{x}}\right )\right )\\ &=-\frac{\left (2 d \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} \left (\frac{d}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int x^{-1+2 (-1-m)} \sqrt{1+\frac{2 c x}{\sqrt{d} \left (b \sqrt{d}-\sqrt{-4 a c+b^2 d}\right )}} \sqrt{1+\frac{2 c x}{\sqrt{d} \left (b \sqrt{d}+\sqrt{-4 a c+b^2 d}\right )}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{\sqrt{1+\frac{2 c \sqrt{\frac{d}{x}}}{d \left (b-\frac{\sqrt{-4 a c+b^2 d}}{\sqrt{d}}\right )}} \sqrt{1+\frac{2 c \sqrt{\frac{d}{x}}}{d \left (b+\frac{\sqrt{-4 a c+b^2 d}}{\sqrt{d}}\right )}}}\\ &=\frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^{1+m} F_1\left (-2 (1+m);-\frac{1}{2},-\frac{1}{2};-1-2 m;-\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d}-\sqrt{-4 a c+b^2 d}\right )},-\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d}+\sqrt{-4 a c+b^2 d}\right )}\right )}{(1+m) \sqrt{1+\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d}-\sqrt{-4 a c+b^2 d}\right )}} \sqrt{1+\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d}+\sqrt{-4 a c+b^2 d}\right )}}}\\ \end{align*}
Mathematica [F] time = 0.146745, size = 0, normalized size = 0. \[ \int \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^m \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.18, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\sqrt{a+{\frac{c}{x}}+b\sqrt{{\frac{d}{x}}}}\, dx \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 \sqrt{b \sqrt{\frac{d}{x}} + a + \frac{c}{x}} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 x^{m} \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 \sqrt{b \sqrt{\frac{d}{x}} + a + \frac{c}{x}} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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