Optimal. Leaf size=386 \[ -\frac{7 b d \left (528 a^2 c^2-680 a b^2 c d+165 b^4 d^2\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{1280 a^6 \sqrt{\frac{d}{x}}}+\frac{x \left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{640 a^5}-\frac{\left (-1680 a^2 b^2 c^2 d+320 a^3 c^3+1260 a b^4 c d^2-231 b^6 d^3\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{512 a^{13/2}}+\frac{b d^2 \left (156 a c-77 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{160 a^4 \left (\frac{d}{x}\right )^{3/2}}-\frac{x^2 \left (100 a c-99 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{240 a^3}-\frac{11 b d^3 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{30 a^2 \left (\frac{d}{x}\right )^{5/2}}+\frac{x^3 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{3 a} \]
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Rubi [A] time = 0.728865, antiderivative size = 386, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {1970, 1357, 744, 834, 806, 724, 206} \[ -\frac{7 b d \left (528 a^2 c^2-680 a b^2 c d+165 b^4 d^2\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{1280 a^6 \sqrt{\frac{d}{x}}}+\frac{x \left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{640 a^5}-\frac{\left (-1680 a^2 b^2 c^2 d+320 a^3 c^3+1260 a b^4 c d^2-231 b^6 d^3\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{512 a^{13/2}}+\frac{b d^2 \left (156 a c-77 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{160 a^4 \left (\frac{d}{x}\right )^{3/2}}-\frac{x^2 \left (100 a c-99 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{240 a^3}-\frac{11 b d^3 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{30 a^2 \left (\frac{d}{x}\right )^{5/2}}+\frac{x^3 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{3 a} \]
Antiderivative was successfully verified.
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Rule 1970
Rule 1357
Rule 744
Rule 834
Rule 806
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}} \, dx &=-\left (d^3 \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{a+b \sqrt{x}+\frac{c x}{d}}} \, dx,x,\frac{d}{x}\right )\right )\\ &=-\left (\left (2 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^7 \sqrt{a+b x+\frac{c x^2}{d}}} \, dx,x,\sqrt{\frac{d}{x}}\right )\right )\\ &=\frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^3}{3 a}+\frac{d^3 \operatorname{Subst}\left (\int \frac{\frac{11 b}{2}+\frac{5 c x}{d}}{x^6 \sqrt{a+b x+\frac{c x^2}{d}}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{3 a}\\ &=-\frac{11 b d^3 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{30 a^2 \left (\frac{d}{x}\right )^{5/2}}+\frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^3}{3 a}-\frac{d^3 \operatorname{Subst}\left (\int \frac{\frac{1}{4} \left (99 b^2-\frac{100 a c}{d}\right )+\frac{22 b c x}{d}}{x^5 \sqrt{a+b x+\frac{c x^2}{d}}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{15 a^2}\\ &=-\frac{11 b d^3 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{30 a^2 \left (\frac{d}{x}\right )^{5/2}}-\frac{\left (100 a c-99 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^2}{240 a^3}+\frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^3}{3 a}+\frac{d^3 \operatorname{Subst}\left (\int \frac{-\frac{9 b \left (156 a c-77 b^2 d\right )}{8 d}-\frac{3 c \left (100 a c-99 b^2 d\right ) x}{4 d^2}}{x^4 \sqrt{a+b x+\frac{c x^2}{d}}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{60 a^3}\\ &=-\frac{11 b d^3 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{30 a^2 \left (\frac{d}{x}\right )^{5/2}}+\frac{b d^2 \left (156 a c-77 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{160 a^4 \left (\frac{d}{x}\right )^{3/2}}-\frac{\left (100 a c-99 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^2}{240 a^3}+\frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^3}{3 a}-\frac{d^3 \operatorname{Subst}\left (\int \frac{\frac{9 \left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right )}{16 d^2}-\frac{9 b c \left (156 a c-77 b^2 d\right ) x}{4 d^2}}{x^3 \sqrt{a+b x+\frac{c x^2}{d}}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{180 a^4}\\ &=-\frac{11 b d^3 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{30 a^2 \left (\frac{d}{x}\right )^{5/2}}+\frac{b d^2 \left (156 a c-77 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{160 a^4 \left (\frac{d}{x}\right )^{3/2}}+\frac{\left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x}{640 a^5}-\frac{\left (100 a c-99 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^2}{240 a^3}+\frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^3}{3 a}+\frac{d^3 \operatorname{Subst}\left (\int \frac{\frac{63 b \left (528 a^2 c^2-680 a b^2 c d+165 b^4 d^2\right )}{32 d^2}+\frac{9 c \left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right ) x}{16 d^3}}{x^2 \sqrt{a+b x+\frac{c x^2}{d}}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{360 a^5}\\ &=-\frac{11 b d^3 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{30 a^2 \left (\frac{d}{x}\right )^{5/2}}+\frac{b d^2 \left (156 a c-77 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{160 a^4 \left (\frac{d}{x}\right )^{3/2}}-\frac{7 b d \left (528 a^2 c^2-680 a b^2 c d+165 b^4 d^2\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{1280 a^6 \sqrt{\frac{d}{x}}}+\frac{\left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x}{640 a^5}-\frac{\left (100 a c-99 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^2}{240 a^3}+\frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^3}{3 a}+\frac{\left (320 a^3 c^3-1680 a^2 b^2 c^2 d+1260 a b^4 c d^2-231 b^6 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+\frac{c x^2}{d}}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{512 a^6}\\ &=-\frac{11 b d^3 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{30 a^2 \left (\frac{d}{x}\right )^{5/2}}+\frac{b d^2 \left (156 a c-77 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{160 a^4 \left (\frac{d}{x}\right )^{3/2}}-\frac{7 b d \left (528 a^2 c^2-680 a b^2 c d+165 b^4 d^2\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{1280 a^6 \sqrt{\frac{d}{x}}}+\frac{\left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x}{640 a^5}-\frac{\left (100 a c-99 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^2}{240 a^3}+\frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^3}{3 a}-\frac{\left (320 a^3 c^3-1680 a^2 b^2 c^2 d+1260 a b^4 c d^2-231 b^6 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b \sqrt{\frac{d}{x}}}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{256 a^6}\\ &=-\frac{11 b d^3 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{30 a^2 \left (\frac{d}{x}\right )^{5/2}}+\frac{b d^2 \left (156 a c-77 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{160 a^4 \left (\frac{d}{x}\right )^{3/2}}-\frac{7 b d \left (528 a^2 c^2-680 a b^2 c d+165 b^4 d^2\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{1280 a^6 \sqrt{\frac{d}{x}}}+\frac{\left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x}{640 a^5}-\frac{\left (100 a c-99 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^2}{240 a^3}+\frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^3}{3 a}-\frac{\left (320 a^3 c^3-1680 a^2 b^2 c^2 d+1260 a b^4 c d^2-231 b^6 d^3\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{512 a^{13/2}}\\ \end{align*}
Mathematica [F] time = 0.302168, size = 0, normalized size = 0. \[ \int \frac{x^2}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.142, size = 655, normalized size = 1.7 \begin{align*} -{\frac{1}{7680}\sqrt{{\frac{1}{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) }}\sqrt{x} \left ( 6930\,{a}^{3/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \left ({\frac{d}{x}} \right ) ^{5/2}{x}^{5/2}{b}^{5}+3696\,{a}^{7/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \left ({\frac{d}{x}} \right ) ^{3/2}{x}^{5/2}{b}^{3}+2816\,{a}^{11/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}{x}^{5/2}b-2560\,{x}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{a}^{13/2}-28560\,{a}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \left ({\frac{d}{x}} \right ) ^{3/2}{x}^{3/2}{b}^{3}c-7488\,{a}^{9/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}{x}^{3/2}bc+3200\,{a}^{11/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{x}^{3/2}c-3168\,{a}^{9/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}d{x}^{3/2}{b}^{2}+22176\,{a}^{7/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}\sqrt{x}b{c}^{2}-4800\,{a}^{9/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{x}{c}^{2}+14112\,{a}^{7/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}d\sqrt{x}{b}^{2}c-4620\,{a}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{d}^{2}\sqrt{x}{b}^{4}-3465\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){d}^{3}a{b}^{6}+18900\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){d}^{2}{a}^{2}{b}^{4}c-25200\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ) d{a}^{3}{b}^{2}{c}^{2}+4800\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){a}^{4}{c}^{3} \right ){\frac{1}{\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}}}{a}^{-{\frac{15}{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{x^{2}}{\sqrt{b \sqrt{\frac{d}{x}} + a + \frac{c}{x}}}\,{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{x^{2}}{\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{x^{2}}{\sqrt{b \sqrt{\frac{d}{x}} + a + \frac{c}{x}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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