Optimal. Leaf size=289 \[ \frac{20 a^4 b x^{3/4} \sqrt{a^2+\frac{2 a b}{\sqrt [4]{x}}+\frac{b^2}{\sqrt{x}}}}{3 \left (a+\frac{b}{\sqrt [4]{x}}\right )}+\frac{a^5 x \sqrt{a^2+\frac{2 a b}{\sqrt [4]{x}}+\frac{b^2}{\sqrt{x}}}}{a+\frac{b}{\sqrt [4]{x}}}+\frac{20 a^3 b^2 \sqrt{x} \sqrt{a^2+\frac{2 a b}{\sqrt [4]{x}}+\frac{b^2}{\sqrt{x}}}}{a+\frac{b}{\sqrt [4]{x}}}+\frac{40 a^2 b^3 \sqrt [4]{x} \sqrt{a^2+\frac{2 a b}{\sqrt [4]{x}}+\frac{b^2}{\sqrt{x}}}}{a+\frac{b}{\sqrt [4]{x}}}-\frac{4 b^5 \sqrt{a^2+\frac{2 a b}{\sqrt [4]{x}}+\frac{b^2}{\sqrt{x}}}}{\sqrt [4]{x} \left (a+\frac{b}{\sqrt [4]{x}}\right )}+\frac{20 a b^4 \log \left (\sqrt [4]{x}\right ) \sqrt{a^2+\frac{2 a b}{\sqrt [4]{x}}+\frac{b^2}{\sqrt{x}}}}{a+\frac{b}{\sqrt [4]{x}}} \]
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Rubi [A] time = 0.138081, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {1341, 1355, 263, 43} \[ \frac{20 a^4 b x^{3/4} \sqrt{a^2+\frac{2 a b}{\sqrt [4]{x}}+\frac{b^2}{\sqrt{x}}}}{3 \left (a+\frac{b}{\sqrt [4]{x}}\right )}+\frac{a^5 x \sqrt{a^2+\frac{2 a b}{\sqrt [4]{x}}+\frac{b^2}{\sqrt{x}}}}{a+\frac{b}{\sqrt [4]{x}}}+\frac{20 a^3 b^2 \sqrt{x} \sqrt{a^2+\frac{2 a b}{\sqrt [4]{x}}+\frac{b^2}{\sqrt{x}}}}{a+\frac{b}{\sqrt [4]{x}}}+\frac{40 a^2 b^3 \sqrt [4]{x} \sqrt{a^2+\frac{2 a b}{\sqrt [4]{x}}+\frac{b^2}{\sqrt{x}}}}{a+\frac{b}{\sqrt [4]{x}}}-\frac{4 b^5 \sqrt{a^2+\frac{2 a b}{\sqrt [4]{x}}+\frac{b^2}{\sqrt{x}}}}{\sqrt [4]{x} \left (a+\frac{b}{\sqrt [4]{x}}\right )}+\frac{20 a b^4 \log \left (\sqrt [4]{x}\right ) \sqrt{a^2+\frac{2 a b}{\sqrt [4]{x}}+\frac{b^2}{\sqrt{x}}}}{a+\frac{b}{\sqrt [4]{x}}} \]
Antiderivative was successfully verified.
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Rule 1341
Rule 1355
Rule 263
Rule 43
Rubi steps
\begin{align*} \int \left (a^2+\frac{b^2}{\sqrt{x}}+\frac{2 a b}{\sqrt [4]{x}}\right )^{5/2} \, dx &=4 \operatorname{Subst}\left (\int \left (a^2+\frac{b^2}{x^2}+\frac{2 a b}{x}\right )^{5/2} x^3 \, dx,x,\sqrt [4]{x}\right )\\ &=\frac{\left (4 \sqrt{a^2+\frac{b^2}{\sqrt{x}}+\frac{2 a b}{\sqrt [4]{x}}}\right ) \operatorname{Subst}\left (\int \left (a b+\frac{b^2}{x}\right )^5 x^3 \, dx,x,\sqrt [4]{x}\right )}{b^4 \left (a b+\frac{b^2}{\sqrt [4]{x}}\right )}\\ &=\frac{\left (4 \sqrt{a^2+\frac{b^2}{\sqrt{x}}+\frac{2 a b}{\sqrt [4]{x}}}\right ) \operatorname{Subst}\left (\int \frac{\left (b^2+a b x\right )^5}{x^2} \, dx,x,\sqrt [4]{x}\right )}{b^4 \left (a b+\frac{b^2}{\sqrt [4]{x}}\right )}\\ &=\frac{\left (4 \sqrt{a^2+\frac{b^2}{\sqrt{x}}+\frac{2 a b}{\sqrt [4]{x}}}\right ) \operatorname{Subst}\left (\int \left (10 a^2 b^8+\frac{b^{10}}{x^2}+\frac{5 a b^9}{x}+10 a^3 b^7 x+5 a^4 b^6 x^2+a^5 b^5 x^3\right ) \, dx,x,\sqrt [4]{x}\right )}{b^4 \left (a b+\frac{b^2}{\sqrt [4]{x}}\right )}\\ &=-\frac{4 b^6 \sqrt{a^2+\frac{b^2}{\sqrt{x}}+\frac{2 a b}{\sqrt [4]{x}}}}{\left (a b+\frac{b^2}{\sqrt [4]{x}}\right ) \sqrt [4]{x}}+\frac{40 a^2 b^4 \sqrt{a^2+\frac{b^2}{\sqrt{x}}+\frac{2 a b}{\sqrt [4]{x}}} \sqrt [4]{x}}{a b+\frac{b^2}{\sqrt [4]{x}}}+\frac{20 a^3 b^3 \sqrt{a^2+\frac{b^2}{\sqrt{x}}+\frac{2 a b}{\sqrt [4]{x}}} \sqrt{x}}{a b+\frac{b^2}{\sqrt [4]{x}}}+\frac{20 a^4 b^2 \sqrt{a^2+\frac{b^2}{\sqrt{x}}+\frac{2 a b}{\sqrt [4]{x}}} x^{3/4}}{3 \left (a b+\frac{b^2}{\sqrt [4]{x}}\right )}+\frac{a^5 \sqrt{a^2+\frac{b^2}{\sqrt{x}}+\frac{2 a b}{\sqrt [4]{x}}} x}{a+\frac{b}{\sqrt [4]{x}}}+\frac{5 a b^5 \sqrt{a^2+\frac{b^2}{\sqrt{x}}+\frac{2 a b}{\sqrt [4]{x}}} \log (x)}{a b+\frac{b^2}{\sqrt [4]{x}}}\\ \end{align*}
Mathematica [A] time = 0.052928, size = 98, normalized size = 0.34 \[ \frac{\sqrt{\frac{\left (a \sqrt [4]{x}+b\right )^2}{\sqrt{x}}} \left (60 a^3 b^2 x^{3/4}+120 a^2 b^3 \sqrt{x}+20 a^4 b x+3 a^5 x^{5/4}+15 a b^4 \sqrt [4]{x} \log (x)-12 b^5\right )}{3 \left (a \sqrt [4]{x}+b\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 94, normalized size = 0.3 \begin{align*}{\frac{1}{3}\sqrt{{ \left ({a}^{2}{x}^{{\frac{3}{4}}}+2\,ab\sqrt{x}+{b}^{2}\sqrt [4]{x} \right ){x}^{-{\frac{3}{4}}}}} \left ( 20\,x{a}^{4}b+15\,\ln \left ( x \right ) \sqrt [4]{x}a{b}^{4}+120\,\sqrt{x}{a}^{2}{b}^{3}+60\,{x}^{3/4}{a}^{3}{b}^{2}+3\,{x}^{5/4}{a}^{5}-12\,{b}^{5} \right ) \left ( a\sqrt [4]{x}+b \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.97714, size = 77, normalized size = 0.27 \begin{align*} 5 \, a b^{4} \log \left (x\right ) + \frac{3 \, a^{5} x^{\frac{5}{4}} + 20 \, a^{4} b x + 60 \, a^{3} b^{2} x^{\frac{3}{4}} + 120 \, a^{2} b^{3} \sqrt{x} - 12 \, b^{5}}{3 \, x^{\frac{1}{4}}} \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(-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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Giac [A] time = 1.23674, size = 170, normalized size = 0.59 \begin{align*} a^{5} x \mathrm{sgn}\left (a x + b x^{\frac{3}{4}}\right ) \mathrm{sgn}\left (x\right ) + 5 \, a b^{4} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (a x + b x^{\frac{3}{4}}\right ) \mathrm{sgn}\left (x\right ) + \frac{20}{3} \, a^{4} b x^{\frac{3}{4}} \mathrm{sgn}\left (a x + b x^{\frac{3}{4}}\right ) \mathrm{sgn}\left (x\right ) + 20 \, a^{3} b^{2} \sqrt{x} \mathrm{sgn}\left (a x + b x^{\frac{3}{4}}\right ) \mathrm{sgn}\left (x\right ) + 40 \, a^{2} b^{3} x^{\frac{1}{4}} \mathrm{sgn}\left (a x + b x^{\frac{3}{4}}\right ) \mathrm{sgn}\left (x\right ) - \frac{4 \, b^{5} \mathrm{sgn}\left (a x + b x^{\frac{3}{4}}\right ) \mathrm{sgn}\left (x\right )}{x^{\frac{1}{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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