Optimal. Leaf size=890 \[ \frac{243 \sqrt [4]{3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) (b c-a d)^{10/3}}{448 b^{2/3} d^4 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{81\ 3^{3/4} \left (1-\sqrt{3}\right ) \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \text{EllipticF}\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right ) (b c-a d)^{10/3}}{896 b^{2/3} d^4 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{243 \left (1+\sqrt{3}\right ) \sqrt{a+b x} \sqrt [6]{c+d x} (b c-a d)^3}{448 b^{2/3} d^3 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac{81 \sqrt{a+b x} (c+d x)^{5/6} (b c-a d)^2}{224 d^3}-\frac{9 (a+b x)^{3/2} (c+d x)^{5/6} (b c-a d)}{28 d^2}+\frac{3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d} \]
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Rubi [A] time = 0.952239, antiderivative size = 890, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {50, 63, 308, 225, 1881} \[ \frac{243 \sqrt [4]{3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) (b c-a d)^{10/3}}{448 b^{2/3} d^4 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{81\ 3^{3/4} \left (1-\sqrt{3}\right ) \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) (b c-a d)^{10/3}}{896 b^{2/3} d^4 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{243 \left (1+\sqrt{3}\right ) \sqrt{a+b x} \sqrt [6]{c+d x} (b c-a d)^3}{448 b^{2/3} d^3 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac{81 \sqrt{a+b x} (c+d x)^{5/6} (b c-a d)^2}{224 d^3}-\frac{9 (a+b x)^{3/2} (c+d x)^{5/6} (b c-a d)}{28 d^2}+\frac{3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 308
Rule 225
Rule 1881
Rubi steps
\begin{align*} \int \frac{(a+b x)^{5/2}}{\sqrt [6]{c+d x}} \, dx &=\frac{3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d}-\frac{(3 (b c-a d)) \int \frac{(a+b x)^{3/2}}{\sqrt [6]{c+d x}} \, dx}{4 d}\\ &=-\frac{9 (b c-a d) (a+b x)^{3/2} (c+d x)^{5/6}}{28 d^2}+\frac{3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d}+\frac{\left (27 (b c-a d)^2\right ) \int \frac{\sqrt{a+b x}}{\sqrt [6]{c+d x}} \, dx}{56 d^2}\\ &=\frac{81 (b c-a d)^2 \sqrt{a+b x} (c+d x)^{5/6}}{224 d^3}-\frac{9 (b c-a d) (a+b x)^{3/2} (c+d x)^{5/6}}{28 d^2}+\frac{3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d}-\frac{\left (81 (b c-a d)^3\right ) \int \frac{1}{\sqrt{a+b x} \sqrt [6]{c+d x}} \, dx}{448 d^3}\\ &=\frac{81 (b c-a d)^2 \sqrt{a+b x} (c+d x)^{5/6}}{224 d^3}-\frac{9 (b c-a d) (a+b x)^{3/2} (c+d x)^{5/6}}{28 d^2}+\frac{3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d}-\frac{\left (243 (b c-a d)^3\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{a-\frac{b c}{d}+\frac{b x^6}{d}}} \, dx,x,\sqrt [6]{c+d x}\right )}{224 d^4}\\ &=\frac{81 (b c-a d)^2 \sqrt{a+b x} (c+d x)^{5/6}}{224 d^3}-\frac{9 (b c-a d) (a+b x)^{3/2} (c+d x)^{5/6}}{28 d^2}+\frac{3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d}+\frac{\left (243 (b c-a d)^3\right ) \operatorname{Subst}\left (\int \frac{\left (-1+\sqrt{3}\right ) (b c-a d)^{2/3}-2 b^{2/3} x^4}{\sqrt{a-\frac{b c}{d}+\frac{b x^6}{d}}} \, dx,x,\sqrt [6]{c+d x}\right )}{448 b^{2/3} d^4}+\frac{\left (243 \left (1-\sqrt{3}\right ) (b c-a d)^{11/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-\frac{b c}{d}+\frac{b x^6}{d}}} \, dx,x,\sqrt [6]{c+d x}\right )}{448 b^{2/3} d^4}\\ &=\frac{81 (b c-a d)^2 \sqrt{a+b x} (c+d x)^{5/6}}{224 d^3}-\frac{9 (b c-a d) (a+b x)^{3/2} (c+d x)^{5/6}}{28 d^2}+\frac{3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d}+\frac{243 \left (1+\sqrt{3}\right ) (b c-a d)^3 \sqrt{a+b x} \sqrt [6]{c+d x}}{448 b^{2/3} d^3 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac{243 \sqrt [4]{3} (b c-a d)^{10/3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{448 b^{2/3} d^4 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{81\ 3^{3/4} \left (1-\sqrt{3}\right ) (b c-a d)^{10/3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{896 b^{2/3} d^4 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}\\ \end{align*}
Mathematica [C] time = 0.0351842, size = 73, normalized size = 0.08 \[ \frac{2 (a+b x)^{7/2} \sqrt [6]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{1}{6},\frac{7}{2};\frac{9}{2};\frac{d (a+b x)}{a d-b c}\right )}{7 b \sqrt [6]{c+d x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt [6]{dx+c}}}}\, 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 \frac{{\left (b x + a\right )}^{\frac{5}{2}}}{{\left (d x + c\right )}^{\frac{1}{6}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b x + a}}{{\left (d x + c\right )}^{\frac{1}{6}}}, x\right ) \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{\left (a + b x\right )^{\frac{5}{2}}}{\sqrt [6]{c + d 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{{\left (b x + a\right )}^{\frac{5}{2}}}{{\left (d x + c\right )}^{\frac{1}{6}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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