Optimal. Leaf size=487 \[ -\frac{81\ 3^{3/4} \sqrt [6]{c+d x} (b c-a d)^{11/3} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \text{EllipticF}\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2816 b d^4 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{81 \sqrt{a+b x} \sqrt [6]{c+d x} (b c-a d)^3}{1408 b d^3}-\frac{9 (a+b x)^{3/2} \sqrt [6]{c+d x} (b c-a d)^2}{352 b d^2}+\frac{3 (a+b x)^{5/2} \sqrt [6]{c+d x} (b c-a d)}{176 b d}+\frac{3 (a+b x)^{7/2} \sqrt [6]{c+d x}}{11 b} \]
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Rubi [A] time = 0.522135, antiderivative size = 487, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {50, 63, 225} \[ -\frac{81\ 3^{3/4} \sqrt [6]{c+d x} (b c-a d)^{11/3} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2816 b d^4 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{81 \sqrt{a+b x} \sqrt [6]{c+d x} (b c-a d)^3}{1408 b d^3}-\frac{9 (a+b x)^{3/2} \sqrt [6]{c+d x} (b c-a d)^2}{352 b d^2}+\frac{3 (a+b x)^{5/2} \sqrt [6]{c+d x} (b c-a d)}{176 b d}+\frac{3 (a+b x)^{7/2} \sqrt [6]{c+d x}}{11 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 225
Rubi steps
\begin{align*} \int (a+b x)^{5/2} \sqrt [6]{c+d x} \, dx &=\frac{3 (a+b x)^{7/2} \sqrt [6]{c+d x}}{11 b}+\frac{(b c-a d) \int \frac{(a+b x)^{5/2}}{(c+d x)^{5/6}} \, dx}{22 b}\\ &=\frac{3 (b c-a d) (a+b x)^{5/2} \sqrt [6]{c+d x}}{176 b d}+\frac{3 (a+b x)^{7/2} \sqrt [6]{c+d x}}{11 b}-\frac{\left (15 (b c-a d)^2\right ) \int \frac{(a+b x)^{3/2}}{(c+d x)^{5/6}} \, dx}{352 b d}\\ &=-\frac{9 (b c-a d)^2 (a+b x)^{3/2} \sqrt [6]{c+d x}}{352 b d^2}+\frac{3 (b c-a d) (a+b x)^{5/2} \sqrt [6]{c+d x}}{176 b d}+\frac{3 (a+b x)^{7/2} \sqrt [6]{c+d x}}{11 b}+\frac{\left (27 (b c-a d)^3\right ) \int \frac{\sqrt{a+b x}}{(c+d x)^{5/6}} \, dx}{704 b d^2}\\ &=\frac{81 (b c-a d)^3 \sqrt{a+b x} \sqrt [6]{c+d x}}{1408 b d^3}-\frac{9 (b c-a d)^2 (a+b x)^{3/2} \sqrt [6]{c+d x}}{352 b d^2}+\frac{3 (b c-a d) (a+b x)^{5/2} \sqrt [6]{c+d x}}{176 b d}+\frac{3 (a+b x)^{7/2} \sqrt [6]{c+d x}}{11 b}-\frac{\left (81 (b c-a d)^4\right ) \int \frac{1}{\sqrt{a+b x} (c+d x)^{5/6}} \, dx}{2816 b d^3}\\ &=\frac{81 (b c-a d)^3 \sqrt{a+b x} \sqrt [6]{c+d x}}{1408 b d^3}-\frac{9 (b c-a d)^2 (a+b x)^{3/2} \sqrt [6]{c+d x}}{352 b d^2}+\frac{3 (b c-a d) (a+b x)^{5/2} \sqrt [6]{c+d x}}{176 b d}+\frac{3 (a+b x)^{7/2} \sqrt [6]{c+d x}}{11 b}-\frac{\left (243 (b c-a d)^4\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 )}{1408 b d^4}\\ &=\frac{81 (b c-a d)^3 \sqrt{a+b x} \sqrt [6]{c+d x}}{1408 b d^3}-\frac{9 (b c-a d)^2 (a+b x)^{3/2} \sqrt [6]{c+d x}}{352 b d^2}+\frac{3 (b c-a d) (a+b x)^{5/2} \sqrt [6]{c+d x}}{176 b d}+\frac{3 (a+b x)^{7/2} \sqrt [6]{c+d x}}{11 b}-\frac{81\ 3^{3/4} (b c-a d)^{11/3} \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 )}{2816 b 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 = 0.042876, size = 73, normalized size = 0.15 \[ \frac{2 (a+b x)^{7/2} \sqrt [6]{c+d x} \, _2F_1\left (-\frac{1}{6},\frac{7}{2};\frac{9}{2};\frac{d (a+b x)}{a d-b c}\right )}{7 b \sqrt [6]{\frac{b (c+d x)}{b c-a d}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.032, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{{\frac{5}{2}}}\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{\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 ({\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 \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{\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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