Optimal. Leaf size=839 \[ \frac{81 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right ) (b c-a d)^{7/3}}{91 b^{2/3} d^3 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{54 \sqrt{2} 3^{3/4} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right ),-7+4 \sqrt{3}\right ) (b c-a d)^{7/3}}{91 b^{2/3} d^3 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{162 \sqrt{a+b x} (b c-a d)^2}{91 b^{2/3} d^2 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}-\frac{54 \sqrt{a+b x} (c+d x)^{2/3} (b c-a d)}{91 d^2}+\frac{6 (a+b x)^{3/2} (c+d x)^{2/3}}{13 d} \]
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Rubi [A] time = 0.937686, antiderivative size = 839, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {50, 63, 304, 219, 1879} \[ \frac{81 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right ) (b c-a d)^{7/3}}{91 b^{2/3} d^3 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{54 \sqrt{2} 3^{3/4} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right ) (b c-a d)^{7/3}}{91 b^{2/3} d^3 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{162 \sqrt{a+b x} (b c-a d)^2}{91 b^{2/3} d^2 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}-\frac{54 \sqrt{a+b x} (c+d x)^{2/3} (b c-a d)}{91 d^2}+\frac{6 (a+b x)^{3/2} (c+d x)^{2/3}}{13 d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 304
Rule 219
Rule 1879
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2}}{\sqrt [3]{c+d x}} \, dx &=\frac{6 (a+b x)^{3/2} (c+d x)^{2/3}}{13 d}-\frac{(9 (b c-a d)) \int \frac{\sqrt{a+b x}}{\sqrt [3]{c+d x}} \, dx}{13 d}\\ &=-\frac{54 (b c-a d) \sqrt{a+b x} (c+d x)^{2/3}}{91 d^2}+\frac{6 (a+b x)^{3/2} (c+d x)^{2/3}}{13 d}+\frac{\left (27 (b c-a d)^2\right ) \int \frac{1}{\sqrt{a+b x} \sqrt [3]{c+d x}} \, dx}{91 d^2}\\ &=-\frac{54 (b c-a d) \sqrt{a+b x} (c+d x)^{2/3}}{91 d^2}+\frac{6 (a+b x)^{3/2} (c+d x)^{2/3}}{13 d}+\frac{\left (81 (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a-\frac{b c}{d}+\frac{b x^3}{d}}} \, dx,x,\sqrt [3]{c+d x}\right )}{91 d^3}\\ &=-\frac{54 (b c-a d) \sqrt{a+b x} (c+d x)^{2/3}}{91 d^2}+\frac{6 (a+b x)^{3/2} (c+d x)^{2/3}}{13 d}-\frac{\left (81 (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} x}{\sqrt{a-\frac{b c}{d}+\frac{b x^3}{d}}} \, dx,x,\sqrt [3]{c+d x}\right )}{91 \sqrt [3]{b} d^3}+\frac{\left (81 \sqrt{2 \left (2+\sqrt{3}\right )} (b c-a d)^{7/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-\frac{b c}{d}+\frac{b x^3}{d}}} \, dx,x,\sqrt [3]{c+d x}\right )}{91 \sqrt [3]{b} d^3}\\ &=-\frac{54 (b c-a d) \sqrt{a+b x} (c+d x)^{2/3}}{91 d^2}+\frac{6 (a+b x)^{3/2} (c+d x)^{2/3}}{13 d}-\frac{162 (b c-a d)^2 \sqrt{a+b x}}{91 b^{2/3} d^2 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac{81 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (b c-a d)^{7/3} \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 (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right )}{91 b^{2/3} d^3 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{54 \sqrt{2} 3^{3/4} (b c-a d)^{7/3} \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 (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right )}{91 b^{2/3} d^3 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}\\ \end{align*}
Mathematica [C] time = 0.0284428, size = 73, normalized size = 0.09 \[ \frac{2 (a+b x)^{5/2} \sqrt [3]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{1}{3},\frac{5}{2};\frac{7}{2};\frac{d (a+b x)}{a d-b c}\right )}{5 b \sqrt [3]{c+d x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt [3]{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{3}{2}}}{{\left (d x + c\right )}^{\frac{1}{3}}}\,{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 x + a\right )}^{\frac{3}{2}}}{{\left (d x + c\right )}^{\frac{1}{3}}}, 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{3}{2}}}{\sqrt [3]{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{3}{2}}}{{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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