Optimal. Leaf size=125 \[ \frac{3 (a+b x)^{4/3} \sqrt{c+d x} \sqrt [4]{e+f x} F_1\left (\frac{4}{3};-\frac{1}{2},-\frac{1}{4};\frac{7}{3};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{4 b \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt [4]{\frac{b (e+f x)}{b e-a f}}} \]
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Rubi [A] time = 0.0727979, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {140, 139, 138} \[ \frac{3 (a+b x)^{4/3} \sqrt{c+d x} \sqrt [4]{e+f x} F_1\left (\frac{4}{3};-\frac{1}{2},-\frac{1}{4};\frac{7}{3};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{4 b \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt [4]{\frac{b (e+f x)}{b e-a f}}} \]
Antiderivative was successfully verified.
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Rule 140
Rule 139
Rule 138
Rubi steps
\begin{align*} \int \sqrt [3]{a+b x} \sqrt{c+d x} \sqrt [4]{e+f x} \, dx &=\frac{\sqrt{c+d x} \int \sqrt [3]{a+b x} \sqrt{\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}} \sqrt [4]{e+f x} \, dx}{\sqrt{\frac{b (c+d x)}{b c-a d}}}\\ &=\frac{\left (\sqrt{c+d x} \sqrt [4]{e+f x}\right ) \int \sqrt [3]{a+b x} \sqrt{\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}} \sqrt [4]{\frac{b e}{b e-a f}+\frac{b f x}{b e-a f}} \, dx}{\sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt [4]{\frac{b (e+f x)}{b e-a f}}}\\ &=\frac{3 (a+b x)^{4/3} \sqrt{c+d x} \sqrt [4]{e+f x} F_1\left (\frac{4}{3};-\frac{1}{2},-\frac{1}{4};\frac{7}{3};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{4 b \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt [4]{\frac{b (e+f x)}{b e-a f}}}\\ \end{align*}
Mathematica [B] time = 2.43141, size = 318, normalized size = 2.54 \[ \frac{12 \sqrt{c+d x} \left (11 d^2 (a+b x) (e+f x) (4 a d f+b (6 c f+3 d e+13 d f x))-6 \left (\frac{d (a+b x)}{b (c+d x)}\right )^{2/3} \left (\frac{d (e+f x)}{f (c+d x)}\right )^{3/4} \left (11 (c+d x) \left (6 a^2 d^2 f^2-4 a b d f (2 c f+d e)+b^2 \left (7 c^2 f^2-6 c d e f+5 d^2 e^2\right )\right ) F_1\left (-\frac{1}{12};\frac{2}{3},\frac{3}{4};\frac{11}{12};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+(b c-a d) (d e-c f) (4 a d f-7 b c f+3 b d e) F_1\left (\frac{11}{12};\frac{2}{3},\frac{3}{4};\frac{23}{12};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )\right )\right )}{3575 b d^3 f (a+b x)^{2/3} (e+f x)^{3/4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.057, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{bx+a}\sqrt{dx+c}\sqrt [4]{fx+e}\, 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{1}{3}} \sqrt{d x + c}{\left (f x + e\right )}^{\frac{1}{4}}\,{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(-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 [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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