Optimal. Leaf size=125 \[ \frac {2 (a+b x)^{3/2} \sqrt [3]{c+d x} \sqrt [4]{e+f x} F_1\left (\frac {3}{2};-\frac {1}{3},-\frac {1}{4};\frac {5}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{3 b \sqrt [3]{\frac {b (c+d x)}{b c-a d}} \sqrt [4]{\frac {b (e+f x)}{b e-a f}}} \]
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Rubi [A]
time = 0.05, antiderivative size = 125, normalized size of antiderivative = 1.00, 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 = {145, 144, 143}
\begin {gather*} \frac {2 (a+b x)^{3/2} \sqrt [3]{c+d x} \sqrt [4]{e+f x} F_1\left (\frac {3}{2};-\frac {1}{3},-\frac {1}{4};\frac {5}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{3 b \sqrt [3]{\frac {b (c+d x)}{b c-a d}} \sqrt [4]{\frac {b (e+f x)}{b e-a f}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 143
Rule 144
Rule 145
Rubi steps
\begin {align*} \int \sqrt {a+b x} \sqrt [3]{c+d x} \sqrt [4]{e+f x} \, dx &=\frac {\sqrt [3]{c+d x} \int \sqrt {a+b x} \sqrt [3]{\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt [4]{e+f x} \, dx}{\sqrt [3]{\frac {b (c+d x)}{b c-a d}}}\\ &=\frac {\left (\sqrt [3]{c+d x} \sqrt [4]{e+f x}\right ) \int \sqrt {a+b x} \sqrt [3]{\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 [3]{\frac {b (c+d x)}{b c-a d}} \sqrt [4]{\frac {b (e+f x)}{b e-a f}}}\\ &=\frac {2 (a+b x)^{3/2} \sqrt [3]{c+d x} \sqrt [4]{e+f x} F_1\left (\frac {3}{2};-\frac {1}{3},-\frac {1}{4};\frac {5}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{3 b \sqrt [3]{\frac {b (c+d x)}{b c-a d}} \sqrt [4]{\frac {b (e+f x)}{b e-a f}}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(473\) vs. \(2(125)=250\).
time = 22.18, size = 473, normalized size = 3.78 \begin {gather*} \frac {12 \left (253 b^2 d f (a+b x) (c+d x) (e+f x) (6 a d f+b (3 d e+4 c f+13 d f x))-276 \left (21 a^3 d^3 f^3-9 a^2 b d^2 f^2 (3 d e+4 c f)+a b^2 d f \left (20 d^2 e^2+14 c d e f+29 c^2 f^2\right )-b^3 \left (5 d^3 e^3+5 c d^2 e^2 f+2 c^2 d e f^2+9 c^3 f^3\right )\right ) (a+b x) \left (\frac {b (c+d x)}{d (a+b x)}\right )^{2/3} \left (\frac {b (e+f x)}{f (a+b x)}\right )^{3/4} F_1\left (\frac {11}{12};\frac {2}{3},\frac {3}{4};\frac {23}{12};\frac {-b c+a d}{d (a+b x)},\frac {-b e+a f}{f (a+b x)}\right )-66 \left (7 a^2 d^2 f^2-2 a b d f (3 d e+4 c f)+b^2 \left (5 d^2 e^2-4 c d e f+6 c^2 f^2\right )\right ) \left (23 b^2 (c+d x) (e+f x)-6 (b c-a d) (b e-a f) \left (\frac {b (c+d x)}{d (a+b x)}\right )^{2/3} \left (\frac {b (e+f x)}{f (a+b x)}\right )^{3/4} F_1\left (\frac {23}{12};\frac {2}{3},\frac {3}{4};\frac {35}{12};\frac {-b c+a d}{d (a+b x)},\frac {-b e+a f}{f (a+b x)}\right )\right )\right )}{82225 b^3 d^2 f^2 \sqrt {a+b x} (c+d x)^{2/3} (e+f x)^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \sqrt {b x +a}\, \left (d x +c \right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a + b x} \sqrt [3]{c + d x} \sqrt [4]{e + f x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (e+f\,x\right )}^{1/4}\,\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{1/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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