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 = 0.05, 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 {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 {gather*}
Antiderivative was successfully verified.
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Rule 143
Rule 144
Rule 145
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*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(318\) vs. \(2(125)=250\).
time = 21.39, size = 318, normalized size = 2.54 \begin {gather*} \frac {12 \sqrt {c+d x} \left (11 d^2 (a+b x) (e+f x) (4 a d f+b (3 d e+6 c f+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 \left (6 a^2 d^2 f^2-4 a b d f (d e+2 c f)+b^2 \left (5 d^2 e^2-6 c d e f+7 c^2 f^2\right )\right ) (c+d x) 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 {-d e+c f}{f (c+d x)}\right )+(b c-a d) (d e-c f) (3 b d e-7 b c f+4 a d f) 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 {-d e+c f}{f (c+d x)}\right )\right )\right )}{3575 b d^3 f (a+b 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 \[\int \left (b x +a \right )^{\frac {1}{3}} \sqrt {d x +c}\, \left (f x +e \right )^{\frac {1}{4}}\, dx\]
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 [3]{a + b x} \sqrt {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}\,{\left (a+b\,x\right )}^{1/3}\,\sqrt {c+d\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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