Optimal. Leaf size=266 \[ \frac {2 \sqrt [4]{d e-c f} \sqrt {-\frac {f (c+d x)}{d e-c f}} \Pi \left (-\frac {\sqrt {b} \sqrt {d e-c f}}{\sqrt {d} \sqrt {b e-a f}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right )\right |-1\right )}{\sqrt {b} \sqrt [4]{d} \sqrt {b e-a f} \sqrt {c+d x}}-\frac {2 \sqrt [4]{d e-c f} \sqrt {-\frac {f (c+d x)}{d e-c f}} \Pi \left (\frac {\sqrt {b} \sqrt {d e-c f}}{\sqrt {d} \sqrt {b e-a f}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right )\right |-1\right )}{\sqrt {b} \sqrt [4]{d} \sqrt {b e-a f} \sqrt {c+d x}} \]
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Rubi [A]
time = 0.24, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {108, 107, 504,
1232} \begin {gather*} \frac {2 \sqrt [4]{d e-c f} \sqrt {-\frac {f (c+d x)}{d e-c f}} \Pi \left (-\frac {\sqrt {b} \sqrt {d e-c f}}{\sqrt {d} \sqrt {b e-a f}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right )\right |-1\right )}{\sqrt {b} \sqrt [4]{d} \sqrt {c+d x} \sqrt {b e-a f}}-\frac {2 \sqrt [4]{d e-c f} \sqrt {-\frac {f (c+d x)}{d e-c f}} \Pi \left (\frac {\sqrt {b} \sqrt {d e-c f}}{\sqrt {d} \sqrt {b e-a f}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right )\right |-1\right )}{\sqrt {b} \sqrt [4]{d} \sqrt {c+d x} \sqrt {b e-a f}} \end {gather*}
Antiderivative was successfully verified.
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Rule 107
Rule 108
Rule 504
Rule 1232
Rubi steps
\begin {align*} \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt [4]{e+f x}} \, dx &=\frac {\sqrt {-\frac {f (c+d x)}{d e-c f}} \int \frac {1}{(a+b x) \sqrt [4]{e+f x} \sqrt {-\frac {c f}{d e-c f}-\frac {d f x}{d e-c f}}} \, dx}{\sqrt {c+d x}}\\ &=-\frac {\left (4 \sqrt {-\frac {f (c+d x)}{d e-c f}}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b e-a f-b x^4\right ) \sqrt {\frac {d e}{d e-c f}-\frac {c f}{d e-c f}-\frac {d x^4}{d e-c f}}} \, dx,x,\sqrt [4]{e+f x}\right )}{\sqrt {c+d x}}\\ &=-\frac {\left (2 \sqrt {-\frac {f (c+d x)}{d e-c f}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {b e-a f}-\sqrt {b} x^2\right ) \sqrt {\frac {d e}{d e-c f}-\frac {c f}{d e-c f}-\frac {d x^4}{d e-c f}}} \, dx,x,\sqrt [4]{e+f x}\right )}{\sqrt {b} \sqrt {c+d x}}+\frac {\left (2 \sqrt {-\frac {f (c+d x)}{d e-c f}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {b e-a f}+\sqrt {b} x^2\right ) \sqrt {\frac {d e}{d e-c f}-\frac {c f}{d e-c f}-\frac {d x^4}{d e-c f}}} \, dx,x,\sqrt [4]{e+f x}\right )}{\sqrt {b} \sqrt {c+d x}}\\ &=\frac {2 \sqrt [4]{d e-c f} \sqrt {-\frac {f (c+d x)}{d e-c f}} \Pi \left (-\frac {\sqrt {b} \sqrt {d e-c f}}{\sqrt {d} \sqrt {b e-a f}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right )\right |-1\right )}{\sqrt {b} \sqrt [4]{d} \sqrt {b e-a f} \sqrt {c+d x}}-\frac {2 \sqrt [4]{d e-c f} \sqrt {-\frac {f (c+d x)}{d e-c f}} \Pi \left (\frac {\sqrt {b} \sqrt {d e-c f}}{\sqrt {d} \sqrt {b e-a f}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right )\right |-1\right )}{\sqrt {b} \sqrt [4]{d} \sqrt {b e-a f} \sqrt {c+d x}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 10.09, size = 118, normalized size = 0.44 \begin {gather*} -\frac {4 \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt [4]{\frac {b (e+f x)}{f (a+b x)}} F_1\left (\frac {3}{4};\frac {1}{2},\frac {1}{4};\frac {7}{4};\frac {-b c+a d}{d (a+b x)},\frac {-b e+a f}{f (a+b x)}\right )}{3 b \sqrt {c+d x} \sqrt [4]{e+f x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b x +a \right ) \left (f x +e \right )^{\frac {1}{4}} \sqrt {d x +c}}\, dx\]
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 \frac {1}{\left (a + b x\right ) \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 = -0.00 \begin {gather*} \int \frac {1}{{\left (e+f\,x\right )}^{1/4}\,\left (a+b\,x\right )\,\sqrt {c+d\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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