Optimal. Leaf size=123 \[ \frac {b (e x)^{1+m} \sqrt {c+d x^4}}{d e (3+m)}-\frac {(b c (1+m)-a d (3+m)) (e x)^{1+m} \sqrt {1+\frac {d x^4}{c}} \, _2F_1\left (\frac {1}{2},\frac {1+m}{4};\frac {5+m}{4};-\frac {d x^4}{c}\right )}{d e (1+m) (3+m) \sqrt {c+d x^4}} \]
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Rubi [A]
time = 0.04, antiderivative size = 115, normalized size of antiderivative = 0.93, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {470, 372, 371}
\begin {gather*} \frac {\sqrt {\frac {d x^4}{c}+1} (e x)^{m+1} \left (\frac {a}{m+1}-\frac {b c}{d (m+3)}\right ) \, _2F_1\left (\frac {1}{2},\frac {m+1}{4};\frac {m+5}{4};-\frac {d x^4}{c}\right )}{e \sqrt {c+d x^4}}+\frac {b \sqrt {c+d x^4} (e x)^{m+1}}{d e (m+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 372
Rule 470
Rubi steps
\begin {align*} \int \frac {(e x)^m \left (a+b x^4\right )}{\sqrt {c+d x^4}} \, dx &=\frac {b (e x)^{1+m} \sqrt {c+d x^4}}{d e (3+m)}-\left (-a+\frac {b c (1+m)}{d (3+m)}\right ) \int \frac {(e x)^m}{\sqrt {c+d x^4}} \, dx\\ &=\frac {b (e x)^{1+m} \sqrt {c+d x^4}}{d e (3+m)}-\frac {\left (\left (-a+\frac {b c (1+m)}{d (3+m)}\right ) \sqrt {1+\frac {d x^4}{c}}\right ) \int \frac {(e x)^m}{\sqrt {1+\frac {d x^4}{c}}} \, dx}{\sqrt {c+d x^4}}\\ &=\frac {b (e x)^{1+m} \sqrt {c+d x^4}}{d e (3+m)}+\frac {\left (a-\frac {b c (1+m)}{d (3+m)}\right ) (e x)^{1+m} \sqrt {1+\frac {d x^4}{c}} \, _2F_1\left (\frac {1}{2},\frac {1+m}{4};\frac {5+m}{4};-\frac {d x^4}{c}\right )}{e (1+m) \sqrt {c+d x^4}}\\ \end {align*}
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Mathematica [A]
time = 1.43, size = 110, normalized size = 0.89 \begin {gather*} \frac {x (e x)^m \sqrt {c+d x^4} \left (b c \, _2F_1\left (-\frac {1}{2},\frac {1+m}{4};\frac {5+m}{4};-\frac {d x^4}{c}\right )+(-b c+a d) \, _2F_1\left (\frac {1}{2},\frac {1+m}{4};\frac {5+m}{4};-\frac {d x^4}{c}\right )\right )}{c d (1+m) \sqrt {1+\frac {d x^4}{c}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m} \left (b \,x^{4}+a \right )}{\sqrt {d \,x^{4}+c}}\, dx\]
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]
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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Sympy [C] Result contains complex when optimal does not.
time = 2.48, size = 119, normalized size = 0.97 \begin {gather*} \frac {a e^{m} x x^{m} \Gamma \left (\frac {m}{4} + \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{4} + \frac {1}{4} \\ \frac {m}{4} + \frac {5}{4} \end {matrix}\middle | {\frac {d x^{4} e^{i \pi }}{c}} \right )}}{4 \sqrt {c} \Gamma \left (\frac {m}{4} + \frac {5}{4}\right )} + \frac {b e^{m} x^{5} x^{m} \Gamma \left (\frac {m}{4} + \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{4} + \frac {5}{4} \\ \frac {m}{4} + \frac {9}{4} \end {matrix}\middle | {\frac {d x^{4} e^{i \pi }}{c}} \right )}}{4 \sqrt {c} \Gamma \left (\frac {m}{4} + \frac {9}{4}\right )} \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 \frac {{\left (e\,x\right )}^m\,\left (b\,x^4+a\right )}{\sqrt {d\,x^4+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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