Optimal. Leaf size=64 \[ \frac {x^5 \sqrt {1+\frac {d x^8}{c}} F_1\left (\frac {5}{8};2,\frac {1}{2};\frac {13}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{5 a^2 \sqrt {c+d x^8}} \]
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Rubi [A]
time = 0.03, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {525, 524}
\begin {gather*} \frac {x^5 \sqrt {\frac {d x^8}{c}+1} F_1\left (\frac {5}{8};2,\frac {1}{2};\frac {13}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{5 a^2 \sqrt {c+d x^8}} \end {gather*}
Antiderivative was successfully verified.
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Rule 524
Rule 525
Rubi steps
\begin {align*} \int \frac {x^4}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx &=\frac {\sqrt {1+\frac {d x^8}{c}} \int \frac {x^4}{\left (a+b x^8\right )^2 \sqrt {1+\frac {d x^8}{c}}} \, dx}{\sqrt {c+d x^8}}\\ &=\frac {x^5 \sqrt {1+\frac {d x^8}{c}} F_1\left (\frac {5}{8};2,\frac {1}{2};\frac {13}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{5 a^2 \sqrt {c+d x^8}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(170\) vs. \(2(64)=128\).
time = 10.14, size = 170, normalized size = 2.66 \begin {gather*} \frac {x^5 \left (65 a b \left (c+d x^8\right )+13 (3 b c-8 a d) \left (a+b x^8\right ) \sqrt {1+\frac {d x^8}{c}} F_1\left (\frac {5}{8};\frac {1}{2},1;\frac {13}{8};-\frac {d x^8}{c},-\frac {b x^8}{a}\right )-5 b d x^8 \left (a+b x^8\right ) \sqrt {1+\frac {d x^8}{c}} F_1\left (\frac {13}{8};\frac {1}{2},1;\frac {21}{8};-\frac {d x^8}{c},-\frac {b x^8}{a}\right )\right )}{520 a^2 (b c-a d) \left (a+b x^8\right ) \sqrt {c+d x^8}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {x^{4}}{\left (b \,x^{8}+a \right )^{2} \sqrt {d \,x^{8}+c}}\, dx\]
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 {x^{4}}{\left (a + b x^{8}\right )^{2} \sqrt {c + d x^{8}}}\, 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 = -0.02 \begin {gather*} \int \frac {x^4}{{\left (b\,x^8+a\right )}^2\,\sqrt {d\,x^8+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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