Optimal. Leaf size=40 \[ -\frac {a \left (a-b x^4\right )^{5/4}}{5 b^2}+\frac {\left (a-b x^4\right )^{9/4}}{9 b^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {272, 45}
\begin {gather*} \frac {\left (a-b x^4\right )^{9/4}}{9 b^2}-\frac {a \left (a-b x^4\right )^{5/4}}{5 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^7 \sqrt [4]{a-b x^4} \, dx &=\frac {1}{4} \text {Subst}\left (\int x \sqrt [4]{a-b x} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (\frac {a \sqrt [4]{a-b x}}{b}-\frac {(a-b x)^{5/4}}{b}\right ) \, dx,x,x^4\right )\\ &=-\frac {a \left (a-b x^4\right )^{5/4}}{5 b^2}+\frac {\left (a-b x^4\right )^{9/4}}{9 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 40, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{a-b x^4} \left (-4 a^2-a b x^4+5 b^2 x^8\right )}{45 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 26, normalized size = 0.65
method | result | size |
gosper | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}} \left (5 b \,x^{4}+4 a \right )}{45 b^{2}}\) | \(26\) |
trager | \(-\frac {\left (-5 b^{2} x^{8}+a b \,x^{4}+4 a^{2}\right ) \left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{45 b^{2}}\) | \(36\) |
risch | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} \left (\left (-b \,x^{4}+a \right )^{3}\right )^{\frac {1}{4}} \left (-5 b^{2} x^{8}+a b \,x^{4}+4 a^{2}\right )}{45 b^{2} \left (-\left (b \,x^{4}-a \right )^{3}\right )^{\frac {1}{4}}}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 32, normalized size = 0.80 \begin {gather*} \frac {{\left (-b x^{4} + a\right )}^{\frac {9}{4}}}{9 \, b^{2}} - \frac {{\left (-b x^{4} + a\right )}^{\frac {5}{4}} a}{5 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 36, normalized size = 0.90 \begin {gather*} \frac {{\left (5 \, b^{2} x^{8} - a b x^{4} - 4 \, a^{2}\right )} {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{45 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs.
\(2 (31) = 62\).
time = 0.22, size = 63, normalized size = 1.58 \begin {gather*} \begin {cases} - \frac {4 a^{2} \sqrt [4]{a - b x^{4}}}{45 b^{2}} - \frac {a x^{4} \sqrt [4]{a - b x^{4}}}{45 b} + \frac {x^{8} \sqrt [4]{a - b x^{4}}}{9} & \text {for}\: b \neq 0 \\\frac {\sqrt [4]{a} x^{8}}{8} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.11, size = 42, normalized size = 1.05 \begin {gather*} \frac {5 \, {\left (b x^{4} - a\right )}^{2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} - 9 \, {\left (-b x^{4} + a\right )}^{\frac {5}{4}} a}{45 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.12, size = 35, normalized size = 0.88 \begin {gather*} -{\left (a-b\,x^4\right )}^{1/4}\,\left (\frac {4\,a^2}{45\,b^2}-\frac {x^8}{9}+\frac {a\,x^4}{45\,b}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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