Optimal. Leaf size=38 \[ \frac {\left (a+b x^4\right )^{9/4}}{9 b^2}-\frac {a \left (a+b x^4\right )^{5/4}}{5 b^2} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac {\left (a+b x^4\right )^{9/4}}{9 b^2}-\frac {a \left (a+b x^4\right )^{5/4}}{5 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^7 \sqrt [4]{a+b x^4} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int x \sqrt [4]{a+b x} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {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 = 0.02, size = 28, normalized size = 0.74 \[ \frac {\left (a+b x^4\right )^{5/4} \left (5 b x^4-4 a\right )}{45 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 34, normalized size = 0.89 \[ \frac {{\left (5 \, b^{2} x^{8} + a b x^{4} - 4 \, a^{2}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{45 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 29, normalized size = 0.76 \[ \frac {5 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} - 9 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} a}{45 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 0.66 \[ -\frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}} \left (-5 b \,x^{4}+4 a \right )}{45 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 30, normalized size = 0.79 \[ \frac {{\left (b x^{4} + a\right )}^{\frac {9}{4}}}{9 \, b^{2}} - \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}} a}{5 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.10, size = 33, normalized size = 0.87 \[ {\left (b\,x^4+a\right )}^{1/4}\,\left (\frac {x^8}{9}-\frac {4\,a^2}{45\,b^2}+\frac {a\,x^4}{45\,b}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.88, size = 63, normalized size = 1.66 \[ \begin {cases} - \frac {4 a^{2} \sqrt [4]{a + b x^{4}}}{45 b^{2}} + \frac {a x^{4} \sqrt [4]{a + b x^{4}}}{45 b} + \frac {x^{8} \sqrt [4]{a + b x^{4}}}{9} & \text {for}\: b \neq 0 \\\frac {\sqrt [4]{a} x^{8}}{8} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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