Optimal. Leaf size=36 \[ -\frac {a \sqrt [4]{a+b x^4}}{b^2}+\frac {\left (a+b x^4\right )^{5/4}}{5 b^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 36, 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 = {272, 45}
\begin {gather*} \frac {\left (a+b x^4\right )^{5/4}}{5 b^2}-\frac {a \sqrt [4]{a+b x^4}}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^7}{\left (a+b x^4\right )^{3/4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x}{(a+b x)^{3/4}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (-\frac {a}{b (a+b x)^{3/4}}+\frac {\sqrt [4]{a+b x}}{b}\right ) \, dx,x,x^4\right )\\ &=-\frac {a \sqrt [4]{a+b x^4}}{b^2}+\frac {\left (a+b x^4\right )^{5/4}}{5 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 27, normalized size = 0.75 \begin {gather*} \frac {\left (-4 a+b x^4\right ) \sqrt [4]{a+b x^4}}{5 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 25, normalized size = 0.69
method | result | size |
gosper | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}} \left (-b \,x^{4}+4 a \right )}{5 b^{2}}\) | \(25\) |
trager | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}} \left (-b \,x^{4}+4 a \right )}{5 b^{2}}\) | \(25\) |
risch | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}} \left (-b \,x^{4}+4 a \right )}{5 b^{2}}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 30, normalized size = 0.83 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}}}{5 \, b^{2}} - \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} a}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 23, normalized size = 0.64 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} {\left (b x^{4} - 4 \, a\right )}}{5 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.31, size = 44, normalized size = 1.22 \begin {gather*} \begin {cases} - \frac {4 a \sqrt [4]{a + b x^{4}}}{5 b^{2}} + \frac {x^{4} \sqrt [4]{a + b x^{4}}}{5 b} & \text {for}\: b \neq 0 \\\frac {x^{8}}{8 a^{\frac {3}{4}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.85, size = 30, normalized size = 0.83 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}}}{5 \, b^{2}} - \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} a}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.13, size = 24, normalized size = 0.67 \begin {gather*} -\frac {{\left (b\,x^4+a\right )}^{1/4}\,\left (4\,a-b\,x^4\right )}{5\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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