Optimal. Leaf size=60 \[ -\frac {a^2 \sqrt [4]{a-b x^4}}{b^3}+\frac {2 a \left (a-b x^4\right )^{5/4}}{5 b^3}-\frac {\left (a-b x^4\right )^{9/4}}{9 b^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 60, 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 {a^2 \sqrt [4]{a-b x^4}}{b^3}-\frac {\left (a-b x^4\right )^{9/4}}{9 b^3}+\frac {2 a \left (a-b x^4\right )^{5/4}}{5 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^{11}}{\left (a-b x^4\right )^{3/4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x^2}{(a-b x)^{3/4}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (\frac {a^2}{b^2 (a-b x)^{3/4}}-\frac {2 a \sqrt [4]{a-b x}}{b^2}+\frac {(a-b x)^{5/4}}{b^2}\right ) \, dx,x,x^4\right )\\ &=-\frac {a^2 \sqrt [4]{a-b x^4}}{b^3}+\frac {2 a \left (a-b x^4\right )^{5/4}}{5 b^3}-\frac {\left (a-b x^4\right )^{9/4}}{9 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 40, normalized size = 0.67 \begin {gather*} \frac {\sqrt [4]{a-b x^4} \left (-32 a^2-8 a b x^4-5 b^2 x^8\right )}{45 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 37, normalized size = 0.62
method | result | size |
gosper | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} \left (5 b^{2} x^{8}+8 a b \,x^{4}+32 a^{2}\right )}{45 b^{3}}\) | \(37\) |
trager | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} \left (5 b^{2} x^{8}+8 a b \,x^{4}+32 a^{2}\right )}{45 b^{3}}\) | \(37\) |
risch | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} \left (\left (-b \,x^{4}+a \right )^{3}\right )^{\frac {1}{4}} \left (5 b^{2} x^{8}+8 a b \,x^{4}+32 a^{2}\right )}{45 b^{3} \left (-\left (b \,x^{4}-a \right )^{3}\right )^{\frac {1}{4}}}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 50, normalized size = 0.83 \begin {gather*} -\frac {{\left (-b x^{4} + a\right )}^{\frac {9}{4}}}{9 \, b^{3}} + \frac {2 \, {\left (-b x^{4} + a\right )}^{\frac {5}{4}} a}{5 \, b^{3}} - \frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{2}}{b^{3}} \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 = 0.60 \begin {gather*} -\frac {{\left (5 \, b^{2} x^{8} + 8 \, a b x^{4} + 32 \, a^{2}\right )} {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{45 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.50, size = 70, normalized size = 1.17 \begin {gather*} \begin {cases} - \frac {32 a^{2} \sqrt [4]{a - b x^{4}}}{45 b^{3}} - \frac {8 a x^{4} \sqrt [4]{a - b x^{4}}}{45 b^{2}} - \frac {x^{8} \sqrt [4]{a - b x^{4}}}{9 b} & \text {for}\: b \neq 0 \\\frac {x^{12}}{12 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 = 1.27, size = 61, normalized size = 1.02 \begin {gather*} -\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{2}}{b^{3}} - \frac {5 \, {\left (b x^{4} - a\right )}^{2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} - 18 \, {\left (-b x^{4} + a\right )}^{\frac {5}{4}} a}{45 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.15, size = 38, normalized size = 0.63 \begin {gather*} -{\left (a-b\,x^4\right )}^{1/4}\,\left (\frac {32\,a^2}{45\,b^3}+\frac {x^8}{9\,b}+\frac {8\,a\,x^4}{45\,b^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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