Optimal. Leaf size=59 \[ \frac {a^2 \left (a+b x^4\right )^{5/4}}{5 b^3}-\frac {2 a \left (a+b x^4\right )^{9/4}}{9 b^3}+\frac {\left (a+b x^4\right )^{13/4}}{13 b^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 59, 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 {a^2 \left (a+b x^4\right )^{5/4}}{5 b^3}+\frac {\left (a+b x^4\right )^{13/4}}{13 b^3}-\frac {2 a \left (a+b x^4\right )^{9/4}}{9 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^{11} \sqrt [4]{a+b x^4} \, dx &=\frac {1}{4} \text {Subst}\left (\int x^2 \sqrt [4]{a+b x} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (\frac {a^2 \sqrt [4]{a+b x}}{b^2}-\frac {2 a (a+b x)^{5/4}}{b^2}+\frac {(a+b x)^{9/4}}{b^2}\right ) \, dx,x,x^4\right )\\ &=\frac {a^2 \left (a+b x^4\right )^{5/4}}{5 b^3}-\frac {2 a \left (a+b x^4\right )^{9/4}}{9 b^3}+\frac {\left (a+b x^4\right )^{13/4}}{13 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 39, normalized size = 0.66 \begin {gather*} \frac {\left (a+b x^4\right )^{5/4} \left (32 a^2-40 a b x^4+45 b^2 x^8\right )}{585 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 36, normalized size = 0.61
method | result | size |
gosper | \(\frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}} \left (45 b^{2} x^{8}-40 a b \,x^{4}+32 a^{2}\right )}{585 b^{3}}\) | \(36\) |
trager | \(\frac {\left (45 b^{3} x^{12}+5 a \,b^{2} x^{8}-8 a^{2} b \,x^{4}+32 a^{3}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{585 b^{3}}\) | \(47\) |
risch | \(\frac {\left (45 b^{3} x^{12}+5 a \,b^{2} x^{8}-8 a^{2} b \,x^{4}+32 a^{3}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{585 b^{3}}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 47, normalized size = 0.80 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{\frac {13}{4}}}{13 \, b^{3}} - \frac {2 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} a}{9 \, b^{3}} + \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}} a^{2}}{5 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 46, normalized size = 0.78 \begin {gather*} \frac {{\left (45 \, b^{3} x^{12} + 5 \, a b^{2} x^{8} - 8 \, a^{2} b x^{4} + 32 \, a^{3}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{585 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.42, size = 87, normalized size = 1.47 \begin {gather*} \begin {cases} \frac {32 a^{3} \sqrt [4]{a + b x^{4}}}{585 b^{3}} - \frac {8 a^{2} x^{4} \sqrt [4]{a + b x^{4}}}{585 b^{2}} + \frac {a x^{8} \sqrt [4]{a + b x^{4}}}{117 b} + \frac {x^{12} \sqrt [4]{a + b x^{4}}}{13} & \text {for}\: b \neq 0 \\\frac {\sqrt [4]{a} x^{12}}{12} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.39, size = 43, normalized size = 0.73 \begin {gather*} \frac {45 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}} - 130 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} a + 117 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} a^{2}}{585 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.12, size = 44, normalized size = 0.75 \begin {gather*} {\left (b\,x^4+a\right )}^{1/4}\,\left (\frac {x^{12}}{13}+\frac {32\,a^3}{585\,b^3}+\frac {a\,x^8}{117\,b}-\frac {8\,a^2\,x^4}{585\,b^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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