Optimal. Leaf size=49 \[ 2 a^3 \sqrt {x}+\frac {2}{3} a^2 c x^{9/2}+\frac {6}{17} a c^2 x^{17/2}+\frac {2}{25} c^3 x^{25/2} \]
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Rubi [A] time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270} \[ \frac {2}{3} a^2 c x^{9/2}+2 a^3 \sqrt {x}+\frac {6}{17} a c^2 x^{17/2}+\frac {2}{25} c^3 x^{25/2} \]
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int \frac {\left (a+c x^4\right )^3}{\sqrt {x}} \, dx &=\int \left (\frac {a^3}{\sqrt {x}}+3 a^2 c x^{7/2}+3 a c^2 x^{15/2}+c^3 x^{23/2}\right ) \, dx\\ &=2 a^3 \sqrt {x}+\frac {2}{3} a^2 c x^{9/2}+\frac {6}{17} a c^2 x^{17/2}+\frac {2}{25} c^3 x^{25/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 41, normalized size = 0.84 \[ \frac {2 \sqrt {x} \left (1275 a^3+425 a^2 c x^4+225 a c^2 x^8+51 c^3 x^{12}\right )}{1275} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 37, normalized size = 0.76 \[ \frac {2}{1275} \, {\left (51 \, c^{3} x^{12} + 225 \, a c^{2} x^{8} + 425 \, a^{2} c x^{4} + 1275 \, a^{3}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 35, normalized size = 0.71 \[ \frac {2}{25} \, c^{3} x^{\frac {25}{2}} + \frac {6}{17} \, a c^{2} x^{\frac {17}{2}} + \frac {2}{3} \, a^{2} c x^{\frac {9}{2}} + 2 \, a^{3} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 38, normalized size = 0.78 \[ \frac {2 \left (51 c^{3} x^{12}+225 a \,c^{2} x^{8}+425 a^{2} c \,x^{4}+1275 a^{3}\right ) \sqrt {x}}{1275} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 35, normalized size = 0.71 \[ \frac {2}{25} \, c^{3} x^{\frac {25}{2}} + \frac {6}{17} \, a c^{2} x^{\frac {17}{2}} + \frac {2}{3} \, a^{2} c x^{\frac {9}{2}} + 2 \, a^{3} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 35, normalized size = 0.71 \[ 2\,a^3\,\sqrt {x}+\frac {2\,c^3\,x^{25/2}}{25}+\frac {2\,a^2\,c\,x^{9/2}}{3}+\frac {6\,a\,c^2\,x^{17/2}}{17} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 24.75, size = 48, normalized size = 0.98 \[ 2 a^{3} \sqrt {x} + \frac {2 a^{2} c x^{\frac {9}{2}}}{3} + \frac {6 a c^{2} x^{\frac {17}{2}}}{17} + \frac {2 c^{3} x^{\frac {25}{2}}}{25} \]
Verification of antiderivative is not currently implemented for this CAS.
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