Optimal. Leaf size=49 \[ -\frac {2 a^3}{5 x^{5/2}}+2 a^2 c x^{3/2}+\frac {6}{11} a c^2 x^{11/2}+\frac {2}{19} c^3 x^{19/2} \]
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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} \[ 2 a^2 c x^{3/2}-\frac {2 a^3}{5 x^{5/2}}+\frac {6}{11} a c^2 x^{11/2}+\frac {2}{19} c^3 x^{19/2} \]
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int \frac {\left (a+c x^4\right )^3}{x^{7/2}} \, dx &=\int \left (\frac {a^3}{x^{7/2}}+3 a^2 c \sqrt {x}+3 a c^2 x^{9/2}+c^3 x^{17/2}\right ) \, dx\\ &=-\frac {2 a^3}{5 x^{5/2}}+2 a^2 c x^{3/2}+\frac {6}{11} a c^2 x^{11/2}+\frac {2}{19} c^3 x^{19/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 41, normalized size = 0.84 \[ \frac {2 \left (-209 a^3+1045 a^2 c x^4+285 a c^2 x^8+55 c^3 x^{12}\right )}{1045 x^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 37, normalized size = 0.76 \[ \frac {2 \, {\left (55 \, c^{3} x^{12} + 285 \, a c^{2} x^{8} + 1045 \, a^{2} c x^{4} - 209 \, a^{3}\right )}}{1045 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 35, normalized size = 0.71 \[ \frac {2}{19} \, c^{3} x^{\frac {19}{2}} + \frac {6}{11} \, a c^{2} x^{\frac {11}{2}} + 2 \, a^{2} c x^{\frac {3}{2}} - \frac {2 \, a^{3}}{5 \, x^{\frac {5}{2}}} \]
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 \[ -\frac {2 \left (-55 c^{3} x^{12}-285 a \,c^{2} x^{8}-1045 a^{2} c \,x^{4}+209 a^{3}\right )}{1045 x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 35, normalized size = 0.71 \[ \frac {2}{19} \, c^{3} x^{\frac {19}{2}} + \frac {6}{11} \, a c^{2} x^{\frac {11}{2}} + 2 \, a^{2} c x^{\frac {3}{2}} - \frac {2 \, a^{3}}{5 \, x^{\frac {5}{2}}} \]
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 \[ \frac {2\,c^3\,x^{19/2}}{19}-\frac {2\,a^3}{5\,x^{5/2}}+2\,a^2\,c\,x^{3/2}+\frac {6\,a\,c^2\,x^{11/2}}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 37.21, size = 48, normalized size = 0.98 \[ - \frac {2 a^{3}}{5 x^{\frac {5}{2}}} + 2 a^{2} c x^{\frac {3}{2}} + \frac {6 a c^{2} x^{\frac {11}{2}}}{11} + \frac {2 c^{3} x^{\frac {19}{2}}}{19} \]
Verification of antiderivative is not currently implemented for this CAS.
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