Optimal. Leaf size=36 \[ -\frac {2 a^2}{5 x^{5/2}}+\frac {4}{3} a c x^{3/2}+\frac {2}{11} c^2 x^{11/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 36, 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 = {276}
\begin {gather*} -\frac {2 a^2}{5 x^{5/2}}+\frac {4}{3} a c x^{3/2}+\frac {2}{11} c^2 x^{11/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rubi steps
\begin {align*} \int \frac {\left (a+c x^4\right )^2}{x^{7/2}} \, dx &=\int \left (\frac {a^2}{x^{7/2}}+2 a c \sqrt {x}+c^2 x^{9/2}\right ) \, dx\\ &=-\frac {2 a^2}{5 x^{5/2}}+\frac {4}{3} a c x^{3/2}+\frac {2}{11} c^2 x^{11/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 30, normalized size = 0.83 \begin {gather*} -\frac {2 \left (33 a^2-110 a c x^4-15 c^2 x^8\right )}{165 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 25, normalized size = 0.69
method | result | size |
derivativedivides | \(-\frac {2 a^{2}}{5 x^{\frac {5}{2}}}+\frac {4 a c \,x^{\frac {3}{2}}}{3}+\frac {2 c^{2} x^{\frac {11}{2}}}{11}\) | \(25\) |
default | \(-\frac {2 a^{2}}{5 x^{\frac {5}{2}}}+\frac {4 a c \,x^{\frac {3}{2}}}{3}+\frac {2 c^{2} x^{\frac {11}{2}}}{11}\) | \(25\) |
gosper | \(-\frac {2 \left (-15 c^{2} x^{8}-110 a c \,x^{4}+33 a^{2}\right )}{165 x^{\frac {5}{2}}}\) | \(27\) |
trager | \(-\frac {2 \left (-15 c^{2} x^{8}-110 a c \,x^{4}+33 a^{2}\right )}{165 x^{\frac {5}{2}}}\) | \(27\) |
risch | \(-\frac {2 \left (-15 c^{2} x^{8}-110 a c \,x^{4}+33 a^{2}\right )}{165 x^{\frac {5}{2}}}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 24, normalized size = 0.67 \begin {gather*} \frac {2}{11} \, c^{2} x^{\frac {11}{2}} + \frac {4}{3} \, a c x^{\frac {3}{2}} - \frac {2 \, a^{2}}{5 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 26, normalized size = 0.72 \begin {gather*} \frac {2 \, {\left (15 \, c^{2} x^{8} + 110 \, a c x^{4} - 33 \, a^{2}\right )}}{165 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.89, size = 34, normalized size = 0.94 \begin {gather*} - \frac {2 a^{2}}{5 x^{\frac {5}{2}}} + \frac {4 a c x^{\frac {3}{2}}}{3} + \frac {2 c^{2} x^{\frac {11}{2}}}{11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.54, size = 24, normalized size = 0.67 \begin {gather*} \frac {2}{11} \, c^{2} x^{\frac {11}{2}} + \frac {4}{3} \, a c x^{\frac {3}{2}} - \frac {2 \, a^{2}}{5 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 26, normalized size = 0.72 \begin {gather*} \frac {-66\,a^2+220\,a\,c\,x^4+30\,c^2\,x^8}{165\,x^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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