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 = {270} \[ -\frac {2 a^2}{5 x^{5/2}}+\frac {4}{3} a c x^{3/2}+\frac {2}{11} c^2 x^{11/2} \]
Antiderivative was successfully verified.
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Rule 270
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 = 0.01, size = 30, normalized size = 0.83 \[ \frac {2 \left (-33 a^2+110 a c x^4+15 c^2 x^8\right )}{165 x^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 26, normalized size = 0.72 \[ \frac {2 \, {\left (15 \, c^{2} x^{8} + 110 \, a c x^{4} - 33 \, a^{2}\right )}}{165 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 24, normalized size = 0.67 \[ \frac {2}{11} \, c^{2} x^{\frac {11}{2}} + \frac {4}{3} \, a c x^{\frac {3}{2}} - \frac {2 \, a^{2}}{5 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 27, normalized size = 0.75 \[ -\frac {2 \left (-15 c^{2} x^{8}-110 a c \,x^{4}+33 a^{2}\right )}{165 x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 24, normalized size = 0.67 \[ \frac {2}{11} \, c^{2} x^{\frac {11}{2}} + \frac {4}{3} \, a c x^{\frac {3}{2}} - \frac {2 \, a^{2}}{5 \, x^{\frac {5}{2}}} \]
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 \[ \frac {-66\,a^2+220\,a\,c\,x^4+30\,c^2\,x^8}{165\,x^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.96, size = 34, normalized size = 0.94 \[ - \frac {2 a^{2}}{5 x^{\frac {5}{2}}} + \frac {4 a c x^{\frac {3}{2}}}{3} + \frac {2 c^{2} x^{\frac {11}{2}}}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
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