Optimal. Leaf size=41 \[ \frac {1}{4} a c^2 x^3 \sqrt {c x^2}+\frac {1}{5} b c^2 x^4 \sqrt {c x^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 45}
\begin {gather*} \frac {1}{4} a c^2 x^3 \sqrt {c x^2}+\frac {1}{5} b c^2 x^4 \sqrt {c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 45
Rubi steps
\begin {align*} \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^2} \, dx &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int x^3 (a+b x) \, dx}{x}\\ &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (a x^3+b x^4\right ) \, dx}{x}\\ &=\frac {1}{4} a c^2 x^3 \sqrt {c x^2}+\frac {1}{5} b c^2 x^4 \sqrt {c x^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 23, normalized size = 0.56 \begin {gather*} \frac {1}{20} c x \left (c x^2\right )^{3/2} (5 a+4 b x) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.86, size = 19, normalized size = 0.46 \begin {gather*} \frac {\left (\frac {a}{4}+\frac {b x}{5}\right ) {\left (c x^2\right )}^{\frac {5}{2}}}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 21, normalized size = 0.51
method | result | size |
gosper | \(\frac {\left (4 b x +5 a \right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{20 x}\) | \(21\) |
default | \(\frac {\left (4 b x +5 a \right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{20 x}\) | \(21\) |
risch | \(\frac {a \,c^{2} x^{3} \sqrt {c \,x^{2}}}{4}+\frac {b \,c^{2} x^{4} \sqrt {c \,x^{2}}}{5}\) | \(34\) |
trager | \(\frac {c^{2} \left (4 b \,x^{4}+5 a \,x^{3}+4 b \,x^{3}+5 a \,x^{2}+4 x^{2} b +5 a x +4 b x +5 a +4 b \right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{20 x}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 24, normalized size = 0.59 \begin {gather*} \frac {1}{5} \, \left (c x^{2}\right )^{\frac {5}{2}} b + \frac {\left (c x^{2}\right )^{\frac {5}{2}} a}{4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.29, size = 28, normalized size = 0.68 \begin {gather*} \frac {1}{20} \, {\left (4 \, b c^{2} x^{4} + 5 \, a c^{2} x^{3}\right )} \sqrt {c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.32, size = 24, normalized size = 0.59 \begin {gather*} \frac {a \left (c x^{2}\right )^{\frac {5}{2}}}{4 x} + \frac {b \left (c x^{2}\right )^{\frac {5}{2}}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 31, normalized size = 0.76 \begin {gather*} \sqrt {c} \left (\frac {1}{4} a c^{2} x^{4} \mathrm {sign}\left (x\right )+\frac {1}{5} b c^{2} x^{5} \mathrm {sign}\left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.28, size = 25, normalized size = 0.61 \begin {gather*} \frac {c^{5/2}\,\left (4\,b\,\sqrt {x^{10}}+5\,a\,x^3\,\sqrt {x^2}\right )}{20} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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