Optimal. Leaf size=29 \[ \frac {c^2 \sqrt {c x^2} (a+b x)^3}{3 b x} \]
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Rubi [A]
time = 0.00, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 32}
\begin {gather*} \frac {c^2 \sqrt {c x^2} (a+b x)^3}{3 b x} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 32
Rubi steps
\begin {align*} \int \frac {\left (c x^2\right )^{5/2} (a+b x)^2}{x^5} \, dx &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int (a+b x)^2 \, dx}{x}\\ &=\frac {c^2 \sqrt {c x^2} (a+b x)^3}{3 b x}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 26, normalized size = 0.90 \begin {gather*} \frac {\left (c x^2\right )^{5/2} (a+b x)^3}{3 b x^5} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.03, size = 27, normalized size = 0.93 \begin {gather*} \frac {\left (a^2+a b x+\frac {b^2 x^2}{3}\right ) {\left (c x^2\right )}^{\frac {5}{2}}}{x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 23, normalized size = 0.79
method | result | size |
default | \(\frac {\left (c \,x^{2}\right )^{\frac {5}{2}} \left (b x +a \right )^{3}}{3 x^{5} b}\) | \(23\) |
risch | \(\frac {c^{2} \left (b x +a \right )^{3} \sqrt {c \,x^{2}}}{3 b x}\) | \(26\) |
gosper | \(\frac {\left (x^{2} b^{2}+3 a b x +3 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{3 x^{4}}\) | \(31\) |
trager | \(\frac {c^{2} \left (x^{2} b^{2}+3 a b x +b^{2} x +3 a^{2}+3 a b +b^{2}\right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{3 x}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.29, size = 36, normalized size = 1.24 \begin {gather*} \frac {1}{3} \, {\left (b^{2} c^{2} x^{2} + 3 \, a b c^{2} x + 3 \, a^{2} c^{2}\right )} \sqrt {c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs.
\(2 (22) = 44\).
time = 0.44, size = 46, normalized size = 1.59 \begin {gather*} \frac {a^{2} \left (c x^{2}\right )^{\frac {5}{2}}}{x^{4}} + \frac {a b \left (c x^{2}\right )^{\frac {5}{2}}}{x^{3}} + \frac {b^{2} \left (c x^{2}\right )^{\frac {5}{2}}}{3 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 42, normalized size = 1.45 \begin {gather*} \sqrt {c} \left (a^{2} c^{2} x \mathrm {sign}\left (x\right )+\frac {1}{3} b^{2} c^{2} x^{3} \mathrm {sign}\left (x\right )+a b c^{2} x^{2} \mathrm {sign}\left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\left (c\,x^2\right )}^{5/2}\,{\left (a+b\,x\right )}^2}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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