Optimal. Leaf size=39 \[ \frac {1}{2} a c^2 x \sqrt {c x^2}+\frac {1}{3} b c^2 x^2 \sqrt {c x^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 39, 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}{2} a c^2 x \sqrt {c x^2}+\frac {1}{3} b c^2 x^2 \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^4} \, dx &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int x (a+b x) \, dx}{x}\\ &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (a x+b x^2\right ) \, dx}{x}\\ &=\frac {1}{2} a c^2 x \sqrt {c x^2}+\frac {1}{3} b c^2 x^2 \sqrt {c x^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 25, normalized size = 0.64 \begin {gather*} \frac {1}{6} c^2 x \sqrt {c x^2} (3 a+2 b x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 21, normalized size = 0.54
method | result | size |
gosper | \(\frac {\left (2 b x +3 a \right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{6 x^{3}}\) | \(21\) |
default | \(\frac {\left (2 b x +3 a \right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{6 x^{3}}\) | \(21\) |
risch | \(\frac {a \,c^{2} x \sqrt {c \,x^{2}}}{2}+\frac {b \,c^{2} x^{2} \sqrt {c \,x^{2}}}{3}\) | \(32\) |
trager | \(\frac {c^{2} \left (2 x^{2} b +3 a x +2 b x +3 a +2 b \right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{6 x}\) | \(40\) |
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 = 1.02, size = 26, normalized size = 0.67 \begin {gather*} \frac {1}{6} \, {\left (2 \, b c^{2} x^{2} + 3 \, a c^{2} x\right )} \sqrt {c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.34, size = 29, normalized size = 0.74 \begin {gather*} \frac {a \left (c x^{2}\right )^{\frac {5}{2}}}{2 x^{3}} + \frac {b \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 = 1.92, size = 28, normalized size = 0.72 \begin {gather*} \frac {1}{6} \, {\left (2 \, b c^{2} x^{3} \mathrm {sgn}\left (x\right ) + 3 \, a c^{2} x^{2} \mathrm {sgn}\left (x\right )\right )} \sqrt {c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.26, size = 20, normalized size = 0.51 \begin {gather*} \frac {c^{5/2}\,\left (2\,b\,\sqrt {x^6}+3\,a\,x\,\left |x\right |\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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