Optimal. Leaf size=25 \[ -\frac {x}{b c \sqrt {c x^2} (a+b x)} \]
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Rubi [A]
time = 0.00, antiderivative size = 25, 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 {x}{b c \sqrt {c x^2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 32
Rubi steps
\begin {align*} \int \frac {x^3}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx &=\frac {x \int \frac {1}{(a+b x)^2} \, dx}{c \sqrt {c x^2}}\\ &=-\frac {x}{b c \sqrt {c x^2} (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 24, normalized size = 0.96 \begin {gather*} -\frac {x^3}{b \left (c x^2\right )^{3/2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 23, normalized size = 0.92
method | result | size |
gosper | \(-\frac {x^{3}}{\left (b x +a \right ) b \left (c \,x^{2}\right )^{\frac {3}{2}}}\) | \(23\) |
default | \(-\frac {x^{3}}{\left (b x +a \right ) b \left (c \,x^{2}\right )^{\frac {3}{2}}}\) | \(23\) |
risch | \(-\frac {x}{b c \left (b x +a \right ) \sqrt {c \,x^{2}}}\) | \(24\) |
trager | \(\frac {\left (-1+x \right ) \sqrt {c \,x^{2}}}{c^{2} \left (b x +a \right ) \left (a +b \right ) x}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 47 vs.
\(2 (23) = 46\).
time = 0.29, size = 47, normalized size = 1.88 \begin {gather*} \frac {a}{\sqrt {c x^{2}} b^{3} c x + \sqrt {c x^{2}} a b^{2} c} - \frac {1}{\sqrt {c x^{2}} b^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.46, size = 29, normalized size = 1.16 \begin {gather*} -\frac {\sqrt {c x^{2}}}{b^{2} c^{2} x^{2} + a b c^{2} x} \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. 73 vs.
\(2 (19) = 38\).
time = 0.55, size = 73, normalized size = 2.92 \begin {gather*} \begin {cases} \frac {\tilde {\infty } x^{2}}{\left (c x^{2}\right )^{\frac {3}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {\tilde {\infty } x^{4}}{\left (c x^{2}\right )^{\frac {3}{2}}} & \text {for}\: a = - b x \\\frac {x^{4}}{a^{2} \left (c x^{2}\right )^{\frac {3}{2}}} & \text {for}\: b = 0 \\- \frac {x^{3}}{a b \left (c x^{2}\right )^{\frac {3}{2}} + b^{2} x \left (c x^{2}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.16, size = 36, normalized size = 1.44 \begin {gather*} \frac {\frac {\mathrm {sgn}\left (x\right )}{a b \sqrt {c}} - \frac {1}{{\left (b x + a\right )} b \sqrt {c} \mathrm {sgn}\left (x\right )}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 25, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {c\,x^2}}{b\,c^2\,x\,\left (a+b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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