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} \[ -\frac {x}{b c \sqrt {c x^2} (a+b x)} \]
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 \[ -\frac {x^3}{b \left (c x^2\right )^{3/2} (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 29, normalized size = 1.16 \[ -\frac {\sqrt {c x^{2}}}{b^{2} c^{2} x^{2} + a b c^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.16, size = 38, normalized size = 1.52 \[ \frac {1}{{\left (b x + a\right )} b c^{\frac {3}{2}} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 23, normalized size = 0.92 \[ -\frac {x^{3}}{\left (b x +a \right ) \left (c \,x^{2}\right )^{\frac {3}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.48, size = 47, normalized size = 1.88 \[ \frac {a}{\sqrt {c x^{2}} b^{3} c x + \sqrt {c x^{2}} a b^{2} c} - \frac {1}{\sqrt {c x^{2}} b^{2} c} \]
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 \[ -\frac {\sqrt {c\,x^2}}{b\,c^2\,x\,\left (a+b\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.94, size = 90, normalized size = 3.60 \[ \begin {cases} \frac {\tilde {\infty } x^{2}}{c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {\tilde {\infty } x^{4}}{c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} & \text {for}\: a = - b x \\\frac {x^{4}}{a^{2} c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} & \text {for}\: b = 0 \\- \frac {x^{3}}{a b c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}} + b^{2} c^{\frac {3}{2}} x \left (x^{2}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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