Optimal. Leaf size=25 \[ -\frac {c \sqrt {c x^2}}{b x (a+b x)} \]
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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} \[ -\frac {c \sqrt {c x^2}}{b x (a+b x)} \]
Antiderivative was successfully verified.
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Rule 15
Rule 32
Rubi steps
\begin {align*} \int \frac {\left (c x^2\right )^{3/2}}{x^3 (a+b x)^2} \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int \frac {1}{(a+b x)^2} \, dx}{x}\\ &=-\frac {c \sqrt {c x^2}}{b x (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 0.96 \[ -\frac {\left (c x^2\right )^{3/2}}{b x^3 (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 24, normalized size = 0.96 \[ -\frac {\sqrt {c x^{2}} c}{b^{2} x^{2} + a b x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.14, size = 29, normalized size = 1.16 \[ -c^{\frac {3}{2}} {\left (\frac {\mathrm {sgn}\relax (x)}{{\left (b x + a\right )} b} - \frac {\mathrm {sgn}\relax (x)}{a b}\right )} \]
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 \[ -\frac {\left (c \,x^{2}\right )^{\frac {3}{2}}}{\left (b x +a \right ) b \,x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 16, normalized size = 0.64 \[ -\frac {c^{\frac {3}{2}}}{b^{2} x + a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 24, normalized size = 0.96 \[ -\frac {c^{3/2}\,\sqrt {x^2}}{b^2\,x^2+a\,b\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.23, size = 44, normalized size = 1.76 \[ \begin {cases} - \frac {c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}}{a b x^{3} + b^{2} x^{4}} & \text {for}\: b \neq 0 \\\frac {c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}}{a^{2} x^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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