Optimal. Leaf size=30 \[ -\frac {1}{2 b c (a+b x) \sqrt {c (a+b x)^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {247, 15, 30} \[ -\frac {1}{2 b c (a+b x) \sqrt {c (a+b x)^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rule 247
Rubi steps
\begin {align*} \int \frac {1}{\left (c (a+b x)^2\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (c x^2\right )^{3/2}} \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \operatorname {Subst}\left (\int \frac {1}{x^3} \, dx,x,a+b x\right )}{b c \sqrt {c (a+b x)^2}}\\ &=-\frac {1}{2 b c (a+b x) \sqrt {c (a+b x)^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.83 \[ -\frac {a+b x}{2 b \left (c (a+b x)^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.84, size = 69, normalized size = 2.30 \[ -\frac {\sqrt {b^{2} c x^{2} + 2 \, a b c x + a^{2} c}}{2 \, {\left (b^{4} c^{2} x^{3} + 3 \, a b^{3} c^{2} x^{2} + 3 \, a^{2} b^{2} c^{2} x + a^{3} b c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 22, normalized size = 0.73 \[ -\frac {b x +a}{2 \left (\left (b x +a \right )^{2} c \right )^{\frac {3}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 17, normalized size = 0.57 \[ -\frac {1}{2 \, b^{3} c^{\frac {3}{2}} {\left (x + \frac {a}{b}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 26, normalized size = 0.87 \[ -\frac {\sqrt {c\,{\left (a+b\,x\right )}^2}}{2\,b\,c^2\,{\left (a+b\,x\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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