Optimal. Leaf size=21 \[ -\frac {2}{b d \sqrt {d (a+b x)+c}} \]
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Rubi [A] time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {33, 32} \begin {gather*} -\frac {2}{b d \sqrt {d (a+b x)+c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 33
Rubi steps
\begin {align*} \int \frac {1}{(c+d (a+b x))^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{(c+d x)^{3/2}} \, dx,x,a+b x\right )}{b}\\ &=-\frac {2}{b d \sqrt {c+d (a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} -\frac {2}{b d \sqrt {d (a+b x)+c}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} -\frac {2}{b d \sqrt {a d+b d x+c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.57, size = 34, normalized size = 1.62 \begin {gather*} -\frac {2 \, \sqrt {b d x + a d + c}}{b^{2} d^{2} x + a b d^{2} + b c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.65, size = 19, normalized size = 0.90 \begin {gather*} -\frac {2}{\sqrt {b d x + a d + c} b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 20, normalized size = 0.95 \begin {gather*} -\frac {2}{\sqrt {b d x +a d +c}\, b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 19, normalized size = 0.90 \begin {gather*} -\frac {2}{\sqrt {{\left (b x + a\right )} d + c} b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 19, normalized size = 0.90 \begin {gather*} -\frac {2}{b\,d\,\sqrt {c+d\,\left (a+b\,x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.79, size = 58, normalized size = 2.76 \begin {gather*} \begin {cases} \frac {x}{c^{\frac {3}{2}}} & \text {for}\: b = 0 \wedge d = 0 \\\frac {x}{\left (a d + c\right )^{\frac {3}{2}}} & \text {for}\: b = 0 \\\frac {x}{c^{\frac {3}{2}}} & \text {for}\: d = 0 \\- \frac {2 \sqrt {a d + b d x + c}}{a b d^{2} + b^{2} d^{2} x + b c d} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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