Optimal. Leaf size=23 \[ -\frac{2}{3 b d (d (a+b x)+c)^{3/2}} \]
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Rubi [A] time = 0.0108823, antiderivative size = 23, normalized size of antiderivative = 1., 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} \[ -\frac{2}{3 b d (d (a+b x)+c)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 33
Rule 32
Rubi steps
\begin{align*} \int \frac{1}{(c+d (a+b x))^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(c+d x)^{5/2}} \, dx,x,a+b x\right )}{b}\\ &=-\frac{2}{3 b d (c+d (a+b x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0304856, size = 23, normalized size = 1. \[ -\frac{2}{3 b d (d (a+b x)+c)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 20, normalized size = 0.9 \begin{align*} -{\frac{2}{3\,bd} \left ( bdx+ad+c \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05801, size = 26, normalized size = 1.13 \begin{align*} -\frac{2}{3 \,{\left ({\left (b x + a\right )} d + c\right )}^{\frac{3}{2}} b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.57646, size = 146, normalized size = 6.35 \begin{align*} -\frac{2 \, \sqrt{b d x + a d + c}}{3 \,{\left (b^{3} d^{3} x^{2} + a^{2} b d^{3} + 2 \, a b c d^{2} + b c^{2} d + 2 \,{\left (a b^{2} d^{3} + b^{2} c d^{2}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.85587, size = 102, normalized size = 4.43 \begin{align*} \begin{cases} \frac{x}{c^{\frac{5}{2}}} & \text{for}\: b = 0 \wedge d = 0 \\\frac{x}{\left (a d + c\right )^{\frac{5}{2}}} & \text{for}\: b = 0 \\\frac{x}{c^{\frac{5}{2}}} & \text{for}\: d = 0 \\- \frac{2 \sqrt{a d + b d x + c}}{3 a^{2} b d^{3} + 6 a b^{2} d^{3} x + 6 a b c d^{2} + 3 b^{3} d^{3} x^{2} + 6 b^{2} c d^{2} x + 3 b c^{2} d} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14405, size = 26, normalized size = 1.13 \begin{align*} -\frac{2}{3 \,{\left ({\left (b x + a\right )} d + c\right )}^{\frac{3}{2}} b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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