Optimal. Leaf size=23 \[ -\frac {2}{3 b d (d (a+b x)+c)^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 23, 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} \[ -\frac {2}{3 b d (d (a+b x)+c)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 32
Rule 33
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*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 1.00 \[ -\frac {2}{3 b d (d (a+b x)+c)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 68, normalized size = 2.96 \[ -\frac {2 \, \sqrt {b d x + a d + c}}{3 \, {\left (b^{3} d^{3} x^{2} + a^{2} b d^{3} + 2 \, a b c d^{2} + b c^{2} d + 2 \, {\left (a b^{2} d^{3} + b^{2} c d^{2}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.32, size = 19, normalized size = 0.83 \[ -\frac {2}{3 \, {\left (b d x + a d + c\right )}^{\frac {3}{2}} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 20, normalized size = 0.87 \[ -\frac {2}{3 \left (b d x +a d +c \right )^{\frac {3}{2}} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.85, size = 19, normalized size = 0.83 \[ -\frac {2}{3 \, {\left ({\left (b x + a\right )} d + c\right )}^{\frac {3}{2}} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 19, normalized size = 0.83 \[ -\frac {2}{3\,b\,d\,{\left (c+d\,\left (a+b\,x\right )\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.79, size = 102, normalized size = 4.43 \[ \begin {cases} \frac {x}{c^{\frac {5}{2}}} & \text {for}\: b = 0 \wedge d = 0 \\\frac {x}{\left (a d + c\right )^{\frac {5}{2}}} & \text {for}\: b = 0 \\\frac {x}{c^{\frac {5}{2}}} & \text {for}\: d = 0 \\- \frac {2 \sqrt {a d + b d x + c}}{3 a^{2} b d^{3} + 6 a b^{2} d^{3} x + 6 a b c d^{2} + 3 b^{3} d^{3} x^{2} + 6 b^{2} c d^{2} x + 3 b c^{2} d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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