Optimal. Leaf size=22 \[ \frac{2 (d \tan (a+b x))^{5/2}}{5 b d} \]
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Rubi [A] time = 0.0432917, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2607, 32} \[ \frac{2 (d \tan (a+b x))^{5/2}}{5 b d} \]
Antiderivative was successfully verified.
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Rule 2607
Rule 32
Rubi steps
\begin{align*} \int \sec ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int (d x)^{3/2} \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac{2 (d \tan (a+b x))^{5/2}}{5 b d}\\ \end{align*}
Mathematica [A] time = 0.0529561, size = 22, normalized size = 1. \[ \frac{2 (d \tan (a+b x))^{5/2}}{5 b d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 19, normalized size = 0.9 \begin{align*}{\frac{2}{5\,bd} \left ( d\tan \left ( bx+a \right ) \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.941616, size = 24, normalized size = 1.09 \begin{align*} \frac{2 \, \left (d \tan \left (b x + a\right )\right )^{\frac{5}{2}}}{5 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.71208, size = 111, normalized size = 5.05 \begin{align*} -\frac{2 \,{\left (d \cos \left (b x + a\right )^{2} - d\right )} \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{5 \, b \cos \left (b x + a\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \tan \left (b x + a\right )\right )^{\frac{3}{2}} \sec \left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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