Optimal. Leaf size=23 \[ \frac{2 (d (a+b x)+c)^{7/2}}{7 b d} \]
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Rubi [A] time = 0.0118938, 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} \[ \frac{2 (d (a+b x)+c)^{7/2}}{7 b d} \]
Antiderivative was successfully verified.
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Rule 33
Rule 32
Rubi steps
\begin{align*} \int (c+d (a+b x))^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int (c+d x)^{5/2} \, dx,x,a+b x\right )}{b}\\ &=\frac{2 (c+d (a+b x))^{7/2}}{7 b d}\\ \end{align*}
Mathematica [A] time = 0.0318637, size = 23, normalized size = 1. \[ \frac{2 (d (a+b x)+c)^{7/2}}{7 b d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 20, normalized size = 0.9 \begin{align*}{\frac{2}{7\,bd} \left ( bdx+ad+c \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00917, size = 26, normalized size = 1.13 \begin{align*} \frac{2 \,{\left ({\left (b x + a\right )} d + c\right )}^{\frac{7}{2}}}{7 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.57616, size = 220, normalized size = 9.57 \begin{align*} \frac{2 \,{\left (b^{3} d^{3} x^{3} + a^{3} d^{3} + 3 \, a^{2} c d^{2} + 3 \, a c^{2} d + c^{3} + 3 \,{\left (a b^{2} d^{3} + b^{2} c d^{2}\right )} x^{2} + 3 \,{\left (a^{2} b d^{3} + 2 \, a b c d^{2} + b c^{2} d\right )} x\right )} \sqrt{b d x + a d + c}}{7 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 81.5393, size = 267, normalized size = 11.61 \begin{align*} \begin{cases} c^{\frac{5}{2}} x & \text{for}\: d = 0 \wedge \left (b = 0 \vee d = 0\right ) \\x \left (a d + c\right )^{\frac{5}{2}} & \text{for}\: b = 0 \\\frac{2 a^{3} d^{2} \sqrt{a d + b d x + c}}{7 b} + \frac{6 a^{2} d^{2} x \sqrt{a d + b d x + c}}{7} + \frac{6 a^{2} c d \sqrt{a d + b d x + c}}{7 b} + \frac{6 a b d^{2} x^{2} \sqrt{a d + b d x + c}}{7} + \frac{12 a c d x \sqrt{a d + b d x + c}}{7} + \frac{6 a c^{2} \sqrt{a d + b d x + c}}{7 b} + \frac{2 b^{2} d^{2} x^{3} \sqrt{a d + b d x + c}}{7} + \frac{6 b c d x^{2} \sqrt{a d + b d x + c}}{7} + \frac{6 c^{2} x \sqrt{a d + b d x + c}}{7} + \frac{2 c^{3} \sqrt{a d + b d x + c}}{7 b d} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28241, size = 263, normalized size = 11.43 \begin{align*} \frac{2 \,{\left (70 \,{\left (b d x + a d + c\right )}^{\frac{3}{2}} a^{2} d^{2} - 42 \,{\left (b d x + a d + c\right )}^{\frac{5}{2}} a d + 140 \,{\left (b d x + a d + c\right )}^{\frac{3}{2}} a c d + 15 \,{\left (b d x + a d + c\right )}^{\frac{7}{2}} - 42 \,{\left (b d x + a d + c\right )}^{\frac{5}{2}} c + 70 \,{\left (b d x + a d + c\right )}^{\frac{3}{2}} c^{2} - 14 \,{\left (5 \,{\left (b d x + a d + c\right )}^{\frac{3}{2}} a d - 3 \,{\left (b d x + a d + c\right )}^{\frac{5}{2}} + 5 \,{\left (b d x + a d + c\right )}^{\frac{3}{2}} c\right )} a d - 14 \,{\left (5 \,{\left (b d x + a d + c\right )}^{\frac{3}{2}} a d - 3 \,{\left (b d x + a d + c\right )}^{\frac{5}{2}} + 5 \,{\left (b d x + a d + c\right )}^{\frac{3}{2}} c\right )} c\right )}}{105 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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