Optimal. Leaf size=30 \[ \frac{c^2 (a+b x)^5 \sqrt{c (a+b x)^2}}{6 b} \]
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Rubi [A] time = 0.0111322, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {247, 15, 30} \[ \frac{c^2 (a+b x)^5 \sqrt{c (a+b x)^2}}{6 b} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 30
Rubi steps
\begin{align*} \int \left (c (a+b x)^2\right )^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (c x^2\right )^{5/2} \, dx,x,a+b x\right )}{b}\\ &=\frac{\left (c^2 \sqrt{c (a+b x)^2}\right ) \operatorname{Subst}\left (\int x^5 \, dx,x,a+b x\right )}{b (a+b x)}\\ &=\frac{c^2 (a+b x)^5 \sqrt{c (a+b x)^2}}{6 b}\\ \end{align*}
Mathematica [A] time = 0.0201105, size = 25, normalized size = 0.83 \[ \frac{(a+b x) \left (c (a+b x)^2\right )^{5/2}}{6 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.004, size = 73, normalized size = 2.4 \begin{align*}{\frac{x \left ({b}^{5}{x}^{5}+6\,a{b}^{4}{x}^{4}+15\,{a}^{2}{b}^{3}{x}^{3}+20\,{a}^{3}{b}^{2}{x}^{2}+15\,{a}^{4}bx+6\,{a}^{5} \right ) }{6\, \left ( bx+a \right ) ^{5}} \left ( c \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.2146, size = 213, normalized size = 7.1 \begin{align*} \frac{{\left (b^{5} c^{2} x^{6} + 6 \, a b^{4} c^{2} x^{5} + 15 \, a^{2} b^{3} c^{2} x^{4} + 20 \, a^{3} b^{2} c^{2} x^{3} + 15 \, a^{4} b c^{2} x^{2} + 6 \, a^{5} c^{2} x\right )} \sqrt{b^{2} c x^{2} + 2 \, a b c x + a^{2} c}}{6 \,{\left (b x + a\right )}} \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 [B] time = 1.12215, size = 174, normalized size = 5.8 \begin{align*} \frac{1}{6} \,{\left (b^{5} c^{2} x^{6} \mathrm{sgn}\left (b x + a\right ) + 6 \, a b^{4} c^{2} x^{5} \mathrm{sgn}\left (b x + a\right ) + 15 \, a^{2} b^{3} c^{2} x^{4} \mathrm{sgn}\left (b x + a\right ) + 20 \, a^{3} b^{2} c^{2} x^{3} \mathrm{sgn}\left (b x + a\right ) + 15 \, a^{4} b c^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{5} c^{2} x \mathrm{sgn}\left (b x + a\right ) + \frac{a^{6} c^{2} \mathrm{sgn}\left (b x + a\right )}{b}\right )} \sqrt{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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