Optimal. Leaf size=103 \[ \frac{a^2 c^2 \sqrt{c x^2} (d x)^{m+6}}{d^6 (m+6) x}+\frac{2 a b c^2 \sqrt{c x^2} (d x)^{m+7}}{d^7 (m+7) x}+\frac{b^2 c^2 \sqrt{c x^2} (d x)^{m+8}}{d^8 (m+8) x} \]
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Rubi [A] time = 0.0489113, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {15, 16, 43} \[ \frac{a^2 c^2 \sqrt{c x^2} (d x)^{m+6}}{d^6 (m+6) x}+\frac{2 a b c^2 \sqrt{c x^2} (d x)^{m+7}}{d^7 (m+7) x}+\frac{b^2 c^2 \sqrt{c x^2} (d x)^{m+8}}{d^8 (m+8) x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 16
Rule 43
Rubi steps
\begin{align*} \int (d x)^m \left (c x^2\right )^{5/2} (a+b x)^2 \, dx &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int x^5 (d x)^m (a+b x)^2 \, dx}{x}\\ &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int (d x)^{5+m} (a+b x)^2 \, dx}{d^5 x}\\ &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int \left (a^2 (d x)^{5+m}+\frac{2 a b (d x)^{6+m}}{d}+\frac{b^2 (d x)^{7+m}}{d^2}\right ) \, dx}{d^5 x}\\ &=\frac{a^2 c^2 (d x)^{6+m} \sqrt{c x^2}}{d^6 (6+m) x}+\frac{2 a b c^2 (d x)^{7+m} \sqrt{c x^2}}{d^7 (7+m) x}+\frac{b^2 c^2 (d x)^{8+m} \sqrt{c x^2}}{d^8 (8+m) x}\\ \end{align*}
Mathematica [A] time = 0.0703456, size = 48, normalized size = 0.47 \[ x \left (c x^2\right )^{5/2} (d x)^m \left (\frac{a^2}{m+6}+\frac{2 a b x}{m+7}+\frac{b^2 x^2}{m+8}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 95, normalized size = 0.9 \begin{align*}{\frac{ \left ({b}^{2}{m}^{2}{x}^{2}+2\,ab{m}^{2}x+13\,{b}^{2}m{x}^{2}+{a}^{2}{m}^{2}+28\,abmx+42\,{b}^{2}{x}^{2}+15\,{a}^{2}m+96\,abx+56\,{a}^{2} \right ) x \left ( dx \right ) ^{m}}{ \left ( 8+m \right ) \left ( 7+m \right ) \left ( 6+m \right ) } \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07237, size = 86, normalized size = 0.83 \begin{align*} \frac{b^{2} c^{\frac{5}{2}} d^{m} x^{8} x^{m}}{m + 8} + \frac{2 \, a b c^{\frac{5}{2}} d^{m} x^{7} x^{m}}{m + 7} + \frac{a^{2} c^{\frac{5}{2}} d^{m} x^{6} x^{m}}{m + 6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37286, size = 265, normalized size = 2.57 \begin{align*} \frac{{\left ({\left (b^{2} c^{2} m^{2} + 13 \, b^{2} c^{2} m + 42 \, b^{2} c^{2}\right )} x^{7} + 2 \,{\left (a b c^{2} m^{2} + 14 \, a b c^{2} m + 48 \, a b c^{2}\right )} x^{6} +{\left (a^{2} c^{2} m^{2} + 15 \, a^{2} c^{2} m + 56 \, a^{2} c^{2}\right )} x^{5}\right )} \sqrt{c x^{2}} \left (d x\right )^{m}}{m^{3} + 21 \, m^{2} + 146 \, m + 336} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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