Optimal. Leaf size=65 \[ \frac{a c^2 \sqrt{c x^2} (d x)^{m+6}}{d^6 (m+6) x}+\frac{b c^2 \sqrt{c x^2} (d x)^{m+7}}{d^7 (m+7) x} \]
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Rubi [A] time = 0.0304476, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {15, 16, 43} \[ \frac{a c^2 \sqrt{c x^2} (d x)^{m+6}}{d^6 (m+6) x}+\frac{b c^2 \sqrt{c x^2} (d x)^{m+7}}{d^7 (m+7) x} \]
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) \, dx &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int x^5 (d x)^m (a+b x) \, dx}{x}\\ &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int (d x)^{5+m} (a+b x) \, dx}{d^5 x}\\ &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int \left (a (d x)^{5+m}+\frac{b (d x)^{6+m}}{d}\right ) \, dx}{d^5 x}\\ &=\frac{a c^2 (d x)^{6+m} \sqrt{c x^2}}{d^6 (6+m) x}+\frac{b c^2 (d x)^{7+m} \sqrt{c x^2}}{d^7 (7+m) x}\\ \end{align*}
Mathematica [A] time = 0.0273481, size = 38, normalized size = 0.58 \[ \frac{x \left (c x^2\right )^{5/2} (d x)^m (a (m+7)+b (m+6) x)}{(m+6) (m+7)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 40, normalized size = 0.6 \begin{align*}{\frac{ \left ( bmx+am+6\,bx+7\,a \right ) x \left ( dx \right ) ^{m}}{ \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 = 1.06589, size = 53, normalized size = 0.82 \begin{align*} \frac{b c^{\frac{5}{2}} d^{m} x^{7} x^{m}}{m + 7} + \frac{a 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 = 1.3552, size = 123, normalized size = 1.89 \begin{align*} \frac{{\left ({\left (b c^{2} m + 6 \, b c^{2}\right )} x^{6} +{\left (a c^{2} m + 7 \, a c^{2}\right )} x^{5}\right )} \sqrt{c x^{2}} \left (d x\right )^{m}}{m^{2} + 13 \, m + 42} \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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