Optimal. Leaf size=97 \[ \frac{a^2 c \sqrt{c x^2} (d x)^{m+4}}{d^4 (m+4) x}+\frac{2 a b c \sqrt{c x^2} (d x)^{m+5}}{d^5 (m+5) x}+\frac{b^2 c \sqrt{c x^2} (d x)^{m+6}}{d^6 (m+6) x} \]
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Rubi [A] time = 0.0446737, antiderivative size = 97, 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} \[ \frac{a^2 c \sqrt{c x^2} (d x)^{m+4}}{d^4 (m+4) x}+\frac{2 a b c \sqrt{c x^2} (d x)^{m+5}}{d^5 (m+5) x}+\frac{b^2 c \sqrt{c x^2} (d x)^{m+6}}{d^6 (m+6) x} \]
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 )^{3/2} (a+b x)^2 \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int x^3 (d x)^m (a+b x)^2 \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int (d x)^{3+m} (a+b x)^2 \, dx}{d^3 x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (a^2 (d x)^{3+m}+\frac{2 a b (d x)^{4+m}}{d}+\frac{b^2 (d x)^{5+m}}{d^2}\right ) \, dx}{d^3 x}\\ &=\frac{a^2 c (d x)^{4+m} \sqrt{c x^2}}{d^4 (4+m) x}+\frac{2 a b c (d x)^{5+m} \sqrt{c x^2}}{d^5 (5+m) x}+\frac{b^2 c (d x)^{6+m} \sqrt{c x^2}}{d^6 (6+m) x}\\ \end{align*}
Mathematica [A] time = 0.0562466, size = 48, normalized size = 0.49 \[ x \left (c x^2\right )^{3/2} (d x)^m \left (\frac{a^2}{m+4}+\frac{2 a b x}{m+5}+\frac{b^2 x^2}{m+6}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 95, normalized size = 1. \begin{align*}{\frac{ \left ({b}^{2}{m}^{2}{x}^{2}+2\,ab{m}^{2}x+9\,{b}^{2}m{x}^{2}+{a}^{2}{m}^{2}+20\,abmx+20\,{b}^{2}{x}^{2}+11\,{a}^{2}m+48\,abx+30\,{a}^{2} \right ) x \left ( dx \right ) ^{m}}{ \left ( 6+m \right ) \left ( 5+m \right ) \left ( 4+m \right ) } \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06926, size = 86, normalized size = 0.89 \begin{align*} \frac{b^{2} c^{\frac{3}{2}} d^{m} x^{6} x^{m}}{m + 6} + \frac{2 \, a b c^{\frac{3}{2}} d^{m} x^{5} x^{m}}{m + 5} + \frac{a^{2} c^{\frac{3}{2}} d^{m} x^{4} x^{m}}{m + 4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.23126, size = 238, normalized size = 2.45 \begin{align*} \frac{{\left ({\left (b^{2} c m^{2} + 9 \, b^{2} c m + 20 \, b^{2} c\right )} x^{5} + 2 \,{\left (a b c m^{2} + 10 \, a b c m + 24 \, a b c\right )} x^{4} +{\left (a^{2} c m^{2} + 11 \, a^{2} c m + 30 \, a^{2} c\right )} x^{3}\right )} \sqrt{c x^{2}} \left (d x\right )^{m}}{m^{3} + 15 \, m^{2} + 74 \, m + 120} \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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