Optimal. Leaf size=73 \[ \frac{3 b^2 c x^9 (c x)^m}{m+9}+\frac{b^3 x^7 (c x)^m}{m+7}+\frac{3 b c^2 x^{11} (c x)^m}{m+11}+\frac{c^3 x^{13} (c x)^m}{m+13} \]
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Rubi [A] time = 0.0498427, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1142, 1584, 270} \[ \frac{3 b^2 c x^9 (c x)^m}{m+9}+\frac{b^3 x^7 (c x)^m}{m+7}+\frac{3 b c^2 x^{11} (c x)^m}{m+11}+\frac{c^3 x^{13} (c x)^m}{m+13} \]
Antiderivative was successfully verified.
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Rule 1142
Rule 1584
Rule 270
Rubi steps
\begin{align*} \int (c x)^m \left (b x^2+c x^4\right )^3 \, dx &=\left (x^{-m} (c x)^m\right ) \operatorname{Subst}\left (\int x^m \left (b x^2+c x^4\right )^3 \, dx,x,x\right )\\ &=\left (x^{-m} (c x)^m\right ) \operatorname{Subst}\left (\int x^{6+m} \left (b+c x^2\right )^3 \, dx,x,x\right )\\ &=\left (x^{-m} (c x)^m\right ) \operatorname{Subst}\left (\int \left (b^3 x^{6+m}+3 b^2 c x^{8+m}+3 b c^2 x^{10+m}+c^3 x^{12+m}\right ) \, dx,x,x\right )\\ &=\frac{b^3 x^7 (c x)^m}{7+m}+\frac{3 b^2 c x^9 (c x)^m}{9+m}+\frac{3 b c^2 x^{11} (c x)^m}{11+m}+\frac{c^3 x^{13} (c x)^m}{13+m}\\ \end{align*}
Mathematica [A] time = 0.0400591, size = 59, normalized size = 0.81 \[ x^7 (c x)^m \left (\frac{3 b^2 c x^2}{m+9}+\frac{b^3}{m+7}+\frac{3 b c^2 x^4}{m+11}+\frac{c^3 x^6}{m+13}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.048, size = 181, normalized size = 2.5 \begin{align*}{\frac{ \left ( cx \right ) ^{m} \left ({c}^{3}{m}^{3}{x}^{6}+27\,{c}^{3}{m}^{2}{x}^{6}+3\,b{c}^{2}{m}^{3}{x}^{4}+239\,{c}^{3}m{x}^{6}+87\,b{c}^{2}{m}^{2}{x}^{4}+693\,{c}^{3}{x}^{6}+3\,{b}^{2}c{m}^{3}{x}^{2}+813\,b{c}^{2}m{x}^{4}+93\,{b}^{2}c{m}^{2}{x}^{2}+2457\,b{c}^{2}{x}^{4}+{b}^{3}{m}^{3}+933\,{b}^{2}cm{x}^{2}+33\,{b}^{3}{m}^{2}+3003\,{b}^{2}c{x}^{2}+359\,{b}^{3}m+1287\,{b}^{3} \right ){x}^{7}}{ \left ( 13+m \right ) \left ( 11+m \right ) \left ( 9+m \right ) \left ( 7+m \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01244, size = 103, normalized size = 1.41 \begin{align*} \frac{c^{m + 3} x^{13} x^{m}}{m + 13} + \frac{3 \, b c^{m + 2} x^{11} x^{m}}{m + 11} + \frac{3 \, b^{2} c^{m + 1} x^{9} x^{m}}{m + 9} + \frac{b^{3} c^{m} x^{7} x^{m}}{m + 7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.58996, size = 375, normalized size = 5.14 \begin{align*} \frac{{\left ({\left (c^{3} m^{3} + 27 \, c^{3} m^{2} + 239 \, c^{3} m + 693 \, c^{3}\right )} x^{13} + 3 \,{\left (b c^{2} m^{3} + 29 \, b c^{2} m^{2} + 271 \, b c^{2} m + 819 \, b c^{2}\right )} x^{11} + 3 \,{\left (b^{2} c m^{3} + 31 \, b^{2} c m^{2} + 311 \, b^{2} c m + 1001 \, b^{2} c\right )} x^{9} +{\left (b^{3} m^{3} + 33 \, b^{3} m^{2} + 359 \, b^{3} m + 1287 \, b^{3}\right )} x^{7}\right )} \left (c x\right )^{m}}{m^{4} + 40 \, m^{3} + 590 \, m^{2} + 3800 \, m + 9009} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.30557, size = 758, normalized size = 10.38 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1681, size = 356, normalized size = 4.88 \begin{align*} \frac{\left (c x\right )^{m} c^{3} m^{3} x^{13} + 27 \, \left (c x\right )^{m} c^{3} m^{2} x^{13} + 3 \, \left (c x\right )^{m} b c^{2} m^{3} x^{11} + 239 \, \left (c x\right )^{m} c^{3} m x^{13} + 87 \, \left (c x\right )^{m} b c^{2} m^{2} x^{11} + 693 \, \left (c x\right )^{m} c^{3} x^{13} + 3 \, \left (c x\right )^{m} b^{2} c m^{3} x^{9} + 813 \, \left (c x\right )^{m} b c^{2} m x^{11} + 93 \, \left (c x\right )^{m} b^{2} c m^{2} x^{9} + 2457 \, \left (c x\right )^{m} b c^{2} x^{11} + \left (c x\right )^{m} b^{3} m^{3} x^{7} + 933 \, \left (c x\right )^{m} b^{2} c m x^{9} + 33 \, \left (c x\right )^{m} b^{3} m^{2} x^{7} + 3003 \, \left (c x\right )^{m} b^{2} c x^{9} + 359 \, \left (c x\right )^{m} b^{3} m x^{7} + 1287 \, \left (c x\right )^{m} b^{3} x^{7}}{m^{4} + 40 \, m^{3} + 590 \, m^{2} + 3800 \, m + 9009} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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