Optimal. Leaf size=81 \[ \frac{3 b^2 c (d x)^{m+5}}{d^5 (m+5)}+\frac{b^3 (d x)^{m+4}}{d^4 (m+4)}+\frac{3 b c^2 (d x)^{m+6}}{d^6 (m+6)}+\frac{c^3 (d x)^{m+7}}{d^7 (m+7)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0666968, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {647, 43} \[ \frac{3 b^2 c (d x)^{m+5}}{d^5 (m+5)}+\frac{b^3 (d x)^{m+4}}{d^4 (m+4)}+\frac{3 b c^2 (d x)^{m+6}}{d^6 (m+6)}+\frac{c^3 (d x)^{m+7}}{d^7 (m+7)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 647
Rule 43
Rubi steps
\begin{align*} \int (d x)^m \left (b x+c x^2\right )^3 \, dx &=\frac{\int (d x)^{3+m} (b+c x)^3 \, dx}{d^3}\\ &=\frac{\int \left (b^3 (d x)^{3+m}+\frac{3 b^2 c (d x)^{4+m}}{d}+\frac{3 b c^2 (d x)^{5+m}}{d^2}+\frac{c^3 (d x)^{6+m}}{d^3}\right ) \, dx}{d^3}\\ &=\frac{b^3 (d x)^{4+m}}{d^4 (4+m)}+\frac{3 b^2 c (d x)^{5+m}}{d^5 (5+m)}+\frac{3 b c^2 (d x)^{6+m}}{d^6 (6+m)}+\frac{c^3 (d x)^{7+m}}{d^7 (7+m)}\\ \end{align*}
Mathematica [A] time = 0.0360907, size = 57, normalized size = 0.7 \[ x^4 (d x)^m \left (\frac{3 b^2 c x}{m+5}+\frac{b^3}{m+4}+\frac{3 b c^2 x^2}{m+6}+\frac{c^3 x^3}{m+7}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.058, size = 173, normalized size = 2.1 \begin{align*}{\frac{ \left ( dx \right ) ^{m} \left ({c}^{3}{m}^{3}{x}^{3}+3\,b{c}^{2}{m}^{3}{x}^{2}+15\,{c}^{3}{m}^{2}{x}^{3}+3\,{b}^{2}c{m}^{3}x+48\,b{c}^{2}{m}^{2}{x}^{2}+74\,{c}^{3}m{x}^{3}+{b}^{3}{m}^{3}+51\,{b}^{2}c{m}^{2}x+249\,b{c}^{2}m{x}^{2}+120\,{x}^{3}{c}^{3}+18\,{b}^{3}{m}^{2}+282\,{b}^{2}cmx+420\,b{x}^{2}{c}^{2}+107\,{b}^{3}m+504\,{b}^{2}xc+210\,{b}^{3} \right ){x}^{4}}{ \left ( 7+m \right ) \left ( 6+m \right ) \left ( 5+m \right ) \left ( 4+m \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.17176, size = 104, normalized size = 1.28 \begin{align*} \frac{c^{3} d^{m} x^{7} x^{m}}{m + 7} + \frac{3 \, b c^{2} d^{m} x^{6} x^{m}}{m + 6} + \frac{3 \, b^{2} c d^{m} x^{5} x^{m}}{m + 5} + \frac{b^{3} d^{m} x^{4} x^{m}}{m + 4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.00345, size = 363, normalized size = 4.48 \begin{align*} \frac{{\left ({\left (c^{3} m^{3} + 15 \, c^{3} m^{2} + 74 \, c^{3} m + 120 \, c^{3}\right )} x^{7} + 3 \,{\left (b c^{2} m^{3} + 16 \, b c^{2} m^{2} + 83 \, b c^{2} m + 140 \, b c^{2}\right )} x^{6} + 3 \,{\left (b^{2} c m^{3} + 17 \, b^{2} c m^{2} + 94 \, b^{2} c m + 168 \, b^{2} c\right )} x^{5} +{\left (b^{3} m^{3} + 18 \, b^{3} m^{2} + 107 \, b^{3} m + 210 \, b^{3}\right )} x^{4}\right )} \left (d x\right )^{m}}{m^{4} + 22 \, m^{3} + 179 \, m^{2} + 638 \, m + 840} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 3.83979, size = 738, normalized size = 9.11 \begin{align*} \begin{cases} \frac{- \frac{b^{3}}{3 x^{3}} - \frac{3 b^{2} c}{2 x^{2}} - \frac{3 b c^{2}}{x} + c^{3} \log{\left (x \right )}}{d^{7}} & \text{for}\: m = -7 \\\frac{- \frac{b^{3}}{2 x^{2}} - \frac{3 b^{2} c}{x} + 3 b c^{2} \log{\left (x \right )} + c^{3} x}{d^{6}} & \text{for}\: m = -6 \\\frac{- \frac{b^{3}}{x} + 3 b^{2} c \log{\left (x \right )} + 3 b c^{2} x + \frac{c^{3} x^{2}}{2}}{d^{5}} & \text{for}\: m = -5 \\\frac{b^{3} \log{\left (x \right )} + 3 b^{2} c x + \frac{3 b c^{2} x^{2}}{2} + \frac{c^{3} x^{3}}{3}}{d^{4}} & \text{for}\: m = -4 \\\frac{b^{3} d^{m} m^{3} x^{4} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{18 b^{3} d^{m} m^{2} x^{4} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{107 b^{3} d^{m} m x^{4} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{210 b^{3} d^{m} x^{4} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{3 b^{2} c d^{m} m^{3} x^{5} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{51 b^{2} c d^{m} m^{2} x^{5} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{282 b^{2} c d^{m} m x^{5} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{504 b^{2} c d^{m} x^{5} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{3 b c^{2} d^{m} m^{3} x^{6} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{48 b c^{2} d^{m} m^{2} x^{6} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{249 b c^{2} d^{m} m x^{6} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{420 b c^{2} d^{m} x^{6} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{c^{3} d^{m} m^{3} x^{7} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{15 c^{3} d^{m} m^{2} x^{7} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{74 c^{3} d^{m} m x^{7} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{120 c^{3} d^{m} x^{7} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.27941, size = 356, normalized size = 4.4 \begin{align*} \frac{\left (d x\right )^{m} c^{3} m^{3} x^{7} + 3 \, \left (d x\right )^{m} b c^{2} m^{3} x^{6} + 15 \, \left (d x\right )^{m} c^{3} m^{2} x^{7} + 3 \, \left (d x\right )^{m} b^{2} c m^{3} x^{5} + 48 \, \left (d x\right )^{m} b c^{2} m^{2} x^{6} + 74 \, \left (d x\right )^{m} c^{3} m x^{7} + \left (d x\right )^{m} b^{3} m^{3} x^{4} + 51 \, \left (d x\right )^{m} b^{2} c m^{2} x^{5} + 249 \, \left (d x\right )^{m} b c^{2} m x^{6} + 120 \, \left (d x\right )^{m} c^{3} x^{7} + 18 \, \left (d x\right )^{m} b^{3} m^{2} x^{4} + 282 \, \left (d x\right )^{m} b^{2} c m x^{5} + 420 \, \left (d x\right )^{m} b c^{2} x^{6} + 107 \, \left (d x\right )^{m} b^{3} m x^{4} + 504 \, \left (d x\right )^{m} b^{2} c x^{5} + 210 \, \left (d x\right )^{m} b^{3} x^{4}}{m^{4} + 22 \, m^{3} + 179 \, m^{2} + 638 \, m + 840} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]