Optimal. Leaf size=58 \[ \frac {b^2 (d x)^{3+m}}{d^3 (3+m)}+\frac {2 b c (d x)^{4+m}}{d^4 (4+m)}+\frac {c^2 (d x)^{5+m}}{d^5 (5+m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 58, normalized size of antiderivative = 1.00, 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 = {661, 45}
\begin {gather*} \frac {b^2 (d x)^{m+3}}{d^3 (m+3)}+\frac {2 b c (d x)^{m+4}}{d^4 (m+4)}+\frac {c^2 (d x)^{m+5}}{d^5 (m+5)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 661
Rubi steps
\begin {align*} \int (d x)^m \left (b x+c x^2\right )^2 \, dx &=\frac {\int (d x)^{2+m} (b+c x)^2 \, dx}{d^2}\\ &=\frac {\int \left (b^2 (d x)^{2+m}+\frac {2 b c (d x)^{3+m}}{d}+\frac {c^2 (d x)^{4+m}}{d^2}\right ) \, dx}{d^2}\\ &=\frac {b^2 (d x)^{3+m}}{d^3 (3+m)}+\frac {2 b c (d x)^{4+m}}{d^4 (4+m)}+\frac {c^2 (d x)^{5+m}}{d^5 (5+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 41, normalized size = 0.71 \begin {gather*} x^3 (d x)^m \left (\frac {b^2}{3+m}+\frac {2 b c x}{4+m}+\frac {c^2 x^2}{5+m}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.44, size = 59, normalized size = 1.02
method | result | size |
norman | \(\frac {b^{2} x^{3} {\mathrm e}^{m \ln \left (d x \right )}}{3+m}+\frac {c^{2} x^{5} {\mathrm e}^{m \ln \left (d x \right )}}{5+m}+\frac {2 b c \,x^{4} {\mathrm e}^{m \ln \left (d x \right )}}{4+m}\) | \(59\) |
gosper | \(\frac {\left (d x \right )^{m} \left (c^{2} m^{2} x^{2}+2 b c \,m^{2} x +7 m \,x^{2} c^{2}+b^{2} m^{2}+16 b c m x +12 c^{2} x^{2}+9 b^{2} m +30 b c x +20 b^{2}\right ) x^{3}}{\left (5+m \right ) \left (4+m \right ) \left (3+m \right )}\) | \(90\) |
risch | \(\frac {\left (d x \right )^{m} \left (c^{2} m^{2} x^{2}+2 b c \,m^{2} x +7 m \,x^{2} c^{2}+b^{2} m^{2}+16 b c m x +12 c^{2} x^{2}+9 b^{2} m +30 b c x +20 b^{2}\right ) x^{3}}{\left (5+m \right ) \left (4+m \right ) \left (3+m \right )}\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 55, normalized size = 0.95 \begin {gather*} \frac {c^{2} d^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, b c d^{m} x^{4} x^{m}}{m + 4} + \frac {b^{2} d^{m} x^{3} x^{m}}{m + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.83, size = 89, normalized size = 1.53 \begin {gather*} \frac {{\left ({\left (c^{2} m^{2} + 7 \, c^{2} m + 12 \, c^{2}\right )} x^{5} + 2 \, {\left (b c m^{2} + 8 \, b c m + 15 \, b c\right )} x^{4} + {\left (b^{2} m^{2} + 9 \, b^{2} m + 20 \, b^{2}\right )} x^{3}\right )} \left (d x\right )^{m}}{m^{3} + 12 \, m^{2} + 47 \, m + 60} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 330 vs.
\(2 (51) = 102\).
time = 0.24, size = 330, normalized size = 5.69 \begin {gather*} \begin {cases} \frac {- \frac {b^{2}}{2 x^{2}} - \frac {2 b c}{x} + c^{2} \log {\left (x \right )}}{d^{5}} & \text {for}\: m = -5 \\\frac {- \frac {b^{2}}{x} + 2 b c \log {\left (x \right )} + c^{2} x}{d^{4}} & \text {for}\: m = -4 \\\frac {b^{2} \log {\left (x \right )} + 2 b c x + \frac {c^{2} x^{2}}{2}}{d^{3}} & \text {for}\: m = -3 \\\frac {b^{2} m^{2} x^{3} \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac {9 b^{2} m x^{3} \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac {20 b^{2} x^{3} \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac {2 b c m^{2} x^{4} \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac {16 b c m x^{4} \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac {30 b c x^{4} \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac {c^{2} m^{2} x^{5} \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac {7 c^{2} m x^{5} \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac {12 c^{2} x^{5} \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 47 m + 60} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 141 vs.
\(2 (58) = 116\).
time = 1.91, size = 141, normalized size = 2.43 \begin {gather*} \frac {\left (d x\right )^{m} c^{2} m^{2} x^{5} + 2 \, \left (d x\right )^{m} b c m^{2} x^{4} + 7 \, \left (d x\right )^{m} c^{2} m x^{5} + \left (d x\right )^{m} b^{2} m^{2} x^{3} + 16 \, \left (d x\right )^{m} b c m x^{4} + 12 \, \left (d x\right )^{m} c^{2} x^{5} + 9 \, \left (d x\right )^{m} b^{2} m x^{3} + 30 \, \left (d x\right )^{m} b c x^{4} + 20 \, \left (d x\right )^{m} b^{2} x^{3}}{m^{3} + 12 \, m^{2} + 47 \, m + 60} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.23, size = 97, normalized size = 1.67 \begin {gather*} {\left (d\,x\right )}^m\,\left (\frac {b^2\,x^3\,\left (m^2+9\,m+20\right )}{m^3+12\,m^2+47\,m+60}+\frac {c^2\,x^5\,\left (m^2+7\,m+12\right )}{m^3+12\,m^2+47\,m+60}+\frac {2\,b\,c\,x^4\,\left (m^2+8\,m+15\right )}{m^3+12\,m^2+47\,m+60}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________