Integrand size = 22, antiderivative size = 97 \[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x)^2 \, dx=\frac {a^2 c (d x)^{4+m} \sqrt {c x^2}}{d^4 (4+m) x}+\frac {2 a b c (d x)^{5+m} \sqrt {c x^2}}{d^5 (5+m) x}+\frac {b^2 c (d x)^{6+m} \sqrt {c x^2}}{d^6 (6+m) x} \]
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Time = 0.04 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {15, 16, 45} \[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x)^2 \, dx=\frac {a^2 c \sqrt {c x^2} (d x)^{m+4}}{d^4 (m+4) x}+\frac {2 a b c \sqrt {c x^2} (d x)^{m+5}}{d^5 (m+5) x}+\frac {b^2 c \sqrt {c x^2} (d x)^{m+6}}{d^6 (m+6) x} \]
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Rule 15
Rule 16
Rule 45
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.05 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.49 \[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x)^2 \, dx=x (d x)^m \left (c x^2\right )^{3/2} \left (\frac {a^2}{4+m}+\frac {2 a b x}{5+m}+\frac {b^2 x^2}{6+m}\right ) \]
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Time = 0.25 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.98
method | result | size |
gosper | \(\frac {x \left (b^{2} m^{2} x^{2}+2 a b \,m^{2} x +9 m \,x^{2} b^{2}+a^{2} m^{2}+20 a b m x +20 b^{2} x^{2}+11 a^{2} m +48 a b x +30 a^{2}\right ) \left (d x \right )^{m} \left (c \,x^{2}\right )^{\frac {3}{2}}}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right )}\) | \(95\) |
risch | \(\frac {c \,x^{3} \sqrt {c \,x^{2}}\, \left (b^{2} m^{2} x^{2}+2 a b \,m^{2} x +9 m \,x^{2} b^{2}+a^{2} m^{2}+20 a b m x +20 b^{2} x^{2}+11 a^{2} m +48 a b x +30 a^{2}\right ) \left (d x \right )^{m}}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right )}\) | \(98\) |
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Time = 0.24 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.08 \[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x)^2 \, dx=\frac {{\left ({\left (b^{2} c m^{2} + 9 \, b^{2} c m + 20 \, b^{2} c\right )} x^{5} + 2 \, {\left (a b c m^{2} + 10 \, a b c m + 24 \, a b c\right )} x^{4} + {\left (a^{2} c m^{2} + 11 \, a^{2} c m + 30 \, a^{2} c\right )} x^{3}\right )} \sqrt {c x^{2}} \left (d x\right )^{m}}{m^{3} + 15 \, m^{2} + 74 \, m + 120} \]
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Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (87) = 174\).
Time = 6.69 (sec) , antiderivative size = 493, normalized size of antiderivative = 5.08 \[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x)^2 \, dx=\begin {cases} \frac {- \frac {a^{2} \left (c x^{2}\right )^{\frac {3}{2}}}{2 x^{5}} - \frac {2 a b \left (c x^{2}\right )^{\frac {3}{2}}}{x^{4}} + \frac {b^{2} \left (c x^{2}\right )^{\frac {3}{2}} \log {\left (x \right )}}{x^{3}}}{d^{6}} & \text {for}\: m = -6 \\\frac {- \frac {a^{2} \left (c x^{2}\right )^{\frac {3}{2}}}{x^{4}} + \frac {2 a b \left (c x^{2}\right )^{\frac {3}{2}} \log {\left (x \right )}}{x^{3}} + \frac {b^{2} \left (c x^{2}\right )^{\frac {3}{2}}}{x^{2}}}{d^{5}} & \text {for}\: m = -5 \\\frac {\frac {a^{2} \left (c x^{2}\right )^{\frac {3}{2}} \log {\left (x \right )}}{x^{3}} + \frac {2 a b \left (c x^{2}\right )^{\frac {3}{2}}}{x^{2}} + \frac {b^{2} \left (c x^{2}\right )^{\frac {3}{2}}}{2 x}}{d^{4}} & \text {for}\: m = -4 \\\frac {a^{2} m^{2} x \left (c x^{2}\right )^{\frac {3}{2}} \left (d x\right )^{m}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {11 a^{2} m x \left (c x^{2}\right )^{\frac {3}{2}} \left (d x\right )^{m}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {30 a^{2} x \left (c x^{2}\right )^{\frac {3}{2}} \left (d x\right )^{m}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {2 a b m^{2} x^{2} \left (c x^{2}\right )^{\frac {3}{2}} \left (d x\right )^{m}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {20 a b m x^{2} \left (c x^{2}\right )^{\frac {3}{2}} \left (d x\right )^{m}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {48 a b x^{2} \left (c x^{2}\right )^{\frac {3}{2}} \left (d x\right )^{m}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {b^{2} m^{2} x^{3} \left (c x^{2}\right )^{\frac {3}{2}} \left (d x\right )^{m}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {9 b^{2} m x^{3} \left (c x^{2}\right )^{\frac {3}{2}} \left (d x\right )^{m}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {20 b^{2} x^{3} \left (c x^{2}\right )^{\frac {3}{2}} \left (d x\right )^{m}}{m^{3} + 15 m^{2} + 74 m + 120} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.66 \[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x)^2 \, dx=\frac {b^{2} c^{\frac {3}{2}} d^{m} x^{6} x^{m}}{m + 6} + \frac {2 \, a b c^{\frac {3}{2}} d^{m} x^{5} x^{m}}{m + 5} + \frac {a^{2} c^{\frac {3}{2}} d^{m} x^{4} x^{m}}{m + 4} \]
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Exception generated. \[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x)^2 \, dx=\text {Exception raised: TypeError} \]
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Time = 0.33 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.25 \[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x)^2 \, dx={\left (d\,x\right )}^m\,\left (\frac {a^2\,c\,x^3\,\sqrt {c\,x^2}\,\left (m^2+11\,m+30\right )}{m^3+15\,m^2+74\,m+120}+\frac {b^2\,c\,x^5\,\sqrt {c\,x^2}\,\left (m^2+9\,m+20\right )}{m^3+15\,m^2+74\,m+120}+\frac {2\,a\,b\,c\,x^4\,\sqrt {c\,x^2}\,\left (m^2+10\,m+24\right )}{m^3+15\,m^2+74\,m+120}\right ) \]
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