Integrand size = 20, antiderivative size = 61 \[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {a c (d x)^{4+m} \sqrt {c x^2}}{d^4 (4+m) x}+\frac {b c (d x)^{5+m} \sqrt {c x^2}}{d^5 (5+m) x} \]
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Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {15, 16, 45} \[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {a c \sqrt {c x^2} (d x)^{m+4}}{d^4 (m+4) x}+\frac {b c \sqrt {c x^2} (d x)^{m+5}}{d^5 (m+5) x} \]
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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) \, dx}{x} \\ & = \frac {\left (c \sqrt {c x^2}\right ) \int (d x)^{3+m} (a+b x) \, dx}{d^3 x} \\ & = \frac {\left (c \sqrt {c x^2}\right ) \int \left (a (d x)^{3+m}+\frac {b (d x)^{4+m}}{d}\right ) \, dx}{d^3 x} \\ & = \frac {a c (d x)^{4+m} \sqrt {c x^2}}{d^4 (4+m) x}+\frac {b c (d x)^{5+m} \sqrt {c x^2}}{d^5 (5+m) x} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.62 \[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {x (d x)^m \left (c x^2\right )^{3/2} (a (5+m)+b (4+m) x)}{(4+m) (5+m)} \]
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Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.66
method | result | size |
gosper | \(\frac {x \left (b m x +a m +4 b x +5 a \right ) \left (d x \right )^{m} \left (c \,x^{2}\right )^{\frac {3}{2}}}{\left (5+m \right ) \left (4+m \right )}\) | \(40\) |
risch | \(\frac {c \,x^{3} \sqrt {c \,x^{2}}\, \left (b m x +a m +4 b x +5 a \right ) \left (d x \right )^{m}}{\left (5+m \right ) \left (4+m \right )}\) | \(43\) |
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Time = 0.23 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.82 \[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {{\left ({\left (b c m + 4 \, b c\right )} x^{4} + {\left (a c m + 5 \, a c\right )} x^{3}\right )} \sqrt {c x^{2}} \left (d x\right )^{m}}{m^{2} + 9 \, m + 20} \]
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Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (53) = 106\).
Time = 3.66 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.95 \[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x) \, dx=\begin {cases} \frac {- \frac {a \left (c x^{2}\right )^{\frac {3}{2}}}{x^{4}} + \frac {b \left (c x^{2}\right )^{\frac {3}{2}} \log {\left (x \right )}}{x^{3}}}{d^{5}} & \text {for}\: m = -5 \\\frac {\frac {a \left (c x^{2}\right )^{\frac {3}{2}} \log {\left (x \right )}}{x^{3}} + \frac {b \left (c x^{2}\right )^{\frac {3}{2}}}{x^{2}}}{d^{4}} & \text {for}\: m = -4 \\\frac {a m x \left (c x^{2}\right )^{\frac {3}{2}} \left (d x\right )^{m}}{m^{2} + 9 m + 20} + \frac {5 a x \left (c x^{2}\right )^{\frac {3}{2}} \left (d x\right )^{m}}{m^{2} + 9 m + 20} + \frac {b m x^{2} \left (c x^{2}\right )^{\frac {3}{2}} \left (d x\right )^{m}}{m^{2} + 9 m + 20} + \frac {4 b x^{2} \left (c x^{2}\right )^{\frac {3}{2}} \left (d x\right )^{m}}{m^{2} + 9 m + 20} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.64 \[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {b c^{\frac {3}{2}} d^{m} x^{5} x^{m}}{m + 5} + \frac {a c^{\frac {3}{2}} d^{m} x^{4} x^{m}}{m + 4} \]
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Exception generated. \[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x) \, dx=\text {Exception raised: TypeError} \]
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Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.69 \[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {c\,x^3\,{\left (d\,x\right )}^m\,\sqrt {c\,x^2}\,\left (5\,a+a\,m+4\,b\,x+b\,m\,x\right )}{m^2+9\,m+20} \]
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