Integrand size = 31, antiderivative size = 109 \[ \int x^3 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {c^2 \left (b c^2+a d^2\right ) (-c+d x)^{3/2} (c+d x)^{3/2}}{3 d^6}+\frac {\left (2 b c^2+a d^2\right ) (-c+d x)^{5/2} (c+d x)^{5/2}}{5 d^6}+\frac {b (-c+d x)^{7/2} (c+d x)^{7/2}}{7 d^6} \]
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Time = 0.06 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {471, 102, 12, 75} \[ \int x^3 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {2 c^2 (d x-c)^{3/2} (c+d x)^{3/2} \left (7 a d^2+4 b c^2\right )}{105 d^6}+\frac {x^2 (d x-c)^{3/2} (c+d x)^{3/2} \left (7 a d^2+4 b c^2\right )}{35 d^4}+\frac {b x^4 (d x-c)^{3/2} (c+d x)^{3/2}}{7 d^2} \]
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Rule 12
Rule 75
Rule 102
Rule 471
Rubi steps \begin{align*} \text {integral}& = \frac {b x^4 (-c+d x)^{3/2} (c+d x)^{3/2}}{7 d^2}-\frac {1}{7} \left (-7 a-\frac {4 b c^2}{d^2}\right ) \int x^3 \sqrt {-c+d x} \sqrt {c+d x} \, dx \\ & = \frac {\left (4 b c^2+7 a d^2\right ) x^2 (-c+d x)^{3/2} (c+d x)^{3/2}}{35 d^4}+\frac {b x^4 (-c+d x)^{3/2} (c+d x)^{3/2}}{7 d^2}+\frac {\left (4 b c^2+7 a d^2\right ) \int 2 c^2 x \sqrt {-c+d x} \sqrt {c+d x} \, dx}{35 d^4} \\ & = \frac {\left (4 b c^2+7 a d^2\right ) x^2 (-c+d x)^{3/2} (c+d x)^{3/2}}{35 d^4}+\frac {b x^4 (-c+d x)^{3/2} (c+d x)^{3/2}}{7 d^2}+\frac {\left (2 c^2 \left (4 b c^2+7 a d^2\right )\right ) \int x \sqrt {-c+d x} \sqrt {c+d x} \, dx}{35 d^4} \\ & = \frac {2 c^2 \left (4 b c^2+7 a d^2\right ) (-c+d x)^{3/2} (c+d x)^{3/2}}{105 d^6}+\frac {\left (4 b c^2+7 a d^2\right ) x^2 (-c+d x)^{3/2} (c+d x)^{3/2}}{35 d^4}+\frac {b x^4 (-c+d x)^{3/2} (c+d x)^{3/2}}{7 d^2} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.69 \[ \int x^3 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {(-c+d x)^{3/2} (c+d x)^{3/2} \left (7 a d^2 \left (2 c^2+3 d^2 x^2\right )+b \left (8 c^4+12 c^2 d^2 x^2+15 d^4 x^4\right )\right )}{105 d^6} \]
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Time = 4.19 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.62
method | result | size |
gosper | \(\frac {\left (d x -c \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}} \left (15 b \,d^{4} x^{4}+21 a \,d^{4} x^{2}+12 b \,c^{2} d^{2} x^{2}+14 a \,c^{2} d^{2}+8 b \,c^{4}\right )}{105 d^{6}}\) | \(68\) |
default | \(-\frac {\sqrt {d x -c}\, \sqrt {d x +c}\, \left (-d^{2} x^{2}+c^{2}\right ) \left (15 b \,d^{4} x^{4}+21 a \,d^{4} x^{2}+12 b \,c^{2} d^{2} x^{2}+14 a \,c^{2} d^{2}+8 b \,c^{4}\right )}{105 d^{6}}\) | \(80\) |
risch | \(\frac {\sqrt {d x +c}\, \left (-15 b \,x^{6} d^{6}-21 a \,d^{6} x^{4}+3 b \,c^{2} d^{4} x^{4}+7 a \,c^{2} d^{4} x^{2}+4 b \,c^{4} d^{2} x^{2}+14 a \,c^{4} d^{2}+8 b \,c^{6}\right ) \left (-d x +c \right )}{105 \sqrt {d x -c}\, d^{6}}\) | \(98\) |
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Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int x^3 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {{\left (15 \, b d^{6} x^{6} - 8 \, b c^{6} - 14 \, a c^{4} d^{2} - 3 \, {\left (b c^{2} d^{4} - 7 \, a d^{6}\right )} x^{4} - {\left (4 \, b c^{4} d^{2} + 7 \, a c^{2} d^{4}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {d x - c}}{105 \, d^{6}} \]
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\[ \int x^3 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\int x^{3} \left (a + b x^{2}\right ) \sqrt {- c + d x} \sqrt {c + d x}\, dx \]
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Time = 0.31 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.14 \[ \int x^3 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} b x^{4}}{7 \, d^{2}} + \frac {4 \, {\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} b c^{2} x^{2}}{35 \, d^{4}} + \frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} a x^{2}}{5 \, d^{2}} + \frac {8 \, {\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} b c^{4}}{105 \, d^{6}} + \frac {2 \, {\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} a c^{2}}{15 \, d^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (91) = 182\).
Time = 0.38 (sec) , antiderivative size = 495, normalized size of antiderivative = 4.54 \[ \int x^3 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {70 \, {\left ({\left ({\left (d x + c\right )} {\left (2 \, {\left (d x + c\right )} {\left (\frac {3 \, {\left (d x + c\right )}}{d^{3}} - \frac {13 \, c}{d^{3}}\right )} + \frac {43 \, c^{2}}{d^{3}}\right )} - \frac {39 \, c^{3}}{d^{3}}\right )} \sqrt {d x + c} \sqrt {d x - c} - \frac {18 \, c^{4} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{3}}\right )} a c + 7 \, {\left ({\left ({\left (2 \, {\left ({\left (d x + c\right )} {\left (4 \, {\left (d x + c\right )} {\left (\frac {5 \, {\left (d x + c\right )}}{d^{5}} - \frac {31 \, c}{d^{5}}\right )} + \frac {321 \, c^{2}}{d^{5}}\right )} - \frac {451 \, c^{3}}{d^{5}}\right )} {\left (d x + c\right )} + \frac {745 \, c^{4}}{d^{5}}\right )} {\left (d x + c\right )} - \frac {405 \, c^{5}}{d^{5}}\right )} \sqrt {d x + c} \sqrt {d x - c} - \frac {150 \, c^{6} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{5}}\right )} b c + 14 \, {\left ({\left ({\left (2 \, {\left (d x + c\right )} {\left (3 \, {\left (d x + c\right )} {\left (\frac {4 \, {\left (d x + c\right )}}{d^{4}} - \frac {21 \, c}{d^{4}}\right )} + \frac {133 \, c^{2}}{d^{4}}\right )} - \frac {295 \, c^{3}}{d^{4}}\right )} {\left (d x + c\right )} + \frac {195 \, c^{4}}{d^{4}}\right )} \sqrt {d x + c} \sqrt {d x - c} + \frac {90 \, c^{5} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{4}}\right )} a d + {\left ({\left ({\left (2 \, {\left ({\left (4 \, {\left (d x + c\right )} {\left (5 \, {\left (d x + c\right )} {\left (\frac {6 \, {\left (d x + c\right )}}{d^{6}} - \frac {43 \, c}{d^{6}}\right )} + \frac {661 \, c^{2}}{d^{6}}\right )} - \frac {4551 \, c^{3}}{d^{6}}\right )} {\left (d x + c\right )} + \frac {4781 \, c^{4}}{d^{6}}\right )} {\left (d x + c\right )} - \frac {6335 \, c^{5}}{d^{6}}\right )} {\left (d x + c\right )} + \frac {2835 \, c^{6}}{d^{6}}\right )} \sqrt {d x + c} \sqrt {d x - c} + \frac {1050 \, c^{7} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{6}}\right )} b d}{1680 \, d} \]
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Time = 5.79 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.08 \[ \int x^3 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=-\sqrt {d\,x-c}\,\left (\frac {\left (8\,b\,c^6+14\,a\,c^4\,d^2\right )\,\sqrt {c+d\,x}}{105\,d^6}-\frac {b\,x^6\,\sqrt {c+d\,x}}{7}+\frac {x^2\,\left (4\,b\,c^4\,d^2+7\,a\,c^2\,d^4\right )\,\sqrt {c+d\,x}}{105\,d^6}-\frac {x^4\,\left (21\,a\,d^6-3\,b\,c^2\,d^4\right )\,\sqrt {c+d\,x}}{105\,d^6}\right ) \]
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