Integrand size = 22, antiderivative size = 315 \[ \int x^2 (a+b x)^{3/2} \sqrt {c+d x} \, dx=-\frac {(b c-a d)^2 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^4}+\frac {(b c-a d) \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^3 d^3}+\frac {\left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{48 b^3 d^2}-\frac {(7 b c+5 a d) (a+b x)^{5/2} (c+d x)^{3/2}}{40 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}+\frac {(b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{7/2} d^{9/2}} \]
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Time = 0.21 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {92, 81, 52, 65, 223, 212} \[ \int x^2 (a+b x)^{3/2} \sqrt {c+d x} \, dx=\frac {\left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) (b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{7/2} d^{9/2}}+\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right )}{48 b^3 d^2}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) (b c-a d)^2}{128 b^3 d^4}+\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) (b c-a d)}{192 b^3 d^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2} (5 a d+7 b c)}{40 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d} \]
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Rule 52
Rule 65
Rule 81
Rule 92
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}+\frac {\int (a+b x)^{3/2} \sqrt {c+d x} \left (-a c-\frac {1}{2} (7 b c+5 a d) x\right ) \, dx}{5 b d} \\ & = -\frac {(7 b c+5 a d) (a+b x)^{5/2} (c+d x)^{3/2}}{40 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}+\frac {\left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \int (a+b x)^{3/2} \sqrt {c+d x} \, dx}{16 b^2 d^2} \\ & = \frac {\left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{48 b^3 d^2}-\frac {(7 b c+5 a d) (a+b x)^{5/2} (c+d x)^{3/2}}{40 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}+\frac {\left ((b c-a d) \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{96 b^3 d^2} \\ & = \frac {(b c-a d) \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^3 d^3}+\frac {\left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{48 b^3 d^2}-\frac {(7 b c+5 a d) (a+b x)^{5/2} (c+d x)^{3/2}}{40 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}-\frac {\left ((b c-a d)^2 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{128 b^3 d^3} \\ & = -\frac {(b c-a d)^2 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^4}+\frac {(b c-a d) \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^3 d^3}+\frac {\left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{48 b^3 d^2}-\frac {(7 b c+5 a d) (a+b x)^{5/2} (c+d x)^{3/2}}{40 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}+\frac {\left ((b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 b^3 d^4} \\ & = -\frac {(b c-a d)^2 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^4}+\frac {(b c-a d) \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^3 d^3}+\frac {\left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{48 b^3 d^2}-\frac {(7 b c+5 a d) (a+b x)^{5/2} (c+d x)^{3/2}}{40 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}+\frac {\left ((b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{128 b^4 d^4} \\ & = -\frac {(b c-a d)^2 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^4}+\frac {(b c-a d) \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^3 d^3}+\frac {\left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{48 b^3 d^2}-\frac {(7 b c+5 a d) (a+b x)^{5/2} (c+d x)^{3/2}}{40 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}+\frac {\left ((b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{128 b^4 d^4} \\ & = -\frac {(b c-a d)^2 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^4}+\frac {(b c-a d) \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^3 d^3}+\frac {\left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{48 b^3 d^2}-\frac {(7 b c+5 a d) (a+b x)^{5/2} (c+d x)^{3/2}}{40 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}+\frac {(b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{7/2} d^{9/2}} \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.77 \[ \int x^2 (a+b x)^{3/2} \sqrt {c+d x} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (45 a^4 d^4-30 a^3 b d^3 (c+d x)+6 a^2 b^2 d^2 \left (-6 c^2+3 c d x+4 d^2 x^2\right )+2 a b^3 d \left (95 c^3-61 c^2 d x+48 c d^2 x^2+264 d^3 x^3\right )+b^4 \left (-105 c^4+70 c^3 d x-56 c^2 d^2 x^2+48 c d^3 x^3+384 d^4 x^4\right )\right )}{1920 b^3 d^4}+\frac {(b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{128 b^{7/2} d^{9/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(787\) vs. \(2(271)=542\).
Time = 0.54 (sec) , antiderivative size = 788, normalized size of antiderivative = 2.50
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (-768 b^{4} d^{4} x^{4} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-1056 a \,b^{3} d^{4} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-96 b^{4} c \,d^{3} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-48 a^{2} b^{2} d^{4} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-192 a \,b^{3} c \,d^{3} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+112 b^{4} c^{2} d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+45 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{5} d^{5}-45 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{4} b c \,d^{4}-30 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} b^{2} c^{2} d^{3}-90 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{3} c^{3} d^{2}+225 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{4} c^{4} d -105 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{5} c^{5}+60 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{3} b \,d^{4} x -36 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b^{2} c \,d^{3} x +244 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{3} c^{2} d^{2} x -140 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{4} c^{3} d x -90 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{4} d^{4}+60 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{3} b c \,d^{3}+72 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b^{2} c^{2} d^{2}-380 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{3} c^{3} d +210 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{4} c^{4}\right )}{3840 b^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, d^{4} \sqrt {b d}}\) | \(788\) |
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Time = 0.27 (sec) , antiderivative size = 704, normalized size of antiderivative = 2.23 \[ \int x^2 (a+b x)^{3/2} \sqrt {c+d x} \, dx=\left [-\frac {15 \, {\left (7 \, b^{5} c^{5} - 15 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} + 3 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (384 \, b^{5} d^{5} x^{4} - 105 \, b^{5} c^{4} d + 190 \, a b^{4} c^{3} d^{2} - 36 \, a^{2} b^{3} c^{2} d^{3} - 30 \, a^{3} b^{2} c d^{4} + 45 \, a^{4} b d^{5} + 48 \, {\left (b^{5} c d^{4} + 11 \, a b^{4} d^{5}\right )} x^{3} - 8 \, {\left (7 \, b^{5} c^{2} d^{3} - 12 \, a b^{4} c d^{4} - 3 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (35 \, b^{5} c^{3} d^{2} - 61 \, a b^{4} c^{2} d^{3} + 9 \, a^{2} b^{3} c d^{4} - 15 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, b^{4} d^{5}}, -\frac {15 \, {\left (7 \, b^{5} c^{5} - 15 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} + 3 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (384 \, b^{5} d^{5} x^{4} - 105 \, b^{5} c^{4} d + 190 \, a b^{4} c^{3} d^{2} - 36 \, a^{2} b^{3} c^{2} d^{3} - 30 \, a^{3} b^{2} c d^{4} + 45 \, a^{4} b d^{5} + 48 \, {\left (b^{5} c d^{4} + 11 \, a b^{4} d^{5}\right )} x^{3} - 8 \, {\left (7 \, b^{5} c^{2} d^{3} - 12 \, a b^{4} c d^{4} - 3 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (35 \, b^{5} c^{3} d^{2} - 61 \, a b^{4} c^{2} d^{3} + 9 \, a^{2} b^{3} c d^{4} - 15 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, b^{4} d^{5}}\right ] \]
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\[ \int x^2 (a+b x)^{3/2} \sqrt {c+d x} \, dx=\int x^{2} \left (a + b x\right )^{\frac {3}{2}} \sqrt {c + d x}\, dx \]
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Exception generated. \[ \int x^2 (a+b x)^{3/2} \sqrt {c+d x} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 878 vs. \(2 (271) = 542\).
Time = 0.40 (sec) , antiderivative size = 878, normalized size of antiderivative = 2.79 \[ \int x^2 (a+b x)^{3/2} \sqrt {c+d x} \, dx=\frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (4 \, {\left (b x + a\right )} {\left (6 \, {\left (b x + a\right )} {\left (\frac {8 \, {\left (b x + a\right )}}{b^{4}} + \frac {b^{20} c d^{7} - 41 \, a b^{19} d^{8}}{b^{23} d^{8}}\right )} - \frac {7 \, b^{21} c^{2} d^{6} + 26 \, a b^{20} c d^{7} - 513 \, a^{2} b^{19} d^{8}}{b^{23} d^{8}}\right )} + \frac {5 \, {\left (7 \, b^{22} c^{3} d^{5} + 19 \, a b^{21} c^{2} d^{6} + 37 \, a^{2} b^{20} c d^{7} - 447 \, a^{3} b^{19} d^{8}\right )}}{b^{23} d^{8}}\right )} {\left (b x + a\right )} - \frac {15 \, {\left (7 \, b^{23} c^{4} d^{4} + 12 \, a b^{22} c^{3} d^{5} + 18 \, a^{2} b^{21} c^{2} d^{6} + 28 \, a^{3} b^{20} c d^{7} - 193 \, a^{4} b^{19} d^{8}\right )}}{b^{23} d^{8}}\right )} \sqrt {b x + a} - \frac {15 \, {\left (7 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} + 10 \, a^{3} b^{2} c^{2} d^{3} + 35 \, a^{4} b c d^{4} - 63 \, a^{5} d^{5}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{3} d^{4}}\right )} {\left | b \right |} + \frac {80 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{2}} + \frac {b^{6} c d^{3} - 13 \, a b^{5} d^{4}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{7} c^{2} d^{2} + 2 \, a b^{6} c d^{3} - 11 \, a^{2} b^{5} d^{4}\right )}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b d^{2}}\right )} a^{2} {\left | b \right |}}{b^{2}} + \frac {20 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )}}{b^{3}} + \frac {b^{12} c d^{5} - 25 \, a b^{11} d^{6}}{b^{14} d^{6}}\right )} - \frac {5 \, b^{13} c^{2} d^{4} + 14 \, a b^{12} c d^{5} - 163 \, a^{2} b^{11} d^{6}}{b^{14} d^{6}}\right )} + \frac {3 \, {\left (5 \, b^{14} c^{3} d^{3} + 9 \, a b^{13} c^{2} d^{4} + 15 \, a^{2} b^{12} c d^{5} - 93 \, a^{3} b^{11} d^{6}\right )}}{b^{14} d^{6}}\right )} \sqrt {b x + a} + \frac {3 \, {\left (5 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 35 \, a^{4} d^{4}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{2} d^{3}}\right )} a {\left | b \right |}}{b}}{1920 \, b} \]
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Timed out. \[ \int x^2 (a+b x)^{3/2} \sqrt {c+d x} \, dx=\int x^2\,{\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x} \,d x \]
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