Integrand size = 22, antiderivative size = 237 \[ \int x^2 \sqrt {a+b x} \sqrt {c+d x} \, dx=-\frac {(b c-a d) \left (4 a b c d-5 (b c+a d)^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^3 d^3}-\frac {\left (4 a b c d-5 (b c+a d)^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{32 b^3 d^2}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{3/2}}{4 b d}+\frac {(b c-a d)^2 \left (4 a b c d-5 (b c+a d)^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{7/2} d^{7/2}} \]
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Time = 0.14 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {92, 81, 52, 65, 223, 212} \[ \int x^2 \sqrt {a+b x} \sqrt {c+d x} \, dx=\frac {\left (4 a b c d-5 (a d+b c)^2\right ) (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{7/2} d^{7/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (4 a b c d-5 (a d+b c)^2\right ) (b c-a d)}{64 b^3 d^3}-\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (4 a b c d-5 (a d+b c)^2\right )}{32 b^3 d^2}-\frac {5 (a+b x)^{3/2} (c+d x)^{3/2} (a d+b c)}{24 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{3/2}}{4 b d} \]
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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)^{3/2} (c+d x)^{3/2}}{4 b d}+\frac {\int \sqrt {a+b x} \sqrt {c+d x} \left (-a c-\frac {5}{2} (b c+a d) x\right ) \, dx}{4 b d} \\ & = -\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{3/2}}{4 b d}-\frac {\left (4 a b c d-5 (b c+a d)^2\right ) \int \sqrt {a+b x} \sqrt {c+d x} \, dx}{16 b^2 d^2} \\ & = -\frac {\left (4 a b c d-5 (b c+a d)^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{32 b^3 d^2}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{3/2}}{4 b d}-\frac {\left ((b c-a d) \left (4 a b c d-5 (b c+a d)^2\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{64 b^3 d^2} \\ & = -\frac {(b c-a d) \left (4 a b c d-5 (b c+a d)^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^3 d^3}-\frac {\left (4 a b c d-5 (b c+a d)^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{32 b^3 d^2}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{3/2}}{4 b d}+\frac {\left ((b c-a d)^2 \left (4 a b c d-5 (b c+a d)^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 b^3 d^3} \\ & = -\frac {(b c-a d) \left (4 a b c d-5 (b c+a d)^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^3 d^3}-\frac {\left (4 a b c d-5 (b c+a d)^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{32 b^3 d^2}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{3/2}}{4 b d}+\frac {\left ((b c-a d)^2 \left (4 a b c d-5 (b c+a 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 )}{64 b^4 d^3} \\ & = -\frac {(b c-a d) \left (4 a b c d-5 (b c+a d)^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^3 d^3}-\frac {\left (4 a b c d-5 (b c+a d)^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{32 b^3 d^2}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{3/2}}{4 b d}+\frac {\left ((b c-a d)^2 \left (4 a b c d-5 (b c+a 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 )}{64 b^4 d^3} \\ & = -\frac {(b c-a d) \left (4 a b c d-5 (b c+a d)^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^3 d^3}-\frac {\left (4 a b c d-5 (b c+a d)^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{32 b^3 d^2}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{3/2}}{4 b d}+\frac {(b c-a d)^2 \left (4 a b c d-5 (b c+a d)^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{7/2} d^{7/2}} \\ \end{align*}
Time = 0.63 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.81 \[ \int x^2 \sqrt {a+b x} \sqrt {c+d x} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (15 a^3 d^3-a^2 b d^2 (7 c+10 d x)+a b^2 d \left (-7 c^2+4 c d x+8 d^2 x^2\right )+b^3 \left (15 c^3-10 c^2 d x+8 c d^2 x^2+48 d^3 x^3\right )\right )}{192 b^3 d^3}-\frac {(b c-a d)^2 \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{64 b^{7/2} d^{7/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(573\) vs. \(2(199)=398\).
Time = 0.53 (sec) , antiderivative size = 574, normalized size of antiderivative = 2.42
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (-96 b^{3} d^{3} x^{3} \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-16 a \,b^{2} d^{3} x^{2} \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-16 b^{3} c \,d^{2} x^{2} \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{4} d^{4}-12 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} b c \,d^{3}-6 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{2} c^{2} d^{2}-12 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{3} c^{3} d +15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{4} c^{4}+20 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b \,d^{3} x -8 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{2} c \,d^{2} x +20 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{3} c^{2} d x -30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{3} d^{3}+14 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b c \,d^{2}+14 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{2} c^{2} d -30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{3} c^{3}\right )}{384 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{3} d^{3} \sqrt {b d}}\) | \(574\) |
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Time = 0.25 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.29 \[ \int x^2 \sqrt {a+b x} \sqrt {c+d x} \, dx=\left [\frac {3 \, {\left (5 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + 5 \, a^{4} d^{4}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (48 \, b^{4} d^{4} x^{3} + 15 \, b^{4} c^{3} d - 7 \, a b^{3} c^{2} d^{2} - 7 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} + 8 \, {\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{2} - 2 \, {\left (5 \, b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + 5 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, b^{4} d^{4}}, \frac {3 \, {\left (5 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + 5 \, a^{4} d^{4}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (48 \, b^{4} d^{4} x^{3} + 15 \, b^{4} c^{3} d - 7 \, a b^{3} c^{2} d^{2} - 7 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} + 8 \, {\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{2} - 2 \, {\left (5 \, b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + 5 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, b^{4} d^{4}}\right ] \]
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\[ \int x^2 \sqrt {a+b x} \sqrt {c+d x} \, dx=\int x^{2} \sqrt {a + b x} \sqrt {c + d x}\, dx \]
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Exception generated. \[ \int x^2 \sqrt {a+b x} \sqrt {c+d x} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (199) = 398\).
Time = 0.36 (sec) , antiderivative size = 500, normalized size of antiderivative = 2.11 \[ \int x^2 \sqrt {a+b x} \sqrt {c+d x} \, dx=\frac {\frac {8 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{2}} + \frac {b^{6} c d^{3} - 13 \, a b^{5} d^{4}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{7} c^{2} d^{2} + 2 \, a b^{6} c d^{3} - 11 \, a^{2} b^{5} d^{4}\right )}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b d^{2}}\right )} a {\left | b \right |}}{b^{2}} + \frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )}}{b^{3}} + \frac {b^{12} c d^{5} - 25 \, a b^{11} d^{6}}{b^{14} d^{6}}\right )} - \frac {5 \, b^{13} c^{2} d^{4} + 14 \, a b^{12} c d^{5} - 163 \, a^{2} b^{11} d^{6}}{b^{14} d^{6}}\right )} + \frac {3 \, {\left (5 \, b^{14} c^{3} d^{3} + 9 \, a b^{13} c^{2} d^{4} + 15 \, a^{2} b^{12} c d^{5} - 93 \, a^{3} b^{11} d^{6}\right )}}{b^{14} d^{6}}\right )} \sqrt {b x + a} + \frac {3 \, {\left (5 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 35 \, a^{4} d^{4}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{2} d^{3}}\right )} {\left | b \right |}}{b}}{192 \, b} \]
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Timed out. \[ \int x^2 \sqrt {a+b x} \sqrt {c+d x} \, dx=\text {Hanged} \]
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