Integrand size = 20, antiderivative size = 163 \[ \int x \sqrt {a+b x} \sqrt {c+d x} \, dx=\frac {1}{8} \left (\frac {a^2}{b^2}-\frac {c^2}{d^2}\right ) \sqrt {a+b x} \sqrt {c+d x}-\frac {(b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b d}+\frac {(b c-a d)^2 (b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{5/2} d^{5/2}} \]
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Time = 0.06 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {81, 52, 65, 223, 212} \[ \int x \sqrt {a+b x} \sqrt {c+d x} \, dx=\frac {1}{8} \sqrt {a+b x} \sqrt {c+d x} \left (\frac {a^2}{b^2}-\frac {c^2}{d^2}\right )+\frac {(a d+b c) (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{5/2} d^{5/2}}-\frac {(a+b x)^{3/2} \sqrt {c+d x} (a d+b c)}{4 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b d} \]
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Rule 52
Rule 65
Rule 81
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b d}-\frac {(b c+a d) \int \sqrt {a+b x} \sqrt {c+d x} \, dx}{2 b d} \\ & = -\frac {(b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b d}-\frac {\left (c^2-\frac {a^2 d^2}{b^2}\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{8 d} \\ & = -\frac {\left (c^2-\frac {a^2 d^2}{b^2}\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 d^2}-\frac {(b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b d}+\frac {\left ((b c-a d)^2 (b c+a d)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 b^2 d^2} \\ & = -\frac {\left (c^2-\frac {a^2 d^2}{b^2}\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 d^2}-\frac {(b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b d}+\frac {\left ((b c-a d)^2 (b c+a d)\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 )}{8 b^3 d^2} \\ & = -\frac {\left (c^2-\frac {a^2 d^2}{b^2}\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 d^2}-\frac {(b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b d}+\frac {\left ((b c-a d)^2 (b c+a d)\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 )}{8 b^3 d^2} \\ & = -\frac {\left (c^2-\frac {a^2 d^2}{b^2}\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 d^2}-\frac {(b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b d}+\frac {(b c-a d)^2 (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{5/2} d^{5/2}} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.81 \[ \int x \sqrt {a+b x} \sqrt {c+d x} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-3 a^2 d^2+2 a b d (c+d x)+b^2 \left (-3 c^2+2 c d x+8 d^2 x^2\right )\right )}{24 b^2 d^2}+\frac {(b c-a d)^2 (b c+a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{8 b^{5/2} d^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(394\) vs. \(2(131)=262\).
Time = 0.53 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.42
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (16 b^{2} d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} d^{3}-3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b c \,d^{2}-3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c^{2} d +3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{3}+4 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,d^{2} x +4 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c d x -6 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{2}+4 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c d -6 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2}\right )}{48 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} d^{2} \sqrt {b d}}\) | \(395\) |
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Time = 0.25 (sec) , antiderivative size = 406, normalized size of antiderivative = 2.49 \[ \int x \sqrt {a+b x} \sqrt {c+d x} \, dx=\left [\frac {3 \, {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (8 \, b^{3} d^{3} x^{2} - 3 \, b^{3} c^{2} d + 2 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3} + 2 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, b^{3} d^{3}}, -\frac {3 \, {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (8 \, b^{3} d^{3} x^{2} - 3 \, b^{3} c^{2} d + 2 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3} + 2 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, b^{3} d^{3}}\right ] \]
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\[ \int x \sqrt {a+b x} \sqrt {c+d x} \, dx=\int x \sqrt {a + b x} \sqrt {c + d x}\, dx \]
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Exception generated. \[ \int x \sqrt {a+b x} \sqrt {c+d x} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (131) = 262\).
Time = 0.33 (sec) , antiderivative size = 345, normalized size of antiderivative = 2.12 \[ \int x \sqrt {a+b x} \sqrt {c+d x} \, dx=\frac {\frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{2}} + \frac {b^{6} c d^{3} - 13 \, a b^{5} d^{4}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{7} c^{2} d^{2} + 2 \, a b^{6} c d^{3} - 11 \, a^{2} b^{5} d^{4}\right )}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b d^{2}}\right )} {\left | b \right |}}{b} + \frac {6 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, b x + 2 \, a + \frac {b c d - 5 \, a d^{2}}{d^{2}}\right )} \sqrt {b x + a} + \frac {{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d}\right )} a {\left | b \right |}}{b^{3}}}{24 \, b} \]
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Time = 77.10 (sec) , antiderivative size = 1077, normalized size of antiderivative = 6.61 \[ \int x \sqrt {a+b x} \sqrt {c+d x} \, dx=\frac {\mathrm {atanh}\left (\frac {\sqrt {d}\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^2\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {b}\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )\,\left (a^3\,d^3-a^2\,b\,c\,d^2-a\,b^2\,c^2\,d+b^3\,c^3\right )}\right )\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^2}{4\,b^{5/2}\,d^{5/2}}-\frac {\frac {\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (\frac {a^3\,b^3\,d^3}{4}-\frac {a^2\,b^4\,c\,d^2}{4}-\frac {a\,b^5\,c^2\,d}{4}+\frac {b^6\,c^3}{4}\right )}{d^8\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (\frac {17\,a^3\,b^2\,d^3}{12}+\frac {101\,a^2\,b^3\,c\,d^2}{4}+\frac {101\,a\,b^4\,c^2\,d}{4}+\frac {17\,b^5\,c^3}{12}\right )}{d^7\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (\frac {19\,a^3\,d^3}{2}+\frac {269\,a^2\,b\,c\,d^2}{2}+\frac {269\,a\,b^2\,c^2\,d}{2}+\frac {19\,b^3\,c^3}{2}\right )}{d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^7}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (\frac {19\,a^3\,b\,d^3}{2}+\frac {269\,a^2\,b^2\,c\,d^2}{2}+\frac {269\,a\,b^3\,c^2\,d}{2}+\frac {19\,b^4\,c^3}{2}\right )}{d^6\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}+\frac {8\,a^{3/2}\,c^{3/2}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{10}}{d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{10}}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{11}\,\left (\frac {a^3\,d^3}{4}-\frac {a^2\,b\,c\,d^2}{4}-\frac {a\,b^2\,c^2\,d}{4}+\frac {b^3\,c^3}{4}\right )}{b^2\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{11}}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^9\,\left (\frac {17\,a^3\,d^3}{12}+\frac {101\,a^2\,b\,c\,d^2}{4}+\frac {101\,a\,b^2\,c^2\,d}{4}+\frac {17\,b^3\,c^3}{12}\right )}{b\,d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^9}+\frac {\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8\,\left (32\,a^2\,d^2+96\,a\,b\,c\,d+32\,b^2\,c^2\right )}{d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}+\frac {\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4\,\left (32\,a^2\,b^2\,d^2+96\,a\,b^3\,c\,d+32\,b^4\,c^2\right )}{d^6\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}+\frac {8\,a^{3/2}\,b^4\,c^{3/2}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{d^6\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}+\frac {\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6\,\left (64\,a^2\,b\,d^2+\frac {656\,a\,b^2\,c\,d}{3}+64\,b^3\,c^2\right )}{d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{12}}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{12}}+\frac {b^6}{d^6}-\frac {6\,b^5\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}+\frac {15\,b^4\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}-\frac {20\,b^3\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}+\frac {15\,b^2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}{d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}-\frac {6\,b\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{10}}{d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{10}}} \]
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