Integrand size = 22, antiderivative size = 137 \[ \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{2 a c x^2}+\frac {3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a^2 c^2 x}-\frac {\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2} c^{5/2}} \]
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Time = 0.05 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {105, 156, 12, 95, 214} \[ \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {3 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{4 a^2 c^2 x}-\frac {\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2} c^{5/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{2 a c x^2} \]
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Rule 12
Rule 95
Rule 105
Rule 156
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a+b x} \sqrt {c+d x}}{2 a c x^2}-\frac {\int \frac {\frac {3}{2} (b c+a d)+b d x}{x^2 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 a c} \\ & = -\frac {\sqrt {a+b x} \sqrt {c+d x}}{2 a c x^2}+\frac {3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a^2 c^2 x}+\frac {\int \frac {3 b^2 c^2+2 a b c d+3 a^2 d^2}{4 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 a^2 c^2} \\ & = -\frac {\sqrt {a+b x} \sqrt {c+d x}}{2 a c x^2}+\frac {3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a^2 c^2 x}+\frac {\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 a^2 c^2} \\ & = -\frac {\sqrt {a+b x} \sqrt {c+d x}}{2 a c x^2}+\frac {3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a^2 c^2 x}+\frac {\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 a^2 c^2} \\ & = -\frac {\sqrt {a+b x} \sqrt {c+d x}}{2 a c x^2}+\frac {3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a^2 c^2 x}-\frac {\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2} c^{5/2}} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} (-2 a c+3 b c x+3 a d x)}{4 a^2 c^2 x^2}-\frac {\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{4 a^{5/2} c^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(257\) vs. \(2(111)=222\).
Time = 1.66 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.88
method | result | size |
default | \(-\frac {\left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} d^{2} x^{2}+2 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b c d \,x^{2}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{2} x^{2}-6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a d x -6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c x +4 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a c \sqrt {a c}\right ) \sqrt {d x +c}\, \sqrt {b x +a}}{8 a^{2} c^{2} \sqrt {a c}\, x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}}\) | \(258\) |
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Time = 0.34 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.42 \[ \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx=\left [\frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt {a c} x^{2} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (2 \, a^{2} c^{2} - 3 \, {\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, a^{3} c^{3} x^{2}}, \frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt {-a c} x^{2} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{2} c^{2} - 3 \, {\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, a^{3} c^{3} x^{2}}\right ] \]
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\[ \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx=\int \frac {1}{x^{3} \sqrt {a + b x} \sqrt {c + d x}}\, dx \]
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Exception generated. \[ \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 951 vs. \(2 (111) = 222\).
Time = 0.59 (sec) , antiderivative size = 951, normalized size of antiderivative = 6.94 \[ \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {\sqrt {b d} b^{6} d^{2} {\left (\frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} a^{2} b^{5} c^{2} d^{2}} - \frac {2 \, {\left (3 \, b^{8} c^{5} - 9 \, a b^{7} c^{4} d + 6 \, a^{2} b^{6} c^{3} d^{2} + 6 \, a^{3} b^{5} c^{2} d^{3} - 9 \, a^{4} b^{4} c d^{4} + 3 \, a^{5} b^{3} d^{5} - 9 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{6} c^{4} - 4 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{5} c^{3} d + 26 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{4} c^{2} d^{2} - 4 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{3} c d^{3} - 9 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{4} b^{2} d^{4} + 9 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{4} c^{3} + 15 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{3} c^{2} d + 15 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{2} c d^{2} + 9 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} b d^{3} - 3 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} b^{2} c^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a b c d - 3 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a^{2} d^{2}\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )}^{2} a^{2} b^{4} c^{2} d^{2}}\right )}}{4 \, {\left | b \right |}} \]
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Time = 20.84 (sec) , antiderivative size = 900, normalized size of antiderivative = 6.57 \[ \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {\ln \left (\frac {\left (\sqrt {c}\,\sqrt {a+b\,x}-\sqrt {a}\,\sqrt {c+d\,x}\right )\,\left (b\,\sqrt {c}-\frac {\sqrt {a}\,d\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {c+d\,x}-\sqrt {c}}\right )}{\sqrt {c+d\,x}-\sqrt {c}}\right )\,\left (3\,\sqrt {a}\,b^2\,c^{5/2}+3\,a^{5/2}\,\sqrt {c}\,d^2+2\,a^{3/2}\,b\,c^{3/2}\,d\right )}{8\,a^3\,c^3}-\frac {\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2\,\left (\frac {11\,a^2\,b^2\,d^2}{32}+\frac {5\,a\,b^3\,c\,d}{8}+\frac {11\,b^4\,c^2}{32}\right )}{a^{5/2}\,c^{5/2}\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}-\frac {b^4}{32\,a^{3/2}\,c^{3/2}\,d^2}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (\frac {a^3\,b\,d^3}{16}-\frac {9\,a^2\,b^2\,c\,d^2}{8}-\frac {9\,a\,b^3\,c^2\,d}{8}+\frac {b^4\,c^3}{16}\right )}{a^3\,c^3\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}-\frac {\left (\frac {c\,b^4}{8}+\frac {a\,d\,b^3}{8}\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{a^2\,c^2\,d^2\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (\frac {a^3\,d^3}{4}-\frac {7\,a^2\,b\,c\,d^2}{16}-\frac {7\,a\,b^2\,c^2\,d}{16}+\frac {b^3\,c^3}{4}\right )}{a^3\,c^3\,d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4\,\left (-\frac {7\,a^4\,d^4}{32}+\frac {a^3\,b\,c\,d^3}{4}+\frac {45\,a^2\,b^2\,c^2\,d^2}{32}+\frac {a\,b^3\,c^3\,d}{4}-\frac {7\,b^4\,c^4}{32}\right )}{a^{7/2}\,c^{7/2}\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}+\frac {b^2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )}{a\,c\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}-\frac {\left (2\,c\,b^2+2\,a\,d\,b\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{\sqrt {a}\,\sqrt {c}\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}-\frac {\left (2\,a\,d+2\,b\,c\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5}{\sqrt {a}\,\sqrt {c}\,d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}}-\frac {\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {c+d\,x}-\sqrt {c}}\right )\,\left (3\,\sqrt {a}\,b^2\,c^{5/2}+3\,a^{5/2}\,\sqrt {c}\,d^2+2\,a^{3/2}\,b\,c^{3/2}\,d\right )}{8\,a^3\,c^3}+\frac {d^2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{32\,a^{3/2}\,c^{3/2}\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}+\frac {3\,d\,\left (a\,d+b\,c\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{16\,a^2\,c^2\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )} \]
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