Integrand size = 22, antiderivative size = 122 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^3} \, dx=-\frac {(b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a c x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 c x^2}+\frac {(b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {96, 95, 214} \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^3} \, dx=\frac {(b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{3/2}}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 c x^2}-\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-a d)}{4 a c x} \]
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Rule 95
Rule 96
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 c x^2}+\frac {(b c-a d) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx}{4 c} \\ & = -\frac {(b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a c x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 c x^2}-\frac {(b c-a d)^2 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 a c} \\ & = -\frac {(b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a c x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 c x^2}-\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 a c} \\ & = -\frac {(b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a c x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 c x^2}+\frac {(b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{3/2}} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^3} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x} (2 a c+b c x+a d x)}{4 a c x^2}+\frac {(b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{4 a^{3/2} c^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(255\) vs. \(2(96)=192\).
Time = 0.54 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.10
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} d^{2} x^{2}-2 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b c d \,x^{2}+\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{2} x^{2}-2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a d x -2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c x -4 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a c \sqrt {a c}\right )}{8 a c \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{2} \sqrt {a c}}\) | \(256\) |
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none
Time = 0.29 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.67 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^3} \, dx=\left [\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a c} x^{2} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (2 \, a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, a^{2} c^{2} x^{2}}, -\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-a c} x^{2} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (2 \, a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, a^{2} c^{2} x^{2}}\right ] \]
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\[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^3} \, dx=\int \frac {\sqrt {a + b x} \sqrt {c + d x}}{x^{3}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^3} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1081 vs. \(2 (96) = 192\).
Time = 0.67 (sec) , antiderivative size = 1081, normalized size of antiderivative = 8.86 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^3} \, dx=\frac {\frac {{\left (\sqrt {b d} b^{3} c^{2} {\left | b \right |} - 2 \, \sqrt {b d} a b^{2} c d {\left | b \right |} + \sqrt {b d} a^{2} b d^{2} {\left | b \right |}\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 b c} - \frac {2 \, {\left (\sqrt {b d} b^{9} c^{5} {\left | b \right |} - 3 \, \sqrt {b d} a b^{8} c^{4} d {\left | b \right |} + 2 \, \sqrt {b d} a^{2} b^{7} c^{3} d^{2} {\left | b \right |} + 2 \, \sqrt {b d} a^{3} b^{6} c^{2} d^{3} {\left | b \right |} - 3 \, \sqrt {b d} a^{4} b^{5} c d^{4} {\left | b \right |} + \sqrt {b d} a^{5} b^{4} d^{5} {\left | b \right |} - 3 \, \sqrt {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} b^{7} c^{4} {\left | b \right |} + 4 \, \sqrt {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} a b^{6} c^{3} d {\left | b \right |} - 2 \, \sqrt {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} a^{2} b^{5} c^{2} d^{2} {\left | b \right |} + 4 \, \sqrt {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} a^{3} b^{4} c d^{3} {\left | b \right |} - 3 \, \sqrt {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} a^{4} b^{3} d^{4} {\left | b \right |} + 3 \, \sqrt {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} b^{5} c^{3} {\left | b \right |} + 5 \, \sqrt {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} a b^{4} c^{2} d {\left | b \right |} + 5 \, \sqrt {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} a^{2} b^{3} c d^{2} {\left | b \right |} + 3 \, \sqrt {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} a^{3} b^{2} d^{3} {\left | b \right |} - \sqrt {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 )}^{6} b^{3} c^{2} {\left | b \right |} - 6 \, \sqrt {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 )}^{6} a b^{2} c d {\left | b \right |} - \sqrt {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 )}^{6} a^{2} b d^{2} {\left | b \right |}\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 c}}{4 \, b} \]
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Time = 21.35 (sec) , antiderivative size = 869, normalized size of antiderivative = 7.12 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^3} \, dx=\frac {\frac {b^4}{32\,\sqrt {a}\,\sqrt {c}\,d^2}-\frac {\left (\frac {c\,b^2}{16}+\frac {a\,d\,b}{16}\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5}{a\,c\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2\,\left (\frac {5\,a^2\,b^2\,d^2}{32}+\frac {3\,a\,b^3\,c\,d}{8}+\frac {5\,b^4\,c^2}{32}\right )}{a^{3/2}\,c^{3/2}\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (\frac {a^3\,b\,d^3}{16}+\frac {3\,a^2\,b^2\,c\,d^2}{8}+\frac {3\,a\,b^3\,c^2\,d}{8}+\frac {b^4\,c^3}{16}\right )}{a^2\,c^2\,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\,c\,d^2\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4\,\left (-\frac {a^4\,d^4}{32}+\frac {a^3\,b\,c\,d^3}{4}+\frac {3\,a^2\,b^2\,c^2\,d^2}{32}+\frac {a\,b^3\,c^3\,d}{4}-\frac {b^4\,c^4}{32}\right )}{a^{5/2}\,c^{5/2}\,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 {\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 (\sqrt {a}\,b^2\,c^{5/2}+a^{5/2}\,\sqrt {c}\,d^2-2\,a^{3/2}\,b\,c^{3/2}\,d\right )}{8\,a^2\,c^2}+\frac {\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {c+d\,x}-\sqrt {c}}\right )\,\left (\sqrt {a}\,b^2\,c^{5/2}+a^{5/2}\,\sqrt {c}\,d^2-2\,a^{3/2}\,b\,c^{3/2}\,d\right )}{8\,a^2\,c^2}+\frac {d^2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{32\,\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}-\frac {d\,\left (a\,d+b\,c\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{16\,a\,c\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )} \]
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