Integrand size = 22, antiderivative size = 124 \[ \int \frac {1}{x^2 \sqrt {a+b x} (c+d x)^{3/2}} \, dx=-\frac {d (b c-3 a d) \sqrt {a+b x}}{a c^2 (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{a c x \sqrt {c+d x}}+\frac {(b c+3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{5/2}} \]
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Time = 0.06 (sec) , antiderivative size = 124, 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, 157, 12, 95, 214} \[ \int \frac {1}{x^2 \sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {(3 a d+b c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{5/2}}-\frac {d \sqrt {a+b x} (b c-3 a d)}{a c^2 \sqrt {c+d x} (b c-a d)}-\frac {\sqrt {a+b x}}{a c x \sqrt {c+d x}} \]
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Rule 12
Rule 95
Rule 105
Rule 157
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a+b x}}{a c x \sqrt {c+d x}}-\frac {\int \frac {\frac {1}{2} (b c+3 a d)+b d x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{a c} \\ & = -\frac {d (b c-3 a d) \sqrt {a+b x}}{a c^2 (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{a c x \sqrt {c+d x}}+\frac {2 \int -\frac {(b c-a d) (b c+3 a d)}{4 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{a c^2 (b c-a d)} \\ & = -\frac {d (b c-3 a d) \sqrt {a+b x}}{a c^2 (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{a c x \sqrt {c+d x}}-\frac {(b c+3 a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 a c^2} \\ & = -\frac {d (b c-3 a d) \sqrt {a+b x}}{a c^2 (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{a c x \sqrt {c+d x}}-\frac {(b c+3 a d) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{a c^2} \\ & = -\frac {d (b c-3 a d) \sqrt {a+b x}}{a c^2 (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{a c x \sqrt {c+d x}}+\frac {(b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{5/2}} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^2 \sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {\frac {\sqrt {a} \sqrt {c} \sqrt {a+b x} (-b c (c+d x)+a d (c+3 d x))}{x \sqrt {c+d x}}+\left (b^2 c^2+2 a b c d-3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{5/2} (b c-a d)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(440\) vs. \(2(104)=208\).
Time = 1.68 (sec) , antiderivative size = 441, normalized size of antiderivative = 3.56
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} d^{3} x^{2}-2 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b c \,d^{2} x^{2}-\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{2} d \,x^{2}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} c \,d^{2} x -2 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b \,c^{2} d x -\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{3} x -6 a \,d^{2} x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c d x -2 a c d \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b \,c^{2}\right )}{2 a \,c^{2} \left (a d -b c \right ) \sqrt {a c}\, x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {d x +c}}\) | \(441\) |
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Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (104) = 208\).
Time = 0.41 (sec) , antiderivative size = 492, normalized size of antiderivative = 3.97 \[ \int \frac {1}{x^2 \sqrt {a+b x} (c+d x)^{3/2}} \, dx=\left [\frac {{\left ({\left (b^{2} c^{2} d + 2 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{2} + {\left (b^{2} c^{3} + 2 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (a b c^{3} - a^{2} c^{2} d + {\left (a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, {\left ({\left (a^{2} b c^{4} d - a^{3} c^{3} d^{2}\right )} x^{2} + {\left (a^{2} b c^{5} - a^{3} c^{4} d\right )} x\right )}}, -\frac {{\left ({\left (b^{2} c^{2} d + 2 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{2} + {\left (b^{2} c^{3} + 2 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (a b c^{3} - a^{2} c^{2} d + {\left (a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left ({\left (a^{2} b c^{4} d - a^{3} c^{3} d^{2}\right )} x^{2} + {\left (a^{2} b c^{5} - a^{3} c^{4} d\right )} x\right )}}\right ] \]
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\[ \int \frac {1}{x^2 \sqrt {a+b x} (c+d x)^{3/2}} \, dx=\int \frac {1}{x^{2} \sqrt {a + b x} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{x^2 \sqrt {a+b x} (c+d x)^{3/2}} \, dx=\int { \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 476 vs. \(2 (104) = 208\).
Time = 0.68 (sec) , antiderivative size = 476, normalized size of antiderivative = 3.84 \[ \int \frac {1}{x^2 \sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {2 \, \sqrt {b x + a} b^{2} d^{2}}{{\left (b c^{3} {\left | b \right |} - a c^{2} d {\left | b \right |}\right )} \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {{\left (\sqrt {b d} b^{3} c + 3 \, \sqrt {b d} a b^{2} d\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} a b c^{2} {\left | b \right |}} - \frac {2 \, {\left (\sqrt {b d} b^{5} c^{2} - 2 \, \sqrt {b d} a b^{4} c d + \sqrt {b d} a^{2} b^{3} d^{2} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{3} c - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{2} d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} a c^{2} {\left | b \right |}} \]
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Timed out. \[ \int \frac {1}{x^2 \sqrt {a+b x} (c+d x)^{3/2}} \, dx=\int \frac {1}{x^2\,\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{3/2}} \,d x \]
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