Integrand size = 22, antiderivative size = 148 \[ \int \frac {\sqrt {a+b x}}{x^2 (c+d x)^{5/2}} \, dx=-\frac {5 d \sqrt {a+b x}}{3 c^2 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{c x (c+d x)^{3/2}}-\frac {d (13 b c-15 a d) \sqrt {a+b x}}{3 c^3 (b c-a d) \sqrt {c+d x}}-\frac {(b c-5 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{7/2}} \]
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Time = 0.09 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {101, 157, 12, 95, 214} \[ \int \frac {\sqrt {a+b x}}{x^2 (c+d x)^{5/2}} \, dx=-\frac {(b c-5 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{7/2}}-\frac {d \sqrt {a+b x} (13 b c-15 a d)}{3 c^3 \sqrt {c+d x} (b c-a d)}-\frac {5 d \sqrt {a+b x}}{3 c^2 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{c x (c+d x)^{3/2}} \]
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Rule 12
Rule 95
Rule 101
Rule 157
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a+b x}}{c x (c+d x)^{3/2}}+\frac {\int \frac {\frac {1}{2} (b c-5 a d)-2 b d x}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{c} \\ & = -\frac {5 d \sqrt {a+b x}}{3 c^2 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{c x (c+d x)^{3/2}}-\frac {2 \int \frac {-\frac {3}{4} (b c-5 a d) (b c-a d)+\frac {5}{2} b d (b c-a d) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 c^2 (b c-a d)} \\ & = -\frac {5 d \sqrt {a+b x}}{3 c^2 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{c x (c+d x)^{3/2}}-\frac {d (13 b c-15 a d) \sqrt {a+b x}}{3 c^3 (b c-a d) \sqrt {c+d x}}+\frac {4 \int \frac {3 (b c-5 a d) (b c-a d)^2}{8 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 c^3 (b c-a d)^2} \\ & = -\frac {5 d \sqrt {a+b x}}{3 c^2 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{c x (c+d x)^{3/2}}-\frac {d (13 b c-15 a d) \sqrt {a+b x}}{3 c^3 (b c-a d) \sqrt {c+d x}}+\frac {(b c-5 a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 c^3} \\ & = -\frac {5 d \sqrt {a+b x}}{3 c^2 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{c x (c+d x)^{3/2}}-\frac {d (13 b c-15 a d) \sqrt {a+b x}}{3 c^3 (b c-a d) \sqrt {c+d x}}+\frac {(b c-5 a d) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{c^3} \\ & = -\frac {5 d \sqrt {a+b x}}{3 c^2 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{c x (c+d x)^{3/2}}-\frac {d (13 b c-15 a d) \sqrt {a+b x}}{3 c^3 (b c-a d) \sqrt {c+d x}}-\frac {(b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{7/2}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {a+b x}}{x^2 (c+d x)^{5/2}} \, dx=\frac {\sqrt {a+b x} \left (-b c \left (3 c^2+18 c d x+13 d^2 x^2\right )+a d \left (3 c^2+20 c d x+15 d^2 x^2\right )\right )}{3 c^3 (b c-a d) x (c+d x)^{3/2}}+\frac {(-b c+5 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{7/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(652\) vs. \(2(120)=240\).
Time = 0.54 (sec) , antiderivative size = 653, normalized size of antiderivative = 4.41
method | result | size |
default | \(\frac {\left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} d^{4} x^{3}-18 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b c \,d^{3} x^{3}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{2} d^{2} x^{3}+30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} c \,d^{3} x^{2}-36 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b \,c^{2} d^{2} x^{2}+6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{3} d \,x^{2}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} c^{2} d^{2} x -18 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b \,c^{3} d x +3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{4} x -30 a \,d^{3} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}+26 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c \,d^{2} x^{2}-40 a c \,d^{2} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}+36 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b \,c^{2} d x -6 a \,c^{2} d \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}+6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b \,c^{3}\right ) \sqrt {b x +a}}{6 c^{3} \left (a d -b c \right ) \sqrt {a c}\, x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \left (d x +c \right )^{\frac {3}{2}}}\) | \(653\) |
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Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (120) = 240\).
Time = 0.58 (sec) , antiderivative size = 670, normalized size of antiderivative = 4.53 \[ \int \frac {\sqrt {a+b x}}{x^2 (c+d x)^{5/2}} \, dx=\left [-\frac {3 \, {\left ({\left (b^{2} c^{2} d^{2} - 6 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} x^{3} + 2 \, {\left (b^{2} c^{3} d - 6 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{2} + {\left (b^{2} c^{4} - 6 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (3 \, a b c^{4} - 3 \, a^{2} c^{3} d + {\left (13 \, a b c^{2} d^{2} - 15 \, a^{2} c d^{3}\right )} x^{2} + 2 \, {\left (9 \, a b c^{3} d - 10 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{12 \, {\left ({\left (a b c^{5} d^{2} - a^{2} c^{4} d^{3}\right )} x^{3} + 2 \, {\left (a b c^{6} d - a^{2} c^{5} d^{2}\right )} x^{2} + {\left (a b c^{7} - a^{2} c^{6} d\right )} x\right )}}, \frac {3 \, {\left ({\left (b^{2} c^{2} d^{2} - 6 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} x^{3} + 2 \, {\left (b^{2} c^{3} d - 6 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{2} + {\left (b^{2} c^{4} - 6 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (3 \, a b c^{4} - 3 \, a^{2} c^{3} d + {\left (13 \, a b c^{2} d^{2} - 15 \, a^{2} c d^{3}\right )} x^{2} + 2 \, {\left (9 \, a b c^{3} d - 10 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left ({\left (a b c^{5} d^{2} - a^{2} c^{4} d^{3}\right )} x^{3} + 2 \, {\left (a b c^{6} d - a^{2} c^{5} d^{2}\right )} x^{2} + {\left (a b c^{7} - a^{2} c^{6} d\right )} x\right )}}\right ] \]
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\[ \int \frac {\sqrt {a+b x}}{x^2 (c+d x)^{5/2}} \, dx=\int \frac {\sqrt {a + b x}}{x^{2} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {a+b x}}{x^2 (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 571 vs. \(2 (120) = 240\).
Time = 1.12 (sec) , antiderivative size = 571, normalized size of antiderivative = 3.86 \[ \int \frac {\sqrt {a+b x}}{x^2 (c+d x)^{5/2}} \, dx=-\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (5 \, b^{4} c^{4} d^{3} {\left | b \right |} - 6 \, a b^{3} c^{3} d^{4} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{3} c^{7} d - a b^{2} c^{6} d^{2}} + \frac {6 \, {\left (b^{5} c^{5} d^{2} {\left | b \right |} - 2 \, a b^{4} c^{4} d^{3} {\left | b \right |} + a^{2} b^{3} c^{3} d^{4} {\left | b \right |}\right )}}{b^{3} c^{7} d - a b^{2} c^{6} d^{2}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {{\left (\sqrt {b d} b^{3} c - 5 \, \sqrt {b d} a b^{2} d\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b c^{3} {\left | b \right |}} - \frac {2 \, {\left (\sqrt {b d} b^{5} c^{2} - 2 \, \sqrt {b d} a b^{4} c d + \sqrt {b d} a^{2} b^{3} d^{2} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{3} c - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{2} d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} c^{3} {\left | b \right |}} \]
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Timed out. \[ \int \frac {\sqrt {a+b x}}{x^2 (c+d x)^{5/2}} \, dx=\int \frac {\sqrt {a+b\,x}}{x^2\,{\left (c+d\,x\right )}^{5/2}} \,d x \]
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