Integrand size = 22, antiderivative size = 234 \[ \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{5/2}} \, dx=-\frac {d (3 b c-35 a d) \sqrt {a+b x}}{12 a c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(b c-7 a d) \sqrt {a+b x}}{4 a c^2 x (c+d x)^{3/2}}-\frac {d \left (3 b^2 c^2-100 a b c d+105 a^2 d^2\right ) \sqrt {a+b x}}{12 a c^4 (b c-a d) \sqrt {c+d x}}+\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{9/2}} \]
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Time = 0.16 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {101, 156, 157, 12, 95, 214} \[ \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{5/2}} \, dx=-\frac {d \sqrt {a+b x} \left (105 a^2 d^2-100 a b c d+3 b^2 c^2\right )}{12 a c^4 \sqrt {c+d x} (b c-a d)}+\frac {\left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{9/2}}-\frac {d \sqrt {a+b x} (3 b c-35 a d)}{12 a c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x} (b c-7 a d)}{4 a c^2 x (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}} \]
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Rule 12
Rule 95
Rule 101
Rule 156
Rule 157
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}+\frac {\int \frac {\frac {1}{2} (b c-7 a d)-3 b d x}{x^2 \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{2 c} \\ & = -\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(b c-7 a d) \sqrt {a+b x}}{4 a c^2 x (c+d x)^{3/2}}-\frac {\int \frac {\frac {1}{4} \left (b^2 c^2+10 a b c d-35 a^2 d^2\right )+b d (b c-7 a d) x}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{2 a c^2} \\ & = -\frac {d (3 b c-35 a d) \sqrt {a+b x}}{12 a c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(b c-7 a d) \sqrt {a+b x}}{4 a c^2 x (c+d x)^{3/2}}+\frac {\int \frac {-\frac {3}{8} (b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right )-\frac {1}{4} b d (3 b c-35 a d) (b c-a d) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 a c^3 (b c-a d)} \\ & = -\frac {d (3 b c-35 a d) \sqrt {a+b x}}{12 a c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(b c-7 a d) \sqrt {a+b x}}{4 a c^2 x (c+d x)^{3/2}}-\frac {d \left (3 b^2 c^2-100 a b c d+105 a^2 d^2\right ) \sqrt {a+b x}}{12 a c^4 (b c-a d) \sqrt {c+d x}}-\frac {2 \int \frac {3 (b c-a d)^2 \left (b^2 c^2+10 a b c d-35 a^2 d^2\right )}{16 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 a c^4 (b c-a d)^2} \\ & = -\frac {d (3 b c-35 a d) \sqrt {a+b x}}{12 a c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(b c-7 a d) \sqrt {a+b x}}{4 a c^2 x (c+d x)^{3/2}}-\frac {d \left (3 b^2 c^2-100 a b c d+105 a^2 d^2\right ) \sqrt {a+b x}}{12 a c^4 (b c-a d) \sqrt {c+d x}}-\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 a c^4} \\ & = -\frac {d (3 b c-35 a d) \sqrt {a+b x}}{12 a c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(b c-7 a d) \sqrt {a+b x}}{4 a c^2 x (c+d x)^{3/2}}-\frac {d \left (3 b^2 c^2-100 a b c d+105 a^2 d^2\right ) \sqrt {a+b x}}{12 a c^4 (b c-a d) \sqrt {c+d x}}-\frac {\left (b^2 c^2+10 a b c d-35 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 c^4} \\ & = -\frac {d (3 b c-35 a d) \sqrt {a+b x}}{12 a c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(b c-7 a d) \sqrt {a+b x}}{4 a c^2 x (c+d x)^{3/2}}-\frac {d \left (3 b^2 c^2-100 a b c d+105 a^2 d^2\right ) \sqrt {a+b x}}{12 a c^4 (b c-a d) \sqrt {c+d x}}+\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{9/2}} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{5/2}} \, dx=-\frac {\sqrt {a+b x} \left (3 b^2 c^2 x (c+d x)^2+2 a b c \left (3 c^3-12 c^2 d x-69 c d^2 x^2-50 d^3 x^3\right )+a^2 d \left (-6 c^3+21 c^2 d x+140 c d^2 x^2+105 d^3 x^3\right )\right )}{12 a c^4 (b c-a d) x^2 (c+d x)^{3/2}}+\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{9/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(987\) vs. \(2(196)=392\).
Time = 0.54 (sec) , antiderivative size = 988, normalized size of antiderivative = 4.22
method | result | size |
default | \(-\frac {\left (105 \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^{3} d^{5} x^{4}-135 \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} b c \,d^{4} x^{4}+27 \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^{2} c^{2} d^{3} x^{4}+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^{3} c^{3} d^{2} x^{4}+210 \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^{3} c \,d^{4} x^{3}-270 \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} b \,c^{2} d^{3} x^{3}+54 \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^{2} c^{3} d^{2} x^{3}+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^{3} c^{4} d \,x^{3}+105 \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^{3} c^{2} d^{3} x^{2}-135 \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} b \,c^{3} d^{2} x^{2}+27 \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^{2} c^{4} 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^{3} c^{5} x^{2}-210 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{4} x^{3}+200 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c \,d^{3} x^{3}-6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} d^{2} x^{3}-280 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c \,d^{3} x^{2}+276 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{2} d^{2} x^{2}-12 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{3} d \,x^{2}-42 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c^{2} d^{2} x +48 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{3} d x -6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{4} x +12 a^{2} c^{3} d \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}-12 a b \,c^{4} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\right ) \sqrt {b x +a}}{24 c^{4} a \left (a d -b c \right ) \sqrt {a c}\, x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \left (d x +c \right )^{\frac {3}{2}}}\) | \(988\) |
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Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (196) = 392\).
Time = 1.39 (sec) , antiderivative size = 904, normalized size of antiderivative = 3.86 \[ \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{5/2}} \, dx=\left [-\frac {3 \, {\left ({\left (b^{3} c^{3} d^{2} + 9 \, a b^{2} c^{2} d^{3} - 45 \, a^{2} b c d^{4} + 35 \, a^{3} d^{5}\right )} x^{4} + 2 \, {\left (b^{3} c^{4} d + 9 \, a b^{2} c^{3} d^{2} - 45 \, a^{2} b c^{2} d^{3} + 35 \, a^{3} c d^{4}\right )} x^{3} + {\left (b^{3} c^{5} + 9 \, a b^{2} c^{4} d - 45 \, a^{2} b c^{3} d^{2} + 35 \, a^{3} c^{2} d^{3}\right )} x^{2}\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 (6 \, a^{2} b c^{5} - 6 \, a^{3} c^{4} d + {\left (3 \, a b^{2} c^{3} d^{2} - 100 \, a^{2} b c^{2} d^{3} + 105 \, a^{3} c d^{4}\right )} x^{3} + 2 \, {\left (3 \, a b^{2} c^{4} d - 69 \, a^{2} b c^{3} d^{2} + 70 \, a^{3} c^{2} d^{3}\right )} x^{2} + 3 \, {\left (a b^{2} c^{5} - 8 \, a^{2} b c^{4} d + 7 \, a^{3} c^{3} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left ({\left (a^{2} b c^{6} d^{2} - a^{3} c^{5} d^{3}\right )} x^{4} + 2 \, {\left (a^{2} b c^{7} d - a^{3} c^{6} d^{2}\right )} x^{3} + {\left (a^{2} b c^{8} - a^{3} c^{7} d\right )} x^{2}\right )}}, -\frac {3 \, {\left ({\left (b^{3} c^{3} d^{2} + 9 \, a b^{2} c^{2} d^{3} - 45 \, a^{2} b c d^{4} + 35 \, a^{3} d^{5}\right )} x^{4} + 2 \, {\left (b^{3} c^{4} d + 9 \, a b^{2} c^{3} d^{2} - 45 \, a^{2} b c^{2} d^{3} + 35 \, a^{3} c d^{4}\right )} x^{3} + {\left (b^{3} c^{5} + 9 \, a b^{2} c^{4} d - 45 \, a^{2} b c^{3} d^{2} + 35 \, a^{3} c^{2} d^{3}\right )} x^{2}\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 (6 \, a^{2} b c^{5} - 6 \, a^{3} c^{4} d + {\left (3 \, a b^{2} c^{3} d^{2} - 100 \, a^{2} b c^{2} d^{3} + 105 \, a^{3} c d^{4}\right )} x^{3} + 2 \, {\left (3 \, a b^{2} c^{4} d - 69 \, a^{2} b c^{3} d^{2} + 70 \, a^{3} c^{2} d^{3}\right )} x^{2} + 3 \, {\left (a b^{2} c^{5} - 8 \, a^{2} b c^{4} d + 7 \, a^{3} c^{3} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{24 \, {\left ({\left (a^{2} b c^{6} d^{2} - a^{3} c^{5} d^{3}\right )} x^{4} + 2 \, {\left (a^{2} b c^{7} d - a^{3} c^{6} d^{2}\right )} x^{3} + {\left (a^{2} b c^{8} - a^{3} c^{7} d\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{5/2}} \, dx=\int \frac {\sqrt {a + b x}}{x^{3} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1204 vs. \(2 (196) = 392\).
Time = 1.43 (sec) , antiderivative size = 1204, normalized size of antiderivative = 5.15 \[ \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{5/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{5/2}} \, dx=\int \frac {\sqrt {a+b\,x}}{x^3\,{\left (c+d\,x\right )}^{5/2}} \,d x \]
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