Integrand size = 17, antiderivative size = 182 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx=-\frac {\sqrt {c+d x}}{4 b (a+b x)^4}-\frac {d \sqrt {c+d x}}{24 b (b c-a d) (a+b x)^3}+\frac {5 d^2 \sqrt {c+d x}}{96 b (b c-a d)^2 (a+b x)^2}-\frac {5 d^3 \sqrt {c+d x}}{64 b (b c-a d)^3 (a+b x)}+\frac {5 d^4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{64 b^{3/2} (b c-a d)^{7/2}} \]
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Time = 0.10 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {43, 44, 65, 214} \[ \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx=\frac {5 d^4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{64 b^{3/2} (b c-a d)^{7/2}}-\frac {5 d^3 \sqrt {c+d x}}{64 b (a+b x) (b c-a d)^3}+\frac {5 d^2 \sqrt {c+d x}}{96 b (a+b x)^2 (b c-a d)^2}-\frac {d \sqrt {c+d x}}{24 b (a+b x)^3 (b c-a d)}-\frac {\sqrt {c+d x}}{4 b (a+b x)^4} \]
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Rule 43
Rule 44
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {c+d x}}{4 b (a+b x)^4}+\frac {d \int \frac {1}{(a+b x)^4 \sqrt {c+d x}} \, dx}{8 b} \\ & = -\frac {\sqrt {c+d x}}{4 b (a+b x)^4}-\frac {d \sqrt {c+d x}}{24 b (b c-a d) (a+b x)^3}-\frac {\left (5 d^2\right ) \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx}{48 b (b c-a d)} \\ & = -\frac {\sqrt {c+d x}}{4 b (a+b x)^4}-\frac {d \sqrt {c+d x}}{24 b (b c-a d) (a+b x)^3}+\frac {5 d^2 \sqrt {c+d x}}{96 b (b c-a d)^2 (a+b x)^2}+\frac {\left (5 d^3\right ) \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx}{64 b (b c-a d)^2} \\ & = -\frac {\sqrt {c+d x}}{4 b (a+b x)^4}-\frac {d \sqrt {c+d x}}{24 b (b c-a d) (a+b x)^3}+\frac {5 d^2 \sqrt {c+d x}}{96 b (b c-a d)^2 (a+b x)^2}-\frac {5 d^3 \sqrt {c+d x}}{64 b (b c-a d)^3 (a+b x)}-\frac {\left (5 d^4\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{128 b (b c-a d)^3} \\ & = -\frac {\sqrt {c+d x}}{4 b (a+b x)^4}-\frac {d \sqrt {c+d x}}{24 b (b c-a d) (a+b x)^3}+\frac {5 d^2 \sqrt {c+d x}}{96 b (b c-a d)^2 (a+b x)^2}-\frac {5 d^3 \sqrt {c+d x}}{64 b (b c-a d)^3 (a+b x)}-\frac {\left (5 d^3\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{64 b (b c-a d)^3} \\ & = -\frac {\sqrt {c+d x}}{4 b (a+b x)^4}-\frac {d \sqrt {c+d x}}{24 b (b c-a d) (a+b x)^3}+\frac {5 d^2 \sqrt {c+d x}}{96 b (b c-a d)^2 (a+b x)^2}-\frac {5 d^3 \sqrt {c+d x}}{64 b (b c-a d)^3 (a+b x)}+\frac {5 d^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{64 b^{3/2} (b c-a d)^{7/2}} \\ \end{align*}
Time = 0.92 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx=\frac {\sqrt {c+d x} \left (-15 a^3 d^3+a^2 b d^2 (118 c+73 d x)+a b^2 d \left (-136 c^2-36 c d x+55 d^2 x^2\right )+b^3 \left (48 c^3+8 c^2 d x-10 c d^2 x^2+15 d^3 x^3\right )\right )}{192 b (-b c+a d)^3 (a+b x)^4}+\frac {5 d^4 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{64 b^{3/2} (-b c+a d)^{7/2}} \]
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Time = 0.33 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.93
method | result | size |
pseudoelliptic | \(\frac {-\frac {5 \sqrt {\left (a d -b c \right ) b}\, \left (\left (-b^{3} x^{3}-\frac {11}{3} a \,b^{2} x^{2}-\frac {73}{15} a^{2} b x +a^{3}\right ) d^{3}-\frac {118 b \left (-\frac {5}{59} b^{2} x^{2}-\frac {18}{59} a b x +a^{2}\right ) c \,d^{2}}{15}+\frac {136 b^{2} \left (-\frac {b x}{17}+a \right ) c^{2} d}{15}-\frac {16 b^{3} c^{3}}{5}\right ) \sqrt {d x +c}}{64}+\frac {5 d^{4} \left (b x +a \right )^{4} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{64}}{\sqrt {\left (a d -b c \right ) b}\, \left (b x +a \right )^{4} b \left (a d -b c \right )^{3}}\) | \(170\) |
derivativedivides | \(2 d^{4} \left (\frac {\frac {5 b^{2} \left (d x +c \right )^{\frac {7}{2}}}{128 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {55 b \left (d x +c \right )^{\frac {5}{2}}}{384 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {73 \left (d x +c \right )^{\frac {3}{2}}}{384 \left (a d -b c \right )}-\frac {5 \sqrt {d x +c}}{128 b}}{\left (\left (d x +c \right ) b +a d -b c \right )^{4}}+\frac {5 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{128 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {\left (a d -b c \right ) b}}\right )\) | \(217\) |
default | \(2 d^{4} \left (\frac {\frac {5 b^{2} \left (d x +c \right )^{\frac {7}{2}}}{128 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {55 b \left (d x +c \right )^{\frac {5}{2}}}{384 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {73 \left (d x +c \right )^{\frac {3}{2}}}{384 \left (a d -b c \right )}-\frac {5 \sqrt {d x +c}}{128 b}}{\left (\left (d x +c \right ) b +a d -b c \right )^{4}}+\frac {5 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{128 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {\left (a d -b c \right ) b}}\right )\) | \(217\) |
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Leaf count of result is larger than twice the leaf count of optimal. 581 vs. \(2 (154) = 308\).
Time = 0.25 (sec) , antiderivative size = 1176, normalized size of antiderivative = 6.46 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx=\left [-\frac {15 \, {\left (b^{4} d^{4} x^{4} + 4 \, a b^{3} d^{4} x^{3} + 6 \, a^{2} b^{2} d^{4} x^{2} + 4 \, a^{3} b d^{4} x + a^{4} d^{4}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) + 2 \, {\left (48 \, b^{5} c^{4} - 184 \, a b^{4} c^{3} d + 254 \, a^{2} b^{3} c^{2} d^{2} - 133 \, a^{3} b^{2} c d^{3} + 15 \, a^{4} b d^{4} + 15 \, {\left (b^{5} c d^{3} - a b^{4} d^{4}\right )} x^{3} - 5 \, {\left (2 \, b^{5} c^{2} d^{2} - 13 \, a b^{4} c d^{3} + 11 \, a^{2} b^{3} d^{4}\right )} x^{2} + {\left (8 \, b^{5} c^{3} d - 44 \, a b^{4} c^{2} d^{2} + 109 \, a^{2} b^{3} c d^{3} - 73 \, a^{3} b^{2} d^{4}\right )} x\right )} \sqrt {d x + c}}{384 \, {\left (a^{4} b^{6} c^{4} - 4 \, a^{5} b^{5} c^{3} d + 6 \, a^{6} b^{4} c^{2} d^{2} - 4 \, a^{7} b^{3} c d^{3} + a^{8} b^{2} d^{4} + {\left (b^{10} c^{4} - 4 \, a b^{9} c^{3} d + 6 \, a^{2} b^{8} c^{2} d^{2} - 4 \, a^{3} b^{7} c d^{3} + a^{4} b^{6} d^{4}\right )} x^{4} + 4 \, {\left (a b^{9} c^{4} - 4 \, a^{2} b^{8} c^{3} d + 6 \, a^{3} b^{7} c^{2} d^{2} - 4 \, a^{4} b^{6} c d^{3} + a^{5} b^{5} d^{4}\right )} x^{3} + 6 \, {\left (a^{2} b^{8} c^{4} - 4 \, a^{3} b^{7} c^{3} d + 6 \, a^{4} b^{6} c^{2} d^{2} - 4 \, a^{5} b^{5} c d^{3} + a^{6} b^{4} d^{4}\right )} x^{2} + 4 \, {\left (a^{3} b^{7} c^{4} - 4 \, a^{4} b^{6} c^{3} d + 6 \, a^{5} b^{5} c^{2} d^{2} - 4 \, a^{6} b^{4} c d^{3} + a^{7} b^{3} d^{4}\right )} x\right )}}, -\frac {15 \, {\left (b^{4} d^{4} x^{4} + 4 \, a b^{3} d^{4} x^{3} + 6 \, a^{2} b^{2} d^{4} x^{2} + 4 \, a^{3} b d^{4} x + a^{4} d^{4}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) + {\left (48 \, b^{5} c^{4} - 184 \, a b^{4} c^{3} d + 254 \, a^{2} b^{3} c^{2} d^{2} - 133 \, a^{3} b^{2} c d^{3} + 15 \, a^{4} b d^{4} + 15 \, {\left (b^{5} c d^{3} - a b^{4} d^{4}\right )} x^{3} - 5 \, {\left (2 \, b^{5} c^{2} d^{2} - 13 \, a b^{4} c d^{3} + 11 \, a^{2} b^{3} d^{4}\right )} x^{2} + {\left (8 \, b^{5} c^{3} d - 44 \, a b^{4} c^{2} d^{2} + 109 \, a^{2} b^{3} c d^{3} - 73 \, a^{3} b^{2} d^{4}\right )} x\right )} \sqrt {d x + c}}{192 \, {\left (a^{4} b^{6} c^{4} - 4 \, a^{5} b^{5} c^{3} d + 6 \, a^{6} b^{4} c^{2} d^{2} - 4 \, a^{7} b^{3} c d^{3} + a^{8} b^{2} d^{4} + {\left (b^{10} c^{4} - 4 \, a b^{9} c^{3} d + 6 \, a^{2} b^{8} c^{2} d^{2} - 4 \, a^{3} b^{7} c d^{3} + a^{4} b^{6} d^{4}\right )} x^{4} + 4 \, {\left (a b^{9} c^{4} - 4 \, a^{2} b^{8} c^{3} d + 6 \, a^{3} b^{7} c^{2} d^{2} - 4 \, a^{4} b^{6} c d^{3} + a^{5} b^{5} d^{4}\right )} x^{3} + 6 \, {\left (a^{2} b^{8} c^{4} - 4 \, a^{3} b^{7} c^{3} d + 6 \, a^{4} b^{6} c^{2} d^{2} - 4 \, a^{5} b^{5} c d^{3} + a^{6} b^{4} d^{4}\right )} x^{2} + 4 \, {\left (a^{3} b^{7} c^{4} - 4 \, a^{4} b^{6} c^{3} d + 6 \, a^{5} b^{5} c^{2} d^{2} - 4 \, a^{6} b^{4} c d^{3} + a^{7} b^{3} d^{4}\right )} x\right )}}\right ] \]
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Timed out. \[ \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (154) = 308\).
Time = 0.31 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.71 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx=-\frac {5 \, d^{4} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{64 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \sqrt {-b^{2} c + a b d}} - \frac {15 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{3} d^{4} - 55 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{3} c d^{4} + 73 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} c^{2} d^{4} + 15 \, \sqrt {d x + c} b^{3} c^{3} d^{4} + 55 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{2} d^{5} - 146 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{2} c d^{5} - 45 \, \sqrt {d x + c} a b^{2} c^{2} d^{5} + 73 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b d^{6} + 45 \, \sqrt {d x + c} a^{2} b c d^{6} - 15 \, \sqrt {d x + c} a^{3} d^{7}}{192 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{4}} \]
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Time = 0.23 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.63 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx=\frac {\frac {73\,d^4\,{\left (c+d\,x\right )}^{3/2}}{192\,\left (a\,d-b\,c\right )}-\frac {5\,d^4\,\sqrt {c+d\,x}}{64\,b}+\frac {5\,b^2\,d^4\,{\left (c+d\,x\right )}^{7/2}}{64\,{\left (a\,d-b\,c\right )}^3}+\frac {55\,b\,d^4\,{\left (c+d\,x\right )}^{5/2}}{192\,{\left (a\,d-b\,c\right )}^2}}{b^4\,{\left (c+d\,x\right )}^4-\left (4\,b^4\,c-4\,a\,b^3\,d\right )\,{\left (c+d\,x\right )}^3-\left (c+d\,x\right )\,\left (-4\,a^3\,b\,d^3+12\,a^2\,b^2\,c\,d^2-12\,a\,b^3\,c^2\,d+4\,b^4\,c^3\right )+a^4\,d^4+b^4\,c^4+{\left (c+d\,x\right )}^2\,\left (6\,a^2\,b^2\,d^2-12\,a\,b^3\,c\,d+6\,b^4\,c^2\right )+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d-4\,a^3\,b\,c\,d^3}+\frac {5\,d^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{64\,b^{3/2}\,{\left (a\,d-b\,c\right )}^{7/2}} \]
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