Integrand size = 17, antiderivative size = 180 \[ \int \frac {1}{(a+b x)^5 \sqrt {c+d x}} \, dx=-\frac {\sqrt {c+d x}}{4 (b c-a d) (a+b x)^4}+\frac {7 d \sqrt {c+d x}}{24 (b c-a d)^2 (a+b x)^3}-\frac {35 d^2 \sqrt {c+d x}}{96 (b c-a d)^3 (a+b x)^2}+\frac {35 d^3 \sqrt {c+d x}}{64 (b c-a d)^4 (a+b x)}-\frac {35 d^4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{64 \sqrt {b} (b c-a d)^{9/2}} \]
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Time = 0.05 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {44, 65, 214} \[ \int \frac {1}{(a+b x)^5 \sqrt {c+d x}} \, dx=-\frac {35 d^4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{64 \sqrt {b} (b c-a d)^{9/2}}+\frac {35 d^3 \sqrt {c+d x}}{64 (a+b x) (b c-a d)^4}-\frac {35 d^2 \sqrt {c+d x}}{96 (a+b x)^2 (b c-a d)^3}+\frac {7 d \sqrt {c+d x}}{24 (a+b x)^3 (b c-a d)^2}-\frac {\sqrt {c+d x}}{4 (a+b x)^4 (b c-a d)} \]
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Rule 44
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {c+d x}}{4 (b c-a d) (a+b x)^4}-\frac {(7 d) \int \frac {1}{(a+b x)^4 \sqrt {c+d x}} \, dx}{8 (b c-a d)} \\ & = -\frac {\sqrt {c+d x}}{4 (b c-a d) (a+b x)^4}+\frac {7 d \sqrt {c+d x}}{24 (b c-a d)^2 (a+b x)^3}+\frac {\left (35 d^2\right ) \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx}{48 (b c-a d)^2} \\ & = -\frac {\sqrt {c+d x}}{4 (b c-a d) (a+b x)^4}+\frac {7 d \sqrt {c+d x}}{24 (b c-a d)^2 (a+b x)^3}-\frac {35 d^2 \sqrt {c+d x}}{96 (b c-a d)^3 (a+b x)^2}-\frac {\left (35 d^3\right ) \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx}{64 (b c-a d)^3} \\ & = -\frac {\sqrt {c+d x}}{4 (b c-a d) (a+b x)^4}+\frac {7 d \sqrt {c+d x}}{24 (b c-a d)^2 (a+b x)^3}-\frac {35 d^2 \sqrt {c+d x}}{96 (b c-a d)^3 (a+b x)^2}+\frac {35 d^3 \sqrt {c+d x}}{64 (b c-a d)^4 (a+b x)}+\frac {\left (35 d^4\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{128 (b c-a d)^4} \\ & = -\frac {\sqrt {c+d x}}{4 (b c-a d) (a+b x)^4}+\frac {7 d \sqrt {c+d x}}{24 (b c-a d)^2 (a+b x)^3}-\frac {35 d^2 \sqrt {c+d x}}{96 (b c-a d)^3 (a+b x)^2}+\frac {35 d^3 \sqrt {c+d x}}{64 (b c-a d)^4 (a+b x)}+\frac {\left (35 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 c-a d)^4} \\ & = -\frac {\sqrt {c+d x}}{4 (b c-a d) (a+b x)^4}+\frac {7 d \sqrt {c+d x}}{24 (b c-a d)^2 (a+b x)^3}-\frac {35 d^2 \sqrt {c+d x}}{96 (b c-a d)^3 (a+b x)^2}+\frac {35 d^3 \sqrt {c+d x}}{64 (b c-a d)^4 (a+b x)}-\frac {35 d^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{64 \sqrt {b} (b c-a d)^{9/2}} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(a+b x)^5 \sqrt {c+d x}} \, dx=\frac {1}{192} \left (\frac {\sqrt {c+d x} \left (279 a^3 d^3+a^2 b d^2 (-326 c+511 d x)+a b^2 d \left (200 c^2-252 c d x+385 d^2 x^2\right )+b^3 \left (-48 c^3+56 c^2 d x-70 c d^2 x^2+105 d^3 x^3\right )\right )}{(b c-a d)^4 (a+b x)^4}+\frac {105 d^4 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{\sqrt {b} (-b c+a d)^{9/2}}\right ) \]
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Time = 0.79 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.94
method | result | size |
pseudoelliptic | \(\frac {\frac {35 d^{4} \left (b x +a \right )^{4} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{64}+\frac {93 \sqrt {\left (a d -b c \right ) b}\, \sqrt {d x +c}\, \left (\frac {\left (35 d^{3} x^{3}-\frac {70}{3} c \,d^{2} x^{2}+\frac {56}{3} c^{2} d x -16 c^{3}\right ) b^{3}}{93}+\frac {200 \left (\frac {77}{40} d^{2} x^{2}-\frac {63}{50} c d x +c^{2}\right ) d a \,b^{2}}{279}-\frac {326 d^{2} \left (-\frac {511 d x}{326}+c \right ) a^{2} b}{279}+a^{3} d^{3}\right )}{64}}{\left (a d -b c \right )^{4} \left (b x +a \right )^{4} \sqrt {\left (a d -b c \right ) b}}\) | \(169\) |
derivativedivides | \(2 d^{4} \left (\frac {\sqrt {d x +c}}{8 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )^{4}}+\frac {\frac {7 \sqrt {d x +c}}{48 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )^{3}}+\frac {7 \left (\frac {5 \sqrt {d x +c}}{24 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )^{2}}+\frac {5 \left (\frac {3 \sqrt {d x +c}}{8 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {3 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}\right )}{6 \left (a d -b c \right )}\right )}{8 \left (a d -b c \right )}}{a d -b c}\right )\) | \(236\) |
default | \(2 d^{4} \left (\frac {\sqrt {d x +c}}{8 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )^{4}}+\frac {\frac {7 \sqrt {d x +c}}{48 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )^{3}}+\frac {7 \left (\frac {5 \sqrt {d x +c}}{24 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )^{2}}+\frac {5 \left (\frac {3 \sqrt {d x +c}}{8 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {3 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}\right )}{6 \left (a d -b c \right )}\right )}{8 \left (a d -b c \right )}}{a d -b c}\right )\) | \(236\) |
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Leaf count of result is larger than twice the leaf count of optimal. 656 vs. \(2 (152) = 304\).
Time = 0.25 (sec) , antiderivative size = 1325, normalized size of antiderivative = 7.36 \[ \int \frac {1}{(a+b x)^5 \sqrt {c+d x}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {1}{(a+b x)^5 \sqrt {c+d x}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {1}{(a+b x)^5 \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 331 vs. \(2 (152) = 304\).
Time = 0.30 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.84 \[ \int \frac {1}{(a+b x)^5 \sqrt {c+d x}} \, dx=\frac {35 \, d^{4} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{64 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {-b^{2} c + a b d}} + \frac {105 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{3} d^{4} - 385 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{3} c d^{4} + 511 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} c^{2} d^{4} - 279 \, \sqrt {d x + c} b^{3} c^{3} d^{4} + 385 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{2} d^{5} - 1022 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{2} c d^{5} + 837 \, \sqrt {d x + c} a b^{2} c^{2} d^{5} + 511 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b d^{6} - 837 \, \sqrt {d x + c} a^{2} b c d^{6} + 279 \, \sqrt {d x + c} a^{3} d^{7}}{192 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{4}} \]
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Time = 0.46 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.71 \[ \int \frac {1}{(a+b x)^5 \sqrt {c+d x}} \, dx=\frac {\frac {93\,d^4\,\sqrt {c+d\,x}}{64\,\left (a\,d-b\,c\right )}+\frac {385\,b^2\,d^4\,{\left (c+d\,x\right )}^{5/2}}{192\,{\left (a\,d-b\,c\right )}^3}+\frac {35\,b^3\,d^4\,{\left (c+d\,x\right )}^{7/2}}{64\,{\left (a\,d-b\,c\right )}^4}+\frac {511\,b\,d^4\,{\left (c+d\,x\right )}^{3/2}}{192\,{\left (a\,d-b\,c\right )}^2}}{b^4\,{\left (c+d\,x\right )}^4-\left (4\,b^4\,c-4\,a\,b^3\,d\right )\,{\left (c+d\,x\right )}^3-\left (c+d\,x\right )\,\left (-4\,a^3\,b\,d^3+12\,a^2\,b^2\,c\,d^2-12\,a\,b^3\,c^2\,d+4\,b^4\,c^3\right )+a^4\,d^4+b^4\,c^4+{\left (c+d\,x\right )}^2\,\left (6\,a^2\,b^2\,d^2-12\,a\,b^3\,c\,d+6\,b^4\,c^2\right )+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d-4\,a^3\,b\,c\,d^3}+\frac {35\,d^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{64\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{9/2}} \]
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