Integrand size = 17, antiderivative size = 147 \[ \int \frac {1}{(a+b x)^4 \sqrt {c+d x}} \, dx=-\frac {\sqrt {c+d x}}{3 (b c-a d) (a+b x)^3}+\frac {5 d \sqrt {c+d x}}{12 (b c-a d)^2 (a+b x)^2}-\frac {5 d^2 \sqrt {c+d x}}{8 (b c-a d)^3 (a+b x)}+\frac {5 d^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 \sqrt {b} (b c-a d)^{7/2}} \]
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Time = 0.04 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {44, 65, 214} \[ \int \frac {1}{(a+b x)^4 \sqrt {c+d x}} \, dx=\frac {5 d^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 \sqrt {b} (b c-a d)^{7/2}}-\frac {5 d^2 \sqrt {c+d x}}{8 (a+b x) (b c-a d)^3}+\frac {5 d \sqrt {c+d x}}{12 (a+b x)^2 (b c-a d)^2}-\frac {\sqrt {c+d x}}{3 (a+b x)^3 (b c-a d)} \]
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Rule 44
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {c+d x}}{3 (b c-a d) (a+b x)^3}-\frac {(5 d) \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx}{6 (b c-a d)} \\ & = -\frac {\sqrt {c+d x}}{3 (b c-a d) (a+b x)^3}+\frac {5 d \sqrt {c+d x}}{12 (b c-a d)^2 (a+b x)^2}+\frac {\left (5 d^2\right ) \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx}{8 (b c-a d)^2} \\ & = -\frac {\sqrt {c+d x}}{3 (b c-a d) (a+b x)^3}+\frac {5 d \sqrt {c+d x}}{12 (b c-a d)^2 (a+b x)^2}-\frac {5 d^2 \sqrt {c+d x}}{8 (b c-a d)^3 (a+b x)}-\frac {\left (5 d^3\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{16 (b c-a d)^3} \\ & = -\frac {\sqrt {c+d x}}{3 (b c-a d) (a+b x)^3}+\frac {5 d \sqrt {c+d x}}{12 (b c-a d)^2 (a+b x)^2}-\frac {5 d^2 \sqrt {c+d x}}{8 (b c-a d)^3 (a+b x)}-\frac {\left (5 d^2\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{8 (b c-a d)^3} \\ & = -\frac {\sqrt {c+d x}}{3 (b c-a d) (a+b x)^3}+\frac {5 d \sqrt {c+d x}}{12 (b c-a d)^2 (a+b x)^2}-\frac {5 d^2 \sqrt {c+d x}}{8 (b c-a d)^3 (a+b x)}+\frac {5 d^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 \sqrt {b} (b c-a d)^{7/2}} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(a+b x)^4 \sqrt {c+d x}} \, dx=\frac {\sqrt {c+d x} \left (33 a^2 d^2+2 a b d (-13 c+20 d x)+b^2 \left (8 c^2-10 c d x+15 d^2 x^2\right )\right )}{24 (-b c+a d)^3 (a+b x)^3}+\frac {5 d^3 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{8 \sqrt {b} (-b c+a d)^{7/2}} \]
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Time = 0.79 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(\frac {33 \left (\left (\frac {5}{11} d^{2} x^{2}-\frac {10}{33} c d x +\frac {8}{33} c^{2}\right ) b^{2}-\frac {26 d \left (-\frac {20 d x}{13}+c \right ) a b}{33}+a^{2} d^{2}\right ) \sqrt {\left (a d -b c \right ) b}\, \sqrt {d x +c}+15 d^{3} \left (b x +a \right )^{3} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{24 \sqrt {\left (a d -b c \right ) b}\, \left (a d -b c \right )^{3} \left (b x +a \right )^{3}}\) | \(130\) |
derivativedivides | \(2 d^{3} \left (\frac {\sqrt {d x +c}}{6 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )^{3}}+\frac {\frac {5 \sqrt {d x +c}}{24 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )^{2}}+\frac {5 \left (\frac {3 \sqrt {d x +c}}{8 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {3 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}\right )}{6 \left (a d -b c \right )}}{a d -b c}\right )\) | \(187\) |
default | \(2 d^{3} \left (\frac {\sqrt {d x +c}}{6 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )^{3}}+\frac {\frac {5 \sqrt {d x +c}}{24 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )^{2}}+\frac {5 \left (\frac {3 \sqrt {d x +c}}{8 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {3 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}\right )}{6 \left (a d -b c \right )}}{a d -b c}\right )\) | \(187\) |
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Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (123) = 246\).
Time = 0.25 (sec) , antiderivative size = 884, normalized size of antiderivative = 6.01 \[ \int \frac {1}{(a+b x)^4 \sqrt {c+d x}} \, dx=\left [-\frac {15 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\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 (8 \, b^{4} c^{3} - 34 \, a b^{3} c^{2} d + 59 \, a^{2} b^{2} c d^{2} - 33 \, a^{3} b d^{3} + 15 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{2} - 10 \, {\left (b^{4} c^{2} d - 5 \, a b^{3} c d^{2} + 4 \, a^{2} b^{2} d^{3}\right )} x\right )} \sqrt {d x + c}}{48 \, {\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4} + {\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}\right )} x^{3} + 3 \, {\left (a b^{7} c^{4} - 4 \, a^{2} b^{6} c^{3} d + 6 \, a^{3} b^{5} c^{2} d^{2} - 4 \, a^{4} b^{4} c d^{3} + a^{5} b^{3} d^{4}\right )} x^{2} + 3 \, {\left (a^{2} b^{6} c^{4} - 4 \, a^{3} b^{5} c^{3} d + 6 \, a^{4} b^{4} c^{2} d^{2} - 4 \, a^{5} b^{3} c d^{3} + a^{6} b^{2} d^{4}\right )} x\right )}}, -\frac {15 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\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 (8 \, b^{4} c^{3} - 34 \, a b^{3} c^{2} d + 59 \, a^{2} b^{2} c d^{2} - 33 \, a^{3} b d^{3} + 15 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{2} - 10 \, {\left (b^{4} c^{2} d - 5 \, a b^{3} c d^{2} + 4 \, a^{2} b^{2} d^{3}\right )} x\right )} \sqrt {d x + c}}{24 \, {\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4} + {\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}\right )} x^{3} + 3 \, {\left (a b^{7} c^{4} - 4 \, a^{2} b^{6} c^{3} d + 6 \, a^{3} b^{5} c^{2} d^{2} - 4 \, a^{4} b^{4} c d^{3} + a^{5} b^{3} d^{4}\right )} x^{2} + 3 \, {\left (a^{2} b^{6} c^{4} - 4 \, a^{3} b^{5} c^{3} d + 6 \, a^{4} b^{4} c^{2} d^{2} - 4 \, a^{5} b^{3} c d^{3} + a^{6} b^{2} d^{4}\right )} x\right )}}\right ] \]
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Timed out. \[ \int \frac {1}{(a+b x)^4 \sqrt {c+d x}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {1}{(a+b x)^4 \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.31 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.57 \[ \int \frac {1}{(a+b x)^4 \sqrt {c+d x}} \, dx=-\frac {5 \, d^{3} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{8 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-b^{2} c + a b d}} - \frac {15 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d^{3} - 40 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c d^{3} + 33 \, \sqrt {d x + c} b^{2} c^{2} d^{3} + 40 \, {\left (d x + c\right )}^{\frac {3}{2}} a b d^{4} - 66 \, \sqrt {d x + c} a b c d^{4} + 33 \, \sqrt {d x + c} a^{2} d^{5}}{24 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{3}} \]
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Time = 0.42 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.48 \[ \int \frac {1}{(a+b x)^4 \sqrt {c+d x}} \, dx=\frac {\frac {11\,d^3\,\sqrt {c+d\,x}}{8\,\left (a\,d-b\,c\right )}+\frac {5\,b^2\,d^3\,{\left (c+d\,x\right )}^{5/2}}{8\,{\left (a\,d-b\,c\right )}^3}+\frac {5\,b\,d^3\,{\left (c+d\,x\right )}^{3/2}}{3\,{\left (a\,d-b\,c\right )}^2}}{\left (c+d\,x\right )\,\left (3\,a^2\,b\,d^2-6\,a\,b^2\,c\,d+3\,b^3\,c^2\right )+b^3\,{\left (c+d\,x\right )}^3-\left (3\,b^3\,c-3\,a\,b^2\,d\right )\,{\left (c+d\,x\right )}^2+a^3\,d^3-b^3\,c^3+3\,a\,b^2\,c^2\,d-3\,a^2\,b\,c\,d^2}+\frac {5\,d^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{8\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{7/2}} \]
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