Integrand size = 17, antiderivative size = 146 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^4} \, dx=-\frac {\sqrt {c+d x}}{3 b (a+b x)^3}-\frac {d \sqrt {c+d x}}{12 b (b c-a d) (a+b x)^2}+\frac {d^2 \sqrt {c+d x}}{8 b (b c-a d)^2 (a+b x)}-\frac {d^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 b^{3/2} (b c-a d)^{5/2}} \]
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Time = 0.07 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {43, 44, 65, 214} \[ \int \frac {\sqrt {c+d x}}{(a+b x)^4} \, dx=-\frac {d^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 b^{3/2} (b c-a d)^{5/2}}+\frac {d^2 \sqrt {c+d x}}{8 b (a+b x) (b c-a d)^2}-\frac {d \sqrt {c+d x}}{12 b (a+b x)^2 (b c-a d)}-\frac {\sqrt {c+d x}}{3 b (a+b x)^3} \]
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Rule 43
Rule 44
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {c+d x}}{3 b (a+b x)^3}+\frac {d \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx}{6 b} \\ & = -\frac {\sqrt {c+d x}}{3 b (a+b x)^3}-\frac {d \sqrt {c+d x}}{12 b (b c-a d) (a+b x)^2}-\frac {d^2 \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx}{8 b (b c-a d)} \\ & = -\frac {\sqrt {c+d x}}{3 b (a+b x)^3}-\frac {d \sqrt {c+d x}}{12 b (b c-a d) (a+b x)^2}+\frac {d^2 \sqrt {c+d x}}{8 b (b c-a d)^2 (a+b x)}+\frac {d^3 \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{16 b (b c-a d)^2} \\ & = -\frac {\sqrt {c+d x}}{3 b (a+b x)^3}-\frac {d \sqrt {c+d x}}{12 b (b c-a d) (a+b x)^2}+\frac {d^2 \sqrt {c+d x}}{8 b (b c-a d)^2 (a+b x)}+\frac {d^2 \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 (b c-a d)^2} \\ & = -\frac {\sqrt {c+d x}}{3 b (a+b x)^3}-\frac {d \sqrt {c+d x}}{12 b (b c-a d) (a+b x)^2}+\frac {d^2 \sqrt {c+d x}}{8 b (b c-a d)^2 (a+b x)}-\frac {d^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 b^{3/2} (b c-a d)^{5/2}} \\ \end{align*}
Time = 0.57 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^4} \, dx=\frac {\sqrt {c+d x} \left (-3 a^2 d^2+2 a b d (7 c+4 d x)+b^2 \left (-8 c^2-2 c d x+3 d^2 x^2\right )\right )}{24 b (b c-a d)^2 (a+b x)^3}+\frac {d^3 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{8 b^{3/2} (-b c+a d)^{5/2}} \]
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Time = 0.53 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.81
method | result | size |
pseudoelliptic | \(\frac {d^{3} \left (b x +a \right )^{3} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )-\left (\frac {\left (d x -2 c \right ) b}{3}+a d \right ) \sqrt {d x +c}\, \left (\left (-3 d x -4 c \right ) b +a d \right ) \sqrt {\left (a d -b c \right ) b}}{8 \sqrt {\left (a d -b c \right ) b}\, \left (b x +a \right )^{3} \left (a d -b c \right )^{2} b}\) | \(118\) |
derivativedivides | \(2 d^{3} \left (\frac {\frac {b \left (d x +c \right )^{\frac {5}{2}}}{16 a^{2} d^{2}-32 a b c d +16 b^{2} c^{2}}+\frac {\left (d x +c \right )^{\frac {3}{2}}}{6 a d -6 b c}-\frac {\sqrt {d x +c}}{16 b}}{\left (\left (d x +c \right ) b +a d -b c \right )^{3}}+\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{16 b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {\left (a d -b c \right ) b}}\right )\) | \(152\) |
default | \(2 d^{3} \left (\frac {\frac {b \left (d x +c \right )^{\frac {5}{2}}}{16 a^{2} d^{2}-32 a b c d +16 b^{2} c^{2}}+\frac {\left (d x +c \right )^{\frac {3}{2}}}{6 a d -6 b c}-\frac {\sqrt {d x +c}}{16 b}}{\left (\left (d x +c \right ) b +a d -b c \right )^{3}}+\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{16 b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {\left (a d -b c \right ) b}}\right )\) | \(152\) |
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Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (122) = 244\).
Time = 0.24 (sec) , antiderivative size = 785, normalized size of antiderivative = 5.38 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^4} \, dx=\left [\frac {3 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) - 2 \, {\left (8 \, b^{4} c^{3} - 22 \, a b^{3} c^{2} d + 17 \, a^{2} b^{2} c d^{2} - 3 \, a^{3} b d^{3} - 3 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{2} + 2 \, {\left (b^{4} c^{2} d - 5 \, a b^{3} c d^{2} + 4 \, a^{2} b^{2} d^{3}\right )} x\right )} \sqrt {d x + c}}{48 \, {\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3} + {\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} x^{3} + 3 \, {\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} x^{2} + 3 \, {\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} x\right )}}, \frac {3 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) - {\left (8 \, b^{4} c^{3} - 22 \, a b^{3} c^{2} d + 17 \, a^{2} b^{2} c d^{2} - 3 \, a^{3} b d^{3} - 3 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{2} + 2 \, {\left (b^{4} c^{2} d - 5 \, a b^{3} c d^{2} + 4 \, a^{2} b^{2} d^{3}\right )} x\right )} \sqrt {d x + c}}{24 \, {\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3} + {\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} x^{3} + 3 \, {\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} x^{2} + 3 \, {\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} x\right )}}\right ] \]
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Timed out. \[ \int \frac {\sqrt {c+d x}}{(a+b x)^4} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\sqrt {c+d x}}{(a+b x)^4} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.30 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.42 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^4} \, dx=\frac {d^{3} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{8 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \sqrt {-b^{2} c + a b d}} + \frac {3 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d^{3} - 8 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c d^{3} - 3 \, \sqrt {d x + c} b^{2} c^{2} d^{3} + 8 \, {\left (d x + c\right )}^{\frac {3}{2}} a b d^{4} + 6 \, \sqrt {d x + c} a b c d^{4} - 3 \, \sqrt {d x + c} a^{2} d^{5}}{24 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{3}} \]
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Time = 0.59 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.42 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^4} \, dx=\frac {\frac {d^3\,{\left (c+d\,x\right )}^{3/2}}{3\,\left (a\,d-b\,c\right )}-\frac {d^3\,\sqrt {c+d\,x}}{8\,b}+\frac {b\,d^3\,{\left (c+d\,x\right )}^{5/2}}{8\,{\left (a\,d-b\,c\right )}^2}}{\left (c+d\,x\right )\,\left (3\,a^2\,b\,d^2-6\,a\,b^2\,c\,d+3\,b^3\,c^2\right )+b^3\,{\left (c+d\,x\right )}^3-\left (3\,b^3\,c-3\,a\,b^2\,d\right )\,{\left (c+d\,x\right )}^2+a^3\,d^3-b^3\,c^3+3\,a\,b^2\,c^2\,d-3\,a^2\,b\,c\,d^2}+\frac {d^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{8\,b^{3/2}\,{\left (a\,d-b\,c\right )}^{5/2}} \]
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