Integrand size = 17, antiderivative size = 126 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^4} \, dx=-\frac {5 d^2 \sqrt {c+d x}}{8 b^3 (a+b x)}-\frac {5 d (c+d x)^{3/2}}{12 b^2 (a+b x)^2}-\frac {(c+d x)^{5/2}}{3 b (a+b x)^3}-\frac {5 d^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 b^{7/2} \sqrt {b c-a d}} \]
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Time = 0.03 (sec) , antiderivative size = 126, 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 = {43, 65, 214} \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^4} \, dx=-\frac {5 d^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 b^{7/2} \sqrt {b c-a d}}-\frac {5 d^2 \sqrt {c+d x}}{8 b^3 (a+b x)}-\frac {5 d (c+d x)^{3/2}}{12 b^2 (a+b x)^2}-\frac {(c+d x)^{5/2}}{3 b (a+b x)^3} \]
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Rule 43
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x)^{5/2}}{3 b (a+b x)^3}+\frac {(5 d) \int \frac {(c+d x)^{3/2}}{(a+b x)^3} \, dx}{6 b} \\ & = -\frac {5 d (c+d x)^{3/2}}{12 b^2 (a+b x)^2}-\frac {(c+d x)^{5/2}}{3 b (a+b x)^3}+\frac {\left (5 d^2\right ) \int \frac {\sqrt {c+d x}}{(a+b x)^2} \, dx}{8 b^2} \\ & = -\frac {5 d^2 \sqrt {c+d x}}{8 b^3 (a+b x)}-\frac {5 d (c+d x)^{3/2}}{12 b^2 (a+b x)^2}-\frac {(c+d x)^{5/2}}{3 b (a+b x)^3}+\frac {\left (5 d^3\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{16 b^3} \\ & = -\frac {5 d^2 \sqrt {c+d x}}{8 b^3 (a+b x)}-\frac {5 d (c+d x)^{3/2}}{12 b^2 (a+b x)^2}-\frac {(c+d x)^{5/2}}{3 b (a+b x)^3}+\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^3} \\ & = -\frac {5 d^2 \sqrt {c+d x}}{8 b^3 (a+b x)}-\frac {5 d (c+d x)^{3/2}}{12 b^2 (a+b x)^2}-\frac {(c+d x)^{5/2}}{3 b (a+b x)^3}-\frac {5 d^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 b^{7/2} \sqrt {b c-a d}} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.94 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^4} \, dx=-\frac {\sqrt {c+d x} \left (15 a^2 d^2+10 a b d (c+4 d x)+b^2 \left (8 c^2+26 c d x+33 d^2 x^2\right )\right )}{24 b^3 (a+b x)^3}+\frac {5 d^3 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{8 b^{7/2} \sqrt {-b c+a d}} \]
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Time = 0.58 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.97
method | result | size |
pseudoelliptic | \(\frac {\frac {5 d^{3} \left (b x +a \right )^{3} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8}-\frac {5 \left (\left (\frac {11}{5} d^{2} x^{2}+\frac {26}{15} c d x +\frac {8}{15} c^{2}\right ) b^{2}+\frac {2 a d \left (4 d x +c \right ) b}{3}+a^{2} d^{2}\right ) \sqrt {d x +c}\, \sqrt {\left (a d -b c \right ) b}}{8}}{b^{3} \left (b x +a \right )^{3} \sqrt {\left (a d -b c \right ) b}}\) | \(122\) |
derivativedivides | \(2 d^{3} \left (\frac {-\frac {11 \left (d x +c \right )^{\frac {5}{2}}}{16 b}-\frac {5 \left (a d -b c \right ) \left (d x +c \right )^{\frac {3}{2}}}{6 b^{2}}-\frac {5 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {d x +c}}{16 b^{3}}}{\left (\left (d x +c \right ) b +a d -b c \right )^{3}}+\frac {5 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{16 b^{3} \sqrt {\left (a d -b c \right ) b}}\right )\) | \(130\) |
default | \(2 d^{3} \left (\frac {-\frac {11 \left (d x +c \right )^{\frac {5}{2}}}{16 b}-\frac {5 \left (a d -b c \right ) \left (d x +c \right )^{\frac {3}{2}}}{6 b^{2}}-\frac {5 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {d x +c}}{16 b^{3}}}{\left (\left (d x +c \right ) b +a d -b c \right )^{3}}+\frac {5 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{16 b^{3} \sqrt {\left (a d -b c \right ) b}}\right )\) | \(130\) |
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Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (102) = 204\).
Time = 0.24 (sec) , antiderivative size = 563, normalized size of antiderivative = 4.47 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^4} \, dx=\left [\frac {15 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) - 2 \, {\left (8 \, b^{4} c^{3} + 2 \, a b^{3} c^{2} d + 5 \, a^{2} b^{2} c d^{2} - 15 \, a^{3} b d^{3} + 33 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{2} + 2 \, {\left (13 \, b^{4} c^{2} d + 7 \, a b^{3} c d^{2} - 20 \, a^{2} b^{2} d^{3}\right )} x\right )} \sqrt {d x + c}}{48 \, {\left (a^{3} b^{5} c - a^{4} b^{4} d + {\left (b^{8} c - a b^{7} d\right )} x^{3} + 3 \, {\left (a b^{7} c - a^{2} b^{6} d\right )} x^{2} + 3 \, {\left (a^{2} b^{6} c - a^{3} b^{5} d\right )} x\right )}}, \frac {15 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) - {\left (8 \, b^{4} c^{3} + 2 \, a b^{3} c^{2} d + 5 \, a^{2} b^{2} c d^{2} - 15 \, a^{3} b d^{3} + 33 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{2} + 2 \, {\left (13 \, b^{4} c^{2} d + 7 \, a b^{3} c d^{2} - 20 \, a^{2} b^{2} d^{3}\right )} x\right )} \sqrt {d x + c}}{24 \, {\left (a^{3} b^{5} c - a^{4} b^{4} d + {\left (b^{8} c - a b^{7} d\right )} x^{3} + 3 \, {\left (a b^{7} c - a^{2} b^{6} d\right )} x^{2} + 3 \, {\left (a^{2} b^{6} c - a^{3} b^{5} d\right )} x\right )}}\right ] \]
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Timed out. \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^4} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^4} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.30 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.28 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^4} \, dx=\frac {5 \, d^{3} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{8 \, \sqrt {-b^{2} c + a b d} b^{3}} - \frac {33 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d^{3} - 40 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c d^{3} + 15 \, \sqrt {d x + c} b^{2} c^{2} d^{3} + 40 \, {\left (d x + c\right )}^{\frac {3}{2}} a b d^{4} - 30 \, \sqrt {d x + c} a b c d^{4} + 15 \, \sqrt {d x + c} a^{2} d^{5}}{24 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}^{3} b^{3}} \]
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Time = 0.39 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.76 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^4} \, dx=\frac {5\,d^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{8\,b^{7/2}\,\sqrt {a\,d-b\,c}}-\frac {\frac {11\,d^3\,{\left (c+d\,x\right )}^{5/2}}{8\,b}+\frac {5\,d^3\,\sqrt {c+d\,x}\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{8\,b^3}+\frac {5\,d^3\,\left (a\,d-b\,c\right )\,{\left (c+d\,x\right )}^{3/2}}{3\,b^2}}{\left (c+d\,x\right )\,\left (3\,a^2\,b\,d^2-6\,a\,b^2\,c\,d+3\,b^3\,c^2\right )+b^3\,{\left (c+d\,x\right )}^3-\left (3\,b^3\,c-3\,a\,b^2\,d\right )\,{\left (c+d\,x\right )}^2+a^3\,d^3-b^3\,c^3+3\,a\,b^2\,c^2\,d-3\,a^2\,b\,c\,d^2} \]
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