Integrand size = 20, antiderivative size = 207 \[ \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)} \, dx=\frac {c (2 b c-3 a d) \sqrt {c+d x}}{4 a^2 x^2}-\frac {\left (8 b^2 c^2-18 a b c d+11 a^2 d^2\right ) \sqrt {c+d x}}{8 a^3 x}-\frac {c (c+d x)^{3/2}}{3 a x^3}+\frac {\left (16 b^3 c^3-40 a b^2 c^2 d+30 a^2 b c d^2-5 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{8 a^4 \sqrt {c}}-\frac {2 \sqrt {b} (b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^4} \]
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Time = 0.18 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {100, 154, 156, 162, 65, 214} \[ \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)} \, dx=-\frac {2 \sqrt {b} (b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^4}+\frac {c \sqrt {c+d x} (2 b c-3 a d)}{4 a^2 x^2}-\frac {\sqrt {c+d x} \left (11 a^2 d^2-18 a b c d+8 b^2 c^2\right )}{8 a^3 x}+\frac {\left (-5 a^3 d^3+30 a^2 b c d^2-40 a b^2 c^2 d+16 b^3 c^3\right ) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{8 a^4 \sqrt {c}}-\frac {c (c+d x)^{3/2}}{3 a x^3} \]
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Rule 65
Rule 100
Rule 154
Rule 156
Rule 162
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {c (c+d x)^{3/2}}{3 a x^3}-\frac {\int \frac {\sqrt {c+d x} \left (\frac {3}{2} c (2 b c-3 a d)+\frac {3}{2} d (b c-2 a d) x\right )}{x^3 (a+b x)} \, dx}{3 a} \\ & = \frac {c (2 b c-3 a d) \sqrt {c+d x}}{4 a^2 x^2}-\frac {c (c+d x)^{3/2}}{3 a x^3}-\frac {\int \frac {-\frac {3}{4} c \left (8 b^2 c^2-18 a b c d+11 a^2 d^2\right )-\frac {3}{4} d \left (6 b^2 c^2-13 a b c d+8 a^2 d^2\right ) x}{x^2 (a+b x) \sqrt {c+d x}} \, dx}{6 a^2} \\ & = \frac {c (2 b c-3 a d) \sqrt {c+d x}}{4 a^2 x^2}-\frac {\left (8 b^2 c^2-18 a b c d+11 a^2 d^2\right ) \sqrt {c+d x}}{8 a^3 x}-\frac {c (c+d x)^{3/2}}{3 a x^3}+\frac {\int \frac {-\frac {3}{8} c \left (16 b^3 c^3-40 a b^2 c^2 d+30 a^2 b c d^2-5 a^3 d^3\right )-\frac {3}{8} b c d \left (8 b^2 c^2-18 a b c d+11 a^2 d^2\right ) x}{x (a+b x) \sqrt {c+d x}} \, dx}{6 a^3 c} \\ & = \frac {c (2 b c-3 a d) \sqrt {c+d x}}{4 a^2 x^2}-\frac {\left (8 b^2 c^2-18 a b c d+11 a^2 d^2\right ) \sqrt {c+d x}}{8 a^3 x}-\frac {c (c+d x)^{3/2}}{3 a x^3}+\frac {\left (b (b c-a d)^3\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{a^4}-\frac {\left (16 b^3 c^3-40 a b^2 c^2 d+30 a^2 b c d^2-5 a^3 d^3\right ) \int \frac {1}{x \sqrt {c+d x}} \, dx}{16 a^4} \\ & = \frac {c (2 b c-3 a d) \sqrt {c+d x}}{4 a^2 x^2}-\frac {\left (8 b^2 c^2-18 a b c d+11 a^2 d^2\right ) \sqrt {c+d x}}{8 a^3 x}-\frac {c (c+d x)^{3/2}}{3 a x^3}+\frac {\left (2 b (b c-a 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 )}{a^4 d}-\frac {\left (16 b^3 c^3-40 a b^2 c^2 d+30 a^2 b c d^2-5 a^3 d^3\right ) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{8 a^4 d} \\ & = \frac {c (2 b c-3 a d) \sqrt {c+d x}}{4 a^2 x^2}-\frac {\left (8 b^2 c^2-18 a b c d+11 a^2 d^2\right ) \sqrt {c+d x}}{8 a^3 x}-\frac {c (c+d x)^{3/2}}{3 a x^3}+\frac {\left (16 b^3 c^3-40 a b^2 c^2 d+30 a^2 b c d^2-5 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{8 a^4 \sqrt {c}}-\frac {2 \sqrt {b} (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^4} \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.86 \[ \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)} \, dx=-\frac {\frac {a \sqrt {c+d x} \left (24 b^2 c^2 x^2-6 a b c x (2 c+9 d x)+a^2 \left (8 c^2+26 c d x+33 d^2 x^2\right )\right )}{x^3}+48 \sqrt {b} (-b c+a d)^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )-\frac {3 \left (16 b^3 c^3-40 a b^2 c^2 d+30 a^2 b c d^2-5 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{\sqrt {c}}}{24 a^4} \]
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Time = 0.65 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.85
method | result | size |
pseudoelliptic | \(\frac {-\frac {2 \left (a d -b c \right )^{3} b \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}-\frac {\sqrt {d x +c}\, a \left (33 a^{2} d^{2} x^{2}-54 a b c d \,x^{2}+24 b^{2} c^{2} x^{2}+26 a^{2} c d x -12 a b \,c^{2} x +8 a^{2} c^{2}\right )}{24 x^{3}}-\frac {\left (5 a^{3} d^{3}-30 a^{2} b c \,d^{2}+40 a \,b^{2} c^{2} d -16 b^{3} c^{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{8 \sqrt {c}}}{a^{4}}\) | \(176\) |
risch | \(-\frac {\sqrt {d x +c}\, \left (33 a^{2} d^{2} x^{2}-54 a b c d \,x^{2}+24 b^{2} c^{2} x^{2}+26 a^{2} c d x -12 a b \,c^{2} x +8 a^{2} c^{2}\right )}{24 a^{3} x^{3}}-\frac {d \left (\frac {16 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{a d \sqrt {\left (a d -b c \right ) b}}-\frac {\left (-5 a^{3} d^{3}+30 a^{2} b c \,d^{2}-40 a \,b^{2} c^{2} d +16 b^{3} c^{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a d \sqrt {c}}\right )}{8 a^{3}}\) | \(219\) |
derivativedivides | \(2 d^{4} \left (\frac {-\frac {\left (\frac {11}{16} a^{3} d^{3}-\frac {9}{8} a^{2} b c \,d^{2}+\frac {1}{2} a \,b^{2} c^{2} d \right ) \left (d x +c \right )^{\frac {5}{2}}+\left (-\frac {5}{6} a^{3} c \,d^{3}+2 a^{2} d^{2} b \,c^{2}-a \,b^{2} c^{3} d \right ) \left (d x +c \right )^{\frac {3}{2}}+\left (-\frac {7}{8} a^{2} b \,c^{3} d^{2}+\frac {1}{2} a \,b^{2} c^{4} d +\frac {5}{16} a^{3} c^{2} d^{3}\right ) \sqrt {d x +c}}{d^{3} x^{3}}-\frac {\left (5 a^{3} d^{3}-30 a^{2} b c \,d^{2}+40 a \,b^{2} c^{2} d -16 b^{3} c^{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{16 \sqrt {c}}}{a^{4} d^{4}}-\frac {\left (a d -b c \right )^{3} b \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{a^{4} d^{4} \sqrt {\left (a d -b c \right ) b}}\right )\) | \(250\) |
default | \(2 d^{4} \left (\frac {-\frac {\left (\frac {11}{16} a^{3} d^{3}-\frac {9}{8} a^{2} b c \,d^{2}+\frac {1}{2} a \,b^{2} c^{2} d \right ) \left (d x +c \right )^{\frac {5}{2}}+\left (-\frac {5}{6} a^{3} c \,d^{3}+2 a^{2} d^{2} b \,c^{2}-a \,b^{2} c^{3} d \right ) \left (d x +c \right )^{\frac {3}{2}}+\left (-\frac {7}{8} a^{2} b \,c^{3} d^{2}+\frac {1}{2} a \,b^{2} c^{4} d +\frac {5}{16} a^{3} c^{2} d^{3}\right ) \sqrt {d x +c}}{d^{3} x^{3}}-\frac {\left (5 a^{3} d^{3}-30 a^{2} b c \,d^{2}+40 a \,b^{2} c^{2} d -16 b^{3} c^{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{16 \sqrt {c}}}{a^{4} d^{4}}-\frac {\left (a d -b c \right )^{3} b \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{a^{4} d^{4} \sqrt {\left (a d -b c \right ) b}}\right )\) | \(250\) |
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Time = 0.42 (sec) , antiderivative size = 930, normalized size of antiderivative = 4.49 \[ \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)} \, dx=\left [\frac {48 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {b^{2} c - a b d} x^{3} \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 ) - 3 \, {\left (16 \, b^{3} c^{3} - 40 \, a b^{2} c^{2} d + 30 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \sqrt {c} x^{3} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) - 2 \, {\left (8 \, a^{3} c^{3} + 3 \, {\left (8 \, a b^{2} c^{3} - 18 \, a^{2} b c^{2} d + 11 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (6 \, a^{2} b c^{3} - 13 \, a^{3} c^{2} d\right )} x\right )} \sqrt {d x + c}}{48 \, a^{4} c x^{3}}, \frac {96 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {-b^{2} c + a b d} x^{3} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) - 3 \, {\left (16 \, b^{3} c^{3} - 40 \, a b^{2} c^{2} d + 30 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \sqrt {c} x^{3} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) - 2 \, {\left (8 \, a^{3} c^{3} + 3 \, {\left (8 \, a b^{2} c^{3} - 18 \, a^{2} b c^{2} d + 11 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (6 \, a^{2} b c^{3} - 13 \, a^{3} c^{2} d\right )} x\right )} \sqrt {d x + c}}{48 \, a^{4} c x^{3}}, -\frac {3 \, {\left (16 \, b^{3} c^{3} - 40 \, a b^{2} c^{2} d + 30 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) - 24 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {b^{2} c - a b d} x^{3} \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 ) + {\left (8 \, a^{3} c^{3} + 3 \, {\left (8 \, a b^{2} c^{3} - 18 \, a^{2} b c^{2} d + 11 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (6 \, a^{2} b c^{3} - 13 \, a^{3} c^{2} d\right )} x\right )} \sqrt {d x + c}}{24 \, a^{4} c x^{3}}, \frac {48 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {-b^{2} c + a b d} x^{3} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) - 3 \, {\left (16 \, b^{3} c^{3} - 40 \, a b^{2} c^{2} d + 30 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) - {\left (8 \, a^{3} c^{3} + 3 \, {\left (8 \, a b^{2} c^{3} - 18 \, a^{2} b c^{2} d + 11 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (6 \, a^{2} b c^{3} - 13 \, a^{3} c^{2} d\right )} x\right )} \sqrt {d x + c}}{24 \, a^{4} c x^{3}}\right ] \]
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\[ \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}}}{x^{4} \left (a + b x\right )}\, dx \]
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Exception generated. \[ \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.30 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.45 \[ \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)} \, dx=\frac {2 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a^{4}} - \frac {{\left (16 \, b^{3} c^{3} - 40 \, a b^{2} c^{2} d + 30 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{8 \, a^{4} \sqrt {-c}} - \frac {24 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} c^{2} d - 48 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c^{3} d + 24 \, \sqrt {d x + c} b^{2} c^{4} d - 54 \, {\left (d x + c\right )}^{\frac {5}{2}} a b c d^{2} + 96 \, {\left (d x + c\right )}^{\frac {3}{2}} a b c^{2} d^{2} - 42 \, \sqrt {d x + c} a b c^{3} d^{2} + 33 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{2} d^{3} - 40 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} c d^{3} + 15 \, \sqrt {d x + c} a^{2} c^{2} d^{3}}{24 \, a^{3} d^{3} x^{3}} \]
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Time = 0.94 (sec) , antiderivative size = 2147, normalized size of antiderivative = 10.37 \[ \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)} \, dx=\text {Too large to display} \]
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