Integrand size = 20, antiderivative size = 219 \[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^2} \, dx=\frac {\left (12 b^2 c^2-17 a b c d+4 a^2 d^2\right ) \sqrt {c+d x}}{4 a^3 (a+b x)}+\frac {c (6 b c-7 a d) \sqrt {c+d x}}{4 a^2 x (a+b x)}-\frac {c (c+d x)^{3/2}}{2 a x^2 (a+b x)}-\frac {\sqrt {c} \left (24 b^2 c^2-40 a b c d+15 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{4 a^4}+\frac {(b c-a d)^{3/2} (6 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^4 \sqrt {b}} \]
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Time = 0.17 (sec) , antiderivative size = 219, 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} \[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^2} \, dx=\frac {(6 b c-a d) (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^4 \sqrt {b}}+\frac {c \sqrt {c+d x} (6 b c-7 a d)}{4 a^2 x (a+b x)}-\frac {\sqrt {c} \left (15 a^2 d^2-40 a b c d+24 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{4 a^4}+\frac {\sqrt {c+d x} \left (4 a^2 d^2-17 a b c d+12 b^2 c^2\right )}{4 a^3 (a+b x)}-\frac {c (c+d x)^{3/2}}{2 a x^2 (a+b x)} \]
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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}}{2 a x^2 (a+b x)}-\frac {\int \frac {\sqrt {c+d x} \left (\frac {1}{2} c (6 b c-7 a d)+\frac {1}{2} d (3 b c-4 a d) x\right )}{x^2 (a+b x)^2} \, dx}{2 a} \\ & = \frac {c (6 b c-7 a d) \sqrt {c+d x}}{4 a^2 x (a+b x)}-\frac {c (c+d x)^{3/2}}{2 a x^2 (a+b x)}-\frac {\int \frac {-\frac {1}{4} c \left (24 b^2 c^2-40 a b c d+15 a^2 d^2\right )-\frac {1}{4} d \left (18 b^2 c^2-27 a b c d+8 a^2 d^2\right ) x}{x (a+b x)^2 \sqrt {c+d x}} \, dx}{2 a^2} \\ & = \frac {\left (12 b^2 c^2-17 a b c d+4 a^2 d^2\right ) \sqrt {c+d x}}{4 a^3 (a+b x)}+\frac {c (6 b c-7 a d) \sqrt {c+d x}}{4 a^2 x (a+b x)}-\frac {c (c+d x)^{3/2}}{2 a x^2 (a+b x)}-\frac {\int \frac {-\frac {1}{4} c (b c-a d) \left (24 b^2 c^2-40 a b c d+15 a^2 d^2\right )-\frac {1}{4} d (b c-a d) \left (12 b^2 c^2-17 a b c d+4 a^2 d^2\right ) x}{x (a+b x) \sqrt {c+d x}} \, dx}{2 a^3 (b c-a d)} \\ & = \frac {\left (12 b^2 c^2-17 a b c d+4 a^2 d^2\right ) \sqrt {c+d x}}{4 a^3 (a+b x)}+\frac {c (6 b c-7 a d) \sqrt {c+d x}}{4 a^2 x (a+b x)}-\frac {c (c+d x)^{3/2}}{2 a x^2 (a+b x)}-\frac {\left ((b c-a d)^2 (6 b c-a d)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{2 a^4}+\frac {\left (c \left (24 b^2 c^2-40 a b c d+15 a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {c+d x}} \, dx}{8 a^4} \\ & = \frac {\left (12 b^2 c^2-17 a b c d+4 a^2 d^2\right ) \sqrt {c+d x}}{4 a^3 (a+b x)}+\frac {c (6 b c-7 a d) \sqrt {c+d x}}{4 a^2 x (a+b x)}-\frac {c (c+d x)^{3/2}}{2 a x^2 (a+b x)}-\frac {\left ((b c-a d)^2 (6 b c-a d)\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 (c \left (24 b^2 c^2-40 a b c d+15 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{4 a^4 d} \\ & = \frac {\left (12 b^2 c^2-17 a b c d+4 a^2 d^2\right ) \sqrt {c+d x}}{4 a^3 (a+b x)}+\frac {c (6 b c-7 a d) \sqrt {c+d x}}{4 a^2 x (a+b x)}-\frac {c (c+d x)^{3/2}}{2 a x^2 (a+b x)}-\frac {\sqrt {c} \left (24 b^2 c^2-40 a b c d+15 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{4 a^4}+\frac {(b c-a d)^{3/2} (6 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^4 \sqrt {b}} \\ \end{align*}
Time = 0.68 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.82 \[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^2} \, dx=\frac {\frac {a \sqrt {c+d x} \left (12 b^2 c^2 x^2+a b c x (6 c-17 d x)+a^2 \left (-2 c^2-9 c d x+4 d^2 x^2\right )\right )}{x^2 (a+b x)}-\frac {4 (6 b c-a d) (-b c+a d)^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{\sqrt {b}}-\sqrt {c} \left (24 b^2 c^2-40 a b c d+15 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{4 a^4} \]
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Time = 1.61 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.92
method | result | size |
pseudoelliptic | \(-\frac {6 \left (x^{2} \left (b c -\frac {a d}{6}\right ) \left (-a d +b c \right )^{2} \sqrt {c}\, \left (b x +a \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )+\frac {\left (\frac {15 x^{2} \left (a^{2} d^{2}-\frac {8}{3} a b c d +\frac {8}{5} b^{2} c^{2}\right ) \left (b x +a \right ) c \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2}+\sqrt {d x +c}\, \sqrt {c}\, a \left (-6 b^{2} c^{2} x^{2}-3 \left (-\frac {17 d x}{6}+c \right ) x c a b +a^{2} \left (c^{2}+\frac {9}{2} c d x -2 d^{2} x^{2}\right )\right )\right ) \sqrt {\left (a d -b c \right ) b}}{12}\right )}{\sqrt {c}\, \sqrt {\left (a d -b c \right ) b}\, a^{4} \left (b x +a \right ) x^{2}}\) | \(201\) |
derivativedivides | \(2 d^{4} \left (-\frac {c \left (\frac {\left (\frac {9}{8} a^{2} d^{2}-a b c d \right ) \left (d x +c \right )^{\frac {3}{2}}+\left (-\frac {7}{8} c \,a^{2} d^{2}+b \,c^{2} d a \right ) \sqrt {d x +c}}{d^{2} x^{2}}+\frac {\left (15 a^{2} d^{2}-40 a b c d +24 b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{8 \sqrt {c}}\right )}{a^{4} d^{4}}+\frac {\left (a d -b c \right )^{2} \left (\frac {\sqrt {d x +c}\, a d}{2 \left (d x +c \right ) b +2 a d -2 b c}+\frac {\left (a d -6 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a^{4} d^{4}}\right )\) | \(203\) |
default | \(2 d^{4} \left (-\frac {c \left (\frac {\left (\frac {9}{8} a^{2} d^{2}-a b c d \right ) \left (d x +c \right )^{\frac {3}{2}}+\left (-\frac {7}{8} c \,a^{2} d^{2}+b \,c^{2} d a \right ) \sqrt {d x +c}}{d^{2} x^{2}}+\frac {\left (15 a^{2} d^{2}-40 a b c d +24 b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{8 \sqrt {c}}\right )}{a^{4} d^{4}}+\frac {\left (a d -b c \right )^{2} \left (\frac {\sqrt {d x +c}\, a d}{2 \left (d x +c \right ) b +2 a d -2 b c}+\frac {\left (a d -6 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a^{4} d^{4}}\right )\) | \(203\) |
risch | \(-\frac {c \sqrt {d x +c}\, \left (9 a d x -8 b c x +2 a c \right )}{4 a^{3} x^{2}}-\frac {d \left (\frac {\frac {8 \left (-\frac {1}{2} a^{3} d^{3}+a^{2} b c \,d^{2}-\frac {1}{2} a \,b^{2} c^{2} d \right ) \sqrt {d x +c}}{\left (d x +c \right ) b +a d -b c}-\frac {4 \left (a^{3} d^{3}-8 a^{2} b c \,d^{2}+13 a \,b^{2} c^{2} d -6 b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}}{a d}+\frac {\sqrt {c}\, \left (15 a^{2} d^{2}-40 a b c d +24 b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a d}\right )}{4 a^{3}}\) | \(219\) |
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Time = 0.32 (sec) , antiderivative size = 1173, normalized size of antiderivative = 5.36 \[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.28 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.21 \[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^2} \, dx=-\frac {{\left (6 \, b^{3} c^{3} - 13 \, a b^{2} c^{2} d + 8 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a^{4}} + \frac {{\left (24 \, b^{2} c^{3} - 40 \, a b c^{2} d + 15 \, a^{2} c d^{2}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{4 \, a^{4} \sqrt {-c}} + \frac {\sqrt {d x + c} b^{2} c^{2} d - 2 \, \sqrt {d x + c} a b c d^{2} + \sqrt {d x + c} a^{2} d^{3}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} a^{3}} + \frac {8 \, {\left (d x + c\right )}^{\frac {3}{2}} b c^{2} d - 8 \, \sqrt {d x + c} b c^{3} d - 9 \, {\left (d x + c\right )}^{\frac {3}{2}} a c d^{2} + 7 \, \sqrt {d x + c} a c^{2} d^{2}}{4 \, a^{3} d^{2} x^{2}} \]
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Time = 0.93 (sec) , antiderivative size = 670, normalized size of antiderivative = 3.06 \[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^2} \, dx=\frac {\frac {\sqrt {c+d\,x}\,\left (11\,a^2\,c^2\,d^3-23\,a\,b\,c^3\,d^2+12\,b^2\,c^4\,d\right )}{4\,a^3}-\frac {{\left (c+d\,x\right )}^{3/2}\,\left (17\,a^2\,c\,d^3-40\,a\,b\,c^2\,d^2+24\,b^2\,c^3\,d\right )}{4\,a^3}+\frac {d\,{\left (c+d\,x\right )}^{5/2}\,\left (4\,a^2\,d^2-17\,a\,b\,c\,d+12\,b^2\,c^2\right )}{4\,a^3}}{b\,{\left (c+d\,x\right )}^3+\left (a\,d-3\,b\,c\right )\,{\left (c+d\,x\right )}^2-b\,c^3+\left (3\,b\,c^2-2\,a\,c\,d\right )\,\left (c+d\,x\right )+a\,c^2\,d}+\frac {\sqrt {c}\,\ln \left (\sqrt {c+d\,x}-\sqrt {c}\right )\,\left (\frac {15\,a^2\,d^2}{8}-5\,a\,b\,c\,d+3\,b^2\,c^2\right )}{a^4}-\frac {\sqrt {c}\,\ln \left (\sqrt {c+d\,x}+\sqrt {c}\right )\,\left (15\,a^2\,d^2-40\,a\,b\,c\,d+24\,b^2\,c^2\right )}{8\,a^4}+\frac {\mathrm {atan}\left (-\frac {b^2\,c^2\,d^7\,\sqrt {c+d\,x}\,\sqrt {-a^3\,b\,d^3+3\,a^2\,b^2\,c\,d^2-3\,a\,b^3\,c^2\,d+b^4\,c^3}\,111{}\mathrm {i}}{8\,\left (41\,a\,b^3\,c^3\,d^8-\frac {291\,b^4\,c^4\,d^7}{8}-\frac {143\,a^2\,b^2\,c^2\,d^9}{8}+\frac {45\,b^5\,c^5\,d^6}{4\,a}+2\,a^3\,b\,c\,d^{10}\right )}+\frac {b^3\,c^3\,d^6\,\sqrt {c+d\,x}\,\sqrt {-a^3\,b\,d^3+3\,a^2\,b^2\,c\,d^2-3\,a\,b^3\,c^2\,d+b^4\,c^3}\,45{}\mathrm {i}}{4\,\left (2\,a^4\,b\,c\,d^{10}-\frac {143\,a^3\,b^2\,c^2\,d^9}{8}+41\,a^2\,b^3\,c^3\,d^8-\frac {291\,a\,b^4\,c^4\,d^7}{8}+\frac {45\,b^5\,c^5\,d^6}{4}\right )}+\frac {b\,c\,d^8\,\sqrt {c+d\,x}\,\sqrt {-a^3\,b\,d^3+3\,a^2\,b^2\,c\,d^2-3\,a\,b^3\,c^2\,d+b^4\,c^3}\,2{}\mathrm {i}}{41\,b^3\,c^3\,d^8-\frac {143\,a\,b^2\,c^2\,d^9}{8}-\frac {291\,b^4\,c^4\,d^7}{8\,a}+\frac {45\,b^5\,c^5\,d^6}{4\,a^2}+2\,a^2\,b\,c\,d^{10}}\right )\,\left (a\,d-6\,b\,c\right )\,\sqrt {-b\,{\left (a\,d-b\,c\right )}^3}\,1{}\mathrm {i}}{a^4\,b} \]
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