Integrand size = 20, antiderivative size = 149 \[ \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^2} \, dx=-\frac {(2 b c-a d) \sqrt {c+d x}}{a^2 (a+b x)}-\frac {c \sqrt {c+d x}}{a x (a+b x)}+\frac {\sqrt {c} (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3}-\frac {\sqrt {b c-a d} (4 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 \sqrt {b}} \]
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Time = 0.14 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {100, 156, 162, 65, 214} \[ \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^2} \, dx=\frac {\sqrt {c} (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3}-\frac {\sqrt {b c-a d} (4 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 \sqrt {b}}-\frac {\sqrt {c+d x} (2 b c-a d)}{a^2 (a+b x)}-\frac {c \sqrt {c+d x}}{a x (a+b x)} \]
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Rule 65
Rule 100
Rule 156
Rule 162
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {c \sqrt {c+d x}}{a x (a+b x)}-\frac {\int \frac {\frac {1}{2} c (4 b c-3 a d)+\frac {1}{2} d (3 b c-2 a d) x}{x (a+b x)^2 \sqrt {c+d x}} \, dx}{a} \\ & = -\frac {(2 b c-a d) \sqrt {c+d x}}{a^2 (a+b x)}-\frac {c \sqrt {c+d x}}{a x (a+b x)}-\frac {\int \frac {\frac {1}{2} c (4 b c-3 a d) (b c-a d)+\frac {1}{2} d (b c-a d) (2 b c-a d) x}{x (a+b x) \sqrt {c+d x}} \, dx}{a^2 (b c-a d)} \\ & = -\frac {(2 b c-a d) \sqrt {c+d x}}{a^2 (a+b x)}-\frac {c \sqrt {c+d x}}{a x (a+b x)}-\frac {(c (4 b c-3 a d)) \int \frac {1}{x \sqrt {c+d x}} \, dx}{2 a^3}+\frac {((b c-a d) (4 b c-a d)) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{2 a^3} \\ & = -\frac {(2 b c-a d) \sqrt {c+d x}}{a^2 (a+b x)}-\frac {c \sqrt {c+d x}}{a x (a+b x)}-\frac {(c (4 b c-3 a d)) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{a^3 d}+\frac {((b c-a d) (4 b c-a d)) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{a^3 d} \\ & = -\frac {(2 b c-a d) \sqrt {c+d x}}{a^2 (a+b x)}-\frac {c \sqrt {c+d x}}{a x (a+b x)}+\frac {\sqrt {c} (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3}-\frac {\sqrt {b c-a d} (4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 \sqrt {b}} \\ \end{align*}
Time = 0.64 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.93 \[ \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^2} \, dx=\frac {\frac {a \sqrt {c+d x} (-a c-2 b c x+a d x)}{x (a+b x)}+\frac {\left (4 b^2 c^2-5 a b c d+a^2 d^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{\sqrt {b} \sqrt {-b c+a d}}+\sqrt {c} (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3} \]
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Time = 0.62 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(2 d^{3} \left (-\frac {c \left (\frac {a \sqrt {d x +c}}{2 x}+\frac {\left (3 a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{a^{3} d^{3}}+\frac {\left (a d -b c \right ) \left (\frac {\sqrt {d x +c}\, a d}{2 \left (d x +c \right ) b +2 a d -2 b c}+\frac {\left (a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3}}\right )\) | \(144\) |
default | \(2 d^{3} \left (-\frac {c \left (\frac {a \sqrt {d x +c}}{2 x}+\frac {\left (3 a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{a^{3} d^{3}}+\frac {\left (a d -b c \right ) \left (\frac {\sqrt {d x +c}\, a d}{2 \left (d x +c \right ) b +2 a d -2 b c}+\frac {\left (a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3}}\right )\) | \(144\) |
pseudoelliptic | \(\frac {x \left (a d -b c \right ) \left (a d -4 b c \right ) \left (b x +a \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )+4 \left (x \left (c^{\frac {3}{2}} b -\frac {3 a d \sqrt {c}}{4}\right ) \left (b x +a \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )-\frac {\left (2 b c x +a \left (-d x +c \right )\right ) a \sqrt {d x +c}}{4}\right ) \sqrt {\left (a d -b c \right ) b}}{x \sqrt {\left (a d -b c \right ) b}\, \left (b x +a \right ) a^{3}}\) | \(145\) |
risch | \(-\frac {c \sqrt {d x +c}}{a^{2} x}-\frac {d \left (\frac {\frac {2 \left (-\frac {1}{2} a^{2} d^{2}+\frac {1}{2} a b c d \right ) \sqrt {d x +c}}{\left (d x +c \right ) b +a d -b c}-\frac {\left (a^{2} d^{2}-5 a b c d +4 b^{2} c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}}{a d}+\frac {\sqrt {c}\, \left (3 a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a d}\right )}{a^{2}}\) | \(163\) |
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Time = 0.29 (sec) , antiderivative size = 770, normalized size of antiderivative = 5.17 \[ \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^2} \, dx=\left [-\frac {{\left ({\left (4 \, b^{2} c - a b d\right )} x^{2} + {\left (4 \, a b c - a^{2} d\right )} x\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + {\left ({\left (4 \, b^{2} c - 3 \, a b d\right )} x^{2} + {\left (4 \, a b c - 3 \, a^{2} d\right )} x\right )} \sqrt {c} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (a^{2} c + {\left (2 \, a b c - a^{2} d\right )} x\right )} \sqrt {d x + c}}{2 \, {\left (a^{3} b x^{2} + a^{4} x\right )}}, -\frac {2 \, {\left ({\left (4 \, b^{2} c - a b d\right )} x^{2} + {\left (4 \, a b c - a^{2} d\right )} x\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + {\left ({\left (4 \, b^{2} c - 3 \, a b d\right )} x^{2} + {\left (4 \, a b c - 3 \, a^{2} d\right )} x\right )} \sqrt {c} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (a^{2} c + {\left (2 \, a b c - a^{2} d\right )} x\right )} \sqrt {d x + c}}{2 \, {\left (a^{3} b x^{2} + a^{4} x\right )}}, -\frac {2 \, {\left ({\left (4 \, b^{2} c - 3 \, a b d\right )} x^{2} + {\left (4 \, a b c - 3 \, a^{2} d\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left ({\left (4 \, b^{2} c - a b d\right )} x^{2} + {\left (4 \, a b c - a^{2} d\right )} x\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left (a^{2} c + {\left (2 \, a b c - a^{2} d\right )} x\right )} \sqrt {d x + c}}{2 \, {\left (a^{3} b x^{2} + a^{4} x\right )}}, -\frac {{\left ({\left (4 \, b^{2} c - a b d\right )} x^{2} + {\left (4 \, a b c - a^{2} d\right )} x\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + {\left ({\left (4 \, b^{2} c - 3 \, a b d\right )} x^{2} + {\left (4 \, a b c - 3 \, a^{2} d\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left (a^{2} c + {\left (2 \, a b c - a^{2} d\right )} x\right )} \sqrt {d x + c}}{a^{3} b x^{2} + a^{4} x}\right ] \]
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\[ \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^2} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{x^{2} \left (a + b x\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.29 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.32 \[ \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^2} \, dx=\frac {{\left (4 \, b^{2} c^{2} - 5 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a^{3}} - \frac {{\left (4 \, b c^{2} - 3 \, a c d\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a^{3} \sqrt {-c}} - \frac {2 \, {\left (d x + c\right )}^{\frac {3}{2}} b c d - 2 \, \sqrt {d x + c} b c^{2} d - {\left (d x + c\right )}^{\frac {3}{2}} a d^{2} + 2 \, \sqrt {d x + c} a c d^{2}}{{\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} + {\left (d x + c\right )} a d - a c d\right )} a^{2}} \]
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Time = 0.70 (sec) , antiderivative size = 429, normalized size of antiderivative = 2.88 \[ \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^2} \, dx=-\frac {\frac {2\,\left (a\,c\,d^2-b\,c^2\,d\right )\,\sqrt {c+d\,x}}{a^2}-\frac {d\,\left (a\,d-2\,b\,c\right )\,{\left (c+d\,x\right )}^{3/2}}{a^2}}{\left (a\,d-2\,b\,c\right )\,\left (c+d\,x\right )+b\,{\left (c+d\,x\right )}^2+b\,c^2-a\,c\,d}-\frac {\sqrt {c}\,\mathrm {atanh}\left (\frac {6\,b\,\sqrt {c}\,d^7\,\sqrt {c+d\,x}}{6\,b\,c\,d^7-\frac {14\,b^2\,c^2\,d^6}{a}+\frac {8\,b^3\,c^3\,d^5}{a^2}}-\frac {14\,b^2\,c^{3/2}\,d^6\,\sqrt {c+d\,x}}{6\,a\,b\,c\,d^7-14\,b^2\,c^2\,d^6+\frac {8\,b^3\,c^3\,d^5}{a}}+\frac {8\,b^3\,c^{5/2}\,d^5\,\sqrt {c+d\,x}}{6\,a^2\,b\,c\,d^7-14\,a\,b^2\,c^2\,d^6+8\,b^3\,c^3\,d^5}\right )\,\left (3\,a\,d-4\,b\,c\right )}{a^3}-\frac {\mathrm {atanh}\left (\frac {2\,b\,c\,d^6\,\sqrt {b^2\,c-a\,b\,d}\,\sqrt {c+d\,x}}{2\,a\,b\,c\,d^7-10\,b^2\,c^2\,d^6+\frac {8\,b^3\,c^3\,d^5}{a}}-\frac {8\,b^2\,c^2\,d^5\,\sqrt {b^2\,c-a\,b\,d}\,\sqrt {c+d\,x}}{2\,a^2\,b\,c\,d^7-10\,a\,b^2\,c^2\,d^6+8\,b^3\,c^3\,d^5}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-4\,b\,c\right )}{a^3\,b} \]
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