Integrand size = 20, antiderivative size = 147 \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^2} \, dx=-\frac {c^2 \sqrt {c+d x}}{a^2 x}-\frac {(b c-a d)^2 \sqrt {c+d x}}{a^2 b (a+b x)}+\frac {c^{3/2} (4 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3}-\frac {(b c-a d)^{3/2} (4 b c+a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 b^{3/2}} \]
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Time = 0.13 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {100, 154, 162, 65, 214} \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^2} \, dx=-\frac {(b c-a d)^{3/2} (a d+4 b c) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 b^{3/2}}+\frac {c^{3/2} (4 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3}-\frac {\sqrt {c+d x} (b c-a d) (2 b c-a d)}{a^2 b (a+b x)}-\frac {c (c+d x)^{3/2}}{a x (a+b x)} \]
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Rule 65
Rule 100
Rule 154
Rule 162
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {c (c+d x)^{3/2}}{a x (a+b x)}-\frac {\int \frac {\sqrt {c+d x} \left (\frac {1}{2} c (4 b c-5 a d)+\frac {1}{2} d (b c-2 a d) x\right )}{x (a+b x)^2} \, dx}{a} \\ & = -\frac {(b c-a d) (2 b c-a d) \sqrt {c+d x}}{a^2 b (a+b x)}-\frac {c (c+d x)^{3/2}}{a x (a+b x)}+\frac {\int \frac {-\frac {1}{2} b c^2 (4 b c-5 a d)-\frac {1}{2} d \left (2 b^2 c^2-2 a b c d-a^2 d^2\right ) x}{x (a+b x) \sqrt {c+d x}} \, dx}{a^2 b} \\ & = -\frac {(b c-a d) (2 b c-a d) \sqrt {c+d x}}{a^2 b (a+b x)}-\frac {c (c+d x)^{3/2}}{a x (a+b x)}-\frac {\left (c^2 (4 b c-5 a d)\right ) \int \frac {1}{x \sqrt {c+d x}} \, dx}{2 a^3}+\frac {\left ((b c-a d)^2 (4 b c+a d)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{2 a^3 b} \\ & = -\frac {(b c-a d) (2 b c-a d) \sqrt {c+d x}}{a^2 b (a+b x)}-\frac {c (c+d x)^{3/2}}{a x (a+b x)}-\frac {\left (c^2 (4 b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{a^3 d}+\frac {\left ((b c-a d)^2 (4 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^3 b d} \\ & = -\frac {(b c-a d) (2 b c-a d) \sqrt {c+d x}}{a^2 b (a+b x)}-\frac {c (c+d x)^{3/2}}{a x (a+b x)}+\frac {c^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3}-\frac {(b c-a d)^{3/2} (4 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 b^{3/2}} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.97 \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^2} \, dx=\frac {-\frac {a \sqrt {c+d x} \left (2 b^2 c^2 x+a^2 d^2 x+a b c (c-2 d x)\right )}{b x (a+b x)}+\frac {(-b c+a d)^{3/2} (4 b c+a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{3/2}}+c^{3/2} (4 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3} \]
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Time = 0.68 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.05
method | result | size |
derivativedivides | \(2 d^{3} \left (-\frac {c^{2} \left (\frac {a \sqrt {d x +c}}{2 x}+\frac {\left (5 a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{a^{3} d^{3}}+\frac {\left (a d -b c \right )^{2} \left (-\frac {a d \sqrt {d x +c}}{2 b \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {\left (a d +4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 b \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3}}\right )\) | \(154\) |
default | \(2 d^{3} \left (-\frac {c^{2} \left (\frac {a \sqrt {d x +c}}{2 x}+\frac {\left (5 a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{a^{3} d^{3}}+\frac {\left (a d -b c \right )^{2} \left (-\frac {a d \sqrt {d x +c}}{2 b \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {\left (a d +4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 b \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3}}\right )\) | \(154\) |
pseudoelliptic | \(\frac {x \left (a d +4 b c \right ) \left (a d -b c \right )^{2} \left (b x +a \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )+4 \sqrt {\left (a d -b c \right ) b}\, \left (x \left (c^{\frac {5}{2}} b -\frac {5 a d \,c^{\frac {3}{2}}}{4}\right ) b \left (b x +a \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )-\frac {\sqrt {d x +c}\, \left (2 b^{2} c^{2} x +a c \left (-2 d x +c \right ) b +a^{2} d^{2} x \right ) a}{4}\right )}{x \sqrt {\left (a d -b c \right ) b}\, \left (b x +a \right ) a^{3} b}\) | \(165\) |
risch | \(-\frac {c^{2} \sqrt {d x +c}}{a^{2} x}+\frac {d \left (\frac {-\frac {a d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {d x +c}}{b \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {\left (a^{3} d^{3}+2 a^{2} b c \,d^{2}-7 a \,b^{2} c^{2} d +4 b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b \sqrt {\left (a d -b c \right ) b}}}{a d}-\frac {c^{\frac {3}{2}} \left (5 a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a d}\right )}{a^{2}}\) | \(194\) |
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Time = 0.31 (sec) , antiderivative size = 1002, normalized size of antiderivative = 6.82 \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^2} \, dx=\left [-\frac {{\left ({\left (4 \, b^{3} c^{2} - 3 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{2} + {\left (4 \, a b^{2} c^{2} - 3 \, a^{2} b c d - a^{3} d^{2}\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^{3} c^{2} - 5 \, a b^{2} c d\right )} x^{2} + {\left (4 \, a b^{2} c^{2} - 5 \, a^{2} b c 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} b c^{2} + {\left (2 \, a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x\right )} \sqrt {d x + c}}{2 \, {\left (a^{3} b^{2} x^{2} + a^{4} b x\right )}}, -\frac {2 \, {\left ({\left (4 \, b^{3} c^{2} - 3 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{2} + {\left (4 \, a b^{2} c^{2} - 3 \, a^{2} b c d - a^{3} d^{2}\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^{3} c^{2} - 5 \, a b^{2} c d\right )} x^{2} + {\left (4 \, a b^{2} c^{2} - 5 \, a^{2} b c 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} b c^{2} + {\left (2 \, a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x\right )} \sqrt {d x + c}}{2 \, {\left (a^{3} b^{2} x^{2} + a^{4} b x\right )}}, -\frac {2 \, {\left ({\left (4 \, b^{3} c^{2} - 5 \, a b^{2} c d\right )} x^{2} + {\left (4 \, a b^{2} c^{2} - 5 \, a^{2} b c d\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left ({\left (4 \, b^{3} c^{2} - 3 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{2} + {\left (4 \, a b^{2} c^{2} - 3 \, a^{2} b c d - a^{3} d^{2}\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} b c^{2} + {\left (2 \, a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x\right )} \sqrt {d x + c}}{2 \, {\left (a^{3} b^{2} x^{2} + a^{4} b x\right )}}, -\frac {{\left ({\left (4 \, b^{3} c^{2} - 3 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{2} + {\left (4 \, a b^{2} c^{2} - 3 \, a^{2} b c d - a^{3} d^{2}\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^{3} c^{2} - 5 \, a b^{2} c d\right )} x^{2} + {\left (4 \, a b^{2} c^{2} - 5 \, a^{2} b c d\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left (a^{2} b c^{2} + {\left (2 \, a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x\right )} \sqrt {d x + c}}{a^{3} b^{2} x^{2} + a^{4} b x}\right ] \]
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\[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^2} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}}}{x^{2} \left (a + b x\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 260 vs. \(2 (127) = 254\).
Time = 0.29 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.77 \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^2} \, dx=-\frac {{\left (4 \, b c^{3} - 5 \, a c^{2} d\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a^{3} \sqrt {-c}} + \frac {{\left (4 \, b^{3} c^{3} - 7 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a^{3} b} - \frac {2 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c^{2} d - 2 \, \sqrt {d x + c} b^{2} c^{3} d - 2 \, {\left (d x + c\right )}^{\frac {3}{2}} a b c d^{2} + 3 \, \sqrt {d x + c} a b c^{2} d^{2} + {\left (d x + c\right )}^{\frac {3}{2}} a^{2} d^{3} - \sqrt {d x + c} a^{2} c d^{3}}{{\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} + {\left (d x + c\right )} a d - a c d\right )} a^{2} b} \]
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Time = 0.39 (sec) , antiderivative size = 1127, normalized size of antiderivative = 7.67 \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^2} \, dx=\frac {\frac {\sqrt {c+d\,x}\,\left (a^2\,c\,d^3-3\,a\,b\,c^2\,d^2+2\,b^2\,c^3\,d\right )}{a^2\,b}-\frac {d\,{\left (c+d\,x\right )}^{3/2}\,\left (a^2\,d^2-2\,a\,b\,c\,d+2\,b^2\,c^2\right )}{a^2\,b}}{\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 {\mathrm {atanh}\left (\frac {10\,d^9\,\sqrt {c^3}\,\sqrt {c+d\,x}}{10\,c^2\,d^9+\frac {32\,b\,c^3\,d^8}{a}-\frac {132\,b^2\,c^4\,d^7}{a^2}+\frac {130\,b^3\,c^5\,d^6}{a^3}-\frac {40\,b^4\,c^6\,d^5}{a^4}}+\frac {32\,c\,d^8\,\sqrt {c^3}\,\sqrt {c+d\,x}}{32\,c^3\,d^8+\frac {10\,a\,c^2\,d^9}{b}-\frac {132\,b\,c^4\,d^7}{a}+\frac {130\,b^2\,c^5\,d^6}{a^2}-\frac {40\,b^3\,c^6\,d^5}{a^3}}-\frac {132\,b\,c^2\,d^7\,\sqrt {c^3}\,\sqrt {c+d\,x}}{32\,a\,c^3\,d^8-132\,b\,c^4\,d^7+\frac {130\,b^2\,c^5\,d^6}{a}+\frac {10\,a^2\,c^2\,d^9}{b}-\frac {40\,b^3\,c^6\,d^5}{a^2}}+\frac {130\,b^2\,c^3\,d^6\,\sqrt {c^3}\,\sqrt {c+d\,x}}{32\,a^2\,c^3\,d^8+130\,b^2\,c^5\,d^6-\frac {40\,b^3\,c^6\,d^5}{a}+\frac {10\,a^3\,c^2\,d^9}{b}-132\,a\,b\,c^4\,d^7}-\frac {40\,b^3\,c^4\,d^5\,\sqrt {c^3}\,\sqrt {c+d\,x}}{32\,a^3\,c^3\,d^8-40\,b^3\,c^6\,d^5+130\,a\,b^2\,c^5\,d^6-132\,a^2\,b\,c^4\,d^7+\frac {10\,a^4\,c^2\,d^9}{b}}\right )\,\left (5\,a\,d-4\,b\,c\right )\,\sqrt {c^3}}{a^3}-\frac {\mathrm {atanh}\left (\frac {30\,c^3\,d^6\,\sqrt {c+d\,x}\,\sqrt {-a^3\,b^3\,d^3+3\,a^2\,b^4\,c\,d^2-3\,a\,b^5\,c^2\,d+b^6\,c^3}}{14\,a^3\,c^2\,d^9+110\,b^3\,c^5\,d^6-82\,a\,b^2\,c^4\,d^7-4\,a^2\,b\,c^3\,d^8+\frac {2\,a^4\,c\,d^{10}}{b}-\frac {40\,b^4\,c^6\,d^5}{a}}-\frac {2\,c\,d^8\,\sqrt {c+d\,x}\,\sqrt {-a^3\,b^3\,d^3+3\,a^2\,b^4\,c\,d^2-3\,a\,b^5\,c^2\,d+b^6\,c^3}}{4\,b^3\,c^3\,d^8-14\,a\,b^2\,c^2\,d^9+\frac {82\,b^4\,c^4\,d^7}{a}-\frac {110\,b^5\,c^5\,d^6}{a^2}+\frac {40\,b^6\,c^6\,d^5}{a^3}-2\,a^2\,b\,c\,d^{10}}+\frac {18\,c^2\,d^7\,\sqrt {c+d\,x}\,\sqrt {-a^3\,b^3\,d^3+3\,a^2\,b^4\,c\,d^2-3\,a\,b^5\,c^2\,d+b^6\,c^3}}{2\,a^3\,c\,d^{10}-82\,b^3\,c^4\,d^7-4\,a\,b^2\,c^3\,d^8+14\,a^2\,b\,c^2\,d^9+\frac {110\,b^4\,c^5\,d^6}{a}-\frac {40\,b^5\,c^6\,d^5}{a^2}}+\frac {40\,c^4\,d^5\,\sqrt {c+d\,x}\,\sqrt {-a^3\,b^3\,d^3+3\,a^2\,b^4\,c\,d^2-3\,a\,b^5\,c^2\,d+b^6\,c^3}}{4\,a^3\,c^3\,d^8+40\,b^3\,c^6\,d^5-110\,a\,b^2\,c^5\,d^6+82\,a^2\,b\,c^4\,d^7-\frac {2\,a^5\,c\,d^{10}}{b^2}-\frac {14\,a^4\,c^2\,d^9}{b}}\right )\,\sqrt {-b^3\,{\left (a\,d-b\,c\right )}^3}\,\left (a\,d+4\,b\,c\right )}{a^3\,b^3} \]
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