Integrand size = 22, antiderivative size = 142 \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx=-\frac {5 c (b c-a d) \sqrt {c+d x}}{a^3 \sqrt {a+b x}}-\frac {5 (b c-a d) (c+d x)^{3/2}}{3 a^2 (a+b x)^{3/2}}-\frac {(c+d x)^{5/2}}{a x (a+b x)^{3/2}}+\frac {5 c^{3/2} (b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{7/2}} \]
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Time = 0.04 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {96, 95, 214} \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx=\frac {5 c^{3/2} (b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{7/2}}-\frac {5 c \sqrt {c+d x} (b c-a d)}{a^3 \sqrt {a+b x}}-\frac {5 (c+d x)^{3/2} (b c-a d)}{3 a^2 (a+b x)^{3/2}}-\frac {(c+d x)^{5/2}}{a x (a+b x)^{3/2}} \]
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Rule 95
Rule 96
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x)^{5/2}}{a x (a+b x)^{3/2}}-\frac {(5 (b c-a d)) \int \frac {(c+d x)^{3/2}}{x (a+b x)^{5/2}} \, dx}{2 a} \\ & = -\frac {5 (b c-a d) (c+d x)^{3/2}}{3 a^2 (a+b x)^{3/2}}-\frac {(c+d x)^{5/2}}{a x (a+b x)^{3/2}}-\frac {(5 c (b c-a d)) \int \frac {\sqrt {c+d x}}{x (a+b x)^{3/2}} \, dx}{2 a^2} \\ & = -\frac {5 c (b c-a d) \sqrt {c+d x}}{a^3 \sqrt {a+b x}}-\frac {5 (b c-a d) (c+d x)^{3/2}}{3 a^2 (a+b x)^{3/2}}-\frac {(c+d x)^{5/2}}{a x (a+b x)^{3/2}}-\frac {\left (5 c^2 (b c-a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 a^3} \\ & = -\frac {5 c (b c-a d) \sqrt {c+d x}}{a^3 \sqrt {a+b x}}-\frac {5 (b c-a d) (c+d x)^{3/2}}{3 a^2 (a+b x)^{3/2}}-\frac {(c+d x)^{5/2}}{a x (a+b x)^{3/2}}-\frac {\left (5 c^2 (b c-a d)\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{a^3} \\ & = -\frac {5 c (b c-a d) \sqrt {c+d x}}{a^3 \sqrt {a+b x}}-\frac {5 (b c-a d) (c+d x)^{3/2}}{3 a^2 (a+b x)^{3/2}}-\frac {(c+d x)^{5/2}}{a x (a+b x)^{3/2}}+\frac {5 c^{3/2} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{7/2}} \\ \end{align*}
Time = 10.12 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.88 \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx=-\frac {3 a^{5/2} (c+d x)^{5/2}+5 (b c-a d) x \left (\sqrt {a} \sqrt {c+d x} (4 a c+3 b c x+a d x)-3 c^{3/2} (a+b x)^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )}{3 a^{7/2} x (a+b x)^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(501\) vs. \(2(116)=232\).
Time = 1.71 (sec) , antiderivative size = 502, normalized size of antiderivative = 3.54
method | result | size |
default | \(-\frac {\sqrt {d x +c}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d \,x^{3}-15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{3} c^{3} x^{3}+30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b \,c^{2} d \,x^{2}-30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{2} c^{3} x^{2}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} c^{2} d x -15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b \,c^{3} x -4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{2} x^{2}-20 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c d \,x^{2}+30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} x^{2}-28 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c d x +40 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{2} x +6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c^{2} \sqrt {a c}\right )}{6 a^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x \sqrt {a c}\, \left (b x +a \right )^{\frac {3}{2}}}\) | \(502\) |
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Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (116) = 232\).
Time = 0.44 (sec) , antiderivative size = 505, normalized size of antiderivative = 3.56 \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx=\left [-\frac {15 \, {\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{3} + 2 \, {\left (a b^{2} c^{2} - a^{2} b c d\right )} x^{2} + {\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt {\frac {c}{a}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (3 \, a^{2} c^{2} + {\left (15 \, b^{2} c^{2} - 10 \, a b c d - 2 \, a^{2} d^{2}\right )} x^{2} + 2 \, {\left (10 \, a b c^{2} - 7 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{12 \, {\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}}, -\frac {15 \, {\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{3} + 2 \, {\left (a b^{2} c^{2} - a^{2} b c d\right )} x^{2} + {\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt {-\frac {c}{a}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) + 2 \, {\left (3 \, a^{2} c^{2} + {\left (15 \, b^{2} c^{2} - 10 \, a b c d - 2 \, a^{2} d^{2}\right )} x^{2} + 2 \, {\left (10 \, a b c^{2} - 7 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}}\right ] \]
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\[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}}}{x^{2} \left (a + b x\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 990 vs. \(2 (116) = 232\).
Time = 1.73 (sec) , antiderivative size = 990, normalized size of antiderivative = 6.97 \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx=\frac {5 \, {\left (\sqrt {b d} b c^{3} {\left | b \right |} - \sqrt {b d} a c^{2} d {\left | b \right |}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} a^{3} b} - \frac {2 \, {\left (\sqrt {b d} b^{3} c^{4} {\left | b \right |} - 2 \, \sqrt {b d} a b^{2} c^{3} d {\left | b \right |} + \sqrt {b d} a^{2} b c^{2} d^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b c^{3} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a c^{2} d {\left | b \right |}\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} a^{3}} - \frac {4 \, {\left (6 \, \sqrt {b d} b^{7} c^{5} {\left | b \right |} - 23 \, \sqrt {b d} a b^{6} c^{4} d {\left | b \right |} + 32 \, \sqrt {b d} a^{2} b^{5} c^{3} d^{2} {\left | b \right |} - 18 \, \sqrt {b d} a^{3} b^{4} c^{2} d^{3} {\left | b \right |} + 2 \, \sqrt {b d} a^{4} b^{3} c d^{4} {\left | b \right |} + \sqrt {b d} a^{5} b^{2} d^{5} {\left | b \right |} - 12 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{5} c^{4} {\left | b \right |} + 36 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{4} c^{3} d {\left | b \right |} - 36 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{3} c^{2} d^{2} {\left | b \right |} + 12 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{2} c d^{3} {\left | b \right |} + 6 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{3} c^{3} {\left | b \right |} - 9 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{2} c^{2} d {\left | b \right |} + 3 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} d^{3} {\left | b \right |}\right )}}{3 \, {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3} a^{3} b^{2}} \]
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Timed out. \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/2}}{x^2\,{\left (a+b\,x\right )}^{5/2}} \,d x \]
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