Integrand size = 22, antiderivative size = 142 \[ \int \frac {(a+b x)^{5/2}}{x^2 (c+d x)^{5/2}} \, dx=\frac {5 (b c-a d) (a+b x)^{3/2}}{3 c^2 (c+d x)^{3/2}}-\frac {(a+b x)^{5/2}}{c x (c+d x)^{3/2}}+\frac {5 a (b c-a d) \sqrt {a+b x}}{c^3 \sqrt {c+d x}}-\frac {5 a^{3/2} (b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{7/2}} \]
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Time = 0.05 (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} \[ \int \frac {(a+b x)^{5/2}}{x^2 (c+d x)^{5/2}} \, dx=-\frac {5 a^{3/2} (b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{7/2}}+\frac {5 a \sqrt {a+b x} (b c-a d)}{c^3 \sqrt {c+d x}}+\frac {5 (a+b x)^{3/2} (b c-a d)}{3 c^2 (c+d x)^{3/2}}-\frac {(a+b x)^{5/2}}{c x (c+d x)^{3/2}} \]
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Rule 95
Rule 96
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{5/2}}{c x (c+d x)^{3/2}}+\frac {(5 (b c-a d)) \int \frac {(a+b x)^{3/2}}{x (c+d x)^{5/2}} \, dx}{2 c} \\ & = \frac {5 (b c-a d) (a+b x)^{3/2}}{3 c^2 (c+d x)^{3/2}}-\frac {(a+b x)^{5/2}}{c x (c+d x)^{3/2}}+\frac {(5 a (b c-a d)) \int \frac {\sqrt {a+b x}}{x (c+d x)^{3/2}} \, dx}{2 c^2} \\ & = \frac {5 (b c-a d) (a+b x)^{3/2}}{3 c^2 (c+d x)^{3/2}}-\frac {(a+b x)^{5/2}}{c x (c+d x)^{3/2}}+\frac {5 a (b c-a d) \sqrt {a+b x}}{c^3 \sqrt {c+d x}}+\frac {\left (5 a^2 (b c-a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 c^3} \\ & = \frac {5 (b c-a d) (a+b x)^{3/2}}{3 c^2 (c+d x)^{3/2}}-\frac {(a+b x)^{5/2}}{c x (c+d x)^{3/2}}+\frac {5 a (b c-a d) \sqrt {a+b x}}{c^3 \sqrt {c+d x}}+\frac {\left (5 a^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 )}{c^3} \\ & = \frac {5 (b c-a d) (a+b x)^{3/2}}{3 c^2 (c+d x)^{3/2}}-\frac {(a+b x)^{5/2}}{c x (c+d x)^{3/2}}+\frac {5 a (b c-a d) \sqrt {a+b x}}{c^3 \sqrt {c+d x}}-\frac {5 a^{3/2} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{7/2}} \\ \end{align*}
Time = 10.15 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^{5/2}}{x^2 (c+d x)^{5/2}} \, dx=\frac {-3 c^{5/2} (a+b x)^{5/2}+5 (b c-a d) x \left (\sqrt {c} \sqrt {a+b x} (4 a c+b c x+3 a d x)-3 a^{3/2} (c+d x)^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )}{3 c^{7/2} x (c+d x)^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(501\) vs. \(2(116)=232\).
Time = 1.64 (sec) , antiderivative size = 502, normalized size of antiderivative = 3.54
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} d^{3} x^{3}-15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b c \,d^{2} x^{3}+30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} c \,d^{2} x^{2}-30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b \,c^{2} d \,x^{2}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} c^{2} d x -15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b \,c^{3} x -30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{2} x^{2}+20 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c d \,x^{2}+4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} x^{2}-40 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c d x +28 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{2} x -6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c^{2} \sqrt {a c}\right )}{6 c^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x \sqrt {a c}\, \left (d x +c \right )^{\frac {3}{2}}}\) | \(502\) |
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Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (116) = 232\).
Time = 0.72 (sec) , antiderivative size = 507, normalized size of antiderivative = 3.57 \[ \int \frac {(a+b x)^{5/2}}{x^2 (c+d x)^{5/2}} \, dx=\left [-\frac {15 \, {\left ({\left (a b c d^{2} - a^{2} d^{3}\right )} x^{3} + 2 \, {\left (a b c^{2} d - a^{2} c d^{2}\right )} x^{2} + {\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt {\frac {a}{c}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (3 \, a^{2} c^{2} - {\left (2 \, b^{2} c^{2} + 10 \, a b c d - 15 \, a^{2} d^{2}\right )} x^{2} - 2 \, {\left (7 \, a b c^{2} - 10 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{12 \, {\left (c^{3} d^{2} x^{3} + 2 \, c^{4} d x^{2} + c^{5} x\right )}}, \frac {15 \, {\left ({\left (a b c d^{2} - a^{2} d^{3}\right )} x^{3} + 2 \, {\left (a b c^{2} d - a^{2} c d^{2}\right )} x^{2} + {\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) - 2 \, {\left (3 \, a^{2} c^{2} - {\left (2 \, b^{2} c^{2} + 10 \, a b c d - 15 \, a^{2} d^{2}\right )} x^{2} - 2 \, {\left (7 \, a b c^{2} - 10 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (c^{3} d^{2} x^{3} + 2 \, c^{4} d x^{2} + c^{5} x\right )}}\right ] \]
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\[ \int \frac {(a+b x)^{5/2}}{x^2 (c+d x)^{5/2}} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}}}{x^{2} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {(a+b x)^{5/2}}{x^2 (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 633 vs. \(2 (116) = 232\).
Time = 1.12 (sec) , antiderivative size = 633, normalized size of antiderivative = 4.46 \[ \int \frac {(a+b x)^{5/2}}{x^2 (c+d x)^{5/2}} \, dx=\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (b^{6} c^{6} d {\left | b \right |} + 4 \, a b^{5} c^{5} d^{2} {\left | b \right |} - 11 \, a^{2} b^{4} c^{4} d^{3} {\left | b \right |} + 6 \, a^{3} b^{3} c^{3} d^{4} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{3} c^{7} d - a b^{2} c^{6} d^{2}} + \frac {6 \, {\left (a b^{6} c^{6} d {\left | b \right |} - 3 \, a^{2} b^{5} c^{5} d^{2} {\left | b \right |} + 3 \, a^{3} b^{4} c^{4} d^{3} {\left | b \right |} - a^{4} b^{3} c^{3} d^{4} {\left | b \right |}\right )}}{b^{3} c^{7} d - a b^{2} c^{6} d^{2}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {5 \, {\left (\sqrt {b d} a^{2} b^{3} c - \sqrt {b d} a^{3} b^{2} d\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b c^{3} {\left | b \right |}} - \frac {2 \, {\left (\sqrt {b d} a^{2} b^{5} c^{2} - 2 \, \sqrt {b d} a^{3} b^{4} c d + \sqrt {b d} a^{4} b^{3} d^{2} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{3} c - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{2} d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} c^{3} {\left | b \right |}} \]
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Timed out. \[ \int \frac {(a+b x)^{5/2}}{x^2 (c+d x)^{5/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}}{x^2\,{\left (c+d\,x\right )}^{5/2}} \,d x \]
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