Integrand size = 22, antiderivative size = 334 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^2} \, dx=-\frac {5 \left (b^3 c^3-19 a b^2 c^2 d-45 a^2 b c d^2-a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b d}-\frac {5 \left (b^2 c^2-18 a b c d-31 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 d}+\frac {5 b (b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}-5 a^{3/2} c^{3/2} (b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {5 \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{3/2} d^{3/2}} \]
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Time = 0.28 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {99, 159, 163, 65, 223, 212, 95, 214} \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^2} \, dx=-5 a^{3/2} c^{3/2} (a d+b c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {5 \sqrt {a+b x} (c+d x)^{3/2} \left (-31 a^2 d^2-18 a b c d+b^2 c^2\right )}{96 d}-\frac {5 \sqrt {a+b x} \sqrt {c+d x} \left (-a^3 d^3-45 a^2 b c d^2-19 a b^2 c^2 d+b^3 c^3\right )}{64 b d}-\frac {5 \left (a^4 d^4-20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+b^4 c^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{3/2} d^{3/2}}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\frac {5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}+\frac {5 b \sqrt {a+b x} (c+d x)^{5/2} (7 a d+b c)}{24 d} \]
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Rule 65
Rule 95
Rule 99
Rule 159
Rule 163
Rule 212
Rule 214
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\int \frac {(a+b x)^{3/2} (c+d x)^{3/2} \left (\frac {5}{2} (b c+a d)+5 b d x\right )}{x} \, dx \\ & = \frac {5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\frac {\int \frac {\sqrt {a+b x} (c+d x)^{3/2} \left (10 a d (b c+a d)+\frac {5}{2} b d (b c+7 a d) x\right )}{x} \, dx}{4 d} \\ & = \frac {5 b (b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\frac {\int \frac {(c+d x)^{3/2} \left (30 a^2 d^2 (b c+a d)-\frac {5}{4} b d \left (b^2 c^2-18 a b c d-31 a^2 d^2\right ) x\right )}{x \sqrt {a+b x}} \, dx}{12 d^2} \\ & = -\frac {5 \left (b^2 c^2-18 a b c d-31 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 d}+\frac {5 b (b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\frac {\int \frac {\sqrt {c+d x} \left (60 a^2 b c d^2 (b c+a d)-\frac {15}{8} b d \left (b^3 c^3-19 a b^2 c^2 d-45 a^2 b c d^2-a^3 d^3\right ) x\right )}{x \sqrt {a+b x}} \, dx}{24 b d^2} \\ & = -\frac {5 \left (b^3 c^3-19 a b^2 c^2 d-45 a^2 b c d^2-a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b d}-\frac {5 \left (b^2 c^2-18 a b c d-31 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 d}+\frac {5 b (b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\frac {\int \frac {60 a^2 b^2 c^2 d^2 (b c+a d)-\frac {15}{16} b d \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{24 b^2 d^2} \\ & = -\frac {5 \left (b^3 c^3-19 a b^2 c^2 d-45 a^2 b c d^2-a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b d}-\frac {5 \left (b^2 c^2-18 a b c d-31 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 d}+\frac {5 b (b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\frac {1}{2} \left (5 a^2 c^2 (b c+a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx-\frac {\left (5 \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 b d} \\ & = -\frac {5 \left (b^3 c^3-19 a b^2 c^2 d-45 a^2 b c d^2-a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b d}-\frac {5 \left (b^2 c^2-18 a b c d-31 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 d}+\frac {5 b (b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\left (5 a^2 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 )-\frac {\left (5 \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{64 b^2 d} \\ & = -\frac {5 \left (b^3 c^3-19 a b^2 c^2 d-45 a^2 b c d^2-a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b d}-\frac {5 \left (b^2 c^2-18 a b c d-31 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 d}+\frac {5 b (b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}-5 a^{3/2} c^{3/2} (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {\left (5 \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right )\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{64 b^2 d} \\ & = -\frac {5 \left (b^3 c^3-19 a b^2 c^2 d-45 a^2 b c d^2-a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b d}-\frac {5 \left (b^2 c^2-18 a b c d-31 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 d}+\frac {5 b (b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}-5 a^{3/2} c^{3/2} (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {5 \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{3/2} d^{3/2}} \\ \end{align*}
Time = 1.04 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^2} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (15 a^3 d^3 x+a^2 b d \left (-192 c^2+601 c d x+118 d^2 x^2\right )+a b^2 d x \left (601 c^2+452 c d x+136 d^2 x^2\right )+b^3 x \left (15 c^3+118 c^2 d x+136 c d^2 x^2+48 d^3 x^3\right )\right )}{192 b d x}-5 a^{3/2} c^{3/2} (b c+a d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )-\frac {5 \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{64 b^{3/2} d^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(816\) vs. \(2(280)=560\).
Time = 0.57 (sec) , antiderivative size = 817, normalized size of antiderivative = 2.45
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (-96 b^{3} d^{3} x^{4} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}-272 a \,b^{2} d^{3} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}-272 b^{3} c \,d^{2} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}+15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{4} d^{4} x -300 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{3} b c \,d^{3} x -1350 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{2} b^{2} c^{2} d^{2} x -300 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a \,b^{3} c^{3} d x +15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, b^{4} c^{4} x +960 \sqrt {b d}\, \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} b \,c^{2} d^{2} x +960 \sqrt {b d}\, \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^{2} c^{3} d x -236 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b \,d^{3} x^{2}-904 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{2} c \,d^{2} x^{2}-236 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, b^{3} c^{2} d \,x^{2}-30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} d^{3} x -1202 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b c \,d^{2} x -1202 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{2} c^{2} d x -30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, b^{3} c^{3} x +384 a^{2} b \,c^{2} d \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\right )}{384 b d \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, x \sqrt {a c}}\) | \(817\) |
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Time = 8.14 (sec) , antiderivative size = 1613, normalized size of antiderivative = 4.83 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^2} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}{x^{2}}\, dx \]
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Exception generated. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 759 vs. \(2 (280) = 560\).
Time = 1.00 (sec) , antiderivative size = 759, normalized size of antiderivative = 2.27 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^2} \, dx=\frac {2 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )} d^{2} {\left | b \right |}}{b^{2}} + \frac {17 \, b^{3} c d^{7} {\left | b \right |} - a b^{2} d^{8} {\left | b \right |}}{b^{4} d^{6}}\right )} + \frac {59 \, b^{4} c^{2} d^{6} {\left | b \right |} + 90 \, a b^{3} c d^{7} {\left | b \right |} - 5 \, a^{2} b^{2} d^{8} {\left | b \right |}}{b^{4} d^{6}}\right )} {\left (b x + a\right )} + \frac {3 \, {\left (5 \, b^{5} c^{3} d^{5} {\left | b \right |} + 161 \, a b^{4} c^{2} d^{6} {\left | b \right |} + 95 \, a^{2} b^{3} c d^{7} {\left | b \right |} - 5 \, a^{3} b^{2} d^{8} {\left | b \right |}\right )}}{b^{4} d^{6}}\right )} \sqrt {b x + a} - \frac {1920 \, {\left (\sqrt {b d} a^{2} b^{2} c^{3} {\left | b \right |} + \sqrt {b d} a^{3} b 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} b} - \frac {768 \, {\left (\sqrt {b d} a^{2} b^{4} c^{4} {\left | b \right |} - 2 \, \sqrt {b d} a^{3} b^{3} c^{3} d {\left | b \right |} + \sqrt {b d} a^{4} b^{2} 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} a^{2} b^{2} 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^{3} b c^{2} d {\left | b \right |}\right )}}{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}} + \frac {15 \, {\left (b^{4} c^{4} {\left | b \right |} - 20 \, a b^{3} c^{3} d {\left | b \right |} - 90 \, a^{2} b^{2} c^{2} d^{2} {\left | b \right |} - 20 \, a^{3} b c d^{3} {\left | b \right |} + a^{4} d^{4} {\left | b \right |}\right )} \log \left ({\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 )}{\sqrt {b d} b d}}{384 \, b} \]
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Timed out. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^2} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2}}{x^2} \,d x \]
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