Integrand size = 22, antiderivative size = 198 \[ \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^2} \, dx=\frac {d (11 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b}+\frac {3}{2} d \sqrt {a+b x} (c+d x)^{3/2}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{x}-\frac {c^{3/2} (b c+5 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}+\frac {\sqrt {d} \left (15 b^2 c^2+10 a b c d-a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2}} \]
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Time = 0.13 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^2} \, dx=\frac {\sqrt {d} \left (-a^2 d^2+10 a b c d+15 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2}}-\frac {c^{3/2} (5 a d+b c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{x}+\frac {3}{2} d \sqrt {a+b x} (c+d x)^{3/2}+\frac {d \sqrt {a+b x} \sqrt {c+d x} (a d+11 b c)}{4 b} \]
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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 {\sqrt {a+b x} (c+d x)^{5/2}}{x}+\int \frac {(c+d x)^{3/2} \left (\frac {1}{2} (b c+5 a d)+3 b d x\right )}{x \sqrt {a+b x}} \, dx \\ & = \frac {3}{2} d \sqrt {a+b x} (c+d x)^{3/2}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{x}+\frac {\int \frac {\sqrt {c+d x} \left (b c (b c+5 a d)+\frac {1}{2} b d (11 b c+a d) x\right )}{x \sqrt {a+b x}} \, dx}{2 b} \\ & = \frac {d (11 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b}+\frac {3}{2} d \sqrt {a+b x} (c+d x)^{3/2}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{x}+\frac {\int \frac {b^2 c^2 (b c+5 a d)+\frac {1}{4} b d \left (15 b^2 c^2+10 a b c d-a^2 d^2\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 b^2} \\ & = \frac {d (11 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b}+\frac {3}{2} d \sqrt {a+b x} (c+d x)^{3/2}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{x}+\frac {1}{2} \left (c^2 (b c+5 a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {\left (d \left (15 b^2 c^2+10 a b c d-a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 b} \\ & = \frac {d (11 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b}+\frac {3}{2} d \sqrt {a+b x} (c+d x)^{3/2}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{x}+\left (c^2 (b c+5 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 (d \left (15 b^2 c^2+10 a b c d-a^2 d^2\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 )}{4 b^2} \\ & = \frac {d (11 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b}+\frac {3}{2} d \sqrt {a+b x} (c+d x)^{3/2}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{x}-\frac {c^{3/2} (b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}+\frac {\left (d \left (15 b^2 c^2+10 a b c d-a^2 d^2\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 )}{4 b^2} \\ & = \frac {d (11 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b}+\frac {3}{2} d \sqrt {a+b x} (c+d x)^{3/2}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{x}-\frac {c^{3/2} (b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}+\frac {\sqrt {d} \left (15 b^2 c^2+10 a b c d-a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2}} \\ \end{align*}
Time = 0.78 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^2} \, dx=\frac {1}{4} \left (\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-4 b c^2+9 b c d x+a d^2 x+2 b d^2 x^2\right )}{b x}-\frac {4 c^{3/2} (b c+5 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}+\frac {\sqrt {d} \left (15 b^2 c^2+10 a b c d-a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(432\) vs. \(2(156)=312\).
Time = 0.54 (sec) , antiderivative size = 433, normalized size of antiderivative = 2.19
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} d^{3} x \sqrt {a c}-10 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b c \,d^{2} x \sqrt {a c}-15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c^{2} d x \sqrt {a c}+20 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b \,c^{2} d x \sqrt {b d}+4 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{3} x \sqrt {b d}-4 b \,d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}-2 a \,d^{2} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}-18 b c d x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}+8 b \,c^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\right )}{8 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b x}\) | \(433\) |
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Time = 1.55 (sec) , antiderivative size = 1079, normalized size of antiderivative = 5.45 \[ \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^2} \, dx=\int \frac {\sqrt {a + b x} \left (c + d x\right )^{\frac {5}{2}}}{x^{2}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 597 vs. \(2 (156) = 312\).
Time = 0.60 (sec) , antiderivative size = 597, normalized size of antiderivative = 3.02 \[ \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^2} \, dx=\frac {2 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )} d^{2} {\left | b \right |}}{b^{2}} + \frac {9 \, b^{3} c d^{3} {\left | b \right |} - a b^{2} d^{4} {\left | b \right |}}{b^{4} d^{2}}\right )} - \frac {8 \, {\left (\sqrt {b d} b^{2} c^{3} {\left | b \right |} + 5 \, \sqrt {b d} a b c^{2} d {\left | b \right |}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b} - \frac {16 \, {\left (\sqrt {b d} b^{4} c^{4} {\left | b \right |} - 2 \, \sqrt {b d} a b^{3} c^{3} d {\left | b \right |} + \sqrt {b d} a^{2} b^{2} c^{2} d^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c^{3} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b c^{2} d {\left | b \right |}\right )}}{b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}} - \frac {{\left (15 \, \sqrt {b d} b^{2} c^{2} {\left | b \right |} + 10 \, \sqrt {b d} a b c d {\left | b \right |} - \sqrt {b d} a^{2} d^{2} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{b^{2}}}{8 \, b} \]
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Timed out. \[ \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^2} \, dx=\int \frac {\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2}}{x^2} \,d x \]
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