Integrand size = 22, antiderivative size = 177 \[ \int \frac {(a+b x)^{5/2}}{x^3 \sqrt {c+d x}} \, dx=-\frac {a (7 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 c^2 x}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}-\frac {\sqrt {a} \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 c^{5/2}}+\frac {2 b^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {d}} \]
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Time = 0.09 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {100, 154, 163, 65, 223, 212, 95, 214} \[ \int \frac {(a+b x)^{5/2}}{x^3 \sqrt {c+d x}} \, dx=-\frac {\sqrt {a} \left (3 a^2 d^2-10 a b c d+15 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 c^{5/2}}+\frac {2 b^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {d}}-\frac {a \sqrt {a+b x} \sqrt {c+d x} (7 b c-3 a d)}{4 c^2 x}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2} \]
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Rule 65
Rule 95
Rule 100
Rule 154
Rule 163
Rule 212
Rule 214
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}-\frac {\int \frac {\sqrt {a+b x} \left (-\frac {1}{2} a (7 b c-3 a d)-2 b^2 c x\right )}{x^2 \sqrt {c+d x}} \, dx}{2 c} \\ & = -\frac {a (7 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 c^2 x}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}-\frac {\int \frac {-\frac {1}{4} a \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right )-2 b^3 c^2 x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 c^2} \\ & = -\frac {a (7 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 c^2 x}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}+b^3 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {\left (a \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 c^2} \\ & = -\frac {a (7 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 c^2 x}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}+\left (2 b^2\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 )+\frac {\left (a \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 c^2} \\ & = -\frac {a (7 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 c^2 x}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}-\frac {\sqrt {a} \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 c^{5/2}}+\left (2 b^2\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 ) \\ & = -\frac {a (7 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 c^2 x}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}-\frac {\sqrt {a} \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 c^{5/2}}+\frac {2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {d}} \\ \end{align*}
Time = 0.53 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x)^{5/2}}{x^3 \sqrt {c+d x}} \, dx=\frac {1}{4} \left (\frac {a \sqrt {a+b x} \sqrt {c+d x} (-2 a c-9 b c x+3 a d x)}{c^2 x^2}-\frac {\sqrt {a} \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{c^{5/2}}+\frac {8 b^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{\sqrt {d}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(353\) vs. \(2(139)=278\).
Time = 0.56 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.00
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} d^{2} x^{2} \sqrt {b d}-10 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b c d \,x^{2} \sqrt {b d}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} x^{2} \sqrt {b d}-8 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{2} x^{2} \sqrt {a c}-6 a^{2} d x \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+18 a b c x \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+4 a^{2} c \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\right )}{8 c^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{2} \sqrt {b d}\, \sqrt {a c}}\) | \(354\) |
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Time = 1.12 (sec) , antiderivative size = 1035, normalized size of antiderivative = 5.85 \[ \int \frac {(a+b x)^{5/2}}{x^3 \sqrt {c+d x}} \, dx=\left [\frac {8 \, b^{2} c^{2} x^{2} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + {\left (15 \, b^{2} c^{2} - 10 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2} \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 (2 \, a^{2} c + 3 \, {\left (3 \, a b c - a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, c^{2} x^{2}}, -\frac {16 \, b^{2} c^{2} x^{2} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) - {\left (15 \, b^{2} c^{2} - 10 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2} \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 (2 \, a^{2} c + 3 \, {\left (3 \, a b c - a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, c^{2} x^{2}}, \frac {4 \, b^{2} c^{2} x^{2} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + {\left (15 \, b^{2} c^{2} - 10 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2} \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 (2 \, a^{2} c + 3 \, {\left (3 \, a b c - a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, c^{2} x^{2}}, -\frac {8 \, b^{2} c^{2} x^{2} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) - {\left (15 \, b^{2} c^{2} - 10 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2} \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 (2 \, a^{2} c + 3 \, {\left (3 \, a b c - a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, c^{2} x^{2}}\right ] \]
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\[ \int \frac {(a+b x)^{5/2}}{x^3 \sqrt {c+d x}} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}}}{x^{3} \sqrt {c + d x}}\, dx \]
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Exception generated. \[ \int \frac {(a+b x)^{5/2}}{x^3 \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1107 vs. \(2 (139) = 278\).
Time = 0.79 (sec) , antiderivative size = 1107, normalized size of antiderivative = 6.25 \[ \int \frac {(a+b x)^{5/2}}{x^3 \sqrt {c+d x}} \, dx=-\frac {{\left (\frac {4 \, \sqrt {b d} b^{2} \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 )}{d} + \frac {{\left (15 \, \sqrt {b d} a b^{3} c^{2} - 10 \, \sqrt {b d} a^{2} b^{2} c d + 3 \, \sqrt {b d} a^{3} b d^{2}\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^{2}} + \frac {2 \, {\left (9 \, \sqrt {b d} a b^{9} c^{5} - 39 \, \sqrt {b d} a^{2} b^{8} c^{4} d + 66 \, \sqrt {b d} a^{3} b^{7} c^{3} d^{2} - 54 \, \sqrt {b d} a^{4} b^{6} c^{2} d^{3} + 21 \, \sqrt {b d} a^{5} b^{5} c d^{4} - 3 \, \sqrt {b d} a^{6} b^{4} d^{5} - 27 \, \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^{7} c^{4} + 40 \, \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^{6} c^{3} d + 10 \, \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^{5} c^{2} d^{2} - 32 \, \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^{4} b^{4} c d^{3} + 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 )}^{2} a^{5} b^{3} d^{4} + 27 \, \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^{5} c^{3} + 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^{2} b^{4} c^{2} d + 21 \, \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} b^{3} c d^{2} - 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^{4} b^{2} d^{3} - 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 )}^{6} a b^{3} c^{2} - 10 \, \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 )}^{6} a^{2} b^{2} c d + 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 )}^{6} a^{3} b d^{2}\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 )}^{2} c^{2}}\right )} b}{4 \, {\left | b \right |}} \]
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Timed out. \[ \int \frac {(a+b x)^{5/2}}{x^3 \sqrt {c+d x}} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}}{x^3\,\sqrt {c+d\,x}} \,d x \]
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