Integrand size = 20, antiderivative size = 102 \[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=-\frac {2 c (a+b x)^{3/2}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 \sqrt {a+b x}}{d^2 \sqrt {c+d x}}+\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{5/2}} \]
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Time = 0.04 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {79, 49, 65, 223, 212} \[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{5/2}}-\frac {2 \sqrt {a+b x}}{d^2 \sqrt {c+d x}}-\frac {2 c (a+b x)^{3/2}}{3 d (c+d x)^{3/2} (b c-a d)} \]
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Rule 49
Rule 65
Rule 79
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {2 c (a+b x)^{3/2}}{3 d (b c-a d) (c+d x)^{3/2}}+\frac {\int \frac {\sqrt {a+b x}}{(c+d x)^{3/2}} \, dx}{d} \\ & = -\frac {2 c (a+b x)^{3/2}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 \sqrt {a+b x}}{d^2 \sqrt {c+d x}}+\frac {b \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{d^2} \\ & = -\frac {2 c (a+b x)^{3/2}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 \sqrt {a+b x}}{d^2 \sqrt {c+d x}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{d^2} \\ & = -\frac {2 c (a+b x)^{3/2}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 \sqrt {a+b x}}{d^2 \sqrt {c+d x}}+\frac {2 \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{d^2} \\ & = -\frac {2 c (a+b x)^{3/2}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 \sqrt {a+b x}}{d^2 \sqrt {c+d x}}+\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{5/2}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00 \[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\frac {2 \sqrt {a+b x} \left (3 b c-3 a d+\frac {c d (a+b x)}{c+d x}\right )}{3 d^2 (-b c+a d) \sqrt {c+d x}}+\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(441\) vs. \(2(80)=160\).
Time = 1.52 (sec) , antiderivative size = 442, normalized size of antiderivative = 4.33
method | result | size |
default | \(\frac {\left (3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b \,d^{3} x^{2}-3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c \,d^{2} x^{2}+6 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b c \,d^{2} x -6 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c^{2} d x +3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b \,c^{2} d -3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c^{3}-6 a \,d^{2} x \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+8 b c d x \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-4 a c d \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+6 b \,c^{2} \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\right ) \sqrt {b x +a}}{3 \sqrt {b d}\, \left (a d -b c \right ) \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, d^{2} \left (d x +c \right )^{\frac {3}{2}}}\) | \(442\) |
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Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (80) = 160\).
Time = 0.32 (sec) , antiderivative size = 469, normalized size of antiderivative = 4.60 \[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\left [\frac {3 \, {\left (b c^{3} - a c^{2} d + {\left (b c d^{2} - a d^{3}\right )} x^{2} + 2 \, {\left (b c^{2} d - a c d^{2}\right )} x\right )} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (3 \, b c^{2} - 2 \, a c d + {\left (4 \, b c d - 3 \, a d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (b c^{3} d^{2} - a c^{2} d^{3} + {\left (b c d^{4} - a d^{5}\right )} x^{2} + 2 \, {\left (b c^{2} d^{3} - a c d^{4}\right )} x\right )}}, -\frac {3 \, {\left (b c^{3} - a c^{2} d + {\left (b c d^{2} - a d^{3}\right )} x^{2} + 2 \, {\left (b c^{2} d - a c d^{2}\right )} x\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) + 2 \, {\left (3 \, b c^{2} - 2 \, a c d + {\left (4 \, b c d - 3 \, a d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (b c^{3} d^{2} - a c^{2} d^{3} + {\left (b c d^{4} - a d^{5}\right )} x^{2} + 2 \, {\left (b c^{2} d^{3} - a c d^{4}\right )} x\right )}}\right ] \]
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\[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\int \frac {x \sqrt {a + b x}}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (80) = 160\).
Time = 0.35 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.84 \[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=-\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (4 \, b^{4} c d^{2} {\left | b \right |} - 3 \, a b^{3} d^{3} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{3} c d^{3} - a b^{2} d^{4}} + \frac {3 \, {\left (b^{5} c^{2} d {\left | b \right |} - 2 \, a b^{4} c d^{2} {\left | b \right |} + a^{2} b^{3} d^{3} {\left | b \right |}\right )}}{b^{3} c d^{3} - a b^{2} d^{4}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {2 \, {\left | b \right |} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d^{2}} \]
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Timed out. \[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\int \frac {x\,\sqrt {a+b\,x}}{{\left (c+d\,x\right )}^{5/2}} \,d x \]
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