Integrand size = 20, antiderivative size = 127 \[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=-\frac {2 c (a+b x)^{3/2}}{d (b c-a d) \sqrt {c+d x}}+\frac {(3 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{d^2 (b c-a d)}-\frac {(3 b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{5/2}} \]
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Time = 0.05 (sec) , antiderivative size = 127, 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, 52, 65, 223, 212} \[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=-\frac {(3 b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{5/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (3 b c-a d)}{d^2 (b c-a d)}-\frac {2 c (a+b x)^{3/2}}{d \sqrt {c+d x} (b c-a d)} \]
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Rule 52
Rule 65
Rule 79
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {2 c (a+b x)^{3/2}}{d (b c-a d) \sqrt {c+d x}}+\frac {(3 b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{d (b c-a d)} \\ & = -\frac {2 c (a+b x)^{3/2}}{d (b c-a d) \sqrt {c+d x}}+\frac {(3 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{d^2 (b c-a d)}-\frac {(3 b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 d^2} \\ & = -\frac {2 c (a+b x)^{3/2}}{d (b c-a d) \sqrt {c+d x}}+\frac {(3 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{d^2 (b c-a d)}-\frac {(3 b c-a d) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b d^2} \\ & = -\frac {2 c (a+b x)^{3/2}}{d (b c-a d) \sqrt {c+d x}}+\frac {(3 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{d^2 (b c-a d)}-\frac {(3 b c-a d) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{b d^2} \\ & = -\frac {2 c (a+b x)^{3/2}}{d (b c-a d) \sqrt {c+d x}}+\frac {(3 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{d^2 (b c-a d)}-\frac {(3 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{5/2}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.64 \[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=\frac {\sqrt {a+b x} (3 c+d x)}{d^2 \sqrt {c+d x}}-\frac {(3 b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(263\) vs. \(2(107)=214\).
Time = 1.52 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.08
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \left (\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,d^{2} x -3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b c d x +\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a c d -3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b \,c^{2}+2 d x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+6 c \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\right )}{2 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, d^{2} \sqrt {d x +c}}\) | \(264\) |
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Time = 0.28 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.43 \[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=\left [-\frac {{\left (3 \, b c^{2} - a c d + {\left (3 \, b c d - a d^{2}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (b d^{2} x + 3 \, b c d\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, {\left (b d^{4} x + b c d^{3}\right )}}, \frac {{\left (3 \, b c^{2} - a c d + {\left (3 \, b c d - a d^{2}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (b d^{2} x + 3 \, b c d\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b d^{4} x + b c d^{3}\right )}}\right ] \]
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\[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=\int \frac {x \sqrt {a + b x}}{\left (c + d x\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.33 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.06 \[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=\frac {\sqrt {b x + a} {\left (\frac {{\left (b x + a\right )} {\left | b \right |}}{b d} + \frac {3 \, b^{2} c d {\left | b \right |} - a b d^{2} {\left | b \right |}}{b^{2} d^{3}}\right )}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {{\left (3 \, b c {\left | b \right |} - a d {\left | b \right |}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b d^{2}} \]
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Timed out. \[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=\int \frac {x\,\sqrt {a+b\,x}}{{\left (c+d\,x\right )}^{3/2}} \,d x \]
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