Integrand size = 22, antiderivative size = 160 \[ \int \frac {(c+d x)^{5/2}}{x^2 \sqrt {a+b x}} \, dx=\frac {d (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{a b}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{a x}+\frac {c^{3/2} (b c-5 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2}}+\frac {d^{3/2} (5 b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2}} \]
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Time = 0.11 (sec) , antiderivative size = 160, 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, 159, 163, 65, 223, 212, 95, 214} \[ \int \frac {(c+d x)^{5/2}}{x^2 \sqrt {a+b x}} \, dx=\frac {c^{3/2} (b c-5 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2}}+\frac {d^{3/2} (5 b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2}}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{a x}+\frac {d \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{a b} \]
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Rule 65
Rule 95
Rule 100
Rule 159
Rule 163
Rule 212
Rule 214
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{a x}-\frac {\int \frac {\sqrt {c+d x} \left (\frac {1}{2} c (b c-5 a d)-d (b c+a d) x\right )}{x \sqrt {a+b x}} \, dx}{a} \\ & = \frac {d (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{a b}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{a x}-\frac {\int \frac {\frac {1}{2} b c^2 (b c-5 a d)-\frac {1}{2} a d^2 (5 b c-a d) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{a b} \\ & = \frac {d (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{a b}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{a x}-\frac {\left (c^2 (b c-5 a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 a}+\frac {\left (d^2 (5 b c-a d)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 b} \\ & = \frac {d (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{a b}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{a x}-\frac {\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 )}{a}+\frac {\left (d^2 (5 b c-a d)\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 )}{b^2} \\ & = \frac {d (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{a b}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{a 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 )}{a^{3/2}}+\frac {\left (d^2 (5 b c-a d)\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 )}{b^2} \\ & = \frac {d (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{a b}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{a 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 )}{a^{3/2}}+\frac {d^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2}} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.88 \[ \int \frac {(c+d x)^{5/2}}{x^2 \sqrt {a+b x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-b c^2+a d^2 x\right )}{a b x}+\frac {c^{3/2} (b c-5 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2}}+\frac {d^{3/2} (5 b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(319\) vs. \(2(128)=256\).
Time = 1.64 (sec) , antiderivative size = 320, normalized size of antiderivative = 2.00
method | result | size |
default | \(-\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} d^{3} x \sqrt {a c}-5 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b c \,d^{2} x \sqrt {a c}+5 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b \,c^{2} d x \sqrt {b d}-\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{3} x \sqrt {b d}-2 a \,d^{2} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}+2 b \,c^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\right )}{2 a \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x \sqrt {b d}\, \sqrt {a c}\, b}\) | \(320\) |
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none
Time = 1.25 (sec) , antiderivative size = 991, normalized size of antiderivative = 6.19 \[ \int \frac {(c+d x)^{5/2}}{x^2 \sqrt {a+b x}} \, dx=\left [-\frac {{\left (5 \, a b c d - a^{2} d^{2}\right )} x \sqrt {\frac {d}{b}} \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^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + {\left (b^{2} c^{2} - 5 \, a b c d\right )} x \sqrt {\frac {c}{a}} \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^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (a d^{2} x - b c^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, a b x}, -\frac {2 \, {\left (5 \, a b c d - a^{2} d^{2}\right )} x \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right ) + {\left (b^{2} c^{2} - 5 \, a b c d\right )} x \sqrt {\frac {c}{a}} \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^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (a d^{2} x - b c^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, a b x}, -\frac {2 \, {\left (b^{2} c^{2} - 5 \, a b c d\right )} x \sqrt {-\frac {c}{a}} \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 {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) + {\left (5 \, a b c d - a^{2} d^{2}\right )} x \sqrt {\frac {d}{b}} \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^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (a d^{2} x - b c^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, a b x}, -\frac {{\left (b^{2} c^{2} - 5 \, a b c d\right )} x \sqrt {-\frac {c}{a}} \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 {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) + {\left (5 \, a b c d - a^{2} d^{2}\right )} x \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right ) - 2 \, {\left (a d^{2} x - b c^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, a b x}\right ] \]
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\[ \int \frac {(c+d x)^{5/2}}{x^2 \sqrt {a+b x}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}}}{x^{2} \sqrt {a + b x}}\, dx \]
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Exception generated. \[ \int \frac {(c+d x)^{5/2}}{x^2 \sqrt {a+b x}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (128) = 256\).
Time = 0.51 (sec) , antiderivative size = 547, normalized size of antiderivative = 3.42 \[ \int \frac {(c+d x)^{5/2}}{x^2 \sqrt {a+b x}} \, dx=\frac {\frac {2 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} d^{2} {\left | b \right |}}{b^{2}} - \frac {{\left (5 \, \sqrt {b d} b c d {\left | b \right |} - \sqrt {b d} a d^{2} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{b^{2}} + \frac {2 \, {\left (\sqrt {b d} b^{2} c^{3} {\left | b \right |} - 5 \, \sqrt {b d} a b c^{2} d {\left | b \right |}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} a b} - \frac {4 \, {\left (\sqrt {b d} b^{4} c^{4} {\left | b \right |} - 2 \, \sqrt {b d} a b^{3} c^{3} d {\left | b \right |} + \sqrt {b d} a^{2} b^{2} c^{2} d^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c^{3} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b c^{2} d {\left | b \right |}\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} a}}{2 \, b} \]
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Timed out. \[ \int \frac {(c+d x)^{5/2}}{x^2 \sqrt {a+b x}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/2}}{x^2\,\sqrt {a+b\,x}} \,d x \]
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