Integrand size = 22, antiderivative size = 171 \[ \int \frac {(c+d x)^{5/2}}{x \sqrt {a+b x}} \, dx=\frac {d (7 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^2}+\frac {d \sqrt {a+b x} (c+d x)^{3/2}}{2 b}-\frac {2 c^{5/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}+\frac {\sqrt {d} \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2}} \]
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Time = 0.11 (sec) , antiderivative size = 171, 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 = {104, 159, 163, 65, 223, 212, 95, 214} \[ \int \frac {(c+d x)^{5/2}}{x \sqrt {a+b x}} \, dx=\frac {\sqrt {d} \left (3 a^2 d^2-10 a b c d+15 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2}}-\frac {2 c^{5/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}+\frac {d \sqrt {a+b x} \sqrt {c+d x} (7 b c-3 a d)}{4 b^2}+\frac {d \sqrt {a+b x} (c+d x)^{3/2}}{2 b} \]
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Rule 65
Rule 95
Rule 104
Rule 159
Rule 163
Rule 212
Rule 214
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {d \sqrt {a+b x} (c+d x)^{3/2}}{2 b}+\frac {\int \frac {\sqrt {c+d x} \left (2 b c^2+\frac {1}{2} d (7 b c-3 a d) x\right )}{x \sqrt {a+b x}} \, dx}{2 b} \\ & = \frac {d (7 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^2}+\frac {d \sqrt {a+b x} (c+d x)^{3/2}}{2 b}+\frac {\int \frac {2 b^2 c^3+\frac {1}{4} d \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 b^2} \\ & = \frac {d (7 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^2}+\frac {d \sqrt {a+b x} (c+d x)^{3/2}}{2 b}+c^3 \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+3 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 b^2} \\ & = \frac {d (7 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^2}+\frac {d \sqrt {a+b x} (c+d x)^{3/2}}{2 b}+\left (2 c^3\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+3 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^3} \\ & = \frac {d (7 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^2}+\frac {d \sqrt {a+b x} (c+d x)^{3/2}}{2 b}-\frac {2 c^{5/2} \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+3 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^3} \\ & = \frac {d (7 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^2}+\frac {d \sqrt {a+b x} (c+d x)^{3/2}}{2 b}-\frac {2 c^{5/2} \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+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2}} \\ \end{align*}
Time = 0.58 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.87 \[ \int \frac {(c+d x)^{5/2}}{x \sqrt {a+b x}} \, dx=\frac {1}{4} \left (\frac {d \sqrt {a+b x} \sqrt {c+d x} (9 b c-3 a d+2 b d x)}{b^2}-\frac {8 c^{5/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}+\frac {\sqrt {d} \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(341\) vs. \(2(133)=266\).
Time = 0.57 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.00
method | result | size |
default | \(-\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (8 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{3} \sqrt {b d}-4 b \,d^{2} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}-3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} d^{3} \sqrt {a c}+10 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b c \,d^{2} \sqrt {a c}-15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c^{2} d \sqrt {a c}+6 a \,d^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}-18 b c d \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\right )}{8 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} \sqrt {b d}\, \sqrt {a c}}\) | \(342\) |
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Time = 1.53 (sec) , antiderivative size = 987, normalized size of antiderivative = 5.77 \[ \int \frac {(c+d x)^{5/2}}{x \sqrt {a+b x}} \, dx=\left [\frac {8 \, b^{2} c^{2} \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 ) + {\left (15 \, b^{2} c^{2} - 10 \, a b c d + 3 \, a^{2} d^{2}\right )} \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 (2 \, b d^{2} x + 9 \, b c d - 3 \, a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, b^{2}}, \frac {4 \, b^{2} c^{2} \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 ) - {\left (15 \, b^{2} c^{2} - 10 \, a b c d + 3 \, a^{2} d^{2}\right )} \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 (2 \, b d^{2} x + 9 \, b c d - 3 \, a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, b^{2}}, \frac {16 \, b^{2} c^{2} \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 (15 \, b^{2} c^{2} - 10 \, a b c d + 3 \, a^{2} d^{2}\right )} \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 (2 \, b d^{2} x + 9 \, b c d - 3 \, a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, b^{2}}, \frac {8 \, b^{2} c^{2} \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 (15 \, b^{2} c^{2} - 10 \, a b c d + 3 \, a^{2} d^{2}\right )} \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 (2 \, b d^{2} x + 9 \, b c d - 3 \, a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, b^{2}}\right ] \]
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\[ \int \frac {(c+d x)^{5/2}}{x \sqrt {a+b x}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}}}{x \sqrt {a + b x}}\, dx \]
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Exception generated. \[ \int \frac {(c+d x)^{5/2}}{x \sqrt {a+b x}} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {(c+d x)^{5/2}}{x \sqrt {a+b x}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {(c+d x)^{5/2}}{x \sqrt {a+b x}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/2}}{x\,\sqrt {a+b\,x}} \,d x \]
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