Integrand size = 22, antiderivative size = 177 \[ \int \frac {(c+d x)^{5/2}}{x^3 \sqrt {a+b x}} \, dx=\frac {c (3 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a^2 x}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}-\frac {\sqrt {c} \left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2}}+\frac {2 d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}} \]
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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} \[ \int \frac {(c+d x)^{5/2}}{x^3 \sqrt {a+b x}} \, dx=\frac {c \sqrt {a+b x} \sqrt {c+d x} (3 b c-7 a d)}{4 a^2 x}-\frac {\sqrt {c} \left (15 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2}}+\frac {2 d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2} \]
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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 {c \sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}-\frac {\int \frac {\sqrt {c+d x} \left (\frac {1}{2} c (3 b c-7 a d)-2 a d^2 x\right )}{x^2 \sqrt {a+b x}} \, dx}{2 a} \\ & = \frac {c (3 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a^2 x}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}-\frac {\int \frac {-\frac {1}{4} c \left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right )-2 a^2 d^3 x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 a^2} \\ & = \frac {c (3 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a^2 x}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}+d^3 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {\left (c \left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 a^2} \\ & = \frac {c (3 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a^2 x}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}+\frac {\left (2 d^3\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}+\frac {\left (c \left (3 b^2 c^2-10 a b c d+15 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 a^2} \\ & = \frac {c (3 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a^2 x}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}-\frac {\sqrt {c} \left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2}}+\frac {\left (2 d^3\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} \\ & = \frac {c (3 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a^2 x}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}-\frac {\sqrt {c} \left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2}}+\frac {2 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}} \\ \end{align*}
Time = 0.76 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.86 \[ \int \frac {(c+d x)^{5/2}}{x^3 \sqrt {a+b x}} \, dx=\frac {1}{4} \left (\frac {c \sqrt {a+b x} \sqrt {c+d x} (-2 a c+3 b c x-9 a d x)}{a^2 x^2}-\frac {\sqrt {c} \left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2}}+\frac {8 d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(353\) vs. \(2(139)=278\).
Time = 0.58 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.00
method | result | size |
default | \(-\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} c \,d^{2} x^{2} \sqrt {b d}-10 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b \,c^{2} d \,x^{2} \sqrt {b d}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{3} x^{2} \sqrt {b d}-8 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} d^{3} x^{2} \sqrt {a c}+18 a c d x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-6 b \,c^{2} x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+4 a \,c^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\right )}{8 a^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{2} \sqrt {b d}\, \sqrt {a c}}\) | \(354\) |
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Time = 1.00 (sec) , antiderivative size = 1031, normalized size of antiderivative = 5.82 \[ \int \frac {(c+d x)^{5/2}}{x^3 \sqrt {a+b x}} \, dx=\left [\frac {8 \, a^{2} d^{2} x^{2} \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 (3 \, b^{2} c^{2} - 10 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{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 ) - 4 \, {\left (2 \, a c^{2} - 3 \, {\left (b c^{2} - 3 \, a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, a^{2} x^{2}}, -\frac {16 \, a^{2} d^{2} x^{2} \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 (3 \, b^{2} c^{2} - 10 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{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 ) + 4 \, {\left (2 \, a c^{2} - 3 \, {\left (b c^{2} - 3 \, a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, a^{2} x^{2}}, \frac {4 \, a^{2} d^{2} x^{2} \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 (3 \, b^{2} c^{2} - 10 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{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 ) - 2 \, {\left (2 \, a c^{2} - 3 \, {\left (b c^{2} - 3 \, a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, a^{2} x^{2}}, -\frac {8 \, a^{2} d^{2} x^{2} \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 (3 \, b^{2} c^{2} - 10 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{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 ) + 2 \, {\left (2 \, a c^{2} - 3 \, {\left (b c^{2} - 3 \, a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, a^{2} x^{2}}\right ] \]
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\[ \int \frac {(c+d x)^{5/2}}{x^3 \sqrt {a+b x}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}}}{x^{3} \sqrt {a + b x}}\, dx \]
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Exception generated. \[ \int \frac {(c+d x)^{5/2}}{x^3 \sqrt {a+b x}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1149 vs. \(2 (139) = 278\).
Time = 0.92 (sec) , antiderivative size = 1149, normalized size of antiderivative = 6.49 \[ \int \frac {(c+d x)^{5/2}}{x^3 \sqrt {a+b x}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(c+d x)^{5/2}}{x^3 \sqrt {a+b x}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/2}}{x^3\,\sqrt {a+b\,x}} \,d x \]
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