Integrand size = 22, antiderivative size = 170 \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^3} \, dx=-\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 x^2}+\frac {\left (b^2 c^2-6 a b c d-3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} \sqrt {c}}+2 \sqrt {b} d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 170, 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 = {99, 154, 163, 65, 223, 212, 95, 214} \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^3} \, dx=\frac {\left (-3 a^2 d^2-6 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} \sqrt {c}}+2 \sqrt {b} d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 x^2}-\frac {\sqrt {a+b x} \sqrt {c+d x} (3 a d+b c)}{4 a x} \]
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Rule 65
Rule 95
Rule 99
Rule 154
Rule 163
Rule 212
Rule 214
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 x^2}+\frac {1}{2} \int \frac {\sqrt {c+d x} \left (\frac {1}{2} (b c+3 a d)+2 b d x\right )}{x^2 \sqrt {a+b x}} \, dx \\ & = -\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 x^2}+\frac {\int \frac {\frac {1}{4} \left (-b^2 c^2+6 a b c d+3 a^2 d^2\right )+2 a b d^2 x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 a} \\ & = -\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 x^2}+\left (b d^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {\left (-b^2 c^2+6 a b c d+3 a^2 d^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 a} \\ & = -\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 x^2}+\left (2 d^2\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 )+\frac {\left (-b^2 c^2+6 a b c d+3 a^2 d^2\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} \\ & = -\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 x^2}+\frac {\left (b^2 c^2-6 a b c d-3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} \sqrt {c}}+\left (2 d^2\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 ) \\ & = -\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 x^2}+\frac {\left (b^2 c^2-6 a b c d-3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} \sqrt {c}}+2 \sqrt {b} d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^3} \, dx=\frac {1}{4} \left (-\frac {\sqrt {a+b x} \sqrt {c+d x} (2 a c+b c x+5 a d x)}{a x^2}+\frac {\left (b^2 c^2-6 a b c d-3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} \sqrt {c}}+8 \sqrt {b} d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(344\) vs. \(2(132)=264\).
Time = 0.53 (sec) , antiderivative size = 345, normalized size of antiderivative = 2.03
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} d^{2} x^{2} \sqrt {b d}+6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b c d \,x^{2} \sqrt {b d}-\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{2} x^{2} \sqrt {b d}-8 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b \,d^{2} x^{2} \sqrt {a c}+10 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a d x +2 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c x +4 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a c \sqrt {b d}\, \sqrt {a c}\right )}{8 a \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{2} \sqrt {b d}\, \sqrt {a c}}\) | \(345\) |
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none
Time = 0.62 (sec) , antiderivative size = 1023, normalized size of antiderivative = 6.02 \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^3} \, dx=\left [\frac {8 \, \sqrt {b d} a^{2} c d x^{2} \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 ) - {\left (b^{2} c^{2} - 6 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {a c} x^{2} \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 c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (2 \, a^{2} c^{2} + {\left (a b c^{2} + 5 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, a^{2} c x^{2}}, -\frac {16 \, \sqrt {-b d} a^{2} c d x^{2} \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 ) + {\left (b^{2} c^{2} - 6 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {a c} x^{2} \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 c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (2 \, a^{2} c^{2} + {\left (a b c^{2} + 5 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, a^{2} c x^{2}}, \frac {4 \, \sqrt {b d} a^{2} c d x^{2} \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 ) - {\left (b^{2} c^{2} - 6 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {-a c} x^{2} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{2} c^{2} + {\left (a b c^{2} + 5 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, a^{2} c x^{2}}, -\frac {8 \, \sqrt {-b d} a^{2} c d x^{2} \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 ) + {\left (b^{2} c^{2} - 6 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {-a c} x^{2} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (2 \, a^{2} c^{2} + {\left (a b c^{2} + 5 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, a^{2} c x^{2}}\right ] \]
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\[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^3} \, dx=\int \frac {\sqrt {a + b x} \left (c + d x\right )^{\frac {3}{2}}}{x^{3}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^3} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1127 vs. \(2 (132) = 264\).
Time = 0.87 (sec) , antiderivative size = 1127, normalized size of antiderivative = 6.63 \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^3} \, dx=\int \frac {\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{3/2}}{x^3} \,d x \]
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