Integrand size = 22, antiderivative size = 178 \[ \int \frac {(c+d x)^{3/2}}{x^3 (a+b x)^{3/2}} \, dx=\frac {3 (b c-a d) (5 b c-a d) \sqrt {c+d x}}{4 a^3 c \sqrt {a+b x}}+\frac {(5 b c-a d) (c+d x)^{3/2}}{4 a^2 c x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a c x^2 \sqrt {a+b x}}-\frac {3 (b c-a d) (5 b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{7/2} \sqrt {c}} \]
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Time = 0.06 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {98, 96, 95, 214} \[ \int \frac {(c+d x)^{3/2}}{x^3 (a+b x)^{3/2}} \, dx=-\frac {3 (b c-a d) (5 b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{7/2} \sqrt {c}}+\frac {3 \sqrt {c+d x} (b c-a d) (5 b c-a d)}{4 a^3 c \sqrt {a+b x}}+\frac {(c+d x)^{3/2} (5 b c-a d)}{4 a^2 c x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a c x^2 \sqrt {a+b x}} \]
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Rule 95
Rule 96
Rule 98
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x)^{5/2}}{2 a c x^2 \sqrt {a+b x}}-\frac {\left (\frac {5 b c}{2}-\frac {a d}{2}\right ) \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^{3/2}} \, dx}{2 a c} \\ & = \frac {(5 b c-a d) (c+d x)^{3/2}}{4 a^2 c x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a c x^2 \sqrt {a+b x}}+\frac {(3 (b c-a d) (5 b c-a d)) \int \frac {\sqrt {c+d x}}{x (a+b x)^{3/2}} \, dx}{8 a^2 c} \\ & = \frac {3 (b c-a d) (5 b c-a d) \sqrt {c+d x}}{4 a^3 c \sqrt {a+b x}}+\frac {(5 b c-a d) (c+d x)^{3/2}}{4 a^2 c x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a c x^2 \sqrt {a+b x}}+\frac {(3 (b c-a d) (5 b c-a d)) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 a^3} \\ & = \frac {3 (b c-a d) (5 b c-a d) \sqrt {c+d x}}{4 a^3 c \sqrt {a+b x}}+\frac {(5 b c-a d) (c+d x)^{3/2}}{4 a^2 c x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a c x^2 \sqrt {a+b x}}+\frac {(3 (b c-a d) (5 b c-a d)) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 a^3} \\ & = \frac {3 (b c-a d) (5 b c-a d) \sqrt {c+d x}}{4 a^3 c \sqrt {a+b x}}+\frac {(5 b c-a d) (c+d x)^{3/2}}{4 a^2 c x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a c x^2 \sqrt {a+b x}}-\frac {3 (b c-a d) (5 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{7/2} \sqrt {c}} \\ \end{align*}
Time = 10.07 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.73 \[ \int \frac {(c+d x)^{3/2}}{x^3 (a+b x)^{3/2}} \, dx=\frac {\sqrt {c+d x} \left (15 b^2 c x^2+a b x (5 c-13 d x)-a^2 (2 c+5 d x)\right )}{4 a^3 x^2 \sqrt {a+b x}}-\frac {3 \left (5 b^2 c^2-6 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{7/2} \sqrt {c}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(463\) vs. \(2(146)=292\).
Time = 1.70 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.61
method | result | size |
default | \(-\frac {\sqrt {d x +c}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b \,d^{2} x^{3}-18 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{2} c d \,x^{3}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{3} c^{2} x^{3}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} d^{2} x^{2}-18 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b c d \,x^{2}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} x^{2}+26 a b d \,x^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-30 b^{2} c \,x^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+10 a^{2} d x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-10 a b c x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+4 a^{2} c \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\right )}{8 a^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{2} \sqrt {a c}\, \sqrt {b x +a}}\) | \(464\) |
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Time = 0.43 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.66 \[ \int \frac {(c+d x)^{3/2}}{x^3 (a+b x)^{3/2}} \, dx=\left [\frac {3 \, {\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + {\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (2 \, a^{3} c^{2} - {\left (15 \, a b^{2} c^{2} - 13 \, a^{2} b c d\right )} x^{2} - 5 \, {\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, {\left (a^{4} b c x^{3} + a^{5} c x^{2}\right )}}, \frac {3 \, {\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + {\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{3} c^{2} - {\left (15 \, a b^{2} c^{2} - 13 \, a^{2} b c d\right )} x^{2} - 5 \, {\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, {\left (a^{4} b c x^{3} + a^{5} c x^{2}\right )}}\right ] \]
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\[ \int \frac {(c+d x)^{3/2}}{x^3 (a+b x)^{3/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{x^{3} \left (a + b x\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {(c+d x)^{3/2}}{x^3 (a+b x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1167 vs. \(2 (146) = 292\).
Time = 1.83 (sec) , antiderivative size = 1167, normalized size of antiderivative = 6.56 \[ \int \frac {(c+d x)^{3/2}}{x^3 (a+b x)^{3/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(c+d x)^{3/2}}{x^3 (a+b x)^{3/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}}{x^3\,{\left (a+b\,x\right )}^{3/2}} \,d x \]
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