Integrand size = 22, antiderivative size = 154 \[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^{3/2}} \, dx=\frac {15 (b c-a d)^2 \sqrt {c+d x}}{4 a^3 \sqrt {a+b x}}+\frac {5 (b c-a d) (c+d x)^{3/2}}{4 a^2 x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a x^2 \sqrt {a+b x}}-\frac {15 \sqrt {c} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{7/2}} \]
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Time = 0.05 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {96, 95, 214} \[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^{3/2}} \, dx=-\frac {15 \sqrt {c} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{7/2}}+\frac {15 \sqrt {c+d x} (b c-a d)^2}{4 a^3 \sqrt {a+b x}}+\frac {5 (c+d x)^{3/2} (b c-a d)}{4 a^2 x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a x^2 \sqrt {a+b x}} \]
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Rule 95
Rule 96
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x)^{5/2}}{2 a x^2 \sqrt {a+b x}}-\frac {(5 (b c-a d)) \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^{3/2}} \, dx}{4 a} \\ & = \frac {5 (b c-a d) (c+d x)^{3/2}}{4 a^2 x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a x^2 \sqrt {a+b x}}+\frac {\left (15 (b c-a d)^2\right ) \int \frac {\sqrt {c+d x}}{x (a+b x)^{3/2}} \, dx}{8 a^2} \\ & = \frac {15 (b c-a d)^2 \sqrt {c+d x}}{4 a^3 \sqrt {a+b x}}+\frac {5 (b c-a d) (c+d x)^{3/2}}{4 a^2 x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a x^2 \sqrt {a+b x}}+\frac {\left (15 c (b c-a d)^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 a^3} \\ & = \frac {15 (b c-a d)^2 \sqrt {c+d x}}{4 a^3 \sqrt {a+b x}}+\frac {5 (b c-a d) (c+d x)^{3/2}}{4 a^2 x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a x^2 \sqrt {a+b x}}+\frac {\left (15 c (b c-a 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^3} \\ & = \frac {15 (b c-a d)^2 \sqrt {c+d x}}{4 a^3 \sqrt {a+b x}}+\frac {5 (b c-a d) (c+d x)^{3/2}}{4 a^2 x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a x^2 \sqrt {a+b x}}-\frac {15 \sqrt {c} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{7/2}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.84 \[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^{3/2}} \, dx=\frac {\sqrt {c+d x} \left (15 b^2 c^2 x^2+5 a b c x (c-5 d x)+a^2 \left (-2 c^2-9 c d x+8 d^2 x^2\right )\right )}{4 a^3 x^2 \sqrt {a+b x}}-\frac {15 \sqrt {c} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{4 a^{7/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(506\) vs. \(2(122)=244\).
Time = 1.71 (sec) , antiderivative size = 507, normalized size of antiderivative = 3.29
method | result | size |
default | \(-\frac {\sqrt {d x +c}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b c \,d^{2} x^{3}-30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d \,x^{3}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{3} c^{3} x^{3}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} c \,d^{2} x^{2}-30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b \,c^{2} d \,x^{2}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{2} c^{3} x^{2}-16 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{2} x^{2}+50 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c d \,x^{2}-30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} x^{2}+18 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c d x -10 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{2} x +4 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c^{2} \sqrt {a c}\right )}{8 a^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{2} \sqrt {a c}\, \sqrt {b x +a}}\) | \(507\) |
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Time = 0.43 (sec) , antiderivative size = 481, normalized size of antiderivative = 3.12 \[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^{3/2}} \, dx=\left [\frac {15 \, {\left ({\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \sqrt {\frac {c}{a}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (2 \, a^{2} c^{2} - {\left (15 \, b^{2} c^{2} - 25 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{2} - {\left (5 \, a b c^{2} - 9 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}}, \frac {15 \, {\left ({\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \sqrt {-\frac {c}{a}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{2} c^{2} - {\left (15 \, b^{2} c^{2} - 25 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{2} - {\left (5 \, a b c^{2} - 9 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}}\right ] \]
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\[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^{3/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}}}{x^{3} \left (a + b x\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1206 vs. \(2 (122) = 244\).
Time = 1.92 (sec) , antiderivative size = 1206, normalized size of antiderivative = 7.83 \[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^{3/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^{3/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/2}}{x^3\,{\left (a+b\,x\right )}^{3/2}} \,d x \]
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