Integrand size = 22, antiderivative size = 220 \[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^{5/2}} \, dx=\frac {5 (7 b c-3 a d) (b c-a d) \sqrt {c+d x}}{4 a^4 \sqrt {a+b x}}+\frac {5 (7 b c-3 a d) (b c-a d) (c+d x)^{3/2}}{12 a^3 c (a+b x)^{3/2}}+\frac {(7 b c-3 a d) (c+d x)^{5/2}}{4 a^2 c x (a+b x)^{3/2}}-\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}}-\frac {5 \sqrt {c} (7 b c-3 a d) (b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{9/2}} \]
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Time = 0.08 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {98, 96, 95, 214} \[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^{5/2}} \, dx=-\frac {5 \sqrt {c} (7 b c-3 a d) (b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{9/2}}+\frac {5 \sqrt {c+d x} (7 b c-3 a d) (b c-a d)}{4 a^4 \sqrt {a+b x}}+\frac {5 (c+d x)^{3/2} (7 b c-3 a d) (b c-a d)}{12 a^3 c (a+b x)^{3/2}}+\frac {(c+d x)^{5/2} (7 b c-3 a d)}{4 a^2 c x (a+b x)^{3/2}}-\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}} \]
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Rule 95
Rule 96
Rule 98
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}}-\frac {\left (\frac {7 b c}{2}-\frac {3 a d}{2}\right ) \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx}{2 a c} \\ & = \frac {(7 b c-3 a d) (c+d x)^{5/2}}{4 a^2 c x (a+b x)^{3/2}}-\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}}+\frac {(5 (7 b c-3 a d) (b c-a d)) \int \frac {(c+d x)^{3/2}}{x (a+b x)^{5/2}} \, dx}{8 a^2 c} \\ & = \frac {5 (7 b c-3 a d) (b c-a d) (c+d x)^{3/2}}{12 a^3 c (a+b x)^{3/2}}+\frac {(7 b c-3 a d) (c+d x)^{5/2}}{4 a^2 c x (a+b x)^{3/2}}-\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}}+\frac {(5 (7 b c-3 a d) (b c-a d)) \int \frac {\sqrt {c+d x}}{x (a+b x)^{3/2}} \, dx}{8 a^3} \\ & = \frac {5 (7 b c-3 a d) (b c-a d) \sqrt {c+d x}}{4 a^4 \sqrt {a+b x}}+\frac {5 (7 b c-3 a d) (b c-a d) (c+d x)^{3/2}}{12 a^3 c (a+b x)^{3/2}}+\frac {(7 b c-3 a d) (c+d x)^{5/2}}{4 a^2 c x (a+b x)^{3/2}}-\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}}+\frac {(5 c (7 b c-3 a d) (b c-a d)) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 a^4} \\ & = \frac {5 (7 b c-3 a d) (b c-a d) \sqrt {c+d x}}{4 a^4 \sqrt {a+b x}}+\frac {5 (7 b c-3 a d) (b c-a d) (c+d x)^{3/2}}{12 a^3 c (a+b x)^{3/2}}+\frac {(7 b c-3 a d) (c+d x)^{5/2}}{4 a^2 c x (a+b x)^{3/2}}-\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}}+\frac {(5 c (7 b c-3 a d) (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^4} \\ & = \frac {5 (7 b c-3 a d) (b c-a d) \sqrt {c+d x}}{4 a^4 \sqrt {a+b x}}+\frac {5 (7 b c-3 a d) (b c-a d) (c+d x)^{3/2}}{12 a^3 c (a+b x)^{3/2}}+\frac {(7 b c-3 a d) (c+d x)^{5/2}}{4 a^2 c x (a+b x)^{3/2}}-\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}}-\frac {5 \sqrt {c} (7 b c-3 a d) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{9/2}} \\ \end{align*}
Time = 10.20 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.72 \[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^{5/2}} \, dx=\frac {-3 a^{7/2} (c+d x)^{7/2}+\frac {1}{2} (7 b c-3 a d) x \left (3 a^{5/2} (c+d x)^{5/2}+5 (b c-a d) x \left (\sqrt {a} \sqrt {c+d x} (4 a c+3 b c x+a d x)-3 c^{3/2} (a+b x)^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )\right )}{6 a^{9/2} c x^2 (a+b x)^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(757\) vs. \(2(182)=364\).
Time = 0.60 (sec) , antiderivative size = 758, normalized size of antiderivative = 3.45
method | result | size |
default | \(-\frac {\sqrt {d x +c}\, \left (45 \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^{2} c \,d^{2} x^{4}-150 \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^{3} c^{2} d \,x^{4}+105 \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^{4} c^{3} x^{4}+90 \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} b c \,d^{2} x^{3}-300 \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^{2} c^{2} d \,x^{3}+210 \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^{3} c^{3} x^{3}+45 \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^{4} c \,d^{2} x^{2}-150 \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} b \,c^{2} d \,x^{2}+105 \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^{2} c^{3} x^{2}-32 a^{2} b \,d^{2} x^{3} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+230 a \,b^{2} c d \,x^{3} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-210 b^{3} c^{2} x^{3} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-48 a^{3} d^{2} x^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+316 a^{2} b c d \,x^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-280 a \,b^{2} c^{2} x^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+54 a^{3} c d x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-42 a^{2} b \,c^{2} x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+12 a^{3} c^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\right )}{24 a^{4} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{2} \sqrt {a c}\, \left (b x +a \right )^{\frac {3}{2}}}\) | \(758\) |
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Time = 0.95 (sec) , antiderivative size = 659, normalized size of antiderivative = 3.00 \[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^{5/2}} \, dx=\left [\frac {15 \, {\left ({\left (7 \, b^{4} c^{2} - 10 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} x^{4} + 2 \, {\left (7 \, a b^{3} c^{2} - 10 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x^{3} + {\left (7 \, a^{2} b^{2} c^{2} - 10 \, a^{3} b c d + 3 \, a^{4} 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 (6 \, a^{3} c^{2} - {\left (105 \, b^{3} c^{2} - 115 \, a b^{2} c d + 16 \, a^{2} b d^{2}\right )} x^{3} - 2 \, {\left (70 \, a b^{2} c^{2} - 79 \, a^{2} b c d + 12 \, a^{3} d^{2}\right )} x^{2} - 3 \, {\left (7 \, a^{2} b c^{2} - 9 \, a^{3} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}, \frac {15 \, {\left ({\left (7 \, b^{4} c^{2} - 10 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} x^{4} + 2 \, {\left (7 \, a b^{3} c^{2} - 10 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x^{3} + {\left (7 \, a^{2} b^{2} c^{2} - 10 \, a^{3} b c d + 3 \, a^{4} 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 (6 \, a^{3} c^{2} - {\left (105 \, b^{3} c^{2} - 115 \, a b^{2} c d + 16 \, a^{2} b d^{2}\right )} x^{3} - 2 \, {\left (70 \, a b^{2} c^{2} - 79 \, a^{2} b c d + 12 \, a^{3} d^{2}\right )} x^{2} - 3 \, {\left (7 \, a^{2} b c^{2} - 9 \, a^{3} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{24 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}\right ] \]
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\[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^{5/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}}}{x^{3} \left (a + b x\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1694 vs. \(2 (182) = 364\).
Time = 1.51 (sec) , antiderivative size = 1694, normalized size of antiderivative = 7.70 \[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^{5/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^{5/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/2}}{x^3\,{\left (a+b\,x\right )}^{5/2}} \,d x \]
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