Integrand size = 22, antiderivative size = 162 \[ \int \frac {(a+b x)^{5/2}}{x^2 \sqrt {c+d x}} \, dx=\frac {b (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{c d}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{c x}-\frac {a^{3/2} (5 b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}-\frac {b^{3/2} (b c-5 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{3/2}} \]
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Time = 0.12 (sec) , antiderivative size = 162, 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, 159, 163, 65, 223, 212, 95, 214} \[ \int \frac {(a+b x)^{5/2}}{x^2 \sqrt {c+d x}} \, dx=-\frac {a^{3/2} (5 b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}-\frac {b^{3/2} (b c-5 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{3/2}}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{c x}+\frac {b \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{c d} \]
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Rule 65
Rule 95
Rule 100
Rule 159
Rule 163
Rule 212
Rule 214
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{c x}-\frac {\int \frac {\sqrt {a+b x} \left (-\frac {1}{2} a (5 b c-a d)-b (b c+a d) x\right )}{x \sqrt {c+d x}} \, dx}{c} \\ & = \frac {b (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{c d}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{c x}-\frac {\int \frac {-\frac {1}{2} a^2 d (5 b c-a d)+\frac {1}{2} b^2 c (b c-5 a d) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{c d} \\ & = \frac {b (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{c d}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{c x}-\frac {\left (b^2 (b c-5 a d)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 d}+\frac {\left (a^2 (5 b c-a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 c} \\ & = \frac {b (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{c d}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{c x}-\frac {(b (b c-5 a d)) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{d}+\frac {\left (a^2 (5 b c-a d)\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{c} \\ & = \frac {b (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{c d}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{c x}-\frac {a^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}-\frac {(b (b c-5 a d)) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{d} \\ & = \frac {b (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{c d}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{c x}-\frac {a^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}-\frac {b^{3/2} (b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{3/2}} \\ \end{align*}
Time = 0.53 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^{5/2}}{x^2 \sqrt {c+d x}} \, dx=\frac {\sqrt {a+b x} \left (-a^2 d+b^2 c x\right ) \sqrt {c+d x}}{c d x}+\frac {a^{3/2} (-5 b c+a d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{c^{3/2}}-\frac {b^{3/2} (b c-5 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{d^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(319\) vs. \(2(130)=260\).
Time = 1.60 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.98
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (5 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c d x \sqrt {a c}-\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{2} x \sqrt {a c}+\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} d^{2} x \sqrt {b d}-5 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b c d x \sqrt {b d}+2 b^{2} c x \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-2 a^{2} d \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\right )}{2 c \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x \sqrt {b d}\, \sqrt {a c}\, d}\) | \(320\) |
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Time = 1.17 (sec) , antiderivative size = 993, normalized size of antiderivative = 6.13 \[ \int \frac {(a+b x)^{5/2}}{x^2 \sqrt {c+d x}} \, dx=\left [-\frac {{\left (b^{2} c^{2} - 5 \, a b c d\right )} x \sqrt {\frac {b}{d}} \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^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + {\left (5 \, a b c d - a^{2} d^{2}\right )} x \sqrt {\frac {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^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (b^{2} c x - a^{2} d\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, c d x}, \frac {2 \, {\left (b^{2} c^{2} - 5 \, a b c d\right )} x \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) - {\left (5 \, a b c d - a^{2} d^{2}\right )} x \sqrt {\frac {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^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (b^{2} c x - a^{2} d\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, c d x}, \frac {2 \, {\left (5 \, a b c d - a^{2} d^{2}\right )} x \sqrt {-\frac {a}{c}} \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 {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) - {\left (b^{2} c^{2} - 5 \, a b c d\right )} x \sqrt {\frac {b}{d}} \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^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (b^{2} c x - a^{2} d\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, c d x}, \frac {{\left (5 \, a b c d - a^{2} d^{2}\right )} x \sqrt {-\frac {a}{c}} \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 {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) + {\left (b^{2} c^{2} - 5 \, a b c d\right )} x \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) + 2 \, {\left (b^{2} c x - a^{2} d\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, c d x}\right ] \]
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\[ \int \frac {(a+b x)^{5/2}}{x^2 \sqrt {c+d x}} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}}}{x^{2} \sqrt {c + d x}}\, dx \]
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Exception generated. \[ \int \frac {(a+b x)^{5/2}}{x^2 \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (130) = 260\).
Time = 0.45 (sec) , antiderivative size = 526, normalized size of antiderivative = 3.25 \[ \int \frac {(a+b x)^{5/2}}{x^2 \sqrt {c+d x}} \, dx=\frac {b {\left (\frac {2 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} b}{d} + \frac {{\left (\sqrt {b d} b^{2} c - 5 \, \sqrt {b d} a b d\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{d^{2}} - \frac {2 \, {\left (5 \, \sqrt {b d} a^{2} b^{2} c - \sqrt {b d} a^{3} b d\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b c} - \frac {4 \, {\left (\sqrt {b d} a^{2} b^{4} c^{2} - 2 \, \sqrt {b d} a^{3} b^{3} c d + \sqrt {b d} a^{4} b^{2} d^{2} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{2} c - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} c}\right )}}{2 \, {\left | b \right |}} \]
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Timed out. \[ \int \frac {(a+b x)^{5/2}}{x^2 \sqrt {c+d x}} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}}{x^2\,\sqrt {c+d\,x}} \,d x \]
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