Integrand size = 22, antiderivative size = 197 \[ \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^2} \, dx=\frac {b (b c+11 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d}+\frac {3}{2} b (a+b x)^{3/2} \sqrt {c+d x}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}-\frac {a^{3/2} (5 b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}-\frac {\sqrt {b} \left (b^2 c^2-10 a b c d-15 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 d^{3/2}} \]
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Time = 0.14 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {99, 159, 163, 65, 223, 212, 95, 214} \[ \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^2} \, dx=-\frac {a^{3/2} (a d+5 b c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}-\frac {\sqrt {b} \left (-15 a^2 d^2-10 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 d^{3/2}}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}+\frac {3}{2} b (a+b x)^{3/2} \sqrt {c+d x}+\frac {b \sqrt {a+b x} \sqrt {c+d x} (11 a d+b c)}{4 d} \]
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Rule 65
Rule 95
Rule 99
Rule 159
Rule 163
Rule 212
Rule 214
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}+\int \frac {(a+b x)^{3/2} \left (\frac {1}{2} (5 b c+a d)+3 b d x\right )}{x \sqrt {c+d x}} \, dx \\ & = \frac {3}{2} b (a+b x)^{3/2} \sqrt {c+d x}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}+\frac {\int \frac {\sqrt {a+b x} \left (a d (5 b c+a d)+\frac {1}{2} b d (b c+11 a d) x\right )}{x \sqrt {c+d x}} \, dx}{2 d} \\ & = \frac {b (b c+11 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d}+\frac {3}{2} b (a+b x)^{3/2} \sqrt {c+d x}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}+\frac {\int \frac {a^2 d^2 (5 b c+a d)-\frac {1}{4} b d \left (b^2 c^2-10 a b c d-15 a^2 d^2\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 d^2} \\ & = \frac {b (b c+11 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d}+\frac {3}{2} b (a+b x)^{3/2} \sqrt {c+d x}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}+\frac {1}{2} \left (a^2 (5 b c+a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx-\frac {\left (b \left (b^2 c^2-10 a b c d-15 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 d} \\ & = \frac {b (b c+11 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d}+\frac {3}{2} b (a+b x)^{3/2} \sqrt {c+d x}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}+\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 )-\frac {\left (b^2 c^2-10 a b c d-15 a^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 )}{4 d} \\ & = \frac {b (b c+11 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d}+\frac {3}{2} b (a+b x)^{3/2} \sqrt {c+d x}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{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 )}{\sqrt {c}}-\frac {\left (b^2 c^2-10 a b c d-15 a^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 )}{4 d} \\ & = \frac {b (b c+11 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d}+\frac {3}{2} b (a+b x)^{3/2} \sqrt {c+d x}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{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 )}{\sqrt {c}}-\frac {\sqrt {b} \left (b^2 c^2-10 a b c d-15 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 d^{3/2}} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^2} \, dx=\frac {1}{4} \left (\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-4 a^2 d+9 a b d x+b^2 x (c+2 d x)\right )}{d x}-\frac {4 a^{3/2} (5 b c+a d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{\sqrt {c}}-\frac {\sqrt {b} \left (b^2 c^2-10 a b c d-15 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{d^{3/2}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(433\) vs. \(2(155)=310\).
Time = 0.56 (sec) , antiderivative size = 434, normalized size of antiderivative = 2.20
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b \,d^{2} x \sqrt {a c}+10 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c d x \sqrt {a c}-\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{2} x \sqrt {a c}-4 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} d^{2} x \sqrt {b d}-20 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b c d x \sqrt {b d}+4 b^{2} d \,x^{2} \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+18 a b d x \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 b^{2} c x \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-8 a^{2} d \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\right )}{8 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, d x}\) | \(434\) |
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Time = 1.44 (sec) , antiderivative size = 1074, normalized size of antiderivative = 5.45 \[ \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^2} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}} \sqrt {c + d x}}{x^{2}}\, dx \]
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Exception generated. \[ \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 578 vs. \(2 (155) = 310\).
Time = 0.50 (sec) , antiderivative size = 578, normalized size of antiderivative = 2.93 \[ \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^2} \, dx=\frac {2 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left | b \right |} + \frac {b c d {\left | b \right |} + 7 \, a d^{2} {\left | b \right |}}{d^{2}}\right )} - \frac {8 \, {\left (5 \, \sqrt {b d} a^{2} b^{2} c {\left | b \right |} + \sqrt {b d} a^{3} b d {\left | b \right |}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b} - \frac {16 \, {\left (\sqrt {b d} a^{2} b^{4} c^{2} {\left | b \right |} - 2 \, \sqrt {b d} a^{3} b^{3} c d {\left | b \right |} + \sqrt {b d} a^{4} b^{2} d^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{2} c {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b d {\left | b \right |}\right )}}{b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}} + \frac {{\left (\sqrt {b d} b^{2} c^{2} {\left | b \right |} - 10 \, \sqrt {b d} a b c d {\left | b \right |} - 15 \, \sqrt {b d} a^{2} d^{2} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{d^{2}}}{8 \, b} \]
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Timed out. \[ \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^2} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}\,\sqrt {c+d\,x}}{x^2} \,d x \]
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