Integrand size = 22, antiderivative size = 257 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^2} \, dx=\frac {\left (5 b^2 c^2+26 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b}+\frac {1}{12} (5 b c+19 a d) \sqrt {a+b x} (c+d x)^{3/2}+\frac {4}{3} b \sqrt {a+b x} (c+d x)^{5/2}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x}-\sqrt {a} c^{3/2} (3 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {\left (5 b^3 c^3+45 a b^2 c^2 d+15 a^2 b c d^2-a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{3/2} \sqrt {d}} \]
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Time = 0.20 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^2} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (a^2 d^2+26 a b c d+5 b^2 c^2\right )}{8 b}+\frac {\left (-a^3 d^3+15 a^2 b c d^2+45 a b^2 c^2 d+5 b^3 c^3\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{3/2} \sqrt {d}}-\sqrt {a} c^{3/2} (5 a d+3 b c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x}+\frac {4}{3} b \sqrt {a+b x} (c+d x)^{5/2}+\frac {1}{12} \sqrt {a+b x} (c+d x)^{3/2} (19 a d+5 b c) \]
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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)^{3/2} (c+d x)^{5/2}}{x}+\int \frac {\sqrt {a+b x} (c+d x)^{3/2} \left (\frac {1}{2} (3 b c+5 a d)+4 b d x\right )}{x} \, dx \\ & = \frac {4}{3} b \sqrt {a+b x} (c+d x)^{5/2}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x}+\frac {\int \frac {(c+d x)^{3/2} \left (\frac {3}{2} a d (3 b c+5 a d)+\frac {1}{2} b d (5 b c+19 a d) x\right )}{x \sqrt {a+b x}} \, dx}{3 d} \\ & = \frac {1}{12} (5 b c+19 a d) \sqrt {a+b x} (c+d x)^{3/2}+\frac {4}{3} b \sqrt {a+b x} (c+d x)^{5/2}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x}+\frac {\int \frac {\sqrt {c+d x} \left (3 a b c d (3 b c+5 a d)+\frac {3}{4} b d \left (5 b^2 c^2+26 a b c d+a^2 d^2\right ) x\right )}{x \sqrt {a+b x}} \, dx}{6 b d} \\ & = \frac {\left (5 b^2 c^2+26 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b}+\frac {1}{12} (5 b c+19 a d) \sqrt {a+b x} (c+d x)^{3/2}+\frac {4}{3} b \sqrt {a+b x} (c+d x)^{5/2}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x}+\frac {\int \frac {3 a b^2 c^2 d (3 b c+5 a d)+\frac {3}{8} b d \left (5 b^3 c^3+45 a b^2 c^2 d+15 a^2 b c d^2-a^3 d^3\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{6 b^2 d} \\ & = \frac {\left (5 b^2 c^2+26 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b}+\frac {1}{12} (5 b c+19 a d) \sqrt {a+b x} (c+d x)^{3/2}+\frac {4}{3} b \sqrt {a+b x} (c+d x)^{5/2}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x}+\frac {1}{2} \left (a c^2 (3 b c+5 a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {\left (5 b^3 c^3+45 a b^2 c^2 d+15 a^2 b c d^2-a^3 d^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 b} \\ & = \frac {\left (5 b^2 c^2+26 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b}+\frac {1}{12} (5 b c+19 a d) \sqrt {a+b x} (c+d x)^{3/2}+\frac {4}{3} b \sqrt {a+b x} (c+d x)^{5/2}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x}+\left (a c^2 (3 b c+5 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 (5 b^3 c^3+45 a b^2 c^2 d+15 a^2 b c d^2-a^3 d^3\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 )}{8 b^2} \\ & = \frac {\left (5 b^2 c^2+26 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b}+\frac {1}{12} (5 b c+19 a d) \sqrt {a+b x} (c+d x)^{3/2}+\frac {4}{3} b \sqrt {a+b x} (c+d x)^{5/2}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x}-\sqrt {a} c^{3/2} (3 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {\left (5 b^3 c^3+45 a b^2 c^2 d+15 a^2 b c d^2-a^3 d^3\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 )}{8 b^2} \\ & = \frac {\left (5 b^2 c^2+26 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b}+\frac {1}{12} (5 b c+19 a d) \sqrt {a+b x} (c+d x)^{3/2}+\frac {4}{3} b \sqrt {a+b x} (c+d x)^{5/2}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x}-\sqrt {a} c^{3/2} (3 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {\left (5 b^3 c^3+45 a b^2 c^2 d+15 a^2 b c d^2-a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{3/2} \sqrt {d}} \\ \end{align*}
Time = 0.71 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^2} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 a^2 d^2 x+2 a b \left (-12 c^2+34 c d x+7 d^2 x^2\right )+b^2 x \left (33 c^2+26 c d x+8 d^2 x^2\right )\right )}{24 b x}-\sqrt {a} c^{3/2} (3 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )+\frac {\left (5 b^3 c^3+45 a b^2 c^2 d+15 a^2 b c d^2-a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{8 b^{3/2} \sqrt {d}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(597\) vs. \(2(209)=418\).
Time = 0.54 (sec) , antiderivative size = 598, normalized size of antiderivative = 2.33
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (-16 b^{2} d^{2} x^{3} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{3} d^{3} x -45 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{2} b c \,d^{2} x -135 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a \,b^{2} c^{2} d x -15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, b^{3} c^{3} x +120 \sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b \,c^{2} d x +72 \sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{2} c^{3} x -28 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,d^{2} x^{2}-52 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c d \,x^{2}-6 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{2} x -136 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c d x -66 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} x +48 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{2}\right )}{48 b \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, x \sqrt {a c}}\) | \(598\) |
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Time = 3.15 (sec) , antiderivative size = 1337, normalized size of antiderivative = 5.20 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^2} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {5}{2}}}{x^{2}}\, dx \]
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Exception generated. \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 668 vs. \(2 (209) = 418\).
Time = 0.78 (sec) , antiderivative size = 668, normalized size of antiderivative = 2.60 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^2} \, dx=\frac {2 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )} d^{2} {\left | b \right |}}{b^{2}} + \frac {13 \, b^{3} c d^{5} {\left | b \right |} - a b^{2} d^{6} {\left | b \right |}}{b^{4} d^{4}}\right )} + \frac {3 \, {\left (11 \, b^{4} c^{2} d^{4} {\left | b \right |} + 14 \, a b^{3} c d^{5} {\left | b \right |} - a^{2} b^{2} d^{6} {\left | b \right |}\right )}}{b^{4} d^{4}}\right )} \sqrt {b x + a} - \frac {48 \, {\left (3 \, \sqrt {b d} a b^{2} c^{3} {\left | b \right |} + 5 \, \sqrt {b d} a^{2} b c^{2} 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 {3 \, {\left (5 \, b^{3} c^{3} {\left | b \right |} + 45 \, a b^{2} c^{2} d {\left | b \right |} + 15 \, a^{2} b c d^{2} {\left | b \right |} - a^{3} d^{3} {\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 )}{\sqrt {b d} b} - \frac {96 \, {\left (\sqrt {b d} a b^{4} c^{4} {\left | b \right |} - 2 \, \sqrt {b d} a^{2} b^{3} c^{3} d {\left | b \right |} + \sqrt {b d} a^{3} b^{2} c^{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 b^{2} c^{3} {\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 c^{2} 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}}}{48 \, b} \]
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Timed out. \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^2} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/2}}{x^2} \,d x \]
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