Integrand size = 22, antiderivative size = 299 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x} \, dx=\frac {\left (64 a^2 b c d^2+(b c-5 a d) (b c-a d) (3 b c+a d)\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b d^2}-\frac {(b c-5 a d) (3 b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 d^2}+\frac {(3 b c+5 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 d}+\frac {1}{4} (a+b x)^{5/2} (c+d x)^{3/2}-2 a^{5/2} c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {\left (3 b^4 c^4-20 a b^3 c^3 d+90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{3/2} d^{5/2}} \]
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Time = 0.28 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {103, 159, 163, 65, 223, 212, 95, 214} \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x} \, dx=-2 a^{5/2} c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {1}{64} \sqrt {a+b x} \sqrt {c+d x} \left (\frac {5 a^3 d}{b}+73 a^2 c-\frac {17 a b c^2}{d}+\frac {3 b^2 c^3}{d^2}\right )+\frac {\left (-5 a^4 d^4+60 a^3 b c d^3+90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+3 b^4 c^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{3/2} d^{5/2}}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (b c-5 a d) (a d+3 b c)}{32 d^2}+\frac {1}{4} (a+b x)^{5/2} (c+d x)^{3/2}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} (5 a d+3 b c)}{24 d} \]
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Rule 65
Rule 95
Rule 103
Rule 159
Rule 163
Rule 212
Rule 214
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} (a+b x)^{5/2} (c+d x)^{3/2}-\frac {1}{4} \int \frac {(a+b x)^{3/2} \sqrt {c+d x} \left (-4 a c+\frac {1}{2} (-3 b c-5 a d) x\right )}{x} \, dx \\ & = \frac {(3 b c+5 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 d}+\frac {1}{4} (a+b x)^{5/2} (c+d x)^{3/2}-\frac {\int \frac {\sqrt {a+b x} \sqrt {c+d x} \left (-12 a^2 c d+\frac {3}{4} (b c-5 a d) (3 b c+a d) x\right )}{x} \, dx}{12 d} \\ & = -\frac {(b c-5 a d) (3 b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 d^2}+\frac {(3 b c+5 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 d}+\frac {1}{4} (a+b x)^{5/2} (c+d x)^{3/2}-\frac {\int \frac {\sqrt {c+d x} \left (-24 a^3 c d^2-\frac {3}{8} \left (64 a^2 b c d^2+(b c-5 a d) (b c-a d) (3 b c+a d)\right ) x\right )}{x \sqrt {a+b x}} \, dx}{24 d^2} \\ & = \frac {1}{64} \left (73 a^2 c+\frac {3 b^2 c^3}{d^2}-\frac {17 a b c^2}{d}+\frac {5 a^3 d}{b}\right ) \sqrt {a+b x} \sqrt {c+d x}-\frac {(b c-5 a d) (3 b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 d^2}+\frac {(3 b c+5 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 d}+\frac {1}{4} (a+b x)^{5/2} (c+d x)^{3/2}-\frac {\int \frac {-24 a^3 b c^2 d^2-\frac {3}{16} \left (3 b^4 c^4-20 a b^3 c^3 d+90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{24 b d^2} \\ & = \frac {1}{64} \left (73 a^2 c+\frac {3 b^2 c^3}{d^2}-\frac {17 a b c^2}{d}+\frac {5 a^3 d}{b}\right ) \sqrt {a+b x} \sqrt {c+d x}-\frac {(b c-5 a d) (3 b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 d^2}+\frac {(3 b c+5 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 d}+\frac {1}{4} (a+b x)^{5/2} (c+d x)^{3/2}+\left (a^3 c^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {\left (3 b^4 c^4-20 a b^3 c^3 d+90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 b d^2} \\ & = \frac {1}{64} \left (73 a^2 c+\frac {3 b^2 c^3}{d^2}-\frac {17 a b c^2}{d}+\frac {5 a^3 d}{b}\right ) \sqrt {a+b x} \sqrt {c+d x}-\frac {(b c-5 a d) (3 b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 d^2}+\frac {(3 b c+5 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 d}+\frac {1}{4} (a+b x)^{5/2} (c+d x)^{3/2}+\left (2 a^3 c^2\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 (3 b^4 c^4-20 a b^3 c^3 d+90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4\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 )}{64 b^2 d^2} \\ & = \frac {1}{64} \left (73 a^2 c+\frac {3 b^2 c^3}{d^2}-\frac {17 a b c^2}{d}+\frac {5 a^3 d}{b}\right ) \sqrt {a+b x} \sqrt {c+d x}-\frac {(b c-5 a d) (3 b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 d^2}+\frac {(3 b c+5 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 d}+\frac {1}{4} (a+b x)^{5/2} (c+d x)^{3/2}-2 a^{5/2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {\left (3 b^4 c^4-20 a b^3 c^3 d+90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4\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 )}{64 b^2 d^2} \\ & = \frac {1}{64} \left (73 a^2 c+\frac {3 b^2 c^3}{d^2}-\frac {17 a b c^2}{d}+\frac {5 a^3 d}{b}\right ) \sqrt {a+b x} \sqrt {c+d x}-\frac {(b c-5 a d) (3 b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 d^2}+\frac {(3 b c+5 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 d}+\frac {1}{4} (a+b x)^{5/2} (c+d x)^{3/2}-2 a^{5/2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {\left (3 b^4 c^4-20 a b^3 c^3 d+90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{3/2} d^{5/2}} \\ \end{align*}
Time = 0.64 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (15 a^3 d^3+a^2 b d^2 (337 c+118 d x)+a b^2 d \left (57 c^2+244 c d x+136 d^2 x^2\right )+b^3 \left (-9 c^3+6 c^2 d x+72 c d^2 x^2+48 d^3 x^3\right )\right )}{192 b d^2}-2 a^{5/2} c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {\left (3 b^4 c^4-20 a b^3 c^3 d+90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{3/2} d^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(708\) vs. \(2(249)=498\).
Time = 1.59 (sec) , antiderivative size = 709, normalized size of antiderivative = 2.37
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (-96 b^{3} d^{3} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}-272 a \,b^{2} d^{3} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}-144 b^{3} c \,d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}+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}\, a^{4} d^{4}-180 \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} b c \,d^{3}-270 \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^{2} c^{2} d^{2}+60 \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^{3} c^{3} d -9 \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^{4} c^{4}+384 \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^{3} b \,c^{2} d^{2}-236 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b \,d^{3} x -488 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{2} c \,d^{2} x -12 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, b^{3} c^{2} d x -30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} d^{3}-674 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b c \,d^{2}-114 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{2} c^{2} d +18 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, b^{3} c^{3}\right )}{384 b \,d^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}}\) | \(709\) |
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Time = 8.55 (sec) , antiderivative size = 1481, normalized size of antiderivative = 4.95 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x} \, dx=\text {Too large to display} \]
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\[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {3}{2}}}{x}\, dx \]
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Exception generated. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{3/2}}{x} \,d x \]
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