Integrand size = 20, antiderivative size = 158 \[ \int \frac {x}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=-\frac {2 c}{3 d (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 (b c+a d)}{3 d (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}-\frac {8 (b c+a d)}{3 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}-\frac {16 d (b c+a d) \sqrt {a+b x}}{3 (b c-a d)^4 \sqrt {c+d x}} \]
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Time = 0.04 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \[ \int \frac {x}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=-\frac {2 c}{3 d (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac {16 d \sqrt {a+b x} (a d+b c)}{3 \sqrt {c+d x} (b c-a d)^4}-\frac {8 (a d+b c)}{3 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^3}+\frac {2 (a d+b c)}{3 d (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)^2} \]
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Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {2 c}{3 d (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {(b c+a d) \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/2}} \, dx}{d (b c-a d)} \\ & = -\frac {2 c}{3 d (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 (b c+a d)}{3 d (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}+\frac {(4 (b c+a d)) \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx}{3 (b c-a d)^2} \\ & = -\frac {2 c}{3 d (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 (b c+a d)}{3 d (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}-\frac {8 (b c+a d)}{3 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}-\frac {(8 d (b c+a d)) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 (b c-a d)^3} \\ & = -\frac {2 c}{3 d (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 (b c+a d)}{3 d (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}-\frac {8 (b c+a d)}{3 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}-\frac {16 d (b c+a d) \sqrt {a+b x}}{3 (b c-a d)^4 \sqrt {c+d x}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.84 \[ \int \frac {x}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=-\frac {2 (a+b x)^{3/2} \left (-c d^2+\frac {6 b c d (c+d x)}{a+b x}+\frac {3 a d^2 (c+d x)}{a+b x}+\frac {3 b^2 c (c+d x)^2}{(a+b x)^2}+\frac {6 a b d (c+d x)^2}{(a+b x)^2}-\frac {a b^2 (c+d x)^3}{(a+b x)^3}\right )}{3 (b c-a d)^4 (c+d x)^{3/2}} \]
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Time = 0.60 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.99
method | result | size |
default | \(-\frac {2 \left (8 x^{3} a \,b^{2} d^{3}+8 x^{3} b^{3} c \,d^{2}+12 x^{2} a^{2} b \,d^{3}+24 x^{2} a \,b^{2} c \,d^{2}+12 x^{2} b^{3} c^{2} d +3 a^{3} d^{3} x +21 a^{2} b c \,d^{2} x +21 a \,b^{2} c^{2} d x +3 b^{3} c^{3} x +2 a^{3} c \,d^{2}+12 a^{2} b \,c^{2} d +2 b^{2} c^{3} a \right )}{3 \left (a d -b c \right )^{4} \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}}}\) | \(157\) |
gosper | \(-\frac {2 \left (8 x^{3} a \,b^{2} d^{3}+8 x^{3} b^{3} c \,d^{2}+12 x^{2} a^{2} b \,d^{3}+24 x^{2} a \,b^{2} c \,d^{2}+12 x^{2} b^{3} c^{2} d +3 a^{3} d^{3} x +21 a^{2} b c \,d^{2} x +21 a \,b^{2} c^{2} d x +3 b^{3} c^{3} x +2 a^{3} c \,d^{2}+12 a^{2} b \,c^{2} d +2 b^{2} c^{3} a \right )}{3 \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(198\) |
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Leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (134) = 268\).
Time = 0.62 (sec) , antiderivative size = 468, normalized size of antiderivative = 2.96 \[ \int \frac {x}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=-\frac {2 \, {\left (2 \, a b^{2} c^{3} + 12 \, a^{2} b c^{2} d + 2 \, a^{3} c d^{2} + 8 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{3} + 12 \, {\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + 3 \, {\left (b^{3} c^{3} + 7 \, a b^{2} c^{2} d + 7 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} + {\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \, {\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} + {\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \, {\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )}} \]
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\[ \int \frac {x}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\int \frac {x}{\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 829 vs. \(2 (134) = 268\).
Time = 0.53 (sec) , antiderivative size = 829, normalized size of antiderivative = 5.25 \[ \int \frac {x}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=-\frac {2 \, {\left (\frac {\sqrt {b x + a} {\left (\frac {{\left (5 \, b^{8} c^{4} d^{3} {\left | b \right |} - 12 \, a b^{7} c^{3} d^{4} {\left | b \right |} + 6 \, a^{2} b^{6} c^{2} d^{5} {\left | b \right |} + 4 \, a^{3} b^{5} c d^{6} {\left | b \right |} - 3 \, a^{4} b^{4} d^{7} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{9} c^{7} d - 7 \, a b^{8} c^{6} d^{2} + 21 \, a^{2} b^{7} c^{5} d^{3} - 35 \, a^{3} b^{6} c^{4} d^{4} + 35 \, a^{4} b^{5} c^{3} d^{5} - 21 \, a^{5} b^{4} c^{2} d^{6} + 7 \, a^{6} b^{3} c d^{7} - a^{7} b^{2} d^{8}} + \frac {3 \, {\left (2 \, b^{9} c^{5} d^{2} {\left | b \right |} - 7 \, a b^{8} c^{4} d^{3} {\left | b \right |} + 8 \, a^{2} b^{7} c^{3} d^{4} {\left | b \right |} - 2 \, a^{3} b^{6} c^{2} d^{5} {\left | b \right |} - 2 \, a^{4} b^{5} c d^{6} {\left | b \right |} + a^{5} b^{4} d^{7} {\left | b \right |}\right )}}{b^{9} c^{7} d - 7 \, a b^{8} c^{6} d^{2} + 21 \, a^{2} b^{7} c^{5} d^{3} - 35 \, a^{3} b^{6} c^{4} d^{4} + 35 \, a^{4} b^{5} c^{3} d^{5} - 21 \, a^{5} b^{4} c^{2} d^{6} + 7 \, a^{6} b^{3} c d^{7} - a^{7} b^{2} d^{8}}\right )}}{{\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} + \frac {2 \, {\left (3 \, \sqrt {b d} b^{8} c^{3} - \sqrt {b d} a b^{7} c^{2} d - 7 \, \sqrt {b d} a^{2} b^{6} c d^{2} + 5 \, \sqrt {b d} a^{3} b^{5} d^{3} - 6 \, \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} b^{6} c^{2} - 6 \, \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^{5} c d + 12 \, \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^{4} d^{2} + 3 \, \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 )}^{4} b^{4} c + 3 \, \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 )}^{4} a b^{3} d\right )}}{{\left (b^{3} c^{3} {\left | b \right |} - 3 \, a b^{2} c^{2} d {\left | b \right |} + 3 \, a^{2} b c d^{2} {\left | b \right |} - a^{3} d^{3} {\left | b \right |}\right )} {\left (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}\right )}^{3}}\right )}}{3 \, b} \]
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Time = 1.97 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.51 \[ \int \frac {x}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=-\frac {\sqrt {c+d\,x}\,\left (\frac {8\,x^2\,{\left (a\,d+b\,c\right )}^2}{d\,{\left (a\,d-b\,c\right )}^4}+\frac {16\,b\,x^3\,\left (a\,d+b\,c\right )}{3\,{\left (a\,d-b\,c\right )}^4}+\frac {x\,\left (6\,a^3\,d^3+42\,a^2\,b\,c\,d^2+42\,a\,b^2\,c^2\,d+6\,b^3\,c^3\right )}{3\,b\,d^2\,{\left (a\,d-b\,c\right )}^4}+\frac {4\,a\,c\,\left (a^2\,d^2+6\,a\,b\,c\,d+b^2\,c^2\right )}{3\,b\,d^2\,{\left (a\,d-b\,c\right )}^4}\right )}{x^3\,\sqrt {a+b\,x}+\frac {a\,c^2\,\sqrt {a+b\,x}}{b\,d^2}+\frac {x^2\,\left (a\,d+2\,b\,c\right )\,\sqrt {a+b\,x}}{b\,d}+\frac {c\,x\,\left (2\,a\,d+b\,c\right )\,\sqrt {a+b\,x}}{b\,d^2}} \]
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