Integrand size = 20, antiderivative size = 118 \[ \int \frac {x}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\frac {2 a}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 (b c+3 a d) \sqrt {a+b x}}{3 b (b c-a d)^2 (c+d x)^{3/2}}+\frac {4 (b c+3 a d) \sqrt {a+b x}}{3 (b c-a d)^3 \sqrt {c+d x}} \]
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Time = 0.03 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \[ \int \frac {x}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\frac {2 a}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}+\frac {4 \sqrt {a+b x} (3 a d+b c)}{3 \sqrt {c+d x} (b c-a d)^3}+\frac {2 \sqrt {a+b x} (3 a d+b c)}{3 b (c+d x)^{3/2} (b c-a d)^2} \]
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Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = \frac {2 a}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}+\frac {(b c+3 a d) \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx}{b (b c-a d)} \\ & = \frac {2 a}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 (b c+3 a d) \sqrt {a+b x}}{3 b (b c-a d)^2 (c+d x)^{3/2}}+\frac {(2 (b c+3 a d)) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 (b c-a d)^2} \\ & = \frac {2 a}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 (b c+3 a d) \sqrt {a+b x}}{3 b (b c-a d)^2 (c+d x)^{3/2}}+\frac {4 (b c+3 a d) \sqrt {a+b x}}{3 (b c-a d)^3 \sqrt {c+d x}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.70 \[ \int \frac {x}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\frac {2 \left (b^2 c x (3 c+2 d x)+a^2 d (2 c+3 d x)+2 a b \left (3 c^2+5 c d x+3 d^2 x^2\right )\right )}{3 (b c-a d)^3 \sqrt {a+b x} (c+d x)^{3/2}} \]
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Time = 1.73 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.74
method | result | size |
default | \(-\frac {2 \left (6 x^{2} a b \,d^{2}+2 x^{2} b^{2} c d +3 a^{2} d^{2} x +10 a b c d x +3 b^{2} c^{2} x +2 a^{2} c d +6 b \,c^{2} a \right )}{3 \left (d x +c \right )^{\frac {3}{2}} \sqrt {b x +a}\, \left (a d -b c \right )^{3}}\) | \(87\) |
gosper | \(-\frac {2 \left (6 x^{2} a b \,d^{2}+2 x^{2} b^{2} c d +3 a^{2} d^{2} x +10 a b c d x +3 b^{2} c^{2} x +2 a^{2} c d +6 b \,c^{2} a \right )}{3 \sqrt {b x +a}\, \left (d x +c \right )^{\frac {3}{2}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(115\) |
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Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (102) = 204\).
Time = 0.47 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.38 \[ \int \frac {x}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\frac {2 \, {\left (6 \, a b c^{2} + 2 \, a^{2} c d + 2 \, {\left (b^{2} c d + 3 \, a b d^{2}\right )} x^{2} + {\left (3 \, b^{2} c^{2} + 10 \, a b c d + 3 \, a^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{3} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{2} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x\right )}} \]
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\[ \int \frac {x}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\int \frac {x}{\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (102) = 204\).
Time = 0.39 (sec) , antiderivative size = 393, normalized size of antiderivative = 3.33 \[ \int \frac {x}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {6 \, \sqrt {b d} a b^{3}}{{\left (b^{2} c^{2} {\left | b \right |} - 2 \, a b c d {\left | b \right |} + a^{2} d^{2} {\left | b \right |}\right )} {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}} + \frac {\sqrt {b x + a} {\left (\frac {{\left (2 \, b^{7} c^{3} d^{2} {\left | b \right |} - a b^{6} c^{2} d^{3} {\left | b \right |} - 4 \, a^{2} b^{5} c d^{4} {\left | b \right |} + 3 \, a^{3} b^{4} d^{5} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{7} c^{5} d - 5 \, a b^{6} c^{4} d^{2} + 10 \, a^{2} b^{5} c^{3} d^{3} - 10 \, a^{3} b^{4} c^{2} d^{4} + 5 \, a^{4} b^{3} c d^{5} - a^{5} b^{2} d^{6}} + \frac {3 \, {\left (b^{8} c^{4} d {\left | b \right |} - 2 \, a b^{7} c^{3} d^{2} {\left | b \right |} + 2 \, a^{3} b^{5} c d^{4} {\left | b \right |} - a^{4} b^{4} d^{5} {\left | b \right |}\right )}}{b^{7} c^{5} d - 5 \, a b^{6} c^{4} d^{2} + 10 \, a^{2} b^{5} c^{3} d^{3} - 10 \, a^{3} b^{4} c^{2} d^{4} + 5 \, a^{4} b^{3} c d^{5} - a^{5} b^{2} d^{6}}\right )}}{{\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}}\right )}}{3 \, b} \]
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Time = 1.74 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.21 \[ \int \frac {x}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx=-\frac {\sqrt {c+d\,x}\,\left (\frac {x\,\left (6\,a^2\,d^2+20\,a\,b\,c\,d+6\,b^2\,c^2\right )}{3\,d^2\,{\left (a\,d-b\,c\right )}^3}+\frac {4\,b\,x^2\,\left (3\,a\,d+b\,c\right )}{3\,d\,{\left (a\,d-b\,c\right )}^3}+\frac {4\,a\,c\,\left (a\,d+3\,b\,c\right )}{3\,d^2\,{\left (a\,d-b\,c\right )}^3}\right )}{x^2\,\sqrt {a+b\,x}+\frac {c^2\,\sqrt {a+b\,x}}{d^2}+\frac {2\,c\,x\,\sqrt {a+b\,x}}{d}} \]
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