Integrand size = 31, antiderivative size = 311 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {35 (9 A b-a B)}{192 a^4 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (9 A b-a B)}{96 a^3 b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (9 A b-a B) (a+b x)}{64 a^5 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (9 A b-a B) (a+b x) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{11/2} \sqrt {b} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.11 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {784, 79, 44, 53, 65, 211} \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (a+b x) (9 A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{11/2} \sqrt {b} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (a+b x) (9 A b-a B)}{64 a^5 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 (9 A b-a B)}{192 a^4 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (9 A b-a B)}{96 a^3 b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rule 44
Rule 53
Rule 65
Rule 79
Rule 211
Rule 784
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {A+B x}{x^{3/2} \left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {A b-a B}{4 a b \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b^2 (9 A b-a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{3/2} \left (a b+b^2 x\right )^4} \, dx}{8 a \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {A b-a B}{4 a b \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 b (9 A b-a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{3/2} \left (a b+b^2 x\right )^3} \, dx}{48 a^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {A b-a B}{4 a b \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (9 A b-a B)}{96 a^3 b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 (9 A b-a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{3/2} \left (a b+b^2 x\right )^2} \, dx}{192 a^3 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {35 (9 A b-a B)}{192 a^4 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (9 A b-a B)}{96 a^3 b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 (9 A b-a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{3/2} \left (a b+b^2 x\right )} \, dx}{128 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {35 (9 A b-a B)}{192 a^4 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (9 A b-a B)}{96 a^3 b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (9 A b-a B) (a+b x)}{64 a^5 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (35 (9 A b-a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{\sqrt {x} \left (a b+b^2 x\right )} \, dx}{128 a^5 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {35 (9 A b-a B)}{192 a^4 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (9 A b-a B)}{96 a^3 b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (9 A b-a B) (a+b x)}{64 a^5 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (35 (9 A b-a B) \left (a b+b^2 x\right )\right ) \text {Subst}\left (\int \frac {1}{a b+b^2 x^2} \, dx,x,\sqrt {x}\right )}{64 a^5 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {35 (9 A b-a B)}{192 a^4 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (9 A b-a B)}{96 a^3 b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (9 A b-a B) (a+b x)}{64 a^5 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (9 A b-a B) (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{11/2} \sqrt {b} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.48 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {\frac {\sqrt {a} \left (-945 A b^4 x^4+105 a b^3 x^3 (-33 A+B x)+7 a^2 b^2 x^2 (-657 A+55 B x)+a^4 (-384 A+279 B x)+a^3 b x (-2511 A+511 B x)\right )}{\sqrt {x}}+\frac {105 (-9 A b+a B) (a+b x)^4 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {b}}}{192 a^{11/2} (a+b x)^3 \sqrt {(a+b x)^2}} \]
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Time = 0.19 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.52
method | result | size |
risch | \(-\frac {2 A \sqrt {\left (b x +a \right )^{2}}}{a^{5} \sqrt {x}\, \left (b x +a \right )}-\frac {\left (\frac {2 \left (\frac {187}{128} A \,b^{4}-\frac {35}{128} B \,b^{3} a \right ) x^{\frac {7}{2}}+\frac {b^{2} a \left (1929 A b -385 B a \right ) x^{\frac {5}{2}}}{192}+2 \left (\frac {765}{128} A \,a^{2} b^{2}-\frac {511}{384} B \,a^{3} b \right ) x^{\frac {3}{2}}+2 \left (\frac {325}{128} A \,a^{3} b -\frac {93}{128} B \,a^{4}\right ) \sqrt {x}}{\left (b x +a \right )^{4}}+\frac {35 \left (9 A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{64 \sqrt {b a}}\right ) \sqrt {\left (b x +a \right )^{2}}}{a^{5} \left (b x +a \right )}\) | \(161\) |
default | \(-\frac {\left (945 A \,x^{4} \sqrt {b a}\, b^{4}-105 B \,x^{4} \sqrt {b a}\, a \,b^{3}+3465 A \,x^{3} \sqrt {b a}\, a \,b^{3}+945 A \,x^{\frac {9}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) b^{5}-385 B \,x^{3} \sqrt {b a}\, a^{2} b^{2}-105 B \,x^{\frac {9}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a \,b^{4}+3780 A \,x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a \,b^{4}-420 B \,x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{2} b^{3}+4599 A \,x^{2} \sqrt {b a}\, a^{2} b^{2}+5670 A \,x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{2} b^{3}-511 B \,x^{2} \sqrt {b a}\, a^{3} b -630 B \,x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{3} b^{2}+3780 A \,x^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{3} b^{2}-420 B \,x^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{4} b +2511 A x \sqrt {b a}\, a^{3} b +945 A \sqrt {x}\, \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{4} b -279 B x \sqrt {b a}\, a^{4}-105 B \sqrt {x}\, \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{5}+384 A \sqrt {b a}\, a^{4}\right ) \left (b x +a \right )}{192 \sqrt {x}\, \sqrt {b a}\, a^{5} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(374\) |
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Time = 0.34 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.80 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\left [\frac {105 \, {\left ({\left (B a b^{4} - 9 \, A b^{5}\right )} x^{5} + 4 \, {\left (B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4} + 6 \, {\left (B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{3} + 4 \, {\left (B a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{2} + {\left (B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a + 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) - 2 \, {\left (384 \, A a^{5} b - 105 \, {\left (B a^{2} b^{4} - 9 \, A a b^{5}\right )} x^{4} - 385 \, {\left (B a^{3} b^{3} - 9 \, A a^{2} b^{4}\right )} x^{3} - 511 \, {\left (B a^{4} b^{2} - 9 \, A a^{3} b^{3}\right )} x^{2} - 279 \, {\left (B a^{5} b - 9 \, A a^{4} b^{2}\right )} x\right )} \sqrt {x}}{384 \, {\left (a^{6} b^{5} x^{5} + 4 \, a^{7} b^{4} x^{4} + 6 \, a^{8} b^{3} x^{3} + 4 \, a^{9} b^{2} x^{2} + a^{10} b x\right )}}, -\frac {105 \, {\left ({\left (B a b^{4} - 9 \, A b^{5}\right )} x^{5} + 4 \, {\left (B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4} + 6 \, {\left (B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{3} + 4 \, {\left (B a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{2} + {\left (B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (384 \, A a^{5} b - 105 \, {\left (B a^{2} b^{4} - 9 \, A a b^{5}\right )} x^{4} - 385 \, {\left (B a^{3} b^{3} - 9 \, A a^{2} b^{4}\right )} x^{3} - 511 \, {\left (B a^{4} b^{2} - 9 \, A a^{3} b^{3}\right )} x^{2} - 279 \, {\left (B a^{5} b - 9 \, A a^{4} b^{2}\right )} x\right )} \sqrt {x}}{192 \, {\left (a^{6} b^{5} x^{5} + 4 \, a^{7} b^{4} x^{4} + 6 \, a^{8} b^{3} x^{3} + 4 \, a^{9} b^{2} x^{2} + a^{10} b x\right )}}\right ] \]
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Timed out. \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (208) = 416\).
Time = 0.37 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.37 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {35 \, {\left ({\left (B a b^{6} + 9 \, A b^{7}\right )} x^{2} - 27 \, {\left (B a^{2} b^{5} - 11 \, A a b^{6}\right )} x\right )} x^{\frac {9}{2}} + 70 \, {\left ({\left (B a^{2} b^{5} + 9 \, A a b^{6}\right )} x^{2} - 81 \, {\left (B a^{3} b^{4} - 11 \, A a^{2} b^{5}\right )} x\right )} x^{\frac {7}{2}} - 140 \, {\left (2 \, {\left (B a^{3} b^{4} + 9 \, A a^{2} b^{5}\right )} x^{2} + 99 \, {\left (B a^{4} b^{3} - 11 \, A a^{3} b^{4}\right )} x\right )} x^{\frac {5}{2}} - 14 \, {\left (85 \, {\left (B a^{4} b^{3} + 9 \, A a^{3} b^{4}\right )} x^{2} + 1251 \, {\left (B a^{5} b^{2} - 11 \, A a^{4} b^{3}\right )} x\right )} x^{\frac {3}{2}} - {\left (1771 \, {\left (B a^{5} b^{2} + 9 \, A a^{4} b^{3}\right )} x^{2} + 11835 \, {\left (B a^{6} b - 11 \, A a^{5} b^{2}\right )} x\right )} \sqrt {x} - \frac {1280 \, {\left ({\left (B a^{6} b + 9 \, A a^{5} b^{2}\right )} x^{2} + 3 \, {\left (B a^{7} - 11 \, A a^{6} b\right )} x\right )}}{\sqrt {x}} - \frac {3840 \, {\left (A a^{6} b x^{2} - A a^{7} x\right )}}{x^{\frac {3}{2}}}}{1920 \, {\left (a^{7} b^{5} x^{5} + 5 \, a^{8} b^{4} x^{4} + 10 \, a^{9} b^{3} x^{3} + 10 \, a^{10} b^{2} x^{2} + 5 \, a^{11} b x + a^{12}\right )}} + \frac {35 \, {\left (B a - 9 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} a^{5}} + \frac {7 \, {\left ({\left (B a b + 9 \, A b^{2}\right )} x^{\frac {3}{2}} - 30 \, {\left (B a^{2} - 9 \, A a b\right )} \sqrt {x}\right )}}{384 \, a^{7}} \]
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Time = 0.28 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.51 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {35 \, {\left (B a - 9 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} a^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {2 \, A}{a^{5} \sqrt {x} \mathrm {sgn}\left (b x + a\right )} + \frac {105 \, B a b^{3} x^{\frac {7}{2}} - 561 \, A b^{4} x^{\frac {7}{2}} + 385 \, B a^{2} b^{2} x^{\frac {5}{2}} - 1929 \, A a b^{3} x^{\frac {5}{2}} + 511 \, B a^{3} b x^{\frac {3}{2}} - 2295 \, A a^{2} b^{2} x^{\frac {3}{2}} + 279 \, B a^{4} \sqrt {x} - 975 \, A a^{3} b \sqrt {x}}{192 \, {\left (b x + a\right )}^{4} a^{5} \mathrm {sgn}\left (b x + a\right )} \]
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Timed out. \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {A+B\,x}{x^{3/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
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