Integrand size = 31, antiderivative size = 258 \[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {5 (7 A b+a B) \sqrt {x}}{64 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) \sqrt {x}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(7 A b+a B) \sqrt {x}}{24 a^2 b (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 (7 A b+a B) \sqrt {x}}{96 a^3 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 (7 A b+a B) (a+b x) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{9/2} b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.10 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {784, 79, 44, 65, 211} \[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {x} (A b-a B)}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\sqrt {x} (a B+7 A b)}{24 a^2 b (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 (a+b x) (a B+7 A b) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{9/2} b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 \sqrt {x} (a B+7 A b)}{64 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 \sqrt {x} (a B+7 A b)}{96 a^3 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rule 44
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}{\sqrt {x} \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) \sqrt {x}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b^2 (7 A b+a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{\sqrt {x} \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) \sqrt {x}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(7 A b+a B) \sqrt {x}}{24 a^2 b (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 b (7 A b+a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{\sqrt {x} \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) \sqrt {x}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(7 A b+a B) \sqrt {x}}{24 a^2 b (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 (7 A b+a B) \sqrt {x}}{96 a^3 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 (7 A b+a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{\sqrt {x} \left (a b+b^2 x\right )^2} \, dx}{64 a^3 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {5 (7 A b+a B) \sqrt {x}}{64 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) \sqrt {x}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(7 A b+a B) \sqrt {x}}{24 a^2 b (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 (7 A b+a B) \sqrt {x}}{96 a^3 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 (7 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^4 b \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {5 (7 A b+a B) \sqrt {x}}{64 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) \sqrt {x}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(7 A b+a B) \sqrt {x}}{24 a^2 b (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 (7 A b+a B) \sqrt {x}}{96 a^3 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 (7 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^4 b \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {5 (7 A b+a B) \sqrt {x}}{64 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) \sqrt {x}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(7 A b+a B) \sqrt {x}}{24 a^2 b (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 (7 A b+a B) \sqrt {x}}{96 a^3 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 (7 A b+a B) (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{9/2} b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.56 \[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {a} \sqrt {b} \sqrt {x} \left (-15 a^4 B+105 A b^4 x^3+5 a b^3 x^2 (77 A+3 B x)+a^2 b^2 x (511 A+55 B x)+a^3 b (279 A+73 B x)\right )+15 (7 A b+a B) (a+b x)^4 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{192 a^{9/2} b^{3/2} (a+b x)^3 \sqrt {(a+b x)^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(356\) vs. \(2(175)=350\).
Time = 0.16 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.38
method | result | size |
default | \(\frac {\left (105 A \,x^{\frac {7}{2}} \sqrt {b a}\, b^{4}+15 B \,x^{\frac {7}{2}} \sqrt {b a}\, a \,b^{3}+385 A \,x^{\frac {5}{2}} \sqrt {b a}\, a \,b^{3}+105 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) b^{5} x^{4}+55 B \,x^{\frac {5}{2}} \sqrt {b a}\, a^{2} b^{2}+15 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a \,b^{4} x^{4}+420 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a \,b^{4} x^{3}+60 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{2} b^{3} x^{3}+511 A \,x^{\frac {3}{2}} \sqrt {b a}\, a^{2} b^{2}+630 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{2} b^{3} x^{2}+73 B \,x^{\frac {3}{2}} \sqrt {b a}\, a^{3} b +90 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{3} b^{2} x^{2}+420 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{3} b^{2} x +60 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{4} b x +279 A \sqrt {x}\, \sqrt {b a}\, a^{3} b +105 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{4} b -15 B \sqrt {x}\, \sqrt {b a}\, a^{4}+15 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{5}\right ) \left (b x +a \right )}{192 \sqrt {b a}\, b \,a^{4} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(357\) |
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Time = 0.37 (sec) , antiderivative size = 523, normalized size of antiderivative = 2.03 \[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\left [-\frac {15 \, {\left (B a^{5} + 7 \, A a^{4} b + {\left (B a b^{4} + 7 \, A b^{5}\right )} x^{4} + 4 \, {\left (B a^{2} b^{3} + 7 \, A a b^{4}\right )} x^{3} + 6 \, {\left (B a^{3} b^{2} + 7 \, A a^{2} b^{3}\right )} x^{2} + 4 \, {\left (B a^{4} b + 7 \, A a^{3} b^{2}\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) + 2 \, {\left (15 \, B a^{5} b - 279 \, A a^{4} b^{2} - 15 \, {\left (B a^{2} b^{4} + 7 \, A a b^{5}\right )} x^{3} - 55 \, {\left (B a^{3} b^{3} + 7 \, A a^{2} b^{4}\right )} x^{2} - 73 \, {\left (B a^{4} b^{2} + 7 \, A a^{3} b^{3}\right )} x\right )} \sqrt {x}}{384 \, {\left (a^{5} b^{6} x^{4} + 4 \, a^{6} b^{5} x^{3} + 6 \, a^{7} b^{4} x^{2} + 4 \, a^{8} b^{3} x + a^{9} b^{2}\right )}}, -\frac {15 \, {\left (B a^{5} + 7 \, A a^{4} b + {\left (B a b^{4} + 7 \, A b^{5}\right )} x^{4} + 4 \, {\left (B a^{2} b^{3} + 7 \, A a b^{4}\right )} x^{3} + 6 \, {\left (B a^{3} b^{2} + 7 \, A a^{2} b^{3}\right )} x^{2} + 4 \, {\left (B a^{4} b + 7 \, A a^{3} b^{2}\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (15 \, B a^{5} b - 279 \, A a^{4} b^{2} - 15 \, {\left (B a^{2} b^{4} + 7 \, A a b^{5}\right )} x^{3} - 55 \, {\left (B a^{3} b^{3} + 7 \, A a^{2} b^{4}\right )} x^{2} - 73 \, {\left (B a^{4} b^{2} + 7 \, A a^{3} b^{3}\right )} x\right )} \sqrt {x}}{192 \, {\left (a^{5} b^{6} x^{4} + 4 \, a^{6} b^{5} x^{3} + 6 \, a^{7} b^{4} x^{2} + 4 \, a^{8} b^{3} x + a^{9} b^{2}\right )}}\right ] \]
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Timed out. \[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 390 vs. \(2 (175) = 350\).
Time = 0.36 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.51 \[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {5 \, {\left ({\left (3 \, B a b^{5} + 7 \, A b^{6}\right )} x^{2} - 21 \, {\left (B a^{2} b^{4} + 9 \, A a b^{5}\right )} x\right )} x^{\frac {9}{2}} + 10 \, {\left ({\left (3 \, B a^{2} b^{4} + 7 \, A a b^{5}\right )} x^{2} - 63 \, {\left (B a^{3} b^{3} + 9 \, A a^{2} b^{4}\right )} x\right )} x^{\frac {7}{2}} - 20 \, {\left (2 \, {\left (3 \, B a^{3} b^{3} + 7 \, A a^{2} b^{4}\right )} x^{2} + 77 \, {\left (B a^{4} b^{2} + 9 \, A a^{3} b^{3}\right )} x\right )} x^{\frac {5}{2}} - 2 \, {\left (85 \, {\left (3 \, B a^{4} b^{2} + 7 \, A a^{3} b^{3}\right )} x^{2} + 973 \, {\left (B a^{5} b + 9 \, A a^{4} b^{2}\right )} x\right )} x^{\frac {3}{2}} - {\left (253 \, {\left (3 \, B a^{5} b + 7 \, A a^{4} b^{2}\right )} x^{2} + 1315 \, {\left (B a^{6} + 9 \, A a^{5} b\right )} x\right )} \sqrt {x} - \frac {1280 \, {\left (A a^{5} b x^{2} + 3 \, A a^{6} x\right )}}{\sqrt {x}}}{1920 \, {\left (a^{6} b^{5} x^{5} + 5 \, a^{7} b^{4} x^{4} + 10 \, a^{8} b^{3} x^{3} + 10 \, a^{9} b^{2} x^{2} + 5 \, a^{10} b x + a^{11}\right )}} + \frac {5 \, {\left (B a + 7 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} a^{4} b} + \frac {{\left (3 \, B a b + 7 \, A b^{2}\right )} x^{\frac {3}{2}} - 30 \, {\left (B a^{2} + 7 \, A a b\right )} \sqrt {x}}{384 \, a^{6} b} \]
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Time = 0.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.57 \[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {5 \, {\left (B a + 7 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} a^{4} b \mathrm {sgn}\left (b x + a\right )} + \frac {15 \, B a b^{3} x^{\frac {7}{2}} + 105 \, A b^{4} x^{\frac {7}{2}} + 55 \, B a^{2} b^{2} x^{\frac {5}{2}} + 385 \, A a b^{3} x^{\frac {5}{2}} + 73 \, B a^{3} b x^{\frac {3}{2}} + 511 \, A a^{2} b^{2} x^{\frac {3}{2}} - 15 \, B a^{4} \sqrt {x} + 279 \, A a^{3} b \sqrt {x}}{192 \, {\left (b x + a\right )}^{4} a^{4} b \mathrm {sgn}\left (b x + a\right )} \]
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Timed out. \[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {A+B\,x}{\sqrt {x}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
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