Integrand size = 29, antiderivative size = 190 \[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {(A b-a B) \sqrt {x}}{5 a b (a+b x)^5}+\frac {(9 A b+a B) \sqrt {x}}{40 a^2 b (a+b x)^4}+\frac {7 (9 A b+a B) \sqrt {x}}{240 a^3 b (a+b x)^3}+\frac {7 (9 A b+a B) \sqrt {x}}{192 a^4 b (a+b x)^2}+\frac {7 (9 A b+a B) \sqrt {x}}{128 a^5 b (a+b x)}+\frac {7 (9 A b+a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{11/2} b^{3/2}} \]
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Time = 0.06 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {27, 79, 44, 65, 211} \[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {7 (a B+9 A b) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{11/2} b^{3/2}}+\frac {7 \sqrt {x} (a B+9 A b)}{128 a^5 b (a+b x)}+\frac {7 \sqrt {x} (a B+9 A b)}{192 a^4 b (a+b x)^2}+\frac {7 \sqrt {x} (a B+9 A b)}{240 a^3 b (a+b x)^3}+\frac {\sqrt {x} (a B+9 A b)}{40 a^2 b (a+b x)^4}+\frac {\sqrt {x} (A b-a B)}{5 a b (a+b x)^5} \]
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Rule 27
Rule 44
Rule 65
Rule 79
Rule 211
Rubi steps \begin{align*} \text {integral}& = \int \frac {A+B x}{\sqrt {x} (a+b x)^6} \, dx \\ & = \frac {(A b-a B) \sqrt {x}}{5 a b (a+b x)^5}+\frac {(9 A b+a B) \int \frac {1}{\sqrt {x} (a+b x)^5} \, dx}{10 a b} \\ & = \frac {(A b-a B) \sqrt {x}}{5 a b (a+b x)^5}+\frac {(9 A b+a B) \sqrt {x}}{40 a^2 b (a+b x)^4}+\frac {(7 (9 A b+a B)) \int \frac {1}{\sqrt {x} (a+b x)^4} \, dx}{80 a^2 b} \\ & = \frac {(A b-a B) \sqrt {x}}{5 a b (a+b x)^5}+\frac {(9 A b+a B) \sqrt {x}}{40 a^2 b (a+b x)^4}+\frac {7 (9 A b+a B) \sqrt {x}}{240 a^3 b (a+b x)^3}+\frac {(7 (9 A b+a B)) \int \frac {1}{\sqrt {x} (a+b x)^3} \, dx}{96 a^3 b} \\ & = \frac {(A b-a B) \sqrt {x}}{5 a b (a+b x)^5}+\frac {(9 A b+a B) \sqrt {x}}{40 a^2 b (a+b x)^4}+\frac {7 (9 A b+a B) \sqrt {x}}{240 a^3 b (a+b x)^3}+\frac {7 (9 A b+a B) \sqrt {x}}{192 a^4 b (a+b x)^2}+\frac {(7 (9 A b+a B)) \int \frac {1}{\sqrt {x} (a+b x)^2} \, dx}{128 a^4 b} \\ & = \frac {(A b-a B) \sqrt {x}}{5 a b (a+b x)^5}+\frac {(9 A b+a B) \sqrt {x}}{40 a^2 b (a+b x)^4}+\frac {7 (9 A b+a B) \sqrt {x}}{240 a^3 b (a+b x)^3}+\frac {7 (9 A b+a B) \sqrt {x}}{192 a^4 b (a+b x)^2}+\frac {7 (9 A b+a B) \sqrt {x}}{128 a^5 b (a+b x)}+\frac {(7 (9 A b+a B)) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{256 a^5 b} \\ & = \frac {(A b-a B) \sqrt {x}}{5 a b (a+b x)^5}+\frac {(9 A b+a B) \sqrt {x}}{40 a^2 b (a+b x)^4}+\frac {7 (9 A b+a B) \sqrt {x}}{240 a^3 b (a+b x)^3}+\frac {7 (9 A b+a B) \sqrt {x}}{192 a^4 b (a+b x)^2}+\frac {7 (9 A b+a B) \sqrt {x}}{128 a^5 b (a+b x)}+\frac {(7 (9 A b+a B)) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{128 a^5 b} \\ & = \frac {(A b-a B) \sqrt {x}}{5 a b (a+b x)^5}+\frac {(9 A b+a B) \sqrt {x}}{40 a^2 b (a+b x)^4}+\frac {7 (9 A b+a B) \sqrt {x}}{240 a^3 b (a+b x)^3}+\frac {7 (9 A b+a B) \sqrt {x}}{192 a^4 b (a+b x)^2}+\frac {7 (9 A b+a B) \sqrt {x}}{128 a^5 b (a+b x)}+\frac {7 (9 A b+a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{11/2} b^{3/2}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.76 \[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\sqrt {x} \left (-105 a^5 B+945 A b^5 x^4+105 a b^4 x^3 (42 A+B x)+14 a^2 b^3 x^2 (576 A+35 B x)+5 a^4 b (579 A+158 B x)+2 a^3 b^2 x (3555 A+448 B x)\right )}{1920 a^5 b (a+b x)^5}+\frac {7 (9 A b+a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{11/2} b^{3/2}} \]
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Time = 0.14 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(\frac {\frac {7 \left (9 A b +B a \right ) b^{3} x^{\frac {9}{2}}}{128 a^{5}}+\frac {49 b^{2} \left (9 A b +B a \right ) x^{\frac {7}{2}}}{192 a^{4}}+\frac {7 \left (9 A b +B a \right ) b \,x^{\frac {5}{2}}}{15 a^{3}}+\frac {79 \left (9 A b +B a \right ) x^{\frac {3}{2}}}{192 a^{2}}+\frac {\left (193 A b -7 B a \right ) \sqrt {x}}{128 b a}}{\left (b x +a \right )^{5}}+\frac {7 \left (9 A b +B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{128 a^{5} b \sqrt {b a}}\) | \(135\) |
default | \(\frac {\frac {7 \left (9 A b +B a \right ) b^{3} x^{\frac {9}{2}}}{128 a^{5}}+\frac {49 b^{2} \left (9 A b +B a \right ) x^{\frac {7}{2}}}{192 a^{4}}+\frac {7 \left (9 A b +B a \right ) b \,x^{\frac {5}{2}}}{15 a^{3}}+\frac {79 \left (9 A b +B a \right ) x^{\frac {3}{2}}}{192 a^{2}}+\frac {\left (193 A b -7 B a \right ) \sqrt {x}}{128 b a}}{\left (b x +a \right )^{5}}+\frac {7 \left (9 A b +B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{128 a^{5} b \sqrt {b a}}\) | \(135\) |
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Time = 0.42 (sec) , antiderivative size = 637, normalized size of antiderivative = 3.35 \[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\left [-\frac {105 \, {\left (B a^{6} + 9 \, A a^{5} b + {\left (B a b^{5} + 9 \, A b^{6}\right )} x^{5} + 5 \, {\left (B a^{2} b^{4} + 9 \, A a b^{5}\right )} x^{4} + 10 \, {\left (B a^{3} b^{3} + 9 \, A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (B a^{4} b^{2} + 9 \, A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (B a^{5} b + 9 \, A a^{4} b^{2}\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) + 2 \, {\left (105 \, B a^{6} b - 2895 \, A a^{5} b^{2} - 105 \, {\left (B a^{2} b^{5} + 9 \, A a b^{6}\right )} x^{4} - 490 \, {\left (B a^{3} b^{4} + 9 \, A a^{2} b^{5}\right )} x^{3} - 896 \, {\left (B a^{4} b^{3} + 9 \, A a^{3} b^{4}\right )} x^{2} - 790 \, {\left (B a^{5} b^{2} + 9 \, A a^{4} b^{3}\right )} x\right )} \sqrt {x}}{3840 \, {\left (a^{6} b^{7} x^{5} + 5 \, a^{7} b^{6} x^{4} + 10 \, a^{8} b^{5} x^{3} + 10 \, a^{9} b^{4} x^{2} + 5 \, a^{10} b^{3} x + a^{11} b^{2}\right )}}, -\frac {105 \, {\left (B a^{6} + 9 \, A a^{5} b + {\left (B a b^{5} + 9 \, A b^{6}\right )} x^{5} + 5 \, {\left (B a^{2} b^{4} + 9 \, A a b^{5}\right )} x^{4} + 10 \, {\left (B a^{3} b^{3} + 9 \, A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (B a^{4} b^{2} + 9 \, A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (B a^{5} b + 9 \, A a^{4} b^{2}\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (105 \, B a^{6} b - 2895 \, A a^{5} b^{2} - 105 \, {\left (B a^{2} b^{5} + 9 \, A a b^{6}\right )} x^{4} - 490 \, {\left (B a^{3} b^{4} + 9 \, A a^{2} b^{5}\right )} x^{3} - 896 \, {\left (B a^{4} b^{3} + 9 \, A a^{3} b^{4}\right )} x^{2} - 790 \, {\left (B a^{5} b^{2} + 9 \, A a^{4} b^{3}\right )} x\right )} \sqrt {x}}{1920 \, {\left (a^{6} b^{7} x^{5} + 5 \, a^{7} b^{6} x^{4} + 10 \, a^{8} b^{5} x^{3} + 10 \, a^{9} b^{4} x^{2} + 5 \, a^{10} b^{3} x + a^{11} b^{2}\right )}}\right ] \]
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Timed out. \[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.04 \[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {105 \, {\left (B a b^{4} + 9 \, A b^{5}\right )} x^{\frac {9}{2}} + 490 \, {\left (B a^{2} b^{3} + 9 \, A a b^{4}\right )} x^{\frac {7}{2}} + 896 \, {\left (B a^{3} b^{2} + 9 \, A a^{2} b^{3}\right )} x^{\frac {5}{2}} + 790 \, {\left (B a^{4} b + 9 \, A a^{3} b^{2}\right )} x^{\frac {3}{2}} - 15 \, {\left (7 \, B a^{5} - 193 \, A a^{4} b\right )} \sqrt {x}}{1920 \, {\left (a^{5} b^{6} x^{5} + 5 \, a^{6} b^{5} x^{4} + 10 \, a^{7} b^{4} x^{3} + 10 \, a^{8} b^{3} x^{2} + 5 \, a^{9} b^{2} x + a^{10} b\right )}} + \frac {7 \, {\left (B a + 9 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{5} b} \]
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Time = 0.27 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.82 \[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {7 \, {\left (B a + 9 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{5} b} + \frac {105 \, B a b^{4} x^{\frac {9}{2}} + 945 \, A b^{5} x^{\frac {9}{2}} + 490 \, B a^{2} b^{3} x^{\frac {7}{2}} + 4410 \, A a b^{4} x^{\frac {7}{2}} + 896 \, B a^{3} b^{2} x^{\frac {5}{2}} + 8064 \, A a^{2} b^{3} x^{\frac {5}{2}} + 790 \, B a^{4} b x^{\frac {3}{2}} + 7110 \, A a^{3} b^{2} x^{\frac {3}{2}} - 105 \, B a^{5} \sqrt {x} + 2895 \, A a^{4} b \sqrt {x}}{1920 \, {\left (b x + a\right )}^{5} a^{5} b} \]
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Time = 10.18 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.91 \[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {79\,x^{3/2}\,\left (9\,A\,b+B\,a\right )}{192\,a^2}+\frac {49\,b^2\,x^{7/2}\,\left (9\,A\,b+B\,a\right )}{192\,a^4}+\frac {7\,b^3\,x^{9/2}\,\left (9\,A\,b+B\,a\right )}{128\,a^5}+\frac {\sqrt {x}\,\left (193\,A\,b-7\,B\,a\right )}{128\,a\,b}+\frac {7\,b\,x^{5/2}\,\left (9\,A\,b+B\,a\right )}{15\,a^3}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5}+\frac {7\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (9\,A\,b+B\,a\right )}{128\,a^{11/2}\,b^{3/2}} \]
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