Integrand size = 29, antiderivative size = 185 \[ \int \frac {x^{3/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {(A b-a B) x^{5/2}}{5 a b (a+b x)^5}-\frac {(A b+a B) x^{3/2}}{8 a b^2 (a+b x)^4}-\frac {(A b+a B) \sqrt {x}}{16 a b^3 (a+b x)^3}+\frac {(A b+a B) \sqrt {x}}{64 a^2 b^3 (a+b x)^2}+\frac {3 (A b+a B) \sqrt {x}}{128 a^3 b^3 (a+b x)}+\frac {3 (A b+a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{7/2} b^{7/2}} \]
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Time = 0.06 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {27, 79, 43, 44, 65, 211} \[ \int \frac {x^{3/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {3 (a B+A b) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{7/2} b^{7/2}}+\frac {3 \sqrt {x} (a B+A b)}{128 a^3 b^3 (a+b x)}+\frac {\sqrt {x} (a B+A b)}{64 a^2 b^3 (a+b x)^2}-\frac {\sqrt {x} (a B+A b)}{16 a b^3 (a+b x)^3}-\frac {x^{3/2} (a B+A b)}{8 a b^2 (a+b x)^4}+\frac {x^{5/2} (A b-a B)}{5 a b (a+b x)^5} \]
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Rule 27
Rule 43
Rule 44
Rule 65
Rule 79
Rule 211
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^{3/2} (A+B x)}{(a+b x)^6} \, dx \\ & = \frac {(A b-a B) x^{5/2}}{5 a b (a+b x)^5}+\frac {(A b+a B) \int \frac {x^{3/2}}{(a+b x)^5} \, dx}{2 a b} \\ & = \frac {(A b-a B) x^{5/2}}{5 a b (a+b x)^5}-\frac {(A b+a B) x^{3/2}}{8 a b^2 (a+b x)^4}+\frac {(3 (A b+a B)) \int \frac {\sqrt {x}}{(a+b x)^4} \, dx}{16 a b^2} \\ & = \frac {(A b-a B) x^{5/2}}{5 a b (a+b x)^5}-\frac {(A b+a B) x^{3/2}}{8 a b^2 (a+b x)^4}-\frac {(A b+a B) \sqrt {x}}{16 a b^3 (a+b x)^3}+\frac {(A b+a B) \int \frac {1}{\sqrt {x} (a+b x)^3} \, dx}{32 a b^3} \\ & = \frac {(A b-a B) x^{5/2}}{5 a b (a+b x)^5}-\frac {(A b+a B) x^{3/2}}{8 a b^2 (a+b x)^4}-\frac {(A b+a B) \sqrt {x}}{16 a b^3 (a+b x)^3}+\frac {(A b+a B) \sqrt {x}}{64 a^2 b^3 (a+b x)^2}+\frac {(3 (A b+a B)) \int \frac {1}{\sqrt {x} (a+b x)^2} \, dx}{128 a^2 b^3} \\ & = \frac {(A b-a B) x^{5/2}}{5 a b (a+b x)^5}-\frac {(A b+a B) x^{3/2}}{8 a b^2 (a+b x)^4}-\frac {(A b+a B) \sqrt {x}}{16 a b^3 (a+b x)^3}+\frac {(A b+a B) \sqrt {x}}{64 a^2 b^3 (a+b x)^2}+\frac {3 (A b+a B) \sqrt {x}}{128 a^3 b^3 (a+b x)}+\frac {(3 (A b+a B)) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{256 a^3 b^3} \\ & = \frac {(A b-a B) x^{5/2}}{5 a b (a+b x)^5}-\frac {(A b+a B) x^{3/2}}{8 a b^2 (a+b x)^4}-\frac {(A b+a B) \sqrt {x}}{16 a b^3 (a+b x)^3}+\frac {(A b+a B) \sqrt {x}}{64 a^2 b^3 (a+b x)^2}+\frac {3 (A b+a B) \sqrt {x}}{128 a^3 b^3 (a+b x)}+\frac {(3 (A b+a B)) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{128 a^3 b^3} \\ & = \frac {(A b-a B) x^{5/2}}{5 a b (a+b x)^5}-\frac {(A b+a B) x^{3/2}}{8 a b^2 (a+b x)^4}-\frac {(A b+a B) \sqrt {x}}{16 a b^3 (a+b x)^3}+\frac {(A b+a B) \sqrt {x}}{64 a^2 b^3 (a+b x)^2}+\frac {3 (A b+a B) \sqrt {x}}{128 a^3 b^3 (a+b x)}+\frac {3 (A b+a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{7/2} b^{7/2}} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.78 \[ \int \frac {x^{3/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\sqrt {x} \left (-15 a^5 B+15 A b^5 x^4+5 a b^4 x^3 (14 A+3 B x)-5 a^4 b (3 A+14 B x)+2 a^2 b^3 x^2 (64 A+35 B x)-2 a^3 b^2 x (35 A+64 B x)\right )}{640 a^3 b^3 (a+b x)^5}+\frac {3 (A b+a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{7/2} b^{7/2}} \]
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Time = 0.14 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.67
method | result | size |
derivativedivides | \(\frac {\frac {3 \left (A b +B a \right ) b \,x^{\frac {9}{2}}}{128 a^{3}}+\frac {7 \left (A b +B a \right ) x^{\frac {7}{2}}}{64 a^{2}}+\frac {\left (A b -B a \right ) x^{\frac {5}{2}}}{5 b a}-\frac {7 \left (A b +B a \right ) x^{\frac {3}{2}}}{64 b^{2}}-\frac {3 a \left (A b +B a \right ) \sqrt {x}}{128 b^{3}}}{\left (b x +a \right )^{5}}+\frac {3 \left (A b +B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{128 a^{3} b^{3} \sqrt {b a}}\) | \(124\) |
default | \(\frac {\frac {3 \left (A b +B a \right ) b \,x^{\frac {9}{2}}}{128 a^{3}}+\frac {7 \left (A b +B a \right ) x^{\frac {7}{2}}}{64 a^{2}}+\frac {\left (A b -B a \right ) x^{\frac {5}{2}}}{5 b a}-\frac {7 \left (A b +B a \right ) x^{\frac {3}{2}}}{64 b^{2}}-\frac {3 a \left (A b +B a \right ) \sqrt {x}}{128 b^{3}}}{\left (b x +a \right )^{5}}+\frac {3 \left (A b +B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{128 a^{3} b^{3} \sqrt {b a}}\) | \(124\) |
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Time = 0.40 (sec) , antiderivative size = 619, normalized size of antiderivative = 3.35 \[ \int \frac {x^{3/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\left [-\frac {15 \, {\left (B a^{6} + A a^{5} b + {\left (B a b^{5} + A b^{6}\right )} x^{5} + 5 \, {\left (B a^{2} b^{4} + A a b^{5}\right )} x^{4} + 10 \, {\left (B a^{3} b^{3} + A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (B a^{4} b^{2} + A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (B a^{5} b + A a^{4} b^{2}\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) + 2 \, {\left (15 \, B a^{6} b + 15 \, A a^{5} b^{2} - 15 \, {\left (B a^{2} b^{5} + A a b^{6}\right )} x^{4} - 70 \, {\left (B a^{3} b^{4} + A a^{2} b^{5}\right )} x^{3} + 128 \, {\left (B a^{4} b^{3} - A a^{3} b^{4}\right )} x^{2} + 70 \, {\left (B a^{5} b^{2} + A a^{4} b^{3}\right )} x\right )} \sqrt {x}}{1280 \, {\left (a^{4} b^{9} x^{5} + 5 \, a^{5} b^{8} x^{4} + 10 \, a^{6} b^{7} x^{3} + 10 \, a^{7} b^{6} x^{2} + 5 \, a^{8} b^{5} x + a^{9} b^{4}\right )}}, -\frac {15 \, {\left (B a^{6} + A a^{5} b + {\left (B a b^{5} + A b^{6}\right )} x^{5} + 5 \, {\left (B a^{2} b^{4} + A a b^{5}\right )} x^{4} + 10 \, {\left (B a^{3} b^{3} + A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (B a^{4} b^{2} + A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (B a^{5} b + A a^{4} b^{2}\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (15 \, B a^{6} b + 15 \, A a^{5} b^{2} - 15 \, {\left (B a^{2} b^{5} + A a b^{6}\right )} x^{4} - 70 \, {\left (B a^{3} b^{4} + A a^{2} b^{5}\right )} x^{3} + 128 \, {\left (B a^{4} b^{3} - A a^{3} b^{4}\right )} x^{2} + 70 \, {\left (B a^{5} b^{2} + A a^{4} b^{3}\right )} x\right )} \sqrt {x}}{640 \, {\left (a^{4} b^{9} x^{5} + 5 \, a^{5} b^{8} x^{4} + 10 \, a^{6} b^{7} x^{3} + 10 \, a^{7} b^{6} x^{2} + 5 \, a^{8} b^{5} x + a^{9} b^{4}\right )}}\right ] \]
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Timed out. \[ \int \frac {x^{3/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.05 \[ \int \frac {x^{3/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {15 \, {\left (B a b^{4} + A b^{5}\right )} x^{\frac {9}{2}} + 70 \, {\left (B a^{2} b^{3} + A a b^{4}\right )} x^{\frac {7}{2}} - 128 \, {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{\frac {5}{2}} - 70 \, {\left (B a^{4} b + A a^{3} b^{2}\right )} x^{\frac {3}{2}} - 15 \, {\left (B a^{5} + A a^{4} b\right )} \sqrt {x}}{640 \, {\left (a^{3} b^{8} x^{5} + 5 \, a^{4} b^{7} x^{4} + 10 \, a^{5} b^{6} x^{3} + 10 \, a^{6} b^{5} x^{2} + 5 \, a^{7} b^{4} x + a^{8} b^{3}\right )}} + \frac {3 \, {\left (B a + A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{3} b^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.83 \[ \int \frac {x^{3/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {3 \, {\left (B a + A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{3} b^{3}} + \frac {15 \, B a b^{4} x^{\frac {9}{2}} + 15 \, A b^{5} x^{\frac {9}{2}} + 70 \, B a^{2} b^{3} x^{\frac {7}{2}} + 70 \, A a b^{4} x^{\frac {7}{2}} - 128 \, B a^{3} b^{2} x^{\frac {5}{2}} + 128 \, A a^{2} b^{3} x^{\frac {5}{2}} - 70 \, B a^{4} b x^{\frac {3}{2}} - 70 \, A a^{3} b^{2} x^{\frac {3}{2}} - 15 \, B a^{5} \sqrt {x} - 15 \, A a^{4} b \sqrt {x}}{640 \, {\left (b x + a\right )}^{5} a^{3} b^{3}} \]
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Time = 10.10 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.87 \[ \int \frac {x^{3/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {7\,x^{7/2}\,\left (A\,b+B\,a\right )}{64\,a^2}-\frac {7\,x^{3/2}\,\left (A\,b+B\,a\right )}{64\,b^2}+\frac {x^{5/2}\,\left (A\,b-B\,a\right )}{5\,a\,b}-\frac {3\,a\,\sqrt {x}\,\left (A\,b+B\,a\right )}{128\,b^3}+\frac {3\,b\,x^{9/2}\,\left (A\,b+B\,a\right )}{128\,a^3}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5}+\frac {3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (A\,b+B\,a\right )}{128\,a^{7/2}\,b^{7/2}} \]
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