Integrand size = 33, antiderivative size = 256 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {(b d-a e)^2 (2 b B d+7 A b e-9 a B e) \sqrt {d+e x}}{b^5}+\frac {(b d-a e) (2 b B d+7 A b e-9 a B e) (d+e x)^{3/2}}{3 b^4}+\frac {(2 b B d+7 A b e-9 a B e) (d+e x)^{5/2}}{5 b^3}+\frac {(2 b B d+7 A b e-9 a B e) (d+e x)^{7/2}}{7 b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}-\frac {(b d-a e)^{5/2} (2 b B d+7 A b e-9 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}} \]
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Time = 0.21 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {27, 79, 52, 65, 214} \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=-\frac {(b d-a e)^{5/2} (-9 a B e+7 A b e+2 b B d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}}+\frac {\sqrt {d+e x} (b d-a e)^2 (-9 a B e+7 A b e+2 b B d)}{b^5}+\frac {(d+e x)^{3/2} (b d-a e) (-9 a B e+7 A b e+2 b B d)}{3 b^4}+\frac {(d+e x)^{5/2} (-9 a B e+7 A b e+2 b B d)}{5 b^3}+\frac {(d+e x)^{7/2} (-9 a B e+7 A b e+2 b B d)}{7 b^2 (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{b (a+b x) (b d-a e)} \]
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Rule 27
Rule 52
Rule 65
Rule 79
Rule 214
Rubi steps \begin{align*} \text {integral}& = \int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^2} \, dx \\ & = -\frac {(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}+\frac {(2 b B d+7 A b e-9 a B e) \int \frac {(d+e x)^{7/2}}{a+b x} \, dx}{2 b (b d-a e)} \\ & = \frac {(2 b B d+7 A b e-9 a B e) (d+e x)^{7/2}}{7 b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}+\frac {(2 b B d+7 A b e-9 a B e) \int \frac {(d+e x)^{5/2}}{a+b x} \, dx}{2 b^2} \\ & = \frac {(2 b B d+7 A b e-9 a B e) (d+e x)^{5/2}}{5 b^3}+\frac {(2 b B d+7 A b e-9 a B e) (d+e x)^{7/2}}{7 b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}+\frac {((b d-a e) (2 b B d+7 A b e-9 a B e)) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{2 b^3} \\ & = \frac {(b d-a e) (2 b B d+7 A b e-9 a B e) (d+e x)^{3/2}}{3 b^4}+\frac {(2 b B d+7 A b e-9 a B e) (d+e x)^{5/2}}{5 b^3}+\frac {(2 b B d+7 A b e-9 a B e) (d+e x)^{7/2}}{7 b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}+\frac {\left ((b d-a e)^2 (2 b B d+7 A b e-9 a B e)\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{2 b^4} \\ & = \frac {(b d-a e)^2 (2 b B d+7 A b e-9 a B e) \sqrt {d+e x}}{b^5}+\frac {(b d-a e) (2 b B d+7 A b e-9 a B e) (d+e x)^{3/2}}{3 b^4}+\frac {(2 b B d+7 A b e-9 a B e) (d+e x)^{5/2}}{5 b^3}+\frac {(2 b B d+7 A b e-9 a B e) (d+e x)^{7/2}}{7 b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}+\frac {\left ((b d-a e)^3 (2 b B d+7 A b e-9 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 b^5} \\ & = \frac {(b d-a e)^2 (2 b B d+7 A b e-9 a B e) \sqrt {d+e x}}{b^5}+\frac {(b d-a e) (2 b B d+7 A b e-9 a B e) (d+e x)^{3/2}}{3 b^4}+\frac {(2 b B d+7 A b e-9 a B e) (d+e x)^{5/2}}{5 b^3}+\frac {(2 b B d+7 A b e-9 a B e) (d+e x)^{7/2}}{7 b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}+\frac {\left ((b d-a e)^3 (2 b B d+7 A b e-9 a B e)\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{b^5 e} \\ & = \frac {(b d-a e)^2 (2 b B d+7 A b e-9 a B e) \sqrt {d+e x}}{b^5}+\frac {(b d-a e) (2 b B d+7 A b e-9 a B e) (d+e x)^{3/2}}{3 b^4}+\frac {(2 b B d+7 A b e-9 a B e) (d+e x)^{5/2}}{5 b^3}+\frac {(2 b B d+7 A b e-9 a B e) (d+e x)^{7/2}}{7 b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}-\frac {(b d-a e)^{5/2} (2 b B d+7 A b e-9 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.19 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {\sqrt {d+e x} \left (-7 A b \left (-105 a^3 e^3+35 a^2 b e^2 (7 d-2 e x)+7 a b^2 e \left (-23 d^2+24 d e x+2 e^2 x^2\right )+b^3 \left (15 d^3-116 d^2 e x-32 d e^2 x^2-6 e^3 x^3\right )\right )+B \left (-945 a^4 e^3+105 a^3 b e^2 (23 d-6 e x)+7 a^2 b^2 e \left (-277 d^2+236 d e x+18 e^2 x^2\right )+a b^3 \left (457 d^3-1380 d^2 e x-316 d e^2 x^2-54 e^3 x^3\right )+2 b^4 x \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )\right )\right )}{105 b^5 (a+b x)}-\frac {(-b d+a e)^{5/2} (2 b B d+7 A b e-9 a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{b^{11/2}} \]
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Time = 0.54 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.09
method | result | size |
pseudoelliptic | \(\frac {-7 \left (a e -b d \right )^{3} \left (b x +a \right ) \left (b \left (A e +\frac {2 B d}{7}\right )-\frac {9 B a e}{7}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )+7 \sqrt {\left (a e -b d \right ) b}\, \left (\left (\frac {2 x^{3} \left (\frac {5 B x}{7}+A \right ) e^{3}}{35}+\frac {32 x^{2} d \left (\frac {33 B x}{56}+A \right ) e^{2}}{105}+\frac {116 x \left (\frac {61 B x}{203}+A \right ) d^{2} e}{105}-\frac {\left (-\frac {352 B x}{105}+A \right ) d^{3}}{7}\right ) b^{4}+\frac {23 \left (-\frac {2 x^{2} \left (\frac {27 B x}{49}+A \right ) e^{3}}{23}-\frac {24 x \left (\frac {79 B x}{294}+A \right ) d \,e^{2}}{23}+d^{2} \left (-\frac {60 B x}{49}+A \right ) e +\frac {457 B \,d^{3}}{1127}\right ) a \,b^{3}}{15}-\frac {7 \left (-\frac {2 \left (\frac {9 B x}{35}+A \right ) x \,e^{2}}{7}+d \left (-\frac {236 B x}{245}+A \right ) e +\frac {277 B \,d^{2}}{245}\right ) e \,a^{2} b^{2}}{3}+\left (\left (-\frac {6 B x}{7}+A \right ) e +\frac {23 B d}{7}\right ) e^{2} a^{3} b -\frac {9 B \,e^{3} a^{4}}{7}\right ) \sqrt {e x +d}}{b^{5} \left (b x +a \right ) \sqrt {\left (a e -b d \right ) b}}\) | \(278\) |
risch | \(\frac {2 \left (15 x^{3} B \,b^{3} e^{3}+21 A \,b^{3} e^{3} x^{2}-42 B a \,b^{2} e^{3} x^{2}+66 B \,b^{3} d \,e^{2} x^{2}-70 A x a \,b^{2} e^{3}+112 A \,b^{3} d \,e^{2} x +105 B x \,a^{2} b \,e^{3}-224 B a \,b^{2} d \,e^{2} x +122 B \,b^{3} d^{2} e x +315 A \,a^{2} b \,e^{3}-700 A a \,b^{2} d \,e^{2}+406 A \,b^{3} d^{2} e -420 B \,e^{3} a^{3}+1050 B \,a^{2} b d \,e^{2}-812 B a \,b^{2} d^{2} e +176 B \,b^{3} d^{3}\right ) \sqrt {e x +d}}{105 b^{5}}-\frac {\left (2 a^{3} e^{3}-6 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -2 b^{3} d^{3}\right ) \left (\frac {\left (-\frac {1}{2} A b e +\frac {1}{2} B a e \right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {\left (7 A b e -9 B a e +2 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}\right )}{b^{5}}\) | \(323\) |
derivativedivides | \(\frac {\frac {2 B \left (e x +d \right )^{\frac {7}{2}} b^{3}}{7}+\frac {2 A \,b^{3} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {4 B a \,b^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 B \,b^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {4 A a \,b^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {4 A \,b^{3} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+2 B \,a^{2} b \,e^{2} \left (e x +d \right )^{\frac {3}{2}}-\frac {8 B a \,b^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 B \,b^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+6 A \,a^{2} b \,e^{3} \sqrt {e x +d}-12 A a \,b^{2} d \,e^{2} \sqrt {e x +d}+6 A \,b^{3} d^{2} e \sqrt {e x +d}-8 B \,a^{3} e^{3} \sqrt {e x +d}+18 B \,a^{2} b d \,e^{2} \sqrt {e x +d}-12 B a \,b^{2} d^{2} e \sqrt {e x +d}+2 B \,b^{3} d^{3} \sqrt {e x +d}}{b^{5}}-\frac {2 \left (\frac {\left (-\frac {1}{2} A \,a^{3} b \,e^{4}+\frac {3}{2} A \,a^{2} b^{2} d \,e^{3}-\frac {3}{2} A a \,b^{3} d^{2} e^{2}+\frac {1}{2} A \,b^{4} d^{3} e +\frac {1}{2} B \,a^{4} e^{4}-\frac {3}{2} B \,a^{3} b d \,e^{3}+\frac {3}{2} B \,a^{2} b^{2} d^{2} e^{2}-\frac {1}{2} B a \,b^{3} d^{3} e \right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {\left (7 A \,a^{3} b \,e^{4}-21 A \,a^{2} b^{2} d \,e^{3}+21 A a \,b^{3} d^{2} e^{2}-7 A \,b^{4} d^{3} e -9 B \,a^{4} e^{4}+29 B \,a^{3} b d \,e^{3}-33 B \,a^{2} b^{2} d^{2} e^{2}+15 B a \,b^{3} d^{3} e -2 b^{4} B \,d^{4}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}\right )}{b^{5}}\) | \(525\) |
default | \(\frac {\frac {2 B \left (e x +d \right )^{\frac {7}{2}} b^{3}}{7}+\frac {2 A \,b^{3} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {4 B a \,b^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 B \,b^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {4 A a \,b^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {4 A \,b^{3} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+2 B \,a^{2} b \,e^{2} \left (e x +d \right )^{\frac {3}{2}}-\frac {8 B a \,b^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 B \,b^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+6 A \,a^{2} b \,e^{3} \sqrt {e x +d}-12 A a \,b^{2} d \,e^{2} \sqrt {e x +d}+6 A \,b^{3} d^{2} e \sqrt {e x +d}-8 B \,a^{3} e^{3} \sqrt {e x +d}+18 B \,a^{2} b d \,e^{2} \sqrt {e x +d}-12 B a \,b^{2} d^{2} e \sqrt {e x +d}+2 B \,b^{3} d^{3} \sqrt {e x +d}}{b^{5}}-\frac {2 \left (\frac {\left (-\frac {1}{2} A \,a^{3} b \,e^{4}+\frac {3}{2} A \,a^{2} b^{2} d \,e^{3}-\frac {3}{2} A a \,b^{3} d^{2} e^{2}+\frac {1}{2} A \,b^{4} d^{3} e +\frac {1}{2} B \,a^{4} e^{4}-\frac {3}{2} B \,a^{3} b d \,e^{3}+\frac {3}{2} B \,a^{2} b^{2} d^{2} e^{2}-\frac {1}{2} B a \,b^{3} d^{3} e \right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {\left (7 A \,a^{3} b \,e^{4}-21 A \,a^{2} b^{2} d \,e^{3}+21 A a \,b^{3} d^{2} e^{2}-7 A \,b^{4} d^{3} e -9 B \,a^{4} e^{4}+29 B \,a^{3} b d \,e^{3}-33 B \,a^{2} b^{2} d^{2} e^{2}+15 B a \,b^{3} d^{3} e -2 b^{4} B \,d^{4}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}\right )}{b^{5}}\) | \(525\) |
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Leaf count of result is larger than twice the leaf count of optimal. 498 vs. \(2 (229) = 458\).
Time = 0.48 (sec) , antiderivative size = 1006, normalized size of antiderivative = 3.93 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\left [\frac {105 \, {\left (2 \, B a b^{3} d^{3} - {\left (13 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} d^{2} e + 2 \, {\left (10 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} d e^{2} - {\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} e^{3} + {\left (2 \, B b^{4} d^{3} - {\left (13 \, B a b^{3} - 7 \, A b^{4}\right )} d^{2} e + 2 \, {\left (10 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} d e^{2} - {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) + 2 \, {\left (30 \, B b^{4} e^{3} x^{4} + {\left (457 \, B a b^{3} - 105 \, A b^{4}\right )} d^{3} - 7 \, {\left (277 \, B a^{2} b^{2} - 161 \, A a b^{3}\right )} d^{2} e + 35 \, {\left (69 \, B a^{3} b - 49 \, A a^{2} b^{2}\right )} d e^{2} - 105 \, {\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} e^{3} + 6 \, {\left (22 \, B b^{4} d e^{2} - {\left (9 \, B a b^{3} - 7 \, A b^{4}\right )} e^{3}\right )} x^{3} + 2 \, {\left (122 \, B b^{4} d^{2} e - 2 \, {\left (79 \, B a b^{3} - 56 \, A b^{4}\right )} d e^{2} + 7 \, {\left (9 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} e^{3}\right )} x^{2} + 2 \, {\left (176 \, B b^{4} d^{3} - 2 \, {\left (345 \, B a b^{3} - 203 \, A b^{4}\right )} d^{2} e + 14 \, {\left (59 \, B a^{2} b^{2} - 42 \, A a b^{3}\right )} d e^{2} - 35 \, {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{210 \, {\left (b^{6} x + a b^{5}\right )}}, -\frac {105 \, {\left (2 \, B a b^{3} d^{3} - {\left (13 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} d^{2} e + 2 \, {\left (10 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} d e^{2} - {\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} e^{3} + {\left (2 \, B b^{4} d^{3} - {\left (13 \, B a b^{3} - 7 \, A b^{4}\right )} d^{2} e + 2 \, {\left (10 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} d e^{2} - {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (30 \, B b^{4} e^{3} x^{4} + {\left (457 \, B a b^{3} - 105 \, A b^{4}\right )} d^{3} - 7 \, {\left (277 \, B a^{2} b^{2} - 161 \, A a b^{3}\right )} d^{2} e + 35 \, {\left (69 \, B a^{3} b - 49 \, A a^{2} b^{2}\right )} d e^{2} - 105 \, {\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} e^{3} + 6 \, {\left (22 \, B b^{4} d e^{2} - {\left (9 \, B a b^{3} - 7 \, A b^{4}\right )} e^{3}\right )} x^{3} + 2 \, {\left (122 \, B b^{4} d^{2} e - 2 \, {\left (79 \, B a b^{3} - 56 \, A b^{4}\right )} d e^{2} + 7 \, {\left (9 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} e^{3}\right )} x^{2} + 2 \, {\left (176 \, B b^{4} d^{3} - 2 \, {\left (345 \, B a b^{3} - 203 \, A b^{4}\right )} d^{2} e + 14 \, {\left (59 \, B a^{2} b^{2} - 42 \, A a b^{3}\right )} d e^{2} - 35 \, {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{105 \, {\left (b^{6} x + a b^{5}\right )}}\right ] \]
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Timed out. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 583 vs. \(2 (229) = 458\).
Time = 0.29 (sec) , antiderivative size = 583, normalized size of antiderivative = 2.28 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {{\left (2 \, B b^{4} d^{4} - 15 \, B a b^{3} d^{3} e + 7 \, A b^{4} d^{3} e + 33 \, B a^{2} b^{2} d^{2} e^{2} - 21 \, A a b^{3} d^{2} e^{2} - 29 \, B a^{3} b d e^{3} + 21 \, A a^{2} b^{2} d e^{3} + 9 \, B a^{4} e^{4} - 7 \, A a^{3} b e^{4}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{5}} + \frac {\sqrt {e x + d} B a b^{3} d^{3} e - \sqrt {e x + d} A b^{4} d^{3} e - 3 \, \sqrt {e x + d} B a^{2} b^{2} d^{2} e^{2} + 3 \, \sqrt {e x + d} A a b^{3} d^{2} e^{2} + 3 \, \sqrt {e x + d} B a^{3} b d e^{3} - 3 \, \sqrt {e x + d} A a^{2} b^{2} d e^{3} - \sqrt {e x + d} B a^{4} e^{4} + \sqrt {e x + d} A a^{3} b e^{4}}{{\left ({\left (e x + d\right )} b - b d + a e\right )} b^{5}} + \frac {2 \, {\left (15 \, {\left (e x + d\right )}^{\frac {7}{2}} B b^{12} + 21 \, {\left (e x + d\right )}^{\frac {5}{2}} B b^{12} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{12} d^{2} + 105 \, \sqrt {e x + d} B b^{12} d^{3} - 42 \, {\left (e x + d\right )}^{\frac {5}{2}} B a b^{11} e + 21 \, {\left (e x + d\right )}^{\frac {5}{2}} A b^{12} e - 140 \, {\left (e x + d\right )}^{\frac {3}{2}} B a b^{11} d e + 70 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{12} d e - 630 \, \sqrt {e x + d} B a b^{11} d^{2} e + 315 \, \sqrt {e x + d} A b^{12} d^{2} e + 105 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{2} b^{10} e^{2} - 70 \, {\left (e x + d\right )}^{\frac {3}{2}} A a b^{11} e^{2} + 945 \, \sqrt {e x + d} B a^{2} b^{10} d e^{2} - 630 \, \sqrt {e x + d} A a b^{11} d e^{2} - 420 \, \sqrt {e x + d} B a^{3} b^{9} e^{3} + 315 \, \sqrt {e x + d} A a^{2} b^{10} e^{3}\right )}}{105 \, b^{14}} \]
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Time = 0.16 (sec) , antiderivative size = 562, normalized size of antiderivative = 2.20 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\left (\frac {2\,A\,e-2\,B\,d}{5\,b^2}+\frac {2\,B\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{5\,b^4}\right )\,{\left (d+e\,x\right )}^{5/2}+\left (\frac {\left (\frac {\left (2\,b^2\,d-2\,a\,b\,e\right )\,\left (\frac {2\,A\,e-2\,B\,d}{b^2}+\frac {2\,B\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{b^4}\right )}{b^2}-\frac {2\,B\,{\left (a\,e-b\,d\right )}^2}{b^4}\right )\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{b^2}-\frac {{\left (a\,e-b\,d\right )}^2\,\left (\frac {2\,A\,e-2\,B\,d}{b^2}+\frac {2\,B\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{b^4}\right )}{b^2}\right )\,\sqrt {d+e\,x}+\left (\frac {\left (2\,b^2\,d-2\,a\,b\,e\right )\,\left (\frac {2\,A\,e-2\,B\,d}{b^2}+\frac {2\,B\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{b^4}\right )}{3\,b^2}-\frac {2\,B\,{\left (a\,e-b\,d\right )}^2}{3\,b^4}\right )\,{\left (d+e\,x\right )}^{3/2}-\frac {\sqrt {d+e\,x}\,\left (B\,a^4\,e^4-3\,B\,a^3\,b\,d\,e^3-A\,a^3\,b\,e^4+3\,B\,a^2\,b^2\,d^2\,e^2+3\,A\,a^2\,b^2\,d\,e^3-B\,a\,b^3\,d^3\,e-3\,A\,a\,b^3\,d^2\,e^2+A\,b^4\,d^3\,e\right )}{b^6\,\left (d+e\,x\right )-b^6\,d+a\,b^5\,e}+\frac {2\,B\,{\left (d+e\,x\right )}^{7/2}}{7\,b^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,{\left (a\,e-b\,d\right )}^{5/2}\,\sqrt {d+e\,x}\,\left (7\,A\,b\,e-9\,B\,a\,e+2\,B\,b\,d\right )}{9\,B\,a^4\,e^4-29\,B\,a^3\,b\,d\,e^3-7\,A\,a^3\,b\,e^4+33\,B\,a^2\,b^2\,d^2\,e^2+21\,A\,a^2\,b^2\,d\,e^3-15\,B\,a\,b^3\,d^3\,e-21\,A\,a\,b^3\,d^2\,e^2+2\,B\,b^4\,d^4+7\,A\,b^4\,d^3\,e}\right )\,{\left (a\,e-b\,d\right )}^{5/2}\,\left (7\,A\,b\,e-9\,B\,a\,e+2\,B\,b\,d\right )}{b^{11/2}} \]
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