Integrand size = 33, antiderivative size = 435 \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {3003 e^4 (2 b B d-3 A b e+a B e)}{640 b (b d-a e)^6 (d+e x)^{5/2}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac {429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac {1001 e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^7 (d+e x)^{3/2}}+\frac {3003 b e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^8 \sqrt {d+e x}}-\frac {3003 b^{3/2} e^4 (2 b B d-3 A b e+a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{17/2}} \]
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Time = 0.34 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {27, 79, 44, 53, 65, 214} \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {3003 b^{3/2} e^4 (a B e-3 A b e+2 b B d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{17/2}}+\frac {3003 b e^4 (a B e-3 A b e+2 b B d)}{128 \sqrt {d+e x} (b d-a e)^8}+\frac {1001 e^4 (a B e-3 A b e+2 b B d)}{128 (d+e x)^{3/2} (b d-a e)^7}+\frac {3003 e^4 (a B e-3 A b e+2 b B d)}{640 b (d+e x)^{5/2} (b d-a e)^6}+\frac {429 e^3 (a B e-3 A b e+2 b B d)}{128 b (a+b x) (d+e x)^{5/2} (b d-a e)^5}-\frac {143 e^2 (a B e-3 A b e+2 b B d)}{192 b (a+b x)^2 (d+e x)^{5/2} (b d-a e)^4}+\frac {13 e (a B e-3 A b e+2 b B d)}{48 b (a+b x)^3 (d+e x)^{5/2} (b d-a e)^3}-\frac {a B e-3 A b e+2 b B d}{8 b (a+b x)^4 (d+e x)^{5/2} (b d-a e)^2}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{5/2} (b d-a e)} \]
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Rule 27
Rule 44
Rule 53
Rule 65
Rule 79
Rule 214
Rubi steps \begin{align*} \text {integral}& = \int \frac {A+B x}{(a+b x)^6 (d+e x)^{7/2}} \, dx \\ & = -\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac {(2 b B d-3 A b e+a B e) \int \frac {1}{(a+b x)^5 (d+e x)^{7/2}} \, dx}{2 b (b d-a e)} \\ & = -\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac {(13 e (2 b B d-3 A b e+a B e)) \int \frac {1}{(a+b x)^4 (d+e x)^{7/2}} \, dx}{16 b (b d-a e)^2} \\ & = -\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac {\left (143 e^2 (2 b B d-3 A b e+a B e)\right ) \int \frac {1}{(a+b x)^3 (d+e x)^{7/2}} \, dx}{96 b (b d-a e)^3} \\ & = -\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}-\frac {\left (429 e^3 (2 b B d-3 A b e+a B e)\right ) \int \frac {1}{(a+b x)^2 (d+e x)^{7/2}} \, dx}{128 b (b d-a e)^4} \\ & = -\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac {429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac {\left (3003 e^4 (2 b B d-3 A b e+a B e)\right ) \int \frac {1}{(a+b x) (d+e x)^{7/2}} \, dx}{256 b (b d-a e)^5} \\ & = \frac {3003 e^4 (2 b B d-3 A b e+a B e)}{640 b (b d-a e)^6 (d+e x)^{5/2}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac {429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac {\left (3003 e^4 (2 b B d-3 A b e+a B e)\right ) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{256 (b d-a e)^6} \\ & = \frac {3003 e^4 (2 b B d-3 A b e+a B e)}{640 b (b d-a e)^6 (d+e x)^{5/2}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac {429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac {1001 e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^7 (d+e x)^{3/2}}+\frac {\left (3003 b e^4 (2 b B d-3 A b e+a B e)\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{256 (b d-a e)^7} \\ & = \frac {3003 e^4 (2 b B d-3 A b e+a B e)}{640 b (b d-a e)^6 (d+e x)^{5/2}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac {429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac {1001 e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^7 (d+e x)^{3/2}}+\frac {3003 b e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^8 \sqrt {d+e x}}+\frac {\left (3003 b^2 e^4 (2 b B d-3 A b e+a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 (b d-a e)^8} \\ & = \frac {3003 e^4 (2 b B d-3 A b e+a B e)}{640 b (b d-a e)^6 (d+e x)^{5/2}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac {429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac {1001 e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^7 (d+e x)^{3/2}}+\frac {3003 b e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^8 \sqrt {d+e x}}+\frac {\left (3003 b^2 e^3 (2 b B d-3 A b e+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 )}{128 (b d-a e)^8} \\ & = \frac {3003 e^4 (2 b B d-3 A b e+a B e)}{640 b (b d-a e)^6 (d+e x)^{5/2}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac {429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac {1001 e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^7 (d+e x)^{3/2}}+\frac {3003 b e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^8 \sqrt {d+e x}}-\frac {3003 b^{3/2} e^4 (2 b B d-3 A b e+a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{17/2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(884\) vs. \(2(435)=870\).
Time = 5.46 (sec) , antiderivative size = 884, normalized size of antiderivative = 2.03 \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {-3 A \left (256 a^7 e^7-256 a^6 b e^6 (12 d+5 e x)+256 a^5 b^2 e^5 \left (116 d^2+160 d e x+65 e^2 x^2\right )+5 a^4 b^3 e^4 \left (5327 d^3+45677 d^2 e x+66157 d e^2 x^2+27599 e^3 x^3\right )+10 a^3 b^4 e^3 \left (-1211 d^4+5810 d^3 e x+54392 d^2 e^2 x^2+80366 d e^3 x^3+33891 e^4 x^4\right )+2 a^2 b^5 e^2 \left (2324 d^5-6545 d^4 e x+30485 d^3 e^2 x^2+302445 d^2 e^3 x^3+452595 d e^4 x^4+192192 e^5 x^5\right )+2 a b^6 e \left (-568 d^6+1240 d^5 e x-3445 d^4 e^2 x^2+15730 d^3 e^3 x^3+163020 d^2 e^4 x^4+246246 d e^5 x^5+105105 e^6 x^6\right )+b^7 \left (128 d^7-240 d^6 e x+520 d^5 e^2 x^2-1430 d^4 e^3 x^3+6435 d^3 e^4 x^4+69069 d^2 e^5 x^5+105105 d e^6 x^6+45045 e^7 x^7\right )\right )+B \left (-256 a^7 e^6 (2 d+5 e x)+256 a^6 b e^5 \left (64 d^2+150 d e x+65 e^2 x^2\right )+a^5 b^2 e^4 \left (100363 d^3+310305 d^2 e x+364065 d e^2 x^2+137995 e^3 x^3\right )+10 a^4 b^3 e^3 \left (2324 d^4+51487 d^3 e x+120549 d^2 e^2 x^2+107965 d e^3 x^3+33891 e^4 x^4\right )+2 a^3 b^4 e^2 \left (-2618 d^5+51555 d^4 e x+574405 d^3 e^2 x^2+1106105 d^2 e^3 x^3+791505 d e^4 x^4+192192 e^5 x^5\right )+2 b^7 d x \left (-240 d^6+520 d^5 e x-1430 d^4 e^2 x^2+6435 d^3 e^3 x^3+69069 d^2 e^4 x^4+105105 d e^5 x^5+45045 e^6 x^6\right )+2 a^2 b^5 e \left (496 d^6-11850 d^5 e x+57525 d^4 e^2 x^2+620620 d^3 e^3 x^3+1068210 d^2 e^4 x^4+630630 d e^5 x^5+105105 e^6 x^6\right )+a b^6 \left (-96 d^7+4720 d^6 e x-13260 d^5 e^2 x^2+61490 d^4 e^3 x^3+658515 d^3 e^4 x^4+1054053 d^2 e^5 x^5+525525 d e^6 x^6+45045 e^7 x^7\right )\right )}{(b d-a e)^8 (a+b x)^5 (d+e x)^{5/2}}+\frac {45045 b^{3/2} e^4 (2 b B d-3 A b e+a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{17/2}}}{1920} \]
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Time = 0.50 (sec) , antiderivative size = 618, normalized size of antiderivative = 1.42
method | result | size |
derivativedivides | \(2 e^{4} \left (-\frac {A e -B d}{5 \left (a e -b d \right )^{6} \left (e x +d \right )^{\frac {5}{2}}}-\frac {-6 A b e +B a e +5 B b d}{3 \left (a e -b d \right )^{7} \left (e x +d \right )^{\frac {3}{2}}}-\frac {3 b \left (7 A b e -2 B a e -5 B b d \right )}{\left (a e -b d \right )^{8} \sqrt {e x +d}}-\frac {b^{2} \left (\frac {\left (\frac {3633}{256} A e \,b^{5}-\frac {1467}{256} B e \,b^{4} a -\frac {1083}{128} B d \,b^{5}\right ) \left (e x +d \right )^{\frac {9}{2}}+\frac {b^{3} \left (23511 A a b \,e^{2}-23511 A \,b^{2} d e -9629 a^{2} B \,e^{2}-4253 B a b d e +13882 B \,b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{384}+\left (\frac {1001}{10} A \,a^{2} b^{3} e^{3}-\frac {1001}{5} A a \,b^{4} d \,e^{2}+\frac {1001}{10} A \,b^{5} d^{2} e -\frac {1253}{30} B \,e^{3} a^{3} b^{2}+\frac {749}{10} B a \,b^{4} d^{2} e -\frac {175}{3} B \,b^{5} d^{3}+\frac {126}{5} B \,a^{2} b^{3} d \,e^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (\frac {9443}{128} A \,a^{3} b^{2} e^{4}-\frac {28329}{128} A \,a^{2} b^{3} d \,e^{3}+\frac {28329}{128} A a \,b^{4} d^{2} e^{2}-\frac {9443}{128} A \,b^{5} d^{3} e -\frac {12131}{384} B \,e^{4} a^{4} b +\frac {20195}{384} B \,a^{3} b^{2} d \,e^{3}+\frac {4067}{128} B \,a^{2} b^{3} d^{2} e^{2}-\frac {36463}{384} B a \,b^{4} d^{3} e +\frac {8099}{192} B \,b^{5} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {5327}{256} A \,a^{4} b \,e^{5}-\frac {5327}{64} A \,a^{3} b^{2} d \,e^{4}+\frac {15981}{128} A \,a^{2} b^{3} d^{2} e^{3}-\frac {5327}{64} A a \,b^{4} d^{3} e^{2}+\frac {5327}{256} A \,b^{5} d^{4} e -\frac {2373}{256} B \,a^{5} e^{5}+\frac {3269}{128} B \,a^{4} b d \,e^{4}-\frac {1211}{128} B \,a^{3} b^{2} d^{2} e^{3}-\frac {1029}{32} B \,a^{2} b^{3} d^{3} e^{2}+\frac {9443}{256} B a \,b^{4} d^{4} e -\frac {1477}{128} B \,b^{5} d^{5}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {3003 \left (3 A b e -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 )}{256 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{8}}\right )\) | \(618\) |
default | \(2 e^{4} \left (-\frac {A e -B d}{5 \left (a e -b d \right )^{6} \left (e x +d \right )^{\frac {5}{2}}}-\frac {-6 A b e +B a e +5 B b d}{3 \left (a e -b d \right )^{7} \left (e x +d \right )^{\frac {3}{2}}}-\frac {3 b \left (7 A b e -2 B a e -5 B b d \right )}{\left (a e -b d \right )^{8} \sqrt {e x +d}}-\frac {b^{2} \left (\frac {\left (\frac {3633}{256} A e \,b^{5}-\frac {1467}{256} B e \,b^{4} a -\frac {1083}{128} B d \,b^{5}\right ) \left (e x +d \right )^{\frac {9}{2}}+\frac {b^{3} \left (23511 A a b \,e^{2}-23511 A \,b^{2} d e -9629 a^{2} B \,e^{2}-4253 B a b d e +13882 B \,b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{384}+\left (\frac {1001}{10} A \,a^{2} b^{3} e^{3}-\frac {1001}{5} A a \,b^{4} d \,e^{2}+\frac {1001}{10} A \,b^{5} d^{2} e -\frac {1253}{30} B \,e^{3} a^{3} b^{2}+\frac {749}{10} B a \,b^{4} d^{2} e -\frac {175}{3} B \,b^{5} d^{3}+\frac {126}{5} B \,a^{2} b^{3} d \,e^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (\frac {9443}{128} A \,a^{3} b^{2} e^{4}-\frac {28329}{128} A \,a^{2} b^{3} d \,e^{3}+\frac {28329}{128} A a \,b^{4} d^{2} e^{2}-\frac {9443}{128} A \,b^{5} d^{3} e -\frac {12131}{384} B \,e^{4} a^{4} b +\frac {20195}{384} B \,a^{3} b^{2} d \,e^{3}+\frac {4067}{128} B \,a^{2} b^{3} d^{2} e^{2}-\frac {36463}{384} B a \,b^{4} d^{3} e +\frac {8099}{192} B \,b^{5} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {5327}{256} A \,a^{4} b \,e^{5}-\frac {5327}{64} A \,a^{3} b^{2} d \,e^{4}+\frac {15981}{128} A \,a^{2} b^{3} d^{2} e^{3}-\frac {5327}{64} A a \,b^{4} d^{3} e^{2}+\frac {5327}{256} A \,b^{5} d^{4} e -\frac {2373}{256} B \,a^{5} e^{5}+\frac {3269}{128} B \,a^{4} b d \,e^{4}-\frac {1211}{128} B \,a^{3} b^{2} d^{2} e^{3}-\frac {1029}{32} B \,a^{2} b^{3} d^{3} e^{2}+\frac {9443}{256} B a \,b^{4} d^{4} e -\frac {1477}{128} B \,b^{5} d^{5}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {3003 \left (3 A b e -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 )}{256 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{8}}\right )\) | \(618\) |
pseudoelliptic | \(\text {Expression too large to display}\) | \(735\) |
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Leaf count of result is larger than twice the leaf count of optimal. 3011 vs. \(2 (391) = 782\).
Time = 5.64 (sec) , antiderivative size = 6033, normalized size of antiderivative = 13.87 \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1700 vs. \(2 (391) = 782\).
Time = 0.31 (sec) , antiderivative size = 1700, normalized size of antiderivative = 3.91 \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \]
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Time = 12.61 (sec) , antiderivative size = 802, normalized size of antiderivative = 1.84 \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {27599\,{\left (d+e\,x\right )}^3\,\left (-3\,A\,b^3\,e^5+2\,B\,d\,b^3\,e^4+B\,a\,b^2\,e^5\right )}{384\,{\left (a\,e-b\,d\right )}^4}-\frac {2\,\left (A\,e^5-B\,d\,e^4\right )}{5\,\left (a\,e-b\,d\right )}+\frac {11297\,{\left (d+e\,x\right )}^4\,\left (-3\,A\,b^4\,e^5+2\,B\,d\,b^4\,e^4+B\,a\,b^3\,e^5\right )}{64\,{\left (a\,e-b\,d\right )}^5}+\frac {7007\,{\left (d+e\,x\right )}^6\,\left (-3\,A\,b^6\,e^5+2\,B\,d\,b^6\,e^4+B\,a\,b^5\,e^5\right )}{64\,{\left (a\,e-b\,d\right )}^7}-\frac {2\,\left (d+e\,x\right )\,\left (B\,a\,e^5-3\,A\,b\,e^5+2\,B\,b\,d\,e^4\right )}{3\,{\left (a\,e-b\,d\right )}^2}+\frac {26\,b\,{\left (d+e\,x\right )}^2\,\left (B\,a\,e^5-3\,A\,b\,e^5+2\,B\,b\,d\,e^4\right )}{3\,{\left (a\,e-b\,d\right )}^3}+\frac {1001\,b^4\,{\left (d+e\,x\right )}^5\,\left (B\,a\,e^5-3\,A\,b\,e^5+2\,B\,b\,d\,e^4\right )}{5\,{\left (a\,e-b\,d\right )}^6}+\frac {3003\,b^6\,{\left (d+e\,x\right )}^7\,\left (B\,a\,e^5-3\,A\,b\,e^5+2\,B\,b\,d\,e^4\right )}{128\,{\left (a\,e-b\,d\right )}^8}}{{\left (d+e\,x\right )}^{5/2}\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )-{\left (d+e\,x\right )}^{9/2}\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+{\left (d+e\,x\right )}^{7/2}\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )+b^5\,{\left (d+e\,x\right )}^{15/2}-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{13/2}+{\left (d+e\,x\right )}^{11/2}\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )}+\frac {3003\,b^{3/2}\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {d+e\,x}\,\left (B\,a\,e-3\,A\,b\,e+2\,B\,b\,d\right )\,\left (a^8\,e^8-8\,a^7\,b\,d\,e^7+28\,a^6\,b^2\,d^2\,e^6-56\,a^5\,b^3\,d^3\,e^5+70\,a^4\,b^4\,d^4\,e^4-56\,a^3\,b^5\,d^5\,e^3+28\,a^2\,b^6\,d^6\,e^2-8\,a\,b^7\,d^7\,e+b^8\,d^8\right )}{{\left (a\,e-b\,d\right )}^{17/2}\,\left (B\,a\,e^5-3\,A\,b\,e^5+2\,B\,b\,d\,e^4\right )}\right )\,\left (B\,a\,e-3\,A\,b\,e+2\,B\,b\,d\right )}{128\,{\left (a\,e-b\,d\right )}^{17/2}} \]
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