\(\int \frac {A+B x}{(d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\) [377]

Optimal result

Integrand size = 33, antiderivative size = 428 \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {2 e^4 (B d-A e)}{5 (b d-a e)^6 (d+e x)^{5/2}}+\frac {2 e^4 (5 b B d-6 A b e+a B e)}{3 (b d-a e)^7 (d+e x)^{3/2}}+\frac {6 b e^4 (5 b B d-7 A b e+2 a B e)}{(b d-a e)^8 \sqrt {d+e x}}-\frac {b^2 (A b-a B) \sqrt {d+e x}}{5 (b d-a e)^4 (a+b x)^5}-\frac {b^2 (10 b B d-39 A b e+29 a B e) \sqrt {d+e x}}{40 (b d-a e)^5 (a+b x)^4}+\frac {b^2 e (310 b B d-753 A b e+443 a B e) \sqrt {d+e x}}{240 (b d-a e)^6 (a+b x)^3}-\frac {b^2 e^2 (886 b B d-1713 A b e+827 a B e) \sqrt {d+e x}}{192 (b d-a e)^7 (a+b x)^2}+\frac {3 b^2 e^3 (722 b B d-1211 A b e+489 a B e) \sqrt {d+e x}}{128 (b d-a e)^8 (a+b x)}-\frac {3003 b^{3/2} e^4 (2 b B d-3 A b e+a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{17/2}} \] Output:

2/5*e^4*(-A*e+B*d)/(-a*e+b*d)^6/(e*x+d)^(5/2)+2/3*e^4*(-6*A*b*e+B*a*e+5*B* 
b*d)/(-a*e+b*d)^7/(e*x+d)^(3/2)+6*b*e^4*(-7*A*b*e+2*B*a*e+5*B*b*d)/(-a*e+b 
*d)^8/(e*x+d)^(1/2)-1/5*b^2*(A*b-B*a)*(e*x+d)^(1/2)/(-a*e+b*d)^4/(b*x+a)^5 
-1/40*b^2*(-39*A*b*e+29*B*a*e+10*B*b*d)*(e*x+d)^(1/2)/(-a*e+b*d)^5/(b*x+a) 
^4+1/240*b^2*e*(-753*A*b*e+443*B*a*e+310*B*b*d)*(e*x+d)^(1/2)/(-a*e+b*d)^6 
/(b*x+a)^3-1/192*b^2*e^2*(-1713*A*b*e+827*B*a*e+886*B*b*d)*(e*x+d)^(1/2)/( 
-a*e+b*d)^7/(b*x+a)^2+3/128*b^2*e^3*(-1211*A*b*e+489*B*a*e+722*B*b*d)*(e*x 
+d)^(1/2)/(-a*e+b*d)^8/(b*x+a)-3003/128*b^(3/2)*e^4*(-3*A*b*e+B*a*e+2*B*b* 
d)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/(-a*e+b*d)^(17/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(884\) vs. \(2(428)=856\).

Time = 9.60 (sec) , antiderivative size = 884, normalized size of antiderivative = 2.07 \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {-3 A \left (256 a^7 e^7-256 a^6 b e^6 (12 d+5 e x)+256 a^5 b^2 e^5 \left (116 d^2+160 d e x+65 e^2 x^2\right )+5 a^4 b^3 e^4 \left (5327 d^3+45677 d^2 e x+66157 d e^2 x^2+27599 e^3 x^3\right )+10 a^3 b^4 e^3 \left (-1211 d^4+5810 d^3 e x+54392 d^2 e^2 x^2+80366 d e^3 x^3+33891 e^4 x^4\right )+2 a^2 b^5 e^2 \left (2324 d^5-6545 d^4 e x+30485 d^3 e^2 x^2+302445 d^2 e^3 x^3+452595 d e^4 x^4+192192 e^5 x^5\right )+2 a b^6 e \left (-568 d^6+1240 d^5 e x-3445 d^4 e^2 x^2+15730 d^3 e^3 x^3+163020 d^2 e^4 x^4+246246 d e^5 x^5+105105 e^6 x^6\right )+b^7 \left (128 d^7-240 d^6 e x+520 d^5 e^2 x^2-1430 d^4 e^3 x^3+6435 d^3 e^4 x^4+69069 d^2 e^5 x^5+105105 d e^6 x^6+45045 e^7 x^7\right )\right )+B \left (-256 a^7 e^6 (2 d+5 e x)+256 a^6 b e^5 \left (64 d^2+150 d e x+65 e^2 x^2\right )+a^5 b^2 e^4 \left (100363 d^3+310305 d^2 e x+364065 d e^2 x^2+137995 e^3 x^3\right )+10 a^4 b^3 e^3 \left (2324 d^4+51487 d^3 e x+120549 d^2 e^2 x^2+107965 d e^3 x^3+33891 e^4 x^4\right )+2 a^3 b^4 e^2 \left (-2618 d^5+51555 d^4 e x+574405 d^3 e^2 x^2+1106105 d^2 e^3 x^3+791505 d e^4 x^4+192192 e^5 x^5\right )+2 b^7 d x \left (-240 d^6+520 d^5 e x-1430 d^4 e^2 x^2+6435 d^3 e^3 x^3+69069 d^2 e^4 x^4+105105 d e^5 x^5+45045 e^6 x^6\right )+2 a^2 b^5 e \left (496 d^6-11850 d^5 e x+57525 d^4 e^2 x^2+620620 d^3 e^3 x^3+1068210 d^2 e^4 x^4+630630 d e^5 x^5+105105 e^6 x^6\right )+a b^6 \left (-96 d^7+4720 d^6 e x-13260 d^5 e^2 x^2+61490 d^4 e^3 x^3+658515 d^3 e^4 x^4+1054053 d^2 e^5 x^5+525525 d e^6 x^6+45045 e^7 x^7\right )\right )}{(b d-a e)^8 (a+b x)^5 (d+e x)^{5/2}}+\frac {45045 b^{3/2} e^4 (2 b B d-3 A b e+a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{17/2}}}{1920} \] Input:

Integrate[(A + B*x)/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
 

Output:

((-3*A*(256*a^7*e^7 - 256*a^6*b*e^6*(12*d + 5*e*x) + 256*a^5*b^2*e^5*(116* 
d^2 + 160*d*e*x + 65*e^2*x^2) + 5*a^4*b^3*e^4*(5327*d^3 + 45677*d^2*e*x + 
66157*d*e^2*x^2 + 27599*e^3*x^3) + 10*a^3*b^4*e^3*(-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) + 2*a^2*b^5*e^2*( 
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) + 2*a*b^6*e*(-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 + 10 
5105*e^6*x^6) + b^7*(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)) + B*(-256*a^7*e^6*(2*d + 5*e*x) + 256*a^6*b*e^5*(64*d^2 + 150*d*e* 
x + 65*e^2*x^2) + a^5*b^2*e^4*(100363*d^3 + 310305*d^2*e*x + 364065*d*e^2* 
x^2 + 137995*e^3*x^3) + 10*a^4*b^3*e^3*(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) + 2*a^3*b^4*e^2*(-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) + 2*b^7*d*x*(-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) + 2*a^2*b^5*e*(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) + a* 
b^6*(-96*d^7 + 4720*d^6*e*x - 13260*d^5*e^2*x^2 + 61490*d^4*e^3*x^3 + 6585 
15*d^3*e^4*x^4 + 1054053*d^2*e^5*x^5 + 525525*d*e^6*x^6 + 45045*e^7*x^7...
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 397, normalized size of antiderivative = 0.93, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1184, 27, 87, 52, 52, 52, 52, 61, 61, 61, 73, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {A+B x}{\left (a^2+2 a b x+b^2 x^2\right )^3 (d+e x)^{7/2}} \, dx\)

\(\Big \downarrow \) 1184

\(\displaystyle b^6 \int \frac {A+B x}{b^6 (a+b x)^6 (d+e x)^{7/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {A+B x}{(a+b x)^6 (d+e x)^{7/2}}dx\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {(a B e-3 A b e+2 b B d) \int \frac {1}{(a+b x)^5 (d+e x)^{7/2}}dx}{2 b (b d-a e)}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{5/2} (b d-a e)}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(a B e-3 A b e+2 b B d) \left (-\frac {13 e \int \frac {1}{(a+b x)^4 (d+e x)^{7/2}}dx}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{5/2} (b d-a e)}\right )}{2 b (b d-a e)}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{5/2} (b d-a e)}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(a B e-3 A b e+2 b B d) \left (-\frac {13 e \left (-\frac {11 e \int \frac {1}{(a+b x)^3 (d+e x)^{7/2}}dx}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{5/2} (b d-a e)}\right )}{2 b (b d-a e)}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{5/2} (b d-a e)}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(a B e-3 A b e+2 b B d) \left (-\frac {13 e \left (-\frac {11 e \left (-\frac {9 e \int \frac {1}{(a+b x)^2 (d+e x)^{7/2}}dx}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{5/2} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{5/2} (b d-a e)}\right )}{2 b (b d-a e)}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{5/2} (b d-a e)}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(a B e-3 A b e+2 b B d) \left (-\frac {13 e \left (-\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \int \frac {1}{(a+b x) (d+e x)^{7/2}}dx}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{5/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{5/2} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{5/2} (b d-a e)}\right )}{2 b (b d-a e)}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{5/2} (b d-a e)}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {(a B e-3 A b e+2 b B d) \left (-\frac {13 e \left (-\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \left (\frac {b \int \frac {1}{(a+b x) (d+e x)^{5/2}}dx}{b d-a e}+\frac {2}{5 (d+e x)^{5/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{5/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{5/2} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{5/2} (b d-a e)}\right )}{2 b (b d-a e)}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{5/2} (b d-a e)}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {(a B e-3 A b e+2 b B d) \left (-\frac {13 e \left (-\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \left (\frac {b \left (\frac {b \int \frac {1}{(a+b x) (d+e x)^{3/2}}dx}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{b d-a e}+\frac {2}{5 (d+e x)^{5/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{5/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{5/2} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{5/2} (b d-a e)}\right )}{2 b (b d-a e)}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{5/2} (b d-a e)}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {(a B e-3 A b e+2 b B d) \left (-\frac {13 e \left (-\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \left (\frac {b \left (\frac {b \left (\frac {b \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b d-a e}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{b d-a e}+\frac {2}{5 (d+e x)^{5/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{5/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{5/2} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{5/2} (b d-a e)}\right )}{2 b (b d-a e)}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{5/2} (b d-a e)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {(a B e-3 A b e+2 b B d) \left (-\frac {13 e \left (-\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \left (\frac {b \left (\frac {b \left (\frac {2 b \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{e (b d-a e)}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{b d-a e}+\frac {2}{5 (d+e x)^{5/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{5/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{5/2} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{5/2} (b d-a e)}\right )}{2 b (b d-a e)}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{5/2} (b d-a e)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(a B e-3 A b e+2 b B d) \left (-\frac {13 e \left (-\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \left (\frac {b \left (\frac {b \left (\frac {2}{\sqrt {d+e x} (b d-a e)}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{b d-a e}+\frac {2}{5 (d+e x)^{5/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{5/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{5/2} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{5/2} (b d-a e)}\right )}{2 b (b d-a e)}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{5/2} (b d-a e)}\)

Input:

Int[(A + B*x)/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
 

Output:

-1/5*(A*b - a*B)/(b*(b*d - a*e)*(a + b*x)^5*(d + e*x)^(5/2)) + ((2*b*B*d - 
 3*A*b*e + a*B*e)*(-1/4*1/((b*d - a*e)*(a + b*x)^4*(d + e*x)^(5/2)) - (13* 
e*(-1/3*1/((b*d - a*e)*(a + b*x)^3*(d + e*x)^(5/2)) - (11*e*(-1/2*1/((b*d 
- a*e)*(a + b*x)^2*(d + e*x)^(5/2)) - (9*e*(-(1/((b*d - a*e)*(a + b*x)*(d 
+ e*x)^(5/2))) - (7*e*(2/(5*(b*d - a*e)*(d + e*x)^(5/2)) + (b*(2/(3*(b*d - 
 a*e)*(d + e*x)^(3/2)) + (b*(2/((b*d - a*e)*Sqrt[d + e*x]) - (2*Sqrt[b]*Ar 
cTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(3/2)))/(b*d - 
 a*e)))/(b*d - a*e)))/(2*(b*d - a*e))))/(4*(b*d - a*e))))/(6*(b*d - a*e))) 
)/(8*(b*d - a*e))))/(2*b*(b*d - a*e))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
 

rule 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0 
] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d 
, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 1184

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p   Int[(d + e*x)^m*(f + g*x 
)^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E 
qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

Maple [A] (verified)

Time = 1.76 (sec) , antiderivative size = 618, normalized size of antiderivative = 1.44

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 \,b^{5} e -\frac {1467}{256} B a \,b^{4} e -\frac {1083}{128} B \,b^{5} d \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 B \,e^{2} a^{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 \,a^{3} b^{2} e^{3}+\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 {b \left (a e -b d \right )}}\right )}{256 \sqrt {b \left (a e -b d \right )}}\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 \,b^{5} e -\frac {1467}{256} B a \,b^{4} e -\frac {1083}{128} B \,b^{5} d \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 B \,e^{2} a^{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 \,a^{3} b^{2} e^{3}+\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 {b \left (a e -b d \right )}}\right )}{256 \sqrt {b \left (a e -b d \right )}}\right )}{\left (a e -b d \right )^{8}}\right )\)	\(618\)
pseudoelliptic	\(\text {Expression too large to display}\)	\(735\)

Input:

int((B*x+A)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)
 

Output:

2*e^4*(-1/5*(A*e-B*d)/(a*e-b*d)^6/(e*x+d)^(5/2)-1/3*(-6*A*b*e+B*a*e+5*B*b* 
d)/(a*e-b*d)^7/(e*x+d)^(3/2)-3*b*(7*A*b*e-2*B*a*e-5*B*b*d)/(a*e-b*d)^8/(e* 
x+d)^(1/2)-1/(a*e-b*d)^8*b^2*(((3633/256*A*b^5*e-1467/256*B*a*b^4*e-1083/1 
28*B*b^5*d)*(e*x+d)^(9/2)+1/384*b^3*(23511*A*a*b*e^2-23511*A*b^2*d*e-9629* 
B*a^2*e^2-4253*B*a*b*d*e+13882*B*b^2*d^2)*(e*x+d)^(7/2)+(1001/10*A*a^2*b^3 
*e^3-1001/5*A*a*b^4*d*e^2+1001/10*A*b^5*d^2*e-1253/30*B*a^3*b^2*e^3+749/10 
*B*a*b^4*d^2*e-175/3*B*b^5*d^3+126/5*B*a^2*b^3*d*e^2)*(e*x+d)^(5/2)+(9443/ 
128*A*a^3*b^2*e^4-28329/128*A*a^2*b^3*d*e^3+28329/128*A*a*b^4*d^2*e^2-9443 
/128*A*b^5*d^3*e-12131/384*B*e^4*a^4*b+20195/384*B*a^3*b^2*d*e^3+4067/128* 
B*a^2*b^3*d^2*e^2-36463/384*B*a*b^4*d^3*e+8099/192*B*b^5*d^4)*(e*x+d)^(3/2 
)+(5327/256*A*a^4*b*e^5-5327/64*A*a^3*b^2*d*e^4+15981/128*A*a^2*b^3*d^2*e^ 
3-5327/64*A*a*b^4*d^3*e^2+5327/256*A*b^5*d^4*e-2373/256*B*a^5*e^5+3269/128 
*B*a^4*b*d*e^4-1211/128*B*a^3*b^2*d^2*e^3-1029/32*B*a^2*b^3*d^3*e^2+9443/2 
56*B*a*b^4*d^4*e-1477/128*B*b^5*d^5)*(e*x+d)^(1/2))/(b*(e*x+d)+a*e-b*d)^5+ 
3003/256*(3*A*b*e-B*a*e-2*B*b*d)/(b*(a*e-b*d))^(1/2)*arctan(b*(e*x+d)^(1/2 
)/(b*(a*e-b*d))^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2993 vs. \(2 (386) = 772\).

Time = 3.94 (sec) , antiderivative size = 6015, normalized size of antiderivative = 14.05 \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fric 
as")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)/(e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxi 
ma")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m 
ore detail
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1700 vs. \(2 (386) = 772\).

Time = 0.22 (sec) , antiderivative size = 1700, normalized size of antiderivative = 3.97 \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac 
")
 

Output:

3003/128*(2*B*b^3*d*e^4 + B*a*b^2*e^5 - 3*A*b^3*e^5)*arctan(sqrt(e*x + d)* 
b/sqrt(-b^2*d + a*b*e))/((b^8*d^8 - 8*a*b^7*d^7*e + 28*a^2*b^6*d^6*e^2 - 5 
6*a^3*b^5*d^5*e^3 + 70*a^4*b^4*d^4*e^4 - 56*a^5*b^3*d^3*e^5 + 28*a^6*b^2*d 
^2*e^6 - 8*a^7*b*d*e^7 + a^8*e^8)*sqrt(-b^2*d + a*b*e)) + 1/1920*(90090*(e 
*x + d)^7*B*b^7*d*e^4 - 420420*(e*x + d)^6*B*b^7*d^2*e^4 + 768768*(e*x + d 
)^5*B*b^7*d^3*e^4 - 677820*(e*x + d)^4*B*b^7*d^4*e^4 + 275990*(e*x + d)^3* 
B*b^7*d^5*e^4 - 33280*(e*x + d)^2*B*b^7*d^6*e^4 - 2560*(e*x + d)*B*b^7*d^7 
*e^4 - 768*B*b^7*d^8*e^4 + 45045*(e*x + d)^7*B*a*b^6*e^5 - 135135*(e*x + d 
)^7*A*b^7*e^5 + 210210*(e*x + d)^6*B*a*b^6*d*e^5 + 630630*(e*x + d)^6*A*b^ 
7*d*e^5 - 1153152*(e*x + d)^5*B*a*b^6*d^2*e^5 - 1153152*(e*x + d)^5*A*b^7* 
d^2*e^5 + 1694550*(e*x + d)^4*B*a*b^6*d^3*e^5 + 1016730*(e*x + d)^4*A*b^7* 
d^3*e^5 - 965965*(e*x + d)^3*B*a*b^6*d^4*e^5 - 413985*(e*x + d)^3*A*b^7*d^ 
4*e^5 + 149760*(e*x + d)^2*B*a*b^6*d^5*e^5 + 49920*(e*x + d)^2*A*b^7*d^5*e 
^5 + 14080*(e*x + d)*B*a*b^6*d^6*e^5 + 3840*(e*x + d)*A*b^7*d^6*e^5 + 5376 
*B*a*b^6*d^7*e^5 + 768*A*b^7*d^7*e^5 + 210210*(e*x + d)^6*B*a^2*b^5*e^6 - 
630630*(e*x + d)^6*A*a*b^6*e^6 + 2306304*(e*x + d)^5*A*a*b^6*d*e^6 - 10167 
30*(e*x + d)^4*B*a^2*b^5*d^2*e^6 - 3050190*(e*x + d)^4*A*a*b^6*d^2*e^6 + 1 
103960*(e*x + d)^3*B*a^2*b^5*d^3*e^6 + 1655940*(e*x + d)^3*A*a*b^6*d^3*e^6 
 - 249600*(e*x + d)^2*B*a^2*b^5*d^4*e^6 - 249600*(e*x + d)^2*A*a*b^6*d^4*e 
^6 - 30720*(e*x + d)*B*a^2*b^5*d^5*e^6 - 23040*(e*x + d)*A*a*b^6*d^5*e^...
 

Mupad [B] (verification not implemented)

Time = 12.89 (sec) , antiderivative size = 802, normalized size of antiderivative = 1.87 \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((A + B*x)/((d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^3),x)
 

Output:

((27599*(d + e*x)^3*(B*a*b^2*e^5 - 3*A*b^3*e^5 + 2*B*b^3*d*e^4))/(384*(a*e 
 - b*d)^4) - (2*(A*e^5 - B*d*e^4))/(5*(a*e - b*d)) + (11297*(d + e*x)^4*(B 
*a*b^3*e^5 - 3*A*b^4*e^5 + 2*B*b^4*d*e^4))/(64*(a*e - b*d)^5) + (7007*(d + 
 e*x)^6*(B*a*b^5*e^5 - 3*A*b^6*e^5 + 2*B*b^6*d*e^4))/(64*(a*e - b*d)^7) - 
(2*(d + e*x)*(B*a*e^5 - 3*A*b*e^5 + 2*B*b*d*e^4))/(3*(a*e - b*d)^2) + (26* 
b*(d + e*x)^2*(B*a*e^5 - 3*A*b*e^5 + 2*B*b*d*e^4))/(3*(a*e - b*d)^3) + (10 
01*b^4*(d + e*x)^5*(B*a*e^5 - 3*A*b*e^5 + 2*B*b*d*e^4))/(5*(a*e - b*d)^6) 
+ (3003*b^6*(d + e*x)^7*(B*a*e^5 - 3*A*b*e^5 + 2*B*b*d*e^4))/(128*(a*e - b 
*d)^8))/((d + e*x)^(5/2)*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3* 
b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4) - (d + e*x)^(9/2)*(10*b^5*d^3 
 - 10*a^3*b^2*e^3 + 30*a^2*b^3*d*e^2 - 30*a*b^4*d^2*e) + (d + e*x)^(7/2)*( 
5*b^5*d^4 + 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) + b^5*(d + e*x)^(15/2) - (5*b^5*d - 5*a*b^4*e)*(d + e*x)^(13/2) + 
(d + e*x)^(11/2)*(10*b^5*d^2 + 10*a^2*b^3*e^2 - 20*a*b^4*d*e)) + (3003*b^( 
3/2)*e^4*atan((b^(1/2)*e^4*(d + e*x)^(1/2)*(B*a*e - 3*A*b*e + 2*B*b*d)*(a^ 
8*e^8 + b^8*d^8 + 28*a^2*b^6*d^6*e^2 - 56*a^3*b^5*d^5*e^3 + 70*a^4*b^4*d^4 
*e^4 - 56*a^5*b^3*d^3*e^5 + 28*a^6*b^2*d^2*e^6 - 8*a*b^7*d^7*e - 8*a^7*b*d 
*e^7))/((a*e - b*d)^(17/2)*(B*a*e^5 - 3*A*b*e^5 + 2*B*b*d*e^4)))*(B*a*e - 
3*A*b*e + 2*B*b*d))/(128*(a*e - b*d)^(17/2))
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 2284, normalized size of antiderivative = 5.34 \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((B*x+A)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)
 

Output:

( - 45045*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sq 
rt(b)*sqrt(a*e - b*d)))*a**4*b**2*d**2*e**4 - 90090*sqrt(b)*sqrt(d + e*x)* 
sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**4*b** 
2*d*e**5*x - 45045*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - b*d)*atan((sqrt(d + e* 
x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**4*b**2*e**6*x**2 - 180180*sqrt(b)*sqrt 
(d + e*x)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)) 
)*a**3*b**3*d**2*e**4*x - 360360*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - b*d)*ata 
n((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**3*b**3*d*e**5*x**2 - 180 
180*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)* 
sqrt(a*e - b*d)))*a**3*b**3*e**6*x**3 - 270270*sqrt(b)*sqrt(d + e*x)*sqrt( 
a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**2*b**4*d** 
2*e**4*x**2 - 540540*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - b*d)*atan((sqrt(d + 
e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**2*b**4*d*e**5*x**3 - 270270*sqrt(b)* 
sqrt(d + e*x)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b 
*d)))*a**2*b**4*e**6*x**4 - 180180*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - b*d)*a 
tan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a*b**5*d**2*e**4*x**3 - 3 
60360*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b 
)*sqrt(a*e - b*d)))*a*b**5*d*e**5*x**4 - 180180*sqrt(b)*sqrt(d + e*x)*sqrt 
(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a*b**5*e**6* 
x**5 - 45045*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*...