3.1055 \(\int \frac{(a+b x)^6 (A+B x)}{(d+e x)^{13}} \, dx\)

Optimal. Leaf size=292 \[ \frac{b^5 (-6 a B e-A b e+7 b B d)}{6 e^8 (d+e x)^6}-\frac{3 b^4 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{7 e^8 (d+e x)^7}+\frac{5 b^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{8 e^8 (d+e x)^8}-\frac{5 b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{9 e^8 (d+e x)^9}+\frac{3 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{10 e^8 (d+e x)^{10}}-\frac{(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{11 e^8 (d+e x)^{11}}+\frac{(b d-a e)^6 (B d-A e)}{12 e^8 (d+e x)^{12}}-\frac{b^6 B}{5 e^8 (d+e x)^5} \]

[Out]

((b*d - a*e)^6*(B*d - A*e))/(12*e^8*(d + e*x)^12) - ((b*d - a*e)^5*(7*b*B*d - 6*
A*b*e - a*B*e))/(11*e^8*(d + e*x)^11) + (3*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e -
2*a*B*e))/(10*e^8*(d + e*x)^10) - (5*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*
B*e))/(9*e^8*(d + e*x)^9) + (5*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e))/
(8*e^8*(d + e*x)^8) - (3*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e))/(7*e^8*(
d + e*x)^7) + (b^5*(7*b*B*d - A*b*e - 6*a*B*e))/(6*e^8*(d + e*x)^6) - (b^6*B)/(5
*e^8*(d + e*x)^5)

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Rubi [A]  time = 1.02502, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{b^5 (-6 a B e-A b e+7 b B d)}{6 e^8 (d+e x)^6}-\frac{3 b^4 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{7 e^8 (d+e x)^7}+\frac{5 b^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{8 e^8 (d+e x)^8}-\frac{5 b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{9 e^8 (d+e x)^9}+\frac{3 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{10 e^8 (d+e x)^{10}}-\frac{(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{11 e^8 (d+e x)^{11}}+\frac{(b d-a e)^6 (B d-A e)}{12 e^8 (d+e x)^{12}}-\frac{b^6 B}{5 e^8 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^6*(A + B*x))/(d + e*x)^13,x]

[Out]

((b*d - a*e)^6*(B*d - A*e))/(12*e^8*(d + e*x)^12) - ((b*d - a*e)^5*(7*b*B*d - 6*
A*b*e - a*B*e))/(11*e^8*(d + e*x)^11) + (3*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e -
2*a*B*e))/(10*e^8*(d + e*x)^10) - (5*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*
B*e))/(9*e^8*(d + e*x)^9) + (5*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e))/
(8*e^8*(d + e*x)^8) - (3*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e))/(7*e^8*(
d + e*x)^7) + (b^5*(7*b*B*d - A*b*e - 6*a*B*e))/(6*e^8*(d + e*x)^6) - (b^6*B)/(5
*e^8*(d + e*x)^5)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**6*(B*x+A)/(e*x+d)**13,x)

[Out]

Timed out

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Mathematica [B]  time = 0.995513, size = 600, normalized size = 2.05 \[ -\frac{210 a^6 e^6 (11 A e+B (d+12 e x))+252 a^5 b e^5 \left (5 A e (d+12 e x)+B \left (d^2+12 d e x+66 e^2 x^2\right )\right )+210 a^4 b^2 e^4 \left (3 A e \left (d^2+12 d e x+66 e^2 x^2\right )+B \left (d^3+12 d^2 e x+66 d e^2 x^2+220 e^3 x^3\right )\right )+140 a^3 b^3 e^3 \left (2 A e \left (d^3+12 d^2 e x+66 d e^2 x^2+220 e^3 x^3\right )+B \left (d^4+12 d^3 e x+66 d^2 e^2 x^2+220 d e^3 x^3+495 e^4 x^4\right )\right )+15 a^2 b^4 e^2 \left (7 A e \left (d^4+12 d^3 e x+66 d^2 e^2 x^2+220 d e^3 x^3+495 e^4 x^4\right )+5 B \left (d^5+12 d^4 e x+66 d^3 e^2 x^2+220 d^2 e^3 x^3+495 d e^4 x^4+792 e^5 x^5\right )\right )+30 a b^5 e \left (A e \left (d^5+12 d^4 e x+66 d^3 e^2 x^2+220 d^2 e^3 x^3+495 d e^4 x^4+792 e^5 x^5\right )+B \left (d^6+12 d^5 e x+66 d^4 e^2 x^2+220 d^3 e^3 x^3+495 d^2 e^4 x^4+792 d e^5 x^5+924 e^6 x^6\right )\right )+b^6 \left (5 A e \left (d^6+12 d^5 e x+66 d^4 e^2 x^2+220 d^3 e^3 x^3+495 d^2 e^4 x^4+792 d e^5 x^5+924 e^6 x^6\right )+7 B \left (d^7+12 d^6 e x+66 d^5 e^2 x^2+220 d^4 e^3 x^3+495 d^3 e^4 x^4+792 d^2 e^5 x^5+924 d e^6 x^6+792 e^7 x^7\right )\right )}{27720 e^8 (d+e x)^{12}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^13,x]

[Out]

-(210*a^6*e^6*(11*A*e + B*(d + 12*e*x)) + 252*a^5*b*e^5*(5*A*e*(d + 12*e*x) + B*
(d^2 + 12*d*e*x + 66*e^2*x^2)) + 210*a^4*b^2*e^4*(3*A*e*(d^2 + 12*d*e*x + 66*e^2
*x^2) + B*(d^3 + 12*d^2*e*x + 66*d*e^2*x^2 + 220*e^3*x^3)) + 140*a^3*b^3*e^3*(2*
A*e*(d^3 + 12*d^2*e*x + 66*d*e^2*x^2 + 220*e^3*x^3) + B*(d^4 + 12*d^3*e*x + 66*d
^2*e^2*x^2 + 220*d*e^3*x^3 + 495*e^4*x^4)) + 15*a^2*b^4*e^2*(7*A*e*(d^4 + 12*d^3
*e*x + 66*d^2*e^2*x^2 + 220*d*e^3*x^3 + 495*e^4*x^4) + 5*B*(d^5 + 12*d^4*e*x + 6
6*d^3*e^2*x^2 + 220*d^2*e^3*x^3 + 495*d*e^4*x^4 + 792*e^5*x^5)) + 30*a*b^5*e*(A*
e*(d^5 + 12*d^4*e*x + 66*d^3*e^2*x^2 + 220*d^2*e^3*x^3 + 495*d*e^4*x^4 + 792*e^5
*x^5) + B*(d^6 + 12*d^5*e*x + 66*d^4*e^2*x^2 + 220*d^3*e^3*x^3 + 495*d^2*e^4*x^4
+ 792*d*e^5*x^5 + 924*e^6*x^6)) + b^6*(5*A*e*(d^6 + 12*d^5*e*x + 66*d^4*e^2*x^2
+ 220*d^3*e^3*x^3 + 495*d^2*e^4*x^4 + 792*d*e^5*x^5 + 924*e^6*x^6) + 7*B*(d^7 +
12*d^6*e*x + 66*d^5*e^2*x^2 + 220*d^4*e^3*x^3 + 495*d^3*e^4*x^4 + 792*d^2*e^5*x
^5 + 924*d*e^6*x^6 + 792*e^7*x^7)))/(27720*e^8*(d + e*x)^12)

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Maple [B]  time = 0.013, size = 814, normalized size = 2.8 \[ -{\frac{5\,{b}^{2} \left ( 4\,A{a}^{3}b{e}^{4}-12\,A{a}^{2}{b}^{2}d{e}^{3}+12\,Aa{b}^{3}{d}^{2}{e}^{2}-4\,A{b}^{4}{d}^{3}e+3\,B{a}^{4}{e}^{4}-16\,B{a}^{3}bd{e}^{3}+30\,B{a}^{2}{b}^{2}{d}^{2}{e}^{2}-24\,Ba{b}^{3}{d}^{3}e+7\,B{b}^{4}{d}^{4} \right ) }{9\,{e}^{8} \left ( ex+d \right ) ^{9}}}-{\frac{3\,b \left ( 5\,A{a}^{4}b{e}^{5}-20\,A{a}^{3}{b}^{2}d{e}^{4}+30\,A{a}^{2}{b}^{3}{d}^{2}{e}^{3}-20\,Aa{b}^{4}{d}^{3}{e}^{2}+5\,A{b}^{5}{d}^{4}e+2\,B{a}^{5}{e}^{5}-15\,B{a}^{4}bd{e}^{4}+40\,B{a}^{3}{b}^{2}{d}^{2}{e}^{3}-50\,B{a}^{2}{b}^{3}{d}^{3}{e}^{2}+30\,Ba{b}^{4}{d}^{4}e-7\,B{b}^{5}{d}^{5} \right ) }{10\,{e}^{8} \left ( ex+d \right ) ^{10}}}-{\frac{{b}^{5} \left ( Abe+6\,Bae-7\,Bbd \right ) }{6\,{e}^{8} \left ( ex+d \right ) ^{6}}}-{\frac{5\,{b}^{3} \left ( 3\,A{a}^{2}b{e}^{3}-6\,Ada{b}^{2}{e}^{2}+3\,A{b}^{3}{d}^{2}e+4\,B{a}^{3}{e}^{3}-15\,B{a}^{2}bd{e}^{2}+18\,B{d}^{2}a{b}^{2}e-7\,{b}^{3}B{d}^{3} \right ) }{8\,{e}^{8} \left ( ex+d \right ) ^{8}}}-{\frac{3\,{b}^{4} \left ( 2\,Aab{e}^{2}-2\,Ad{b}^{2}e+5\,B{a}^{2}{e}^{2}-12\,Bdabe+7\,{b}^{2}B{d}^{2} \right ) }{7\,{e}^{8} \left ( ex+d \right ) ^{7}}}-{\frac{{a}^{6}A{e}^{7}-6\,Ad{a}^{5}b{e}^{6}+15\,A{d}^{2}{a}^{4}{b}^{2}{e}^{5}-20\,A{d}^{3}{a}^{3}{b}^{3}{e}^{4}+15\,A{d}^{4}{a}^{2}{b}^{4}{e}^{3}-6\,A{d}^{5}a{b}^{5}{e}^{2}+A{d}^{6}{b}^{6}e-Bd{a}^{6}{e}^{6}+6\,B{d}^{2}{a}^{5}b{e}^{5}-15\,B{d}^{3}{a}^{4}{b}^{2}{e}^{4}+20\,B{d}^{4}{a}^{3}{b}^{3}{e}^{3}-15\,B{d}^{5}{a}^{2}{b}^{4}{e}^{2}+6\,B{d}^{6}a{b}^{5}e-{b}^{6}B{d}^{7}}{12\,{e}^{8} \left ( ex+d \right ) ^{12}}}-{\frac{6\,A{a}^{5}b{e}^{6}-30\,Ad{a}^{4}{b}^{2}{e}^{5}+60\,A{d}^{2}{a}^{3}{b}^{3}{e}^{4}-60\,A{d}^{3}{a}^{2}{b}^{4}{e}^{3}+30\,A{d}^{4}a{b}^{5}{e}^{2}-6\,A{d}^{5}{b}^{6}e+{a}^{6}B{e}^{6}-12\,Bd{a}^{5}b{e}^{5}+45\,B{d}^{2}{a}^{4}{b}^{2}{e}^{4}-80\,B{d}^{3}{a}^{3}{b}^{3}{e}^{3}+75\,B{d}^{4}{a}^{2}{b}^{4}{e}^{2}-36\,B{d}^{5}a{b}^{5}e+7\,{b}^{6}B{d}^{6}}{11\,{e}^{8} \left ( ex+d \right ) ^{11}}}-{\frac{B{b}^{6}}{5\,{e}^{8} \left ( ex+d \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^6*(B*x+A)/(e*x+d)^13,x)

[Out]

-5/9*b^2*(4*A*a^3*b*e^4-12*A*a^2*b^2*d*e^3+12*A*a*b^3*d^2*e^2-4*A*b^4*d^3*e+3*B*
a^4*e^4-16*B*a^3*b*d*e^3+30*B*a^2*b^2*d^2*e^2-24*B*a*b^3*d^3*e+7*B*b^4*d^4)/e^8/
(e*x+d)^9-3/10*b*(5*A*a^4*b*e^5-20*A*a^3*b^2*d*e^4+30*A*a^2*b^3*d^2*e^3-20*A*a*b
^4*d^3*e^2+5*A*b^5*d^4*e+2*B*a^5*e^5-15*B*a^4*b*d*e^4+40*B*a^3*b^2*d^2*e^3-50*B*
a^2*b^3*d^3*e^2+30*B*a*b^4*d^4*e-7*B*b^5*d^5)/e^8/(e*x+d)^10-1/6*b^5*(A*b*e+6*B*
a*e-7*B*b*d)/e^8/(e*x+d)^6-5/8*b^3*(3*A*a^2*b*e^3-6*A*a*b^2*d*e^2+3*A*b^3*d^2*e+
4*B*a^3*e^3-15*B*a^2*b*d*e^2+18*B*a*b^2*d^2*e-7*B*b^3*d^3)/e^8/(e*x+d)^8-3/7*b^4
*(2*A*a*b*e^2-2*A*b^2*d*e+5*B*a^2*e^2-12*B*a*b*d*e+7*B*b^2*d^2)/e^8/(e*x+d)^7-1/
12*(A*a^6*e^7-6*A*a^5*b*d*e^6+15*A*a^4*b^2*d^2*e^5-20*A*a^3*b^3*d^3*e^4+15*A*a^2
*b^4*d^4*e^3-6*A*a*b^5*d^5*e^2+A*b^6*d^6*e-B*a^6*d*e^6+6*B*a^5*b*d^2*e^5-15*B*a^
4*b^2*d^3*e^4+20*B*a^3*b^3*d^4*e^3-15*B*a^2*b^4*d^5*e^2+6*B*a*b^5*d^6*e-B*b^6*d^
7)/e^8/(e*x+d)^12-1/11*(6*A*a^5*b*e^6-30*A*a^4*b^2*d*e^5+60*A*a^3*b^3*d^2*e^4-60
*A*a^2*b^4*d^3*e^3+30*A*a*b^5*d^4*e^2-6*A*b^6*d^5*e+B*a^6*e^6-12*B*a^5*b*d*e^5+4
5*B*a^4*b^2*d^2*e^4-80*B*a^3*b^3*d^3*e^3+75*B*a^2*b^4*d^4*e^2-36*B*a*b^5*d^5*e+7
*B*b^6*d^6)/e^8/(e*x+d)^11-1/5*b^6*B/e^8/(e*x+d)^5

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Maxima [A]  time = 1.43153, size = 1207, normalized size = 4.13 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6/(e*x + d)^13,x, algorithm="maxima")

[Out]

-1/27720*(5544*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 2310*A*a^6*e^7 + 5*(6*B*a*b^5 + A*b
^6)*d^6*e + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 35*(4*B*a^3*b^3 + 3*A*a^2*b^4
)*d^4*e^3 + 70*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 126*(2*B*a^5*b + 5*A*a^4*b^
2)*d^2*e^5 + 210*(B*a^6 + 6*A*a^5*b)*d*e^6 + 924*(7*B*b^6*d*e^6 + 5*(6*B*a*b^5 +
A*b^6)*e^7)*x^6 + 792*(7*B*b^6*d^2*e^5 + 5*(6*B*a*b^5 + A*b^6)*d*e^6 + 15*(5*B*
a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 495*(7*B*b^6*d^3*e^4 + 5*(6*B*a*b^5 + A*b^6)*d^2
*e^5 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + 35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*
x^4 + 220*(7*B*b^6*d^4*e^3 + 5*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 15*(5*B*a^2*b^4 + 2
*A*a*b^5)*d^2*e^5 + 35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 70*(3*B*a^4*b^2 + 4*A
*a^3*b^3)*e^7)*x^3 + 66*(7*B*b^6*d^5*e^2 + 5*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 15*(5
*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 + 35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 70*(3
*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 + 126*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 12*(7
*B*b^6*d^6*e + 5*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*
e^3 + 35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 70*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^
2*e^5 + 126*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 210*(B*a^6 + 6*A*a^5*b)*e^7)*x)/(e
^20*x^12 + 12*d*e^19*x^11 + 66*d^2*e^18*x^10 + 220*d^3*e^17*x^9 + 495*d^4*e^16*x
^8 + 792*d^5*e^15*x^7 + 924*d^6*e^14*x^6 + 792*d^7*e^13*x^5 + 495*d^8*e^12*x^4 +
220*d^9*e^11*x^3 + 66*d^10*e^10*x^2 + 12*d^11*e^9*x + d^12*e^8)

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Fricas [A]  time = 0.216972, size = 1207, normalized size = 4.13 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6/(e*x + d)^13,x, algorithm="fricas")

[Out]

-1/27720*(5544*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 2310*A*a^6*e^7 + 5*(6*B*a*b^5 + A*b
^6)*d^6*e + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 35*(4*B*a^3*b^3 + 3*A*a^2*b^4
)*d^4*e^3 + 70*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 126*(2*B*a^5*b + 5*A*a^4*b^
2)*d^2*e^5 + 210*(B*a^6 + 6*A*a^5*b)*d*e^6 + 924*(7*B*b^6*d*e^6 + 5*(6*B*a*b^5 +
A*b^6)*e^7)*x^6 + 792*(7*B*b^6*d^2*e^5 + 5*(6*B*a*b^5 + A*b^6)*d*e^6 + 15*(5*B*
a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 495*(7*B*b^6*d^3*e^4 + 5*(6*B*a*b^5 + A*b^6)*d^2
*e^5 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + 35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*
x^4 + 220*(7*B*b^6*d^4*e^3 + 5*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 15*(5*B*a^2*b^4 + 2
*A*a*b^5)*d^2*e^5 + 35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 70*(3*B*a^4*b^2 + 4*A
*a^3*b^3)*e^7)*x^3 + 66*(7*B*b^6*d^5*e^2 + 5*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 15*(5
*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 + 35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 70*(3
*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 + 126*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 12*(7
*B*b^6*d^6*e + 5*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*
e^3 + 35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 70*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^
2*e^5 + 126*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 210*(B*a^6 + 6*A*a^5*b)*e^7)*x)/(e
^20*x^12 + 12*d*e^19*x^11 + 66*d^2*e^18*x^10 + 220*d^3*e^17*x^9 + 495*d^4*e^16*x
^8 + 792*d^5*e^15*x^7 + 924*d^6*e^14*x^6 + 792*d^7*e^13*x^5 + 495*d^8*e^12*x^4 +
220*d^9*e^11*x^3 + 66*d^10*e^10*x^2 + 12*d^11*e^9*x + d^12*e^8)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**6*(B*x+A)/(e*x+d)**13,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.226434, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6/(e*x + d)^13,x, algorithm="giac")

[Out]

Done