3.1057 \(\int \frac{(a+b x)^6 (A+B x)}{(d+e x)^{15}} \, dx\)

Optimal. Leaf size=292 \[ \frac{b^5 (-6 a B e-A b e+7 b B d)}{8 e^8 (d+e x)^8}-\frac{b^4 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{3 e^8 (d+e x)^9}+\frac{b^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{2 e^8 (d+e x)^{10}}-\frac{5 b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{11 e^8 (d+e x)^{11}}+\frac{b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{4 e^8 (d+e x)^{12}}-\frac{(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{13 e^8 (d+e x)^{13}}+\frac{(b d-a e)^6 (B d-A e)}{14 e^8 (d+e x)^{14}}-\frac{b^6 B}{7 e^8 (d+e x)^7} \]

[Out]

((b*d - a*e)^6*(B*d - A*e))/(14*e^8*(d + e*x)^14) - ((b*d - a*e)^5*(7*b*B*d - 6*
A*b*e - a*B*e))/(13*e^8*(d + e*x)^13) + (b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*
a*B*e))/(4*e^8*(d + e*x)^12) - (5*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e
))/(11*e^8*(d + e*x)^11) + (b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e))/(2*
e^8*(d + e*x)^10) - (b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e))/(3*e^8*(d +
e*x)^9) + (b^5*(7*b*B*d - A*b*e - 6*a*B*e))/(8*e^8*(d + e*x)^8) - (b^6*B)/(7*e^8
*(d + e*x)^7)

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Rubi [A]  time = 1.01193, 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 \[ \frac{b^5 (-6 a B e-A b e+7 b B d)}{8 e^8 (d+e x)^8}-\frac{b^4 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{3 e^8 (d+e x)^9}+\frac{b^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{2 e^8 (d+e x)^{10}}-\frac{5 b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{11 e^8 (d+e x)^{11}}+\frac{b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{4 e^8 (d+e x)^{12}}-\frac{(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{13 e^8 (d+e x)^{13}}+\frac{(b d-a e)^6 (B d-A e)}{14 e^8 (d+e x)^{14}}-\frac{b^6 B}{7 e^8 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^6*(A + B*x))/(d + e*x)^15,x]

[Out]

((b*d - a*e)^6*(B*d - A*e))/(14*e^8*(d + e*x)^14) - ((b*d - a*e)^5*(7*b*B*d - 6*
A*b*e - a*B*e))/(13*e^8*(d + e*x)^13) + (b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*
a*B*e))/(4*e^8*(d + e*x)^12) - (5*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e
))/(11*e^8*(d + e*x)^11) + (b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e))/(2*
e^8*(d + e*x)^10) - (b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e))/(3*e^8*(d +
e*x)^9) + (b^5*(7*b*B*d - A*b*e - 6*a*B*e))/(8*e^8*(d + e*x)^8) - (b^6*B)/(7*e^8
*(d + e*x)^7)

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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)**15,x)

[Out]

Timed out

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Mathematica [B]  time = 0.968726, size = 602, normalized size = 2.06 \[ -\frac{132 a^6 e^6 (13 A e+B (d+14 e x))+132 a^5 b e^5 \left (6 A e (d+14 e x)+B \left (d^2+14 d e x+91 e^2 x^2\right )\right )+30 a^4 b^2 e^4 \left (11 A e \left (d^2+14 d e x+91 e^2 x^2\right )+3 B \left (d^3+14 d^2 e x+91 d e^2 x^2+364 e^3 x^3\right )\right )+24 a^3 b^3 e^3 \left (5 A e \left (d^3+14 d^2 e x+91 d e^2 x^2+364 e^3 x^3\right )+2 B \left (d^4+14 d^3 e x+91 d^2 e^2 x^2+364 d e^3 x^3+1001 e^4 x^4\right )\right )+4 a^2 b^4 e^2 \left (9 A e \left (d^4+14 d^3 e x+91 d^2 e^2 x^2+364 d e^3 x^3+1001 e^4 x^4\right )+5 B \left (d^5+14 d^4 e x+91 d^3 e^2 x^2+364 d^2 e^3 x^3+1001 d e^4 x^4+2002 e^5 x^5\right )\right )+2 a b^5 e \left (4 A e \left (d^5+14 d^4 e x+91 d^3 e^2 x^2+364 d^2 e^3 x^3+1001 d e^4 x^4+2002 e^5 x^5\right )+3 B \left (d^6+14 d^5 e x+91 d^4 e^2 x^2+364 d^3 e^3 x^3+1001 d^2 e^4 x^4+2002 d e^5 x^5+3003 e^6 x^6\right )\right )+b^6 \left (A e \left (d^6+14 d^5 e x+91 d^4 e^2 x^2+364 d^3 e^3 x^3+1001 d^2 e^4 x^4+2002 d e^5 x^5+3003 e^6 x^6\right )+B \left (d^7+14 d^6 e x+91 d^5 e^2 x^2+364 d^4 e^3 x^3+1001 d^3 e^4 x^4+2002 d^2 e^5 x^5+3003 d e^6 x^6+3432 e^7 x^7\right )\right )}{24024 e^8 (d+e x)^{14}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^15,x]

[Out]

-(132*a^6*e^6*(13*A*e + B*(d + 14*e*x)) + 132*a^5*b*e^5*(6*A*e*(d + 14*e*x) + B*
(d^2 + 14*d*e*x + 91*e^2*x^2)) + 30*a^4*b^2*e^4*(11*A*e*(d^2 + 14*d*e*x + 91*e^2
*x^2) + 3*B*(d^3 + 14*d^2*e*x + 91*d*e^2*x^2 + 364*e^3*x^3)) + 24*a^3*b^3*e^3*(5
*A*e*(d^3 + 14*d^2*e*x + 91*d*e^2*x^2 + 364*e^3*x^3) + 2*B*(d^4 + 14*d^3*e*x + 9
1*d^2*e^2*x^2 + 364*d*e^3*x^3 + 1001*e^4*x^4)) + 4*a^2*b^4*e^2*(9*A*e*(d^4 + 14*
d^3*e*x + 91*d^2*e^2*x^2 + 364*d*e^3*x^3 + 1001*e^4*x^4) + 5*B*(d^5 + 14*d^4*e*x
+ 91*d^3*e^2*x^2 + 364*d^2*e^3*x^3 + 1001*d*e^4*x^4 + 2002*e^5*x^5)) + 2*a*b^5*
e*(4*A*e*(d^5 + 14*d^4*e*x + 91*d^3*e^2*x^2 + 364*d^2*e^3*x^3 + 1001*d*e^4*x^4 +
2002*e^5*x^5) + 3*B*(d^6 + 14*d^5*e*x + 91*d^4*e^2*x^2 + 364*d^3*e^3*x^3 + 1001
*d^2*e^4*x^4 + 2002*d*e^5*x^5 + 3003*e^6*x^6)) + b^6*(A*e*(d^6 + 14*d^5*e*x + 91
*d^4*e^2*x^2 + 364*d^3*e^3*x^3 + 1001*d^2*e^4*x^4 + 2002*d*e^5*x^5 + 3003*e^6*x^
6) + B*(d^7 + 14*d^6*e*x + 91*d^5*e^2*x^2 + 364*d^4*e^3*x^3 + 1001*d^3*e^4*x^4 +
2002*d^2*e^5*x^5 + 3003*d*e^6*x^6 + 3432*e^7*x^7)))/(24024*e^8*(d + e*x)^14)

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Maple [B]  time = 0.013, size = 814, normalized size = 2.8 \[ -{\frac{{a}^{6}A{e}^{7}-6\,Ad{a}^{5}b{e}^{6}+15\,A{d}^{2}{a}^{4}{b}^{2}{e}^{5}-20\,A{d}^{3}{a}^{3}{b}^{3}{e}^{4}+15\,A{d}^{4}{a}^{2}{b}^{4}{e}^{3}-6\,A{d}^{5}a{b}^{5}{e}^{2}+A{d}^{6}{b}^{6}e-Bd{a}^{6}{e}^{6}+6\,B{d}^{2}{a}^{5}b{e}^{5}-15\,B{d}^{3}{a}^{4}{b}^{2}{e}^{4}+20\,B{d}^{4}{a}^{3}{b}^{3}{e}^{3}-15\,B{d}^{5}{a}^{2}{b}^{4}{e}^{2}+6\,B{d}^{6}a{b}^{5}e-{b}^{6}B{d}^{7}}{14\,{e}^{8} \left ( ex+d \right ) ^{14}}}-{\frac{{b}^{4} \left ( 2\,Aab{e}^{2}-2\,Ad{b}^{2}e+5\,B{a}^{2}{e}^{2}-12\,Bdabe+7\,{b}^{2}B{d}^{2} \right ) }{3\,{e}^{8} \left ( ex+d \right ) ^{9}}}-{\frac{{b}^{3} \left ( 3\,A{a}^{2}b{e}^{3}-6\,Ada{b}^{2}{e}^{2}+3\,A{b}^{3}{d}^{2}e+4\,B{a}^{3}{e}^{3}-15\,B{a}^{2}bd{e}^{2}+18\,B{d}^{2}a{b}^{2}e-7\,{b}^{3}B{d}^{3} \right ) }{2\,{e}^{8} \left ( ex+d \right ) ^{10}}}-{\frac{{b}^{5} \left ( Abe+6\,Bae-7\,Bbd \right ) }{8\,{e}^{8} \left ( ex+d \right ) ^{8}}}-{\frac{B{b}^{6}}{7\,{e}^{8} \left ( ex+d \right ) ^{7}}}-{\frac{b \left ( 5\,A{a}^{4}b{e}^{5}-20\,A{a}^{3}{b}^{2}d{e}^{4}+30\,A{a}^{2}{b}^{3}{d}^{2}{e}^{3}-20\,Aa{b}^{4}{d}^{3}{e}^{2}+5\,A{b}^{5}{d}^{4}e+2\,B{a}^{5}{e}^{5}-15\,B{a}^{4}bd{e}^{4}+40\,B{a}^{3}{b}^{2}{d}^{2}{e}^{3}-50\,B{a}^{2}{b}^{3}{d}^{3}{e}^{2}+30\,Ba{b}^{4}{d}^{4}e-7\,B{b}^{5}{d}^{5} \right ) }{4\,{e}^{8} \left ( ex+d \right ) ^{12}}}-{\frac{5\,{b}^{2} \left ( 4\,A{a}^{3}b{e}^{4}-12\,A{a}^{2}{b}^{2}d{e}^{3}+12\,Aa{b}^{3}{d}^{2}{e}^{2}-4\,A{b}^{4}{d}^{3}e+3\,B{a}^{4}{e}^{4}-16\,B{a}^{3}bd{e}^{3}+30\,B{a}^{2}{b}^{2}{d}^{2}{e}^{2}-24\,Ba{b}^{3}{d}^{3}e+7\,B{b}^{4}{d}^{4} \right ) }{11\,{e}^{8} \left ( ex+d \right ) ^{11}}}-{\frac{6\,A{a}^{5}b{e}^{6}-30\,Ad{a}^{4}{b}^{2}{e}^{5}+60\,A{d}^{2}{a}^{3}{b}^{3}{e}^{4}-60\,A{d}^{3}{a}^{2}{b}^{4}{e}^{3}+30\,A{d}^{4}a{b}^{5}{e}^{2}-6\,A{d}^{5}{b}^{6}e+{a}^{6}B{e}^{6}-12\,Bd{a}^{5}b{e}^{5}+45\,B{d}^{2}{a}^{4}{b}^{2}{e}^{4}-80\,B{d}^{3}{a}^{3}{b}^{3}{e}^{3}+75\,B{d}^{4}{a}^{2}{b}^{4}{e}^{2}-36\,B{d}^{5}a{b}^{5}e+7\,{b}^{6}B{d}^{6}}{13\,{e}^{8} \left ( ex+d \right ) ^{13}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^6*(B*x+A)/(e*x+d)^15,x)

[Out]

-1/14*(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)^14-1/3*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)^9-1/2*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)^10-1/8*b^5
*(A*b*e+6*B*a*e-7*B*b*d)/e^8/(e*x+d)^8-1/7*b^6*B/e^8/(e*x+d)^7-1/4*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)^12-5/11*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^2-2
4*B*a*b^3*d^3*e+7*B*b^4*d^4)/e^8/(e*x+d)^11-1/13*(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+45*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)^13

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Maxima [A]  time = 1.44844, size = 1218, normalized size = 4.17 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6/(e*x + d)^15,x, algorithm="maxima")

[Out]

-1/24024*(3432*B*b^6*e^7*x^7 + B*b^6*d^7 + 1716*A*a^6*e^7 + (6*B*a*b^5 + A*b^6)*
d^6*e + 4*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 12*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4
*e^3 + 30*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 66*(2*B*a^5*b + 5*A*a^4*b^2)*d^2
*e^5 + 132*(B*a^6 + 6*A*a^5*b)*d*e^6 + 3003*(B*b^6*d*e^6 + (6*B*a*b^5 + A*b^6)*e
^7)*x^6 + 2002*(B*b^6*d^2*e^5 + (6*B*a*b^5 + A*b^6)*d*e^6 + 4*(5*B*a^2*b^4 + 2*A
*a*b^5)*e^7)*x^5 + 1001*(B*b^6*d^3*e^4 + (6*B*a*b^5 + A*b^6)*d^2*e^5 + 4*(5*B*a^
2*b^4 + 2*A*a*b^5)*d*e^6 + 12*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 364*(B*b^6*
d^4*e^3 + (6*B*a*b^5 + A*b^6)*d^3*e^4 + 4*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 12
*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 30*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 9
1*(B*b^6*d^5*e^2 + (6*B*a*b^5 + A*b^6)*d^4*e^3 + 4*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3
*e^4 + 12*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 30*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d
*e^6 + 66*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 14*(B*b^6*d^6*e + (6*B*a*b^5 + A*
b^6)*d^5*e^2 + 4*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 + 12*(4*B*a^3*b^3 + 3*A*a^2*b
^4)*d^3*e^4 + 30*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 + 66*(2*B*a^5*b + 5*A*a^4*b
^2)*d*e^6 + 132*(B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^22*x^14 + 14*d*e^21*x^13 + 91*d^2
*e^20*x^12 + 364*d^3*e^19*x^11 + 1001*d^4*e^18*x^10 + 2002*d^5*e^17*x^9 + 3003*d
^6*e^16*x^8 + 3432*d^7*e^15*x^7 + 3003*d^8*e^14*x^6 + 2002*d^9*e^13*x^5 + 1001*d
^10*e^12*x^4 + 364*d^11*e^11*x^3 + 91*d^12*e^10*x^2 + 14*d^13*e^9*x + d^14*e^8)

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Fricas [A]  time = 0.215072, size = 1218, normalized size = 4.17 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6/(e*x + d)^15,x, algorithm="fricas")

[Out]

-1/24024*(3432*B*b^6*e^7*x^7 + B*b^6*d^7 + 1716*A*a^6*e^7 + (6*B*a*b^5 + A*b^6)*
d^6*e + 4*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 12*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4
*e^3 + 30*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 66*(2*B*a^5*b + 5*A*a^4*b^2)*d^2
*e^5 + 132*(B*a^6 + 6*A*a^5*b)*d*e^6 + 3003*(B*b^6*d*e^6 + (6*B*a*b^5 + A*b^6)*e
^7)*x^6 + 2002*(B*b^6*d^2*e^5 + (6*B*a*b^5 + A*b^6)*d*e^6 + 4*(5*B*a^2*b^4 + 2*A
*a*b^5)*e^7)*x^5 + 1001*(B*b^6*d^3*e^4 + (6*B*a*b^5 + A*b^6)*d^2*e^5 + 4*(5*B*a^
2*b^4 + 2*A*a*b^5)*d*e^6 + 12*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 364*(B*b^6*
d^4*e^3 + (6*B*a*b^5 + A*b^6)*d^3*e^4 + 4*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 12
*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 30*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 9
1*(B*b^6*d^5*e^2 + (6*B*a*b^5 + A*b^6)*d^4*e^3 + 4*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3
*e^4 + 12*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 30*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d
*e^6 + 66*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 14*(B*b^6*d^6*e + (6*B*a*b^5 + A*
b^6)*d^5*e^2 + 4*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 + 12*(4*B*a^3*b^3 + 3*A*a^2*b
^4)*d^3*e^4 + 30*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 + 66*(2*B*a^5*b + 5*A*a^4*b
^2)*d*e^6 + 132*(B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^22*x^14 + 14*d*e^21*x^13 + 91*d^2
*e^20*x^12 + 364*d^3*e^19*x^11 + 1001*d^4*e^18*x^10 + 2002*d^5*e^17*x^9 + 3003*d
^6*e^16*x^8 + 3432*d^7*e^15*x^7 + 3003*d^8*e^14*x^6 + 2002*d^9*e^13*x^5 + 1001*d
^10*e^12*x^4 + 364*d^11*e^11*x^3 + 91*d^12*e^10*x^2 + 14*d^13*e^9*x + d^14*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)**15,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.223274, 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)^15,x, algorithm="giac")

[Out]

Done