3.1056 \(\int \frac{(a+b x)^6 (A+B x)}{(d+e x)^{14}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 1.00359, 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 \[ \frac{b^5 (-6 a B e-A b e+7 b B d)}{7 e^8 (d+e x)^7}-\frac{3 b^4 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{8 e^8 (d+e x)^8}+\frac{5 b^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{9 e^8 (d+e x)^9}-\frac{b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{2 e^8 (d+e x)^{10}}+\frac{3 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{11 e^8 (d+e x)^{11}}-\frac{(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{12 e^8 (d+e x)^{12}}+\frac{(b d-a e)^6 (B d-A e)}{13 e^8 (d+e x)^{13}}-\frac{b^6 B}{6 e^8 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^6*(A + B*x))/(d + e*x)^14,x]

[Out]

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

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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)**14,x)

[Out]

Timed out

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Mathematica [B]  time = 0.968186, size = 605, normalized size = 2.07 \[ -\frac{462 a^6 e^6 (12 A e+B (d+13 e x))+252 a^5 b e^5 \left (11 A e (d+13 e x)+2 B \left (d^2+13 d e x+78 e^2 x^2\right )\right )+126 a^4 b^2 e^4 \left (10 A e \left (d^2+13 d e x+78 e^2 x^2\right )+3 B \left (d^3+13 d^2 e x+78 d e^2 x^2+286 e^3 x^3\right )\right )+56 a^3 b^3 e^3 \left (9 A e \left (d^3+13 d^2 e x+78 d e^2 x^2+286 e^3 x^3\right )+4 B \left (d^4+13 d^3 e x+78 d^2 e^2 x^2+286 d e^3 x^3+715 e^4 x^4\right )\right )+21 a^2 b^4 e^2 \left (8 A e \left (d^4+13 d^3 e x+78 d^2 e^2 x^2+286 d e^3 x^3+715 e^4 x^4\right )+5 B \left (d^5+13 d^4 e x+78 d^3 e^2 x^2+286 d^2 e^3 x^3+715 d e^4 x^4+1287 e^5 x^5\right )\right )+6 a b^5 e \left (7 A e \left (d^5+13 d^4 e x+78 d^3 e^2 x^2+286 d^2 e^3 x^3+715 d e^4 x^4+1287 e^5 x^5\right )+6 B \left (d^6+13 d^5 e x+78 d^4 e^2 x^2+286 d^3 e^3 x^3+715 d^2 e^4 x^4+1287 d e^5 x^5+1716 e^6 x^6\right )\right )+b^6 \left (6 A e \left (d^6+13 d^5 e x+78 d^4 e^2 x^2+286 d^3 e^3 x^3+715 d^2 e^4 x^4+1287 d e^5 x^5+1716 e^6 x^6\right )+7 B \left (d^7+13 d^6 e x+78 d^5 e^2 x^2+286 d^4 e^3 x^3+715 d^3 e^4 x^4+1287 d^2 e^5 x^5+1716 d e^6 x^6+1716 e^7 x^7\right )\right )}{72072 e^8 (d+e x)^{13}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^14,x]

[Out]

-(462*a^6*e^6*(12*A*e + B*(d + 13*e*x)) + 252*a^5*b*e^5*(11*A*e*(d + 13*e*x) + 2
*B*(d^2 + 13*d*e*x + 78*e^2*x^2)) + 126*a^4*b^2*e^4*(10*A*e*(d^2 + 13*d*e*x + 78
*e^2*x^2) + 3*B*(d^3 + 13*d^2*e*x + 78*d*e^2*x^2 + 286*e^3*x^3)) + 56*a^3*b^3*e^
3*(9*A*e*(d^3 + 13*d^2*e*x + 78*d*e^2*x^2 + 286*e^3*x^3) + 4*B*(d^4 + 13*d^3*e*x
+ 78*d^2*e^2*x^2 + 286*d*e^3*x^3 + 715*e^4*x^4)) + 21*a^2*b^4*e^2*(8*A*e*(d^4 +
13*d^3*e*x + 78*d^2*e^2*x^2 + 286*d*e^3*x^3 + 715*e^4*x^4) + 5*B*(d^5 + 13*d^4*
e*x + 78*d^3*e^2*x^2 + 286*d^2*e^3*x^3 + 715*d*e^4*x^4 + 1287*e^5*x^5)) + 6*a*b^
5*e*(7*A*e*(d^5 + 13*d^4*e*x + 78*d^3*e^2*x^2 + 286*d^2*e^3*x^3 + 715*d*e^4*x^4
+ 1287*e^5*x^5) + 6*B*(d^6 + 13*d^5*e*x + 78*d^4*e^2*x^2 + 286*d^3*e^3*x^3 + 715
*d^2*e^4*x^4 + 1287*d*e^5*x^5 + 1716*e^6*x^6)) + b^6*(6*A*e*(d^6 + 13*d^5*e*x +
78*d^4*e^2*x^2 + 286*d^3*e^3*x^3 + 715*d^2*e^4*x^4 + 1287*d*e^5*x^5 + 1716*e^6*x
^6) + 7*B*(d^7 + 13*d^6*e*x + 78*d^5*e^2*x^2 + 286*d^4*e^3*x^3 + 715*d^3*e^4*x^4
+ 1287*d^2*e^5*x^5 + 1716*d*e^6*x^6 + 1716*e^7*x^7)))/(72072*e^8*(d + e*x)^13)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^6*(B*x+A)/(e*x+d)^14,x)

[Out]

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

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Maxima [A]  time = 1.44659, size = 1222, normalized size = 4.18 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6/(e*x + d)^14,x, algorithm="maxima")

[Out]

-1/72072*(12012*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 5544*A*a^6*e^7 + 6*(6*B*a*b^5 + A*
b^6)*d^6*e + 21*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 56*(4*B*a^3*b^3 + 3*A*a^2*b^
4)*d^4*e^3 + 126*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 252*(2*B*a^5*b + 5*A*a^4*
b^2)*d^2*e^5 + 462*(B*a^6 + 6*A*a^5*b)*d*e^6 + 1716*(7*B*b^6*d*e^6 + 6*(6*B*a*b^
5 + A*b^6)*e^7)*x^6 + 1287*(7*B*b^6*d^2*e^5 + 6*(6*B*a*b^5 + A*b^6)*d*e^6 + 21*(
5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 715*(7*B*b^6*d^3*e^4 + 6*(6*B*a*b^5 + A*b^6)
*d^2*e^5 + 21*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + 56*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e
^7)*x^4 + 286*(7*B*b^6*d^4*e^3 + 6*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 21*(5*B*a^2*b^4
+ 2*A*a*b^5)*d^2*e^5 + 56*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 126*(3*B*a^4*b^2
+ 4*A*a^3*b^3)*e^7)*x^3 + 78*(7*B*b^6*d^5*e^2 + 6*(6*B*a*b^5 + A*b^6)*d^4*e^3 +
21*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 + 56*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 +
126*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 + 252*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 +
13*(7*B*b^6*d^6*e + 6*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 21*(5*B*a^2*b^4 + 2*A*a*b^5
)*d^4*e^3 + 56*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 126*(3*B*a^4*b^2 + 4*A*a^3*
b^3)*d^2*e^5 + 252*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 462*(B*a^6 + 6*A*a^5*b)*e^7
)*x)/(e^21*x^13 + 13*d*e^20*x^12 + 78*d^2*e^19*x^11 + 286*d^3*e^18*x^10 + 715*d^
4*e^17*x^9 + 1287*d^5*e^16*x^8 + 1716*d^6*e^15*x^7 + 1716*d^7*e^14*x^6 + 1287*d^
8*e^13*x^5 + 715*d^9*e^12*x^4 + 286*d^10*e^11*x^3 + 78*d^11*e^10*x^2 + 13*d^12*e
^9*x + d^13*e^8)

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Fricas [A]  time = 0.213393, size = 1222, normalized size = 4.18 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6/(e*x + d)^14,x, algorithm="fricas")

[Out]

-1/72072*(12012*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 5544*A*a^6*e^7 + 6*(6*B*a*b^5 + A*
b^6)*d^6*e + 21*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 56*(4*B*a^3*b^3 + 3*A*a^2*b^
4)*d^4*e^3 + 126*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 252*(2*B*a^5*b + 5*A*a^4*
b^2)*d^2*e^5 + 462*(B*a^6 + 6*A*a^5*b)*d*e^6 + 1716*(7*B*b^6*d*e^6 + 6*(6*B*a*b^
5 + A*b^6)*e^7)*x^6 + 1287*(7*B*b^6*d^2*e^5 + 6*(6*B*a*b^5 + A*b^6)*d*e^6 + 21*(
5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 715*(7*B*b^6*d^3*e^4 + 6*(6*B*a*b^5 + A*b^6)
*d^2*e^5 + 21*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + 56*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e
^7)*x^4 + 286*(7*B*b^6*d^4*e^3 + 6*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 21*(5*B*a^2*b^4
+ 2*A*a*b^5)*d^2*e^5 + 56*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 126*(3*B*a^4*b^2
+ 4*A*a^3*b^3)*e^7)*x^3 + 78*(7*B*b^6*d^5*e^2 + 6*(6*B*a*b^5 + A*b^6)*d^4*e^3 +
21*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 + 56*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 +
126*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 + 252*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 +
13*(7*B*b^6*d^6*e + 6*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 21*(5*B*a^2*b^4 + 2*A*a*b^5
)*d^4*e^3 + 56*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 126*(3*B*a^4*b^2 + 4*A*a^3*
b^3)*d^2*e^5 + 252*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 462*(B*a^6 + 6*A*a^5*b)*e^7
)*x)/(e^21*x^13 + 13*d*e^20*x^12 + 78*d^2*e^19*x^11 + 286*d^3*e^18*x^10 + 715*d^
4*e^17*x^9 + 1287*d^5*e^16*x^8 + 1716*d^6*e^15*x^7 + 1716*d^7*e^14*x^6 + 1287*d^
8*e^13*x^5 + 715*d^9*e^12*x^4 + 286*d^10*e^11*x^3 + 78*d^11*e^10*x^2 + 13*d^12*e
^9*x + d^13*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)**14,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.230852, 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)^14,x, algorithm="giac")

[Out]

Done