3.1061 \(\int \frac{(a+b x)^6 (A+B x)}{(d+e x)^2} \, dx\)

Optimal. Leaf size=277 \[ \frac{5 b^2 (d+e x)^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{2 e^8}-\frac{5 b^3 (d+e x)^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{3 e^8}+\frac{3 b^4 (d+e x)^4 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{4 e^8}-\frac{b^5 (d+e x)^5 (-6 a B e-A b e+7 b B d)}{5 e^8}+\frac{(b d-a e)^6 (B d-A e)}{e^8 (d+e x)}-\frac{3 b x (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^7}+\frac{(b d-a e)^5 \log (d+e x) (-a B e-6 A b e+7 b B d)}{e^8}+\frac{b^6 B (d+e x)^6}{6 e^8} \]

[Out]

(-3*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*x)/e^7 + ((b*d - a*e)^6*(B*d - A*e))/(e^8*(d + e*x)) + (5*b^
2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e)*(d + e*x)^2)/(2*e^8) - (5*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e -
4*a*B*e)*(d + e*x)^3)/(3*e^8) + (3*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^4)/(4*e^8) - (b^5*
(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^5)/(5*e^8) + (b^6*B*(d + e*x)^6)/(6*e^8) + ((b*d - a*e)^5*(7*b*B*d - 6*A
*b*e - a*B*e)*Log[d + e*x])/e^8

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Rubi [A]  time = 0.592999, antiderivative size = 277, 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, Rules used = {77} \[ \frac{5 b^2 (d+e x)^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{2 e^8}-\frac{5 b^3 (d+e x)^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{3 e^8}+\frac{3 b^4 (d+e x)^4 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{4 e^8}-\frac{b^5 (d+e x)^5 (-6 a B e-A b e+7 b B d)}{5 e^8}+\frac{(b d-a e)^6 (B d-A e)}{e^8 (d+e x)}-\frac{3 b x (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^7}+\frac{(b d-a e)^5 \log (d+e x) (-a B e-6 A b e+7 b B d)}{e^8}+\frac{b^6 B (d+e x)^6}{6 e^8} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^2,x]

[Out]

(-3*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*x)/e^7 + ((b*d - a*e)^6*(B*d - A*e))/(e^8*(d + e*x)) + (5*b^
2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e)*(d + e*x)^2)/(2*e^8) - (5*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e -
4*a*B*e)*(d + e*x)^3)/(3*e^8) + (3*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^4)/(4*e^8) - (b^5*
(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^5)/(5*e^8) + (b^6*B*(d + e*x)^6)/(6*e^8) + ((b*d - a*e)^5*(7*b*B*d - 6*A
*b*e - a*B*e)*Log[d + e*x])/e^8

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{(a+b x)^6 (A+B x)}{(d+e x)^2} \, dx &=\int \left (\frac{3 b (b d-a e)^4 (-7 b B d+5 A b e+2 a B e)}{e^7}+\frac{(-b d+a e)^6 (-B d+A e)}{e^7 (d+e x)^2}+\frac{(-b d+a e)^5 (-7 b B d+6 A b e+a B e)}{e^7 (d+e x)}-\frac{5 b^2 (b d-a e)^3 (-7 b B d+4 A b e+3 a B e) (d+e x)}{e^7}+\frac{5 b^3 (b d-a e)^2 (-7 b B d+3 A b e+4 a B e) (d+e x)^2}{e^7}-\frac{3 b^4 (b d-a e) (-7 b B d+2 A b e+5 a B e) (d+e x)^3}{e^7}+\frac{b^5 (-7 b B d+A b e+6 a B e) (d+e x)^4}{e^7}+\frac{b^6 B (d+e x)^5}{e^7}\right ) \, dx\\ &=-\frac{3 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e) x}{e^7}+\frac{(b d-a e)^6 (B d-A e)}{e^8 (d+e x)}+\frac{5 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e) (d+e x)^2}{2 e^8}-\frac{5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e) (d+e x)^3}{3 e^8}+\frac{3 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) (d+e x)^4}{4 e^8}-\frac{b^5 (7 b B d-A b e-6 a B e) (d+e x)^5}{5 e^8}+\frac{b^6 B (d+e x)^6}{6 e^8}+\frac{(b d-a e)^5 (7 b B d-6 A b e-a B e) \log (d+e x)}{e^8}\\ \end{align*}

Mathematica [B]  time = 0.278761, size = 643, normalized size = 2.32 \[ \frac{75 a^2 b^4 e^2 \left (4 A e \left (6 d^2 e^2 x^2+9 d^3 e x-3 d^4-2 d e^3 x^3+e^4 x^4\right )+B \left (-30 d^3 e^2 x^2+10 d^2 e^3 x^3-48 d^4 e x+12 d^5-5 d e^4 x^4+3 e^5 x^5\right )\right )+200 a^3 b^3 e^3 \left (3 A e \left (-4 d^2 e x+2 d^3-3 d e^2 x^2+e^3 x^3\right )+2 B \left (6 d^2 e^2 x^2+9 d^3 e x-3 d^4-2 d e^3 x^3+e^4 x^4\right )\right )+450 a^4 b^2 e^4 \left (2 A e \left (-d^2+d e x+e^2 x^2\right )+B \left (-4 d^2 e x+2 d^3-3 d e^2 x^2+e^3 x^3\right )\right )+360 a^5 b e^5 \left (A d e+B \left (-d^2+d e x+e^2 x^2\right )\right )+60 a^6 e^6 (B d-A e)+6 a b^5 e \left (5 A e \left (-30 d^3 e^2 x^2+10 d^2 e^3 x^3-48 d^4 e x+12 d^5-5 d e^4 x^4+3 e^5 x^5\right )-6 B \left (-30 d^4 e^2 x^2+10 d^3 e^3 x^3-5 d^2 e^4 x^4-50 d^5 e x+10 d^6+3 d e^5 x^5-2 e^6 x^6\right )\right )+60 (d+e x) (b d-a e)^5 \log (d+e x) (-a B e-6 A b e+7 b B d)+b^6 \left (6 A e \left (30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4+50 d^5 e x-10 d^6-3 d e^5 x^5+2 e^6 x^6\right )+B \left (-210 d^5 e^2 x^2+70 d^4 e^3 x^3-35 d^3 e^4 x^4+21 d^2 e^5 x^5-360 d^6 e x+60 d^7-14 d e^6 x^6+10 e^7 x^7\right )\right )}{60 e^8 (d+e x)} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^2,x]

[Out]

(60*a^6*e^6*(B*d - A*e) + 360*a^5*b*e^5*(A*d*e + B*(-d^2 + d*e*x + e^2*x^2)) + 450*a^4*b^2*e^4*(2*A*e*(-d^2 +
d*e*x + e^2*x^2) + B*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3)) + 200*a^3*b^3*e^3*(3*A*e*(2*d^3 - 4*d^2*e*x
- 3*d*e^2*x^2 + e^3*x^3) + 2*B*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4)) + 75*a^2*b^4*e^2*
(4*A*e*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + B*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2
+ 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5)) + 6*a*b^5*e*(5*A*e*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2
*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5) - 6*B*(10*d^6 - 50*d^5*e*x - 30*d^4*e^2*x^2 + 10*d^3*e^3*x^3 - 5*d^2*e^4*x
^4 + 3*d*e^5*x^5 - 2*e^6*x^6)) + b^6*(6*A*e*(-10*d^6 + 50*d^5*e*x + 30*d^4*e^2*x^2 - 10*d^3*e^3*x^3 + 5*d^2*e^
4*x^4 - 3*d*e^5*x^5 + 2*e^6*x^6) + B*(60*d^7 - 360*d^6*e*x - 210*d^5*e^2*x^2 + 70*d^4*e^3*x^3 - 35*d^3*e^4*x^4
+ 21*d^2*e^5*x^5 - 14*d*e^6*x^6 + 10*e^7*x^7)) + 60*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d + e*x)*Log[d
+ e*x])/(60*e^8*(d + e*x))

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Maple [B]  time = 0.014, size = 1047, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^6*(B*x+A)/(e*x+d)^2,x)

[Out]

20/3*b^3/e^2*B*x^3*a^3-6*b^6/e^7*B*d^5*x+5*b^4/e^2*A*x^3*a^2-2/5*b^6/e^3*B*x^5*d+6*b/e^2*B*a^5*x-30/e^3*ln(e*x
+d)*A*a^4*b^2*d+60/e^4*ln(e*x+d)*A*a^3*b^3*d^2-60/e^5*ln(e*x+d)*A*a^2*b^4*d^3+30/e^6*ln(e*x+d)*A*a*b^5*d^4-12/
e^3*ln(e*x+d)*B*a^5*b*d+45/e^4*ln(e*x+d)*B*a^4*b^2*d^2-80/e^5*ln(e*x+d)*B*a^3*b^3*d^3+75/e^6*ln(e*x+d)*B*a^2*b
^4*d^4-36/e^7*ln(e*x+d)*B*a*b^5*d^5-24*b^5/e^5*A*a*d^3*x-30*b^2/e^3*B*a^4*d*x+60*b^3/e^4*B*a^3*d^2*x-60*b^4/e^
5*B*a^2*d^3*x-40*b^3/e^3*A*a^3*d*x+1/6*b^6/e^2*B*x^6+1/5*b^6/e^2*A*x^5-1/e/(e*x+d)*a^6*A+1/e^2*ln(e*x+d)*B*a^6
+15/4*b^4/e^2*B*x^4*a^2-1/2*b^6/e^3*A*x^4*d+3/2*b^5/e^2*A*x^4*a+6/5*b^5/e^2*B*x^5*a+5*b^6/e^6*A*d^4*x+1/e^8/(e
*x+d)*b^6*B*d^7-4/3*b^6/e^5*B*x^3*d^3+10*b^3/e^2*A*x^2*a^3-2*b^6/e^5*A*x^2*d^3+15/2*b^2/e^2*B*x^2*a^4+5/2*b^6/
e^6*B*x^2*d^4+b^6/e^4*A*x^3*d^2+3/4*b^6/e^4*B*x^4*d^2+7/e^8*ln(e*x+d)*b^6*B*d^6+15*b^2/e^2*A*a^4*x-1/e^7/(e*x+
d)*A*b^6*d^6+1/e^2/(e*x+d)*B*d*a^6+6/e^2*ln(e*x+d)*A*a^5*b-6/e^7*ln(e*x+d)*A*b^6*d^5-6/e^3/(e*x+d)*B*a^5*b*d^2
+15/e^4/(e*x+d)*B*a^4*b^2*d^3-20/e^5/(e*x+d)*B*a^3*b^3*d^4+15/e^6/(e*x+d)*B*a^2*b^4*d^5-6/e^7/(e*x+d)*B*a*b^5*
d^6-12*b^5/e^5*B*x^2*a*d^3-3*b^5/e^3*B*x^4*a*d-4*b^5/e^3*A*x^3*a*d-10*b^4/e^3*B*x^3*a^2*d+6*b^5/e^4*B*x^3*a*d^
2-15*b^4/e^3*A*x^2*a^2*d+9*b^5/e^4*A*x^2*a*d^2-20*b^3/e^3*B*x^2*a^3*d+45*b^4/e^4*A*a^2*d^2*x+6/e^2/(e*x+d)*A*d
*a^5*b-15/e^3/(e*x+d)*A*a^4*b^2*d^2+20/e^4/(e*x+d)*A*a^3*b^3*d^3-15/e^5/(e*x+d)*A*a^2*b^4*d^4+6/e^6/(e*x+d)*A*
a*b^5*d^5+30*b^5/e^6*B*a*d^4*x+45/2*b^4/e^4*B*x^2*a^2*d^2

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Maxima [B]  time = 1.29218, size = 1041, normalized size = 3.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^2,x, algorithm="maxima")

[Out]

(B*b^6*d^7 - A*a^6*e^7 - (6*B*a*b^5 + A*b^6)*d^6*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 5*(4*B*a^3*b^3 + 3*
A*a^2*b^4)*d^4*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + (B*a^6 + 6*
A*a^5*b)*d*e^6)/(e^9*x + d*e^8) + 1/60*(10*B*b^6*e^5*x^6 - 12*(2*B*b^6*d*e^4 - (6*B*a*b^5 + A*b^6)*e^5)*x^5 +
15*(3*B*b^6*d^2*e^3 - 2*(6*B*a*b^5 + A*b^6)*d*e^4 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*e^5)*x^4 - 20*(4*B*b^6*d^3*e^2
- 3*(6*B*a*b^5 + A*b^6)*d^2*e^3 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^4 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^5)*x^3
+ 30*(5*B*b^6*d^4*e - 4*(6*B*a*b^5 + A*b^6)*d^3*e^2 + 9*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^3 - 10*(4*B*a^3*b^3 +
3*A*a^2*b^4)*d*e^4 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^5)*x^2 - 60*(6*B*b^6*d^5 - 5*(6*B*a*b^5 + A*b^6)*d^4*e +
12*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^2 - 15*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^3 + 10*(3*B*a^4*b^2 + 4*A*a^3*b^3)
*d*e^4 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*e^5)*x)/e^7 + (7*B*b^6*d^6 - 6*(6*B*a*b^5 + A*b^6)*d^5*e + 15*(5*B*a^2*b^
4 + 2*A*a*b^5)*d^4*e^2 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^3 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^4 - 6*(
2*B*a^5*b + 5*A*a^4*b^2)*d*e^5 + (B*a^6 + 6*A*a^5*b)*e^6)*log(e*x + d)/e^8

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Fricas [B]  time = 1.88394, size = 2233, normalized size = 8.06 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^2,x, algorithm="fricas")

[Out]

1/60*(10*B*b^6*e^7*x^7 + 60*B*b^6*d^7 - 60*A*a^6*e^7 - 60*(6*B*a*b^5 + A*b^6)*d^6*e + 180*(5*B*a^2*b^4 + 2*A*a
*b^5)*d^5*e^2 - 300*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 300*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 180*(2*B*a
^5*b + 5*A*a^4*b^2)*d^2*e^5 + 60*(B*a^6 + 6*A*a^5*b)*d*e^6 - 2*(7*B*b^6*d*e^6 - 6*(6*B*a*b^5 + A*b^6)*e^7)*x^6
+ 3*(7*B*b^6*d^2*e^5 - 6*(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 - 5*(7*B*b^6*d^3*e
^4 - 6*(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 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*
x^4 + 10*(7*B*b^6*d^4*e^3 - 6*(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 - 20*(4*B*a^3
*b^3 + 3*A*a^2*b^4)*d*e^6 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 - 30*(7*B*b^6*d^5*e^2 - 6*(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 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 15*(3*B*a^4*b^2
+ 4*A*a^3*b^3)*d*e^6 - 6*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 - 60*(6*B*b^6*d^6*e - 5*(6*B*a*b^5 + A*b^6)*d^5*e^
2 + 12*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 15*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 10*(3*B*a^4*b^2 + 4*A*a^3*
b^3)*d^2*e^5 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6)*x + 60*(7*B*b^6*d^7 - 6*(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 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4
- 6*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + (B*a^6 + 6*A*a^5*b)*d*e^6 + (7*B*b^6*d^6*e - 6*(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 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 15*(3*B*a^4*b^2 + 4*A*
a^3*b^3)*d^2*e^5 - 6*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + (B*a^6 + 6*A*a^5*b)*e^7)*x)*log(e*x + d))/(e^9*x + d*e^
8)

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Sympy [B]  time = 4.61943, size = 755, normalized size = 2.73 \begin{align*} \frac{B b^{6} x^{6}}{6 e^{2}} + \frac{- 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}}{d e^{8} + e^{9} x} + \frac{x^{5} \left (A b^{6} e + 6 B a b^{5} e - 2 B b^{6} d\right )}{5 e^{3}} + \frac{x^{4} \left (6 A a b^{5} e^{2} - 2 A b^{6} d e + 15 B a^{2} b^{4} e^{2} - 12 B a b^{5} d e + 3 B b^{6} d^{2}\right )}{4 e^{4}} + \frac{x^{3} \left (15 A a^{2} b^{4} e^{3} - 12 A a b^{5} d e^{2} + 3 A b^{6} d^{2} e + 20 B a^{3} b^{3} e^{3} - 30 B a^{2} b^{4} d e^{2} + 18 B a b^{5} d^{2} e - 4 B b^{6} d^{3}\right )}{3 e^{5}} + \frac{x^{2} \left (20 A a^{3} b^{3} e^{4} - 30 A a^{2} b^{4} d e^{3} + 18 A a b^{5} d^{2} e^{2} - 4 A b^{6} d^{3} e + 15 B a^{4} b^{2} e^{4} - 40 B a^{3} b^{3} d e^{3} + 45 B a^{2} b^{4} d^{2} e^{2} - 24 B a b^{5} d^{3} e + 5 B b^{6} d^{4}\right )}{2 e^{6}} + \frac{x \left (15 A a^{4} b^{2} e^{5} - 40 A a^{3} b^{3} d e^{4} + 45 A a^{2} b^{4} d^{2} e^{3} - 24 A a b^{5} d^{3} e^{2} + 5 A b^{6} d^{4} e + 6 B a^{5} b e^{5} - 30 B a^{4} b^{2} d e^{4} + 60 B a^{3} b^{3} d^{2} e^{3} - 60 B a^{2} b^{4} d^{3} e^{2} + 30 B a b^{5} d^{4} e - 6 B b^{6} d^{5}\right )}{e^{7}} + \frac{\left (a e - b d\right )^{5} \left (6 A b e + B a e - 7 B b d\right ) \log{\left (d + e x \right )}}{e^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**6*(B*x+A)/(e*x+d)**2,x)

[Out]

B*b**6*x**6/(6*e**2) + (-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
)/(d*e**8 + e**9*x) + x**5*(A*b**6*e + 6*B*a*b**5*e - 2*B*b**6*d)/(5*e**3) + x**4*(6*A*a*b**5*e**2 - 2*A*b**6*
d*e + 15*B*a**2*b**4*e**2 - 12*B*a*b**5*d*e + 3*B*b**6*d**2)/(4*e**4) + x**3*(15*A*a**2*b**4*e**3 - 12*A*a*b**
5*d*e**2 + 3*A*b**6*d**2*e + 20*B*a**3*b**3*e**3 - 30*B*a**2*b**4*d*e**2 + 18*B*a*b**5*d**2*e - 4*B*b**6*d**3)
/(3*e**5) + x**2*(20*A*a**3*b**3*e**4 - 30*A*a**2*b**4*d*e**3 + 18*A*a*b**5*d**2*e**2 - 4*A*b**6*d**3*e + 15*B
*a**4*b**2*e**4 - 40*B*a**3*b**3*d*e**3 + 45*B*a**2*b**4*d**2*e**2 - 24*B*a*b**5*d**3*e + 5*B*b**6*d**4)/(2*e*
*6) + x*(15*A*a**4*b**2*e**5 - 40*A*a**3*b**3*d*e**4 + 45*A*a**2*b**4*d**2*e**3 - 24*A*a*b**5*d**3*e**2 + 5*A*
b**6*d**4*e + 6*B*a**5*b*e**5 - 30*B*a**4*b**2*d*e**4 + 60*B*a**3*b**3*d**2*e**3 - 60*B*a**2*b**4*d**3*e**2 +
30*B*a*b**5*d**4*e - 6*B*b**6*d**5)/e**7 + (a*e - b*d)**5*(6*A*b*e + B*a*e - 7*B*b*d)*log(d + e*x)/e**8

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Giac [B]  time = 2.19599, size = 1264, normalized size = 4.56 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^2,x, algorithm="giac")

[Out]

1/60*(10*B*b^6 - 12*(7*B*b^6*d*e - 6*B*a*b^5*e^2 - A*b^6*e^2)*e^(-1)/(x*e + d) + 45*(7*B*b^6*d^2*e^2 - 12*B*a*
b^5*d*e^3 - 2*A*b^6*d*e^3 + 5*B*a^2*b^4*e^4 + 2*A*a*b^5*e^4)*e^(-2)/(x*e + d)^2 - 100*(7*B*b^6*d^3*e^3 - 18*B*
a*b^5*d^2*e^4 - 3*A*b^6*d^2*e^4 + 15*B*a^2*b^4*d*e^5 + 6*A*a*b^5*d*e^5 - 4*B*a^3*b^3*e^6 - 3*A*a^2*b^4*e^6)*e^
(-3)/(x*e + d)^3 + 150*(7*B*b^6*d^4*e^4 - 24*B*a*b^5*d^3*e^5 - 4*A*b^6*d^3*e^5 + 30*B*a^2*b^4*d^2*e^6 + 12*A*a
*b^5*d^2*e^6 - 16*B*a^3*b^3*d*e^7 - 12*A*a^2*b^4*d*e^7 + 3*B*a^4*b^2*e^8 + 4*A*a^3*b^3*e^8)*e^(-4)/(x*e + d)^4
- 180*(7*B*b^6*d^5*e^5 - 30*B*a*b^5*d^4*e^6 - 5*A*b^6*d^4*e^6 + 50*B*a^2*b^4*d^3*e^7 + 20*A*a*b^5*d^3*e^7 - 4
0*B*a^3*b^3*d^2*e^8 - 30*A*a^2*b^4*d^2*e^8 + 15*B*a^4*b^2*d*e^9 + 20*A*a^3*b^3*d*e^9 - 2*B*a^5*b*e^10 - 5*A*a^
4*b^2*e^10)*e^(-5)/(x*e + d)^5)*(x*e + d)^6*e^(-8) - (7*B*b^6*d^6 - 36*B*a*b^5*d^5*e - 6*A*b^6*d^5*e + 75*B*a^
2*b^4*d^4*e^2 + 30*A*a*b^5*d^4*e^2 - 80*B*a^3*b^3*d^3*e^3 - 60*A*a^2*b^4*d^3*e^3 + 45*B*a^4*b^2*d^2*e^4 + 60*A
*a^3*b^3*d^2*e^4 - 12*B*a^5*b*d*e^5 - 30*A*a^4*b^2*d*e^5 + B*a^6*e^6 + 6*A*a^5*b*e^6)*e^(-8)*log(abs(x*e + d)*
e^(-1)/(x*e + d)^2) + (B*b^6*d^7*e^6/(x*e + d) - 6*B*a*b^5*d^6*e^7/(x*e + d) - A*b^6*d^6*e^7/(x*e + d) + 15*B*
a^2*b^4*d^5*e^8/(x*e + d) + 6*A*a*b^5*d^5*e^8/(x*e + d) - 20*B*a^3*b^3*d^4*e^9/(x*e + d) - 15*A*a^2*b^4*d^4*e^
9/(x*e + d) + 15*B*a^4*b^2*d^3*e^10/(x*e + d) + 20*A*a^3*b^3*d^3*e^10/(x*e + d) - 6*B*a^5*b*d^2*e^11/(x*e + d)
- 15*A*a^4*b^2*d^2*e^11/(x*e + d) + B*a^6*d*e^12/(x*e + d) + 6*A*a^5*b*d*e^12/(x*e + d) - A*a^6*e^13/(x*e + d
))*e^(-14)