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

Optimal. Leaf size=272 \[ -\frac {b^5 x (-6 a B e-A b e+6 b B d)}{e^7}+\frac {3 b^4 (b d-a e) \log (d+e x) (-5 a B e-2 A b e+7 b B d)}{e^8}+\frac {5 b^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{e^8 (d+e x)}-\frac {5 b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{2 e^8 (d+e x)^2}+\frac {b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8 (d+e x)^3}-\frac {(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{4 e^8 (d+e x)^4}+\frac {(b d-a e)^6 (B d-A e)}{5 e^8 (d+e x)^5}+\frac {b^6 B x^2}{2 e^6} \]

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Rubi [A]  time = 0.36, antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \begin {gather*} -\frac {b^5 x (-6 a B e-A b e+6 b B d)}{e^7}+\frac {5 b^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{e^8 (d+e x)}-\frac {5 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)^2}+\frac {3 b^4 (b d-a e) \log (d+e x) (-5 a B e-2 A b e+7 b B d)}{e^8}+\frac {b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8 (d+e x)^3}-\frac {(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{4 e^8 (d+e x)^4}+\frac {(b d-a e)^6 (B d-A e)}{5 e^8 (d+e x)^5}+\frac {b^6 B x^2}{2 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [B]  time = 0.33, size = 633, normalized size = 2.33 \begin {gather*} \frac {-a^6 e^6 (4 A e+B (d+5 e x))-2 a^5 b e^5 \left (3 A e (d+5 e x)+2 B \left (d^2+5 d e x+10 e^2 x^2\right )\right )-5 a^4 b^2 e^4 \left (2 A e \left (d^2+5 d e x+10 e^2 x^2\right )+3 B \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )-20 a^3 b^3 e^3 \left (A e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+4 B \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+5 a^2 b^4 e^2 \left (B d \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )-12 A e \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+2 a b^5 e \left (A d e \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )-6 B \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )\right )+60 b^4 (d+e x)^5 (b d-a e) \log (d+e x) (-5 a B e-2 A b e+7 b B d)+b^6 \left (B \left (459 d^7+1875 d^6 e x+2700 d^5 e^2 x^2+1300 d^4 e^3 x^3-400 d^3 e^4 x^4-500 d^2 e^5 x^5-70 d e^6 x^6+10 e^7 x^7\right )-2 A e \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )\right )}{20 e^8 (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(a^6*e^6*(4*A*e + B*(d + 5*e*x))) - 2*a^5*b*e^5*(3*A*e*(d + 5*e*x) + 2*B*(d^2 + 5*d*e*x + 10*e^2*x^2)) - 5*a
^4*b^2*e^4*(2*A*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*B*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3)) - 20*a^3*b
^3*e^3*(A*e*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 4*B*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x
^3 + 5*e^4*x^4)) + 5*a^2*b^4*e^2*(-12*A*e*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4) + B*d*
(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4)) + 2*a*b^5*e*(A*d*e*(137*d^4 + 625*d^
3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4) - 6*B*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^
3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^6*x^6)) + b^6*(-2*A*e*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2
+ 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^6*x^6) + B*(459*d^7 + 1875*d^6*e*x + 2700*d^5*e^2*x^
2 + 1300*d^4*e^3*x^3 - 400*d^3*e^4*x^4 - 500*d^2*e^5*x^5 - 70*d*e^6*x^6 + 10*e^7*x^7)) + 60*b^4*(b*d - a*e)*(7
*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^5*Log[d + e*x])/(20*e^8*(d + e*x)^5)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 2.31, size = 1157, normalized size = 4.25

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(10*B*b^6*e^7*x^7 + 459*B*b^6*d^7 - 4*A*a^6*e^7 - 174*(6*B*a*b^5 + A*b^6)*d^6*e + 137*(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 - 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 2*(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 - 10*(7*B*b^6*d*e^6 - 2*(6*B*a*b^5 + A*b^6)*e^7)*x^6 - 100
*(5*B*b^6*d^2*e^5 - (6*B*a*b^5 + A*b^6)*d*e^6)*x^5 - 100*(4*B*b^6*d^3*e^4 + (6*B*a*b^5 + A*b^6)*d^2*e^5 - 3*(5
*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 50*(26*B*b^6*d^4*e^3 - 16*(6*B*a*b^5 +
A*b^6)*d^3*e^4 + 18*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 - 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 - (3*B*a^4*b^2 + 4
*A*a^3*b^3)*e^7)*x^3 + 10*(270*B*b^6*d^5*e^2 - 120*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 110*(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 - 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 - 2*(2*B*a^5*b + 5*A*a
^4*b^2)*e^7)*x^2 + 5*(375*B*b^6*d^6*e - 150*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 125*(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 - 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 - 2*(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 + 60*(7*B*b^6*d^7 - 2*(6*B*a*b^5 + A*b^6)*d^6*e + (5*B*a^2*b^4 + 2*A*a*b
^5)*d^5*e^2 + (7*B*b^6*d^2*e^5 - 2*(6*B*a*b^5 + A*b^6)*d*e^6 + (5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 5*(7*B*b^6
*d^3*e^4 - 2*(6*B*a*b^5 + A*b^6)*d^2*e^5 + (5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6)*x^4 + 10*(7*B*b^6*d^4*e^3 - 2*(6*B
*a*b^5 + A*b^6)*d^3*e^4 + (5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5)*x^3 + 10*(7*B*b^6*d^5*e^2 - 2*(6*B*a*b^5 + A*b^6)
*d^4*e^3 + (5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4)*x^2 + 5*(7*B*b^6*d^6*e - 2*(6*B*a*b^5 + A*b^6)*d^5*e^2 + (5*B*a^
2*b^4 + 2*A*a*b^5)*d^4*e^3)*x)*log(e*x + d))/(e^13*x^5 + 5*d*e^12*x^4 + 10*d^2*e^11*x^3 + 10*d^3*e^10*x^2 + 5*
d^4*e^9*x + d^5*e^8)

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giac [B]  time = 1.24, size = 779, normalized size = 2.86 \begin {gather*} 3 \, {\left (7 \, B b^{6} d^{2} - 12 \, B a b^{5} d e - 2 \, A b^{6} d e + 5 \, B a^{2} b^{4} e^{2} + 2 \, A a b^{5} e^{2}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{2} \, {\left (B b^{6} x^{2} e^{6} - 12 \, B b^{6} d x e^{5} + 12 \, B a b^{5} x e^{6} + 2 \, A b^{6} x e^{6}\right )} e^{\left (-12\right )} + \frac {{\left (459 \, B b^{6} d^{7} - 1044 \, B a b^{5} d^{6} e - 174 \, A b^{6} d^{6} e + 685 \, B a^{2} b^{4} d^{5} e^{2} + 274 \, A a b^{5} d^{5} e^{2} - 80 \, B a^{3} b^{3} d^{4} e^{3} - 60 \, A a^{2} b^{4} d^{4} e^{3} - 15 \, B a^{4} b^{2} d^{3} e^{4} - 20 \, A a^{3} b^{3} d^{3} e^{4} - 4 \, B a^{5} b d^{2} e^{5} - 10 \, A a^{4} b^{2} d^{2} e^{5} - B a^{6} d e^{6} - 6 \, A a^{5} b d e^{6} - 4 \, A a^{6} e^{7} + 100 \, {\left (7 \, B b^{6} d^{3} e^{4} - 18 \, B a b^{5} d^{2} e^{5} - 3 \, A b^{6} d^{2} e^{5} + 15 \, B a^{2} b^{4} d e^{6} + 6 \, A a b^{5} d e^{6} - 4 \, B a^{3} b^{3} e^{7} - 3 \, A a^{2} b^{4} e^{7}\right )} x^{4} + 50 \, {\left (49 \, B b^{6} d^{4} e^{3} - 120 \, B a b^{5} d^{3} e^{4} - 20 \, A b^{6} d^{3} e^{4} + 90 \, B a^{2} b^{4} d^{2} e^{5} + 36 \, A a b^{5} d^{2} e^{5} - 16 \, B a^{3} b^{3} d e^{6} - 12 \, A a^{2} b^{4} d e^{6} - 3 \, B a^{4} b^{2} e^{7} - 4 \, A a^{3} b^{3} e^{7}\right )} x^{3} + 10 \, {\left (329 \, B b^{6} d^{5} e^{2} - 780 \, B a b^{5} d^{4} e^{3} - 130 \, A b^{6} d^{4} e^{3} + 550 \, B a^{2} b^{4} d^{3} e^{4} + 220 \, A a b^{5} d^{3} e^{4} - 80 \, B a^{3} b^{3} d^{2} e^{5} - 60 \, A a^{2} b^{4} d^{2} e^{5} - 15 \, B a^{4} b^{2} d e^{6} - 20 \, A a^{3} b^{3} d e^{6} - 4 \, B a^{5} b e^{7} - 10 \, A a^{4} b^{2} e^{7}\right )} x^{2} + 5 \, {\left (399 \, B b^{6} d^{6} e - 924 \, B a b^{5} d^{5} e^{2} - 154 \, A b^{6} d^{5} e^{2} + 625 \, B a^{2} b^{4} d^{4} e^{3} + 250 \, A a b^{5} d^{4} e^{3} - 80 \, B a^{3} b^{3} d^{3} e^{4} - 60 \, A a^{2} b^{4} d^{3} e^{4} - 15 \, B a^{4} b^{2} d^{2} e^{5} - 20 \, A a^{3} b^{3} d^{2} e^{5} - 4 \, B a^{5} b d e^{6} - 10 \, A a^{4} b^{2} d e^{6} - B a^{6} e^{7} - 6 \, A a^{5} b e^{7}\right )} x\right )} e^{\left (-8\right )}}{20 \, {\left (x e + d\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(7*B*b^6*d^2 - 12*B*a*b^5*d*e - 2*A*b^6*d*e + 5*B*a^2*b^4*e^2 + 2*A*a*b^5*e^2)*e^(-8)*log(abs(x*e + d)) + 1/
2*(B*b^6*x^2*e^6 - 12*B*b^6*d*x*e^5 + 12*B*a*b^5*x*e^6 + 2*A*b^6*x*e^6)*e^(-12) + 1/20*(459*B*b^6*d^7 - 1044*B
*a*b^5*d^6*e - 174*A*b^6*d^6*e + 685*B*a^2*b^4*d^5*e^2 + 274*A*a*b^5*d^5*e^2 - 80*B*a^3*b^3*d^4*e^3 - 60*A*a^2
*b^4*d^4*e^3 - 15*B*a^4*b^2*d^3*e^4 - 20*A*a^3*b^3*d^3*e^4 - 4*B*a^5*b*d^2*e^5 - 10*A*a^4*b^2*d^2*e^5 - B*a^6*
d*e^6 - 6*A*a^5*b*d*e^6 - 4*A*a^6*e^7 + 100*(7*B*b^6*d^3*e^4 - 18*B*a*b^5*d^2*e^5 - 3*A*b^6*d^2*e^5 + 15*B*a^2
*b^4*d*e^6 + 6*A*a*b^5*d*e^6 - 4*B*a^3*b^3*e^7 - 3*A*a^2*b^4*e^7)*x^4 + 50*(49*B*b^6*d^4*e^3 - 120*B*a*b^5*d^3
*e^4 - 20*A*b^6*d^3*e^4 + 90*B*a^2*b^4*d^2*e^5 + 36*A*a*b^5*d^2*e^5 - 16*B*a^3*b^3*d*e^6 - 12*A*a^2*b^4*d*e^6
- 3*B*a^4*b^2*e^7 - 4*A*a^3*b^3*e^7)*x^3 + 10*(329*B*b^6*d^5*e^2 - 780*B*a*b^5*d^4*e^3 - 130*A*b^6*d^4*e^3 + 5
50*B*a^2*b^4*d^3*e^4 + 220*A*a*b^5*d^3*e^4 - 80*B*a^3*b^3*d^2*e^5 - 60*A*a^2*b^4*d^2*e^5 - 15*B*a^4*b^2*d*e^6
- 20*A*a^3*b^3*d*e^6 - 4*B*a^5*b*e^7 - 10*A*a^4*b^2*e^7)*x^2 + 5*(399*B*b^6*d^6*e - 924*B*a*b^5*d^5*e^2 - 154*
A*b^6*d^5*e^2 + 625*B*a^2*b^4*d^4*e^3 + 250*A*a*b^5*d^4*e^3 - 80*B*a^3*b^3*d^3*e^4 - 60*A*a^2*b^4*d^3*e^4 - 15
*B*a^4*b^2*d^2*e^5 - 20*A*a^3*b^3*d^2*e^5 - 4*B*a^5*b*d*e^6 - 10*A*a^4*b^2*d*e^6 - B*a^6*e^7 - 6*A*a^5*b*e^7)*
x)*e^(-8)/(x*e + d)^5

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maple [B]  time = 0.02, size = 1202, normalized size = 4.42

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/e^2/(e*x+d)^4*B*a^6-1/5/e/(e*x+d)^5*A*a^6+b^6/e^6*A*x-40*b^3/e^5/(e*x+d)^3*B*a^3*d^2+15/2/e^3/(e*x+d)^4*A
*a^4*b^2*d-15/e^4/(e*x+d)^4*A*a^3*b^3*d^2+15/e^5/(e*x+d)^4*A*a^2*b^4*d^3-15/2/e^6/(e*x+d)^4*A*a*b^5*d^4+3/e^3/
(e*x+d)^4*B*a^5*b*d-45/4/e^4/(e*x+d)^4*B*a^4*b^2*d^2+20/e^5/(e*x+d)^4*B*a^3*b^3*d^3+1/2*b^6*B*x^2/e^6-5*b^2/e^
3/(e*x+d)^3*A*a^4-5*b^6/e^7/(e*x+d)^3*A*d^4-2*b/e^3/(e*x+d)^3*B*a^5+7*b^6/e^8/(e*x+d)^3*B*d^5-20*b^3/e^5/(e*x+
d)*B*a^3+35*b^6/e^8/(e*x+d)*B*d^3-15*b^4/e^5/(e*x+d)*A*a^2-15*b^6/e^7/(e*x+d)*A*d^2+6*b^5/e^6*B*x*a-6*b^6/e^7*
B*x*d-75/4/e^6/(e*x+d)^4*B*a^2*b^4*d^4+6/5/e^2/(e*x+d)^5*A*d*a^5*b-3/e^3/(e*x+d)^5*A*d^2*a^4*b^2+4/e^4/(e*x+d)
^5*A*d^3*a^3*b^3-3/e^5/(e*x+d)^5*A*d^4*a^2*b^4+6/5/e^6/(e*x+d)^5*A*d^5*a*b^5-6/5/e^3/(e*x+d)^5*B*d^2*a^5*b+3/e
^4/(e*x+d)^5*B*d^3*a^4*b^2-4/e^5/(e*x+d)^5*B*d^4*a^3*b^3+3/e^6/(e*x+d)^5*B*d^5*a^2*b^4-6/5/e^7/(e*x+d)^5*B*a*b
^5*d^6-36*b^5/e^7*ln(e*x+d)*B*d*a+30*b^4/e^5/(e*x+d)^2*A*a^2*d-30*b^5/e^6/(e*x+d)^2*A*a*d^2+40*b^3/e^5/(e*x+d)
^2*B*a^3*d-75*b^4/e^6/(e*x+d)^2*B*a^2*d^2+60*b^5/e^7/(e*x+d)^2*B*a*d^3+9/e^7/(e*x+d)^4*B*a*b^5*d^5-35/2*b^6/e^
8/(e*x+d)^2*B*d^4-3/2/e^2/(e*x+d)^4*A*a^5*b+3/2/e^7/(e*x+d)^4*A*b^6*d^5-7/4/e^8/(e*x+d)^4*B*b^6*d^6-1/5/e^7/(e
*x+d)^5*A*b^6*d^6+1/5/e^2/(e*x+d)^5*B*d*a^6+1/5/e^8/(e*x+d)^5*B*b^6*d^7+30*b^5/e^6/(e*x+d)*A*d*a+50*b^4/e^6/(e
*x+d)^3*B*a^2*d^3-30*b^5/e^7/(e*x+d)^3*B*a*d^4-90*b^5/e^7/(e*x+d)*B*d^2*a+20*b^3/e^4/(e*x+d)^3*A*a^3*d+6*b^5/e
^6*ln(e*x+d)*A*a-6*b^6/e^7*ln(e*x+d)*A*d+15*b^4/e^6*ln(e*x+d)*B*a^2+21*b^6/e^8*ln(e*x+d)*B*d^2-10*b^3/e^4/(e*x
+d)^2*A*a^3+10*b^6/e^7/(e*x+d)^2*A*d^3-15/2*b^2/e^4/(e*x+d)^2*B*a^4-30*b^4/e^5/(e*x+d)^3*A*a^2*d^2+20*b^5/e^6/
(e*x+d)^3*A*a*d^3+15*b^2/e^4/(e*x+d)^3*B*a^4*d+75*b^4/e^6/(e*x+d)*B*d*a^2

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maxima [B]  time = 0.91, size = 814, normalized size = 2.99 \begin {gather*} \frac {459 \, B b^{6} d^{7} - 4 \, A a^{6} e^{7} - 174 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 137 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} - 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} - 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} - 2 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} - {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 100 \, {\left (7 \, B b^{6} d^{3} e^{4} - 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} - {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 50 \, {\left (49 \, B b^{6} d^{4} e^{3} - 20 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 18 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} - 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} - {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 10 \, {\left (329 \, B b^{6} d^{5} e^{2} - 130 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 110 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} - 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} - 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} - 2 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 5 \, {\left (399 \, B b^{6} d^{6} e - 154 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 125 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} - 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} - 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} - 2 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} - {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{20 \, {\left (e^{13} x^{5} + 5 \, d e^{12} x^{4} + 10 \, d^{2} e^{11} x^{3} + 10 \, d^{3} e^{10} x^{2} + 5 \, d^{4} e^{9} x + d^{5} e^{8}\right )}} + \frac {B b^{6} e x^{2} - 2 \, {\left (6 \, B b^{6} d - {\left (6 \, B a b^{5} + A b^{6}\right )} e\right )} x}{2 \, e^{7}} + \frac {3 \, {\left (7 \, B b^{6} d^{2} - 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{2}\right )} \log \left (e x + d\right )}{e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(459*B*b^6*d^7 - 4*A*a^6*e^7 - 174*(6*B*a*b^5 + A*b^6)*d^6*e + 137*(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 - 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 2*(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 + 100*(7*B*b^6*d^3*e^4 - 3*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 3*(5*B*a^2*b^4 + 2*A
*a*b^5)*d*e^6 - (4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 50*(49*B*b^6*d^4*e^3 - 20*(6*B*a*b^5 + A*b^6)*d^3*e^4 +
18*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 - 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 - (3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)
*x^3 + 10*(329*B*b^6*d^5*e^2 - 130*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 110*(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 - 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 - 2*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2
+ 5*(399*B*b^6*d^6*e - 154*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 125*(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 - 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 - 2*(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)/(e^13*x^5 + 5*d*e^12*x^4 + 10*d^2*e^11*x^3 + 10*d^3*e^10*x^2 + 5*d^4*e^9*x + d^5*e^8) +
1/2*(B*b^6*e*x^2 - 2*(6*B*b^6*d - (6*B*a*b^5 + A*b^6)*e)*x)/e^7 + 3*(7*B*b^6*d^2 - 2*(6*B*a*b^5 + A*b^6)*d*e
+ (5*B*a^2*b^4 + 2*A*a*b^5)*e^2)*log(e*x + d)/e^8

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mupad [B]  time = 0.24, size = 862, normalized size = 3.17 \begin {gather*} x\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^6}-\frac {6\,B\,b^6\,d}{e^7}\right )-\frac {x^3\,\left (\frac {15\,B\,a^4\,b^2\,e^6}{2}+40\,B\,a^3\,b^3\,d\,e^5+10\,A\,a^3\,b^3\,e^6-225\,B\,a^2\,b^4\,d^2\,e^4+30\,A\,a^2\,b^4\,d\,e^5+300\,B\,a\,b^5\,d^3\,e^3-90\,A\,a\,b^5\,d^2\,e^4-\frac {245\,B\,b^6\,d^4\,e^2}{2}+50\,A\,b^6\,d^3\,e^3\right )+\frac {B\,a^6\,d\,e^6+4\,A\,a^6\,e^7+4\,B\,a^5\,b\,d^2\,e^5+6\,A\,a^5\,b\,d\,e^6+15\,B\,a^4\,b^2\,d^3\,e^4+10\,A\,a^4\,b^2\,d^2\,e^5+80\,B\,a^3\,b^3\,d^4\,e^3+20\,A\,a^3\,b^3\,d^3\,e^4-685\,B\,a^2\,b^4\,d^5\,e^2+60\,A\,a^2\,b^4\,d^4\,e^3+1044\,B\,a\,b^5\,d^6\,e-274\,A\,a\,b^5\,d^5\,e^2-459\,B\,b^6\,d^7+174\,A\,b^6\,d^6\,e}{20\,e}+x\,\left (\frac {B\,a^6\,e^6}{4}+B\,a^5\,b\,d\,e^5+\frac {3\,A\,a^5\,b\,e^6}{2}+\frac {15\,B\,a^4\,b^2\,d^2\,e^4}{4}+\frac {5\,A\,a^4\,b^2\,d\,e^5}{2}+20\,B\,a^3\,b^3\,d^3\,e^3+5\,A\,a^3\,b^3\,d^2\,e^4-\frac {625\,B\,a^2\,b^4\,d^4\,e^2}{4}+15\,A\,a^2\,b^4\,d^3\,e^3+231\,B\,a\,b^5\,d^5\,e-\frac {125\,A\,a\,b^5\,d^4\,e^2}{2}-\frac {399\,B\,b^6\,d^6}{4}+\frac {77\,A\,b^6\,d^5\,e}{2}\right )+x^2\,\left (2\,B\,a^5\,b\,e^6+\frac {15\,B\,a^4\,b^2\,d\,e^5}{2}+5\,A\,a^4\,b^2\,e^6+40\,B\,a^3\,b^3\,d^2\,e^4+10\,A\,a^3\,b^3\,d\,e^5-275\,B\,a^2\,b^4\,d^3\,e^3+30\,A\,a^2\,b^4\,d^2\,e^4+390\,B\,a\,b^5\,d^4\,e^2-110\,A\,a\,b^5\,d^3\,e^3-\frac {329\,B\,b^6\,d^5\,e}{2}+65\,A\,b^6\,d^4\,e^2\right )+x^4\,\left (20\,B\,a^3\,b^3\,e^6-75\,B\,a^2\,b^4\,d\,e^5+15\,A\,a^2\,b^4\,e^6+90\,B\,a\,b^5\,d^2\,e^4-30\,A\,a\,b^5\,d\,e^5-35\,B\,b^6\,d^3\,e^3+15\,A\,b^6\,d^2\,e^4\right )}{d^5\,e^7+5\,d^4\,e^8\,x+10\,d^3\,e^9\,x^2+10\,d^2\,e^{10}\,x^3+5\,d\,e^{11}\,x^4+e^{12}\,x^5}+\frac {\ln \left (d+e\,x\right )\,\left (15\,B\,a^2\,b^4\,e^2-36\,B\,a\,b^5\,d\,e+6\,A\,a\,b^5\,e^2+21\,B\,b^6\,d^2-6\,A\,b^6\,d\,e\right )}{e^8}+\frac {B\,b^6\,x^2}{2\,e^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*((A*b^6 + 6*B*a*b^5)/e^6 - (6*B*b^6*d)/e^7) - (x^3*(10*A*a^3*b^3*e^6 + (15*B*a^4*b^2*e^6)/2 + 50*A*b^6*d^3*e
^3 - (245*B*b^6*d^4*e^2)/2 - 90*A*a*b^5*d^2*e^4 + 30*A*a^2*b^4*d*e^5 + 300*B*a*b^5*d^3*e^3 + 40*B*a^3*b^3*d*e^
5 - 225*B*a^2*b^4*d^2*e^4) + (4*A*a^6*e^7 - 459*B*b^6*d^7 + 174*A*b^6*d^6*e + B*a^6*d*e^6 - 274*A*a*b^5*d^5*e^
2 + 4*B*a^5*b*d^2*e^5 + 60*A*a^2*b^4*d^4*e^3 + 20*A*a^3*b^3*d^3*e^4 + 10*A*a^4*b^2*d^2*e^5 - 685*B*a^2*b^4*d^5
*e^2 + 80*B*a^3*b^3*d^4*e^3 + 15*B*a^4*b^2*d^3*e^4 + 6*A*a^5*b*d*e^6 + 1044*B*a*b^5*d^6*e)/(20*e) + x*((B*a^6*
e^6)/4 - (399*B*b^6*d^6)/4 + (3*A*a^5*b*e^6)/2 + (77*A*b^6*d^5*e)/2 - (125*A*a*b^5*d^4*e^2)/2 + (5*A*a^4*b^2*d
*e^5)/2 + 15*A*a^2*b^4*d^3*e^3 + 5*A*a^3*b^3*d^2*e^4 - (625*B*a^2*b^4*d^4*e^2)/4 + 20*B*a^3*b^3*d^3*e^3 + (15*
B*a^4*b^2*d^2*e^4)/4 + 231*B*a*b^5*d^5*e + B*a^5*b*d*e^5) + x^2*(2*B*a^5*b*e^6 - (329*B*b^6*d^5*e)/2 + 5*A*a^4
*b^2*e^6 + 65*A*b^6*d^4*e^2 - 110*A*a*b^5*d^3*e^3 + 10*A*a^3*b^3*d*e^5 + 390*B*a*b^5*d^4*e^2 + (15*B*a^4*b^2*d
*e^5)/2 + 30*A*a^2*b^4*d^2*e^4 - 275*B*a^2*b^4*d^3*e^3 + 40*B*a^3*b^3*d^2*e^4) + x^4*(15*A*a^2*b^4*e^6 + 20*B*
a^3*b^3*e^6 + 15*A*b^6*d^2*e^4 - 35*B*b^6*d^3*e^3 + 90*B*a*b^5*d^2*e^4 - 75*B*a^2*b^4*d*e^5 - 30*A*a*b^5*d*e^5
))/(d^5*e^7 + e^12*x^5 + 5*d^4*e^8*x + 5*d*e^11*x^4 + 10*d^3*e^9*x^2 + 10*d^2*e^10*x^3) + (log(d + e*x)*(21*B*
b^6*d^2 - 6*A*b^6*d*e + 6*A*a*b^5*e^2 + 15*B*a^2*b^4*e^2 - 36*B*a*b^5*d*e))/e^8 + (B*b^6*x^2)/(2*e^6)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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