3.11.66 \(\int \frac {(a+b x)^6 (A+B x)}{(d+e x)^7} \, dx\) [1066]

Optimal. Leaf size=278 \[ \frac {b^6 B x}{e^7}+\frac {(b d-a e)^6 (B d-A e)}{6 e^8 (d+e x)^6}-\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e)}{5 e^8 (d+e x)^5}+\frac {3 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{4 e^8 (d+e x)^4}-\frac {5 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e)}{3 e^8 (d+e x)^3}+\frac {5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e)}{2 e^8 (d+e x)^2}-\frac {3 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e)}{e^8 (d+e x)}-\frac {b^5 (7 b B d-A b e-6 a B e) \log (d+e x)}{e^8} \]

[Out]

b^6*B*x/e^7+1/6*(-a*e+b*d)^6*(-A*e+B*d)/e^8/(e*x+d)^6-1/5*(-a*e+b*d)^5*(-6*A*b*e-B*a*e+7*B*b*d)/e^8/(e*x+d)^5+
3/4*b*(-a*e+b*d)^4*(-5*A*b*e-2*B*a*e+7*B*b*d)/e^8/(e*x+d)^4-5/3*b^2*(-a*e+b*d)^3*(-4*A*b*e-3*B*a*e+7*B*b*d)/e^
8/(e*x+d)^3+5/2*b^3*(-a*e+b*d)^2*(-3*A*b*e-4*B*a*e+7*B*b*d)/e^8/(e*x+d)^2-3*b^4*(-a*e+b*d)*(-2*A*b*e-5*B*a*e+7
*B*b*d)/e^8/(e*x+d)-b^5*(-A*b*e-6*B*a*e+7*B*b*d)*ln(e*x+d)/e^8

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Rubi [A]
time = 0.22, antiderivative size = 278, 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 = {78} \begin {gather*} -\frac {b^5 \log (d+e x) (-6 a B e-A b e+7 b B d)}{e^8}-\frac {3 b^4 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{e^8 (d+e x)}+\frac {5 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)^2}-\frac {5 b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{3 e^8 (d+e x)^3}+\frac {3 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{4 e^8 (d+e x)^4}-\frac {(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{5 e^8 (d+e x)^5}+\frac {(b d-a e)^6 (B d-A e)}{6 e^8 (d+e x)^6}+\frac {b^6 B x}{e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^7,x]

[Out]

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

Rule 78

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(619\) vs. \(2(278)=556\).
time = 0.23, size = 619, normalized size = 2.23 \begin {gather*} -\frac {2 a^6 e^6 (5 A e+B (d+6 e x))+6 a^5 b e^5 \left (2 A e (d+6 e x)+B \left (d^2+6 d e x+15 e^2 x^2\right )\right )+15 a^4 b^2 e^4 \left (A e \left (d^2+6 d e x+15 e^2 x^2\right )+B \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )+20 a^3 b^3 e^3 \left (A e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 B \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )+30 a^2 b^4 e^2 \left (A e \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+5 B \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )-6 a b^5 e \left (-10 A e \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )+B d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )\right )-b^6 \left (A d e \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )-B \left (669 d^7+3594 d^6 e x+7725 d^5 e^2 x^2+8200 d^4 e^3 x^3+4050 d^3 e^4 x^4+360 d^2 e^5 x^5-360 d e^6 x^6-60 e^7 x^7\right )\right )+60 b^5 (7 b B d-A b e-6 a B e) (d+e x)^6 \log (d+e x)}{60 e^8 (d+e x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^7,x]

[Out]

-1/60*(2*a^6*e^6*(5*A*e + B*(d + 6*e*x)) + 6*a^5*b*e^5*(2*A*e*(d + 6*e*x) + B*(d^2 + 6*d*e*x + 15*e^2*x^2)) +
15*a^4*b^2*e^4*(A*e*(d^2 + 6*d*e*x + 15*e^2*x^2) + B*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3)) + 20*a^3*b
^3*e^3*(A*e*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 2*B*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x
^3 + 15*e^4*x^4)) + 30*a^2*b^4*e^2*(A*e*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) + 5*B*(
d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5)) - 6*a*b^5*e*(-10*A*e*(d^5 + 6*d
^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5) + B*d*(147*d^5 + 822*d^4*e*x + 1875*d^3*e
^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5)) - b^6*(A*d*e*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*
x^2 + 2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5) - B*(669*d^7 + 3594*d^6*e*x + 7725*d^5*e^2*x^2 + 8200*d
^4*e^3*x^3 + 4050*d^3*e^4*x^4 + 360*d^2*e^5*x^5 - 360*d*e^6*x^6 - 60*e^7*x^7)) + 60*b^5*(7*b*B*d - A*b*e - 6*a
*B*e)*(d + e*x)^6*Log[d + e*x])/(e^8*(d + e*x)^6)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(804\) vs. \(2(268)=536\).
time = 0.08, size = 805, normalized size = 2.90 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)^7,x,method=_RETURNVERBOSE)

[Out]

b^6*B*x/e^7-5/2*b^3/e^8*(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*x+d)^2-3/4*b/e^8*(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*x+d)^4-3*b^4/e^8*(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*x+d)-1/6*(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)^6-1/5/e^8*(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*x+d)^5-5/3*b^2/e^8*(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*x+d)^
3+b^5/e^8*(A*b*e+6*B*a*e-7*B*b*d)*ln(e*x+d)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 830 vs. \(2 (286) = 572\).
time = 0.32, size = 830, normalized size = 2.99 \begin {gather*} B b^{6} x e^{\left (-7\right )} - {\left (7 \, B b^{6} d - 6 \, B a b^{5} e - A b^{6} e\right )} e^{\left (-8\right )} \log \left (x e + d\right ) - \frac {669 \, B b^{6} d^{7} + 10 \, A a^{6} e^{7} - 147 \, {\left (6 \, B a b^{5} e + A b^{6} e\right )} d^{6} + 30 \, {\left (5 \, B a^{2} b^{4} e^{2} + 2 \, A a b^{5} e^{2}\right )} d^{5} + 180 \, {\left (7 \, B b^{6} d^{2} e^{5} + 5 \, B a^{2} b^{4} e^{7} + 2 \, A a b^{5} e^{7} - 2 \, {\left (6 \, B a b^{5} e^{6} + A b^{6} e^{6}\right )} d\right )} x^{5} + 10 \, {\left (4 \, B a^{3} b^{3} e^{3} + 3 \, A a^{2} b^{4} e^{3}\right )} d^{4} + 150 \, {\left (35 \, B b^{6} d^{3} e^{4} + 4 \, B a^{3} b^{3} e^{7} + 3 \, A a^{2} b^{4} e^{7} - 9 \, {\left (6 \, B a b^{5} e^{5} + A b^{6} e^{5}\right )} d^{2} + 3 \, {\left (5 \, B a^{2} b^{4} e^{6} + 2 \, A a b^{5} e^{6}\right )} d\right )} x^{4} + 5 \, {\left (3 \, B a^{4} b^{2} e^{4} + 4 \, A a^{3} b^{3} e^{4}\right )} d^{3} + 100 \, {\left (91 \, B b^{6} d^{4} e^{3} + 3 \, B a^{4} b^{2} e^{7} + 4 \, A a^{3} b^{3} e^{7} - 22 \, {\left (6 \, B a b^{5} e^{4} + A b^{6} e^{4}\right )} d^{3} + 6 \, {\left (5 \, B a^{2} b^{4} e^{5} + 2 \, A a b^{5} e^{5}\right )} d^{2} + 2 \, {\left (4 \, B a^{3} b^{3} e^{6} + 3 \, A a^{2} b^{4} e^{6}\right )} d\right )} x^{3} + 3 \, {\left (2 \, B a^{5} b e^{5} + 5 \, A a^{4} b^{2} e^{5}\right )} d^{2} + 15 \, {\left (539 \, B b^{6} d^{5} e^{2} + 6 \, B a^{5} b e^{7} + 15 \, A a^{4} b^{2} e^{7} - 125 \, {\left (6 \, B a b^{5} e^{3} + A b^{6} e^{3}\right )} d^{4} + 30 \, {\left (5 \, B a^{2} b^{4} e^{4} + 2 \, A a b^{5} e^{4}\right )} d^{3} + 10 \, {\left (4 \, B a^{3} b^{3} e^{5} + 3 \, A a^{2} b^{4} e^{5}\right )} d^{2} + 5 \, {\left (3 \, B a^{4} b^{2} e^{6} + 4 \, A a^{3} b^{3} e^{6}\right )} d\right )} x^{2} + 2 \, {\left (B a^{6} e^{6} + 6 \, A a^{5} b e^{6}\right )} d + 6 \, {\left (609 \, B b^{6} d^{6} e + 2 \, B a^{6} e^{7} + 12 \, A a^{5} b e^{7} - 137 \, {\left (6 \, B a b^{5} e^{2} + A b^{6} e^{2}\right )} d^{5} + 30 \, {\left (5 \, B a^{2} b^{4} e^{3} + 2 \, A a b^{5} e^{3}\right )} d^{4} + 10 \, {\left (4 \, B a^{3} b^{3} e^{4} + 3 \, A a^{2} b^{4} e^{4}\right )} d^{3} + 5 \, {\left (3 \, B a^{4} b^{2} e^{5} + 4 \, A a^{3} b^{3} e^{5}\right )} d^{2} + 3 \, {\left (2 \, B a^{5} b e^{6} + 5 \, A a^{4} b^{2} e^{6}\right )} d\right )} x}{60 \, {\left (x^{6} e^{14} + 6 \, d x^{5} e^{13} + 15 \, d^{2} x^{4} e^{12} + 20 \, d^{3} x^{3} e^{11} + 15 \, d^{4} x^{2} e^{10} + 6 \, d^{5} x e^{9} + d^{6} e^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^7,x, algorithm="maxima")

[Out]

B*b^6*x*e^(-7) - (7*B*b^6*d - 6*B*a*b^5*e - A*b^6*e)*e^(-8)*log(x*e + d) - 1/60*(669*B*b^6*d^7 + 10*A*a^6*e^7
- 147*(6*B*a*b^5*e + A*b^6*e)*d^6 + 30*(5*B*a^2*b^4*e^2 + 2*A*a*b^5*e^2)*d^5 + 180*(7*B*b^6*d^2*e^5 + 5*B*a^2*
b^4*e^7 + 2*A*a*b^5*e^7 - 2*(6*B*a*b^5*e^6 + A*b^6*e^6)*d)*x^5 + 10*(4*B*a^3*b^3*e^3 + 3*A*a^2*b^4*e^3)*d^4 +
150*(35*B*b^6*d^3*e^4 + 4*B*a^3*b^3*e^7 + 3*A*a^2*b^4*e^7 - 9*(6*B*a*b^5*e^5 + A*b^6*e^5)*d^2 + 3*(5*B*a^2*b^4
*e^6 + 2*A*a*b^5*e^6)*d)*x^4 + 5*(3*B*a^4*b^2*e^4 + 4*A*a^3*b^3*e^4)*d^3 + 100*(91*B*b^6*d^4*e^3 + 3*B*a^4*b^2
*e^7 + 4*A*a^3*b^3*e^7 - 22*(6*B*a*b^5*e^4 + A*b^6*e^4)*d^3 + 6*(5*B*a^2*b^4*e^5 + 2*A*a*b^5*e^5)*d^2 + 2*(4*B
*a^3*b^3*e^6 + 3*A*a^2*b^4*e^6)*d)*x^3 + 3*(2*B*a^5*b*e^5 + 5*A*a^4*b^2*e^5)*d^2 + 15*(539*B*b^6*d^5*e^2 + 6*B
*a^5*b*e^7 + 15*A*a^4*b^2*e^7 - 125*(6*B*a*b^5*e^3 + A*b^6*e^3)*d^4 + 30*(5*B*a^2*b^4*e^4 + 2*A*a*b^5*e^4)*d^3
+ 10*(4*B*a^3*b^3*e^5 + 3*A*a^2*b^4*e^5)*d^2 + 5*(3*B*a^4*b^2*e^6 + 4*A*a^3*b^3*e^6)*d)*x^2 + 2*(B*a^6*e^6 +
6*A*a^5*b*e^6)*d + 6*(609*B*b^6*d^6*e + 2*B*a^6*e^7 + 12*A*a^5*b*e^7 - 137*(6*B*a*b^5*e^2 + A*b^6*e^2)*d^5 + 3
0*(5*B*a^2*b^4*e^3 + 2*A*a*b^5*e^3)*d^4 + 10*(4*B*a^3*b^3*e^4 + 3*A*a^2*b^4*e^4)*d^3 + 5*(3*B*a^4*b^2*e^5 + 4*
A*a^3*b^3*e^5)*d^2 + 3*(2*B*a^5*b*e^6 + 5*A*a^4*b^2*e^6)*d)*x)/(x^6*e^14 + 6*d*x^5*e^13 + 15*d^2*x^4*e^12 + 20
*d^3*x^3*e^11 + 15*d^4*x^2*e^10 + 6*d^5*x*e^9 + d^6*e^8)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1022 vs. \(2 (286) = 572\).
time = 1.00, size = 1022, normalized size = 3.68 \begin {gather*} -\frac {669 \, B b^{6} d^{7} - {\left (60 \, B b^{6} x^{7} - 10 \, A a^{6} - 180 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} - 150 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} - 100 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 45 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 12 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )} e^{7} - {\left (360 \, B b^{6} d x^{6} + 360 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d x^{5} - 450 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d x^{4} - 200 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d x^{3} - 75 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d x^{2} - 18 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d x - 2 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d\right )} e^{6} + 3 \, {\left (120 \, B b^{6} d^{2} x^{5} - 450 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} x^{4} + 200 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} x^{3} + 50 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} x^{2} + 10 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} x + {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2}\right )} e^{5} + 5 \, {\left (810 \, B b^{6} d^{3} x^{4} - 440 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} x^{3} + 90 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} x^{2} + 12 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} x + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3}\right )} e^{4} + 5 \, {\left (1640 \, B b^{6} d^{4} x^{3} - 375 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} x^{2} + 36 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} x + 2 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4}\right )} e^{3} + 3 \, {\left (2575 \, B b^{6} d^{5} x^{2} - 274 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} x + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5}\right )} e^{2} + 3 \, {\left (1198 \, B b^{6} d^{6} x - 49 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6}\right )} e + 60 \, {\left (7 \, B b^{6} d^{7} - {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} e^{7} + {\left (7 \, B b^{6} d x^{6} - 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d x^{5}\right )} e^{6} + 3 \, {\left (14 \, B b^{6} d^{2} x^{5} - 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} x^{4}\right )} e^{5} + 5 \, {\left (21 \, B b^{6} d^{3} x^{4} - 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} x^{3}\right )} e^{4} + 5 \, {\left (28 \, B b^{6} d^{4} x^{3} - 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} x^{2}\right )} e^{3} + 3 \, {\left (35 \, B b^{6} d^{5} x^{2} - 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} x\right )} e^{2} + {\left (42 \, B b^{6} d^{6} x - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6}\right )} e\right )} \log \left (x e + d\right )}{60 \, {\left (x^{6} e^{14} + 6 \, d x^{5} e^{13} + 15 \, d^{2} x^{4} e^{12} + 20 \, d^{3} x^{3} e^{11} + 15 \, d^{4} x^{2} e^{10} + 6 \, d^{5} x e^{9} + d^{6} e^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^7,x, algorithm="fricas")

[Out]

-1/60*(669*B*b^6*d^7 - (60*B*b^6*x^7 - 10*A*a^6 - 180*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 - 150*(4*B*a^3*b^3 + 3*A*a
^2*b^4)*x^4 - 100*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 - 45*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 - 12*(B*a^6 + 6*A*a^5*b)*
x)*e^7 - (360*B*b^6*d*x^6 + 360*(6*B*a*b^5 + A*b^6)*d*x^5 - 450*(5*B*a^2*b^4 + 2*A*a*b^5)*d*x^4 - 200*(4*B*a^3
*b^3 + 3*A*a^2*b^4)*d*x^3 - 75*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*x^2 - 18*(2*B*a^5*b + 5*A*a^4*b^2)*d*x - 2*(B*a^6
+ 6*A*a^5*b)*d)*e^6 + 3*(120*B*b^6*d^2*x^5 - 450*(6*B*a*b^5 + A*b^6)*d^2*x^4 + 200*(5*B*a^2*b^4 + 2*A*a*b^5)*
d^2*x^3 + 50*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*x^2 + 10*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*x + (2*B*a^5*b + 5*A*a^4
*b^2)*d^2)*e^5 + 5*(810*B*b^6*d^3*x^4 - 440*(6*B*a*b^5 + A*b^6)*d^3*x^3 + 90*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*x^2
+ 12*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*x + (3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3)*e^4 + 5*(1640*B*b^6*d^4*x^3 - 375*(
6*B*a*b^5 + A*b^6)*d^4*x^2 + 36*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*x + 2*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4)*e^3 + 3*(
2575*B*b^6*d^5*x^2 - 274*(6*B*a*b^5 + A*b^6)*d^5*x + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5)*e^2 + 3*(1198*B*b^6*d^6
*x - 49*(6*B*a*b^5 + A*b^6)*d^6)*e + 60*(7*B*b^6*d^7 - (6*B*a*b^5 + A*b^6)*x^6*e^7 + (7*B*b^6*d*x^6 - 6*(6*B*a
*b^5 + A*b^6)*d*x^5)*e^6 + 3*(14*B*b^6*d^2*x^5 - 5*(6*B*a*b^5 + A*b^6)*d^2*x^4)*e^5 + 5*(21*B*b^6*d^3*x^4 - 4*
(6*B*a*b^5 + A*b^6)*d^3*x^3)*e^4 + 5*(28*B*b^6*d^4*x^3 - 3*(6*B*a*b^5 + A*b^6)*d^4*x^2)*e^3 + 3*(35*B*b^6*d^5*
x^2 - 2*(6*B*a*b^5 + A*b^6)*d^5*x)*e^2 + (42*B*b^6*d^6*x - (6*B*a*b^5 + A*b^6)*d^6)*e)*log(x*e + d))/(x^6*e^14
+ 6*d*x^5*e^13 + 15*d^2*x^4*e^12 + 20*d^3*x^3*e^11 + 15*d^4*x^2*e^10 + 6*d^5*x*e^9 + d^6*e^8)

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Sympy [F(-1)] Timed out
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)**7,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 775 vs. \(2 (286) = 572\).
time = 2.31, size = 775, normalized size = 2.79 \begin {gather*} B b^{6} x e^{\left (-7\right )} - {\left (7 \, B b^{6} d - 6 \, B a b^{5} e - A b^{6} e\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (669 \, B b^{6} d^{7} - 882 \, B a b^{5} d^{6} e - 147 \, A b^{6} d^{6} e + 150 \, B a^{2} b^{4} d^{5} e^{2} + 60 \, A a b^{5} d^{5} e^{2} + 40 \, B a^{3} b^{3} d^{4} e^{3} + 30 \, 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} + 6 \, B a^{5} b d^{2} e^{5} + 15 \, A a^{4} b^{2} d^{2} e^{5} + 2 \, B a^{6} d e^{6} + 12 \, A a^{5} b d e^{6} + 10 \, A a^{6} e^{7} + 180 \, {\left (7 \, B b^{6} d^{2} e^{5} - 12 \, B a b^{5} d e^{6} - 2 \, A b^{6} d e^{6} + 5 \, B a^{2} b^{4} e^{7} + 2 \, A a b^{5} e^{7}\right )} x^{5} + 150 \, {\left (35 \, B b^{6} d^{3} e^{4} - 54 \, B a b^{5} d^{2} e^{5} - 9 \, 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} + 100 \, {\left (91 \, B b^{6} d^{4} e^{3} - 132 \, B a b^{5} d^{3} e^{4} - 22 \, A b^{6} d^{3} e^{4} + 30 \, B a^{2} b^{4} d^{2} e^{5} + 12 \, A a b^{5} d^{2} e^{5} + 8 \, B a^{3} b^{3} d e^{6} + 6 \, 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} + 15 \, {\left (539 \, B b^{6} d^{5} e^{2} - 750 \, B a b^{5} d^{4} e^{3} - 125 \, A b^{6} d^{4} e^{3} + 150 \, B a^{2} b^{4} d^{3} e^{4} + 60 \, A a b^{5} d^{3} e^{4} + 40 \, B a^{3} b^{3} d^{2} e^{5} + 30 \, 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} + 6 \, B a^{5} b e^{7} + 15 \, A a^{4} b^{2} e^{7}\right )} x^{2} + 6 \, {\left (609 \, B b^{6} d^{6} e - 822 \, B a b^{5} d^{5} e^{2} - 137 \, A b^{6} d^{5} e^{2} + 150 \, B a^{2} b^{4} d^{4} e^{3} + 60 \, A a b^{5} d^{4} e^{3} + 40 \, B a^{3} b^{3} d^{3} e^{4} + 30 \, 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} + 6 \, B a^{5} b d e^{6} + 15 \, A a^{4} b^{2} d e^{6} + 2 \, B a^{6} e^{7} + 12 \, A a^{5} b e^{7}\right )} x\right )} e^{\left (-8\right )}}{60 \, {\left (x e + d\right )}^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^7,x, algorithm="giac")

[Out]

B*b^6*x*e^(-7) - (7*B*b^6*d - 6*B*a*b^5*e - A*b^6*e)*e^(-8)*log(abs(x*e + d)) - 1/60*(669*B*b^6*d^7 - 882*B*a*
b^5*d^6*e - 147*A*b^6*d^6*e + 150*B*a^2*b^4*d^5*e^2 + 60*A*a*b^5*d^5*e^2 + 40*B*a^3*b^3*d^4*e^3 + 30*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 + 6*B*a^5*b*d^2*e^5 + 15*A*a^4*b^2*d^2*e^5 + 2*B*a^6*d*
e^6 + 12*A*a^5*b*d*e^6 + 10*A*a^6*e^7 + 180*(7*B*b^6*d^2*e^5 - 12*B*a*b^5*d*e^6 - 2*A*b^6*d*e^6 + 5*B*a^2*b^4*
e^7 + 2*A*a*b^5*e^7)*x^5 + 150*(35*B*b^6*d^3*e^4 - 54*B*a*b^5*d^2*e^5 - 9*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 + 100*(91*B*b^6*d^4*e^3 - 132*B*a*b^5*d^3*e^4 - 22*A
*b^6*d^3*e^4 + 30*B*a^2*b^4*d^2*e^5 + 12*A*a*b^5*d^2*e^5 + 8*B*a^3*b^3*d*e^6 + 6*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 + 15*(539*B*b^6*d^5*e^2 - 750*B*a*b^5*d^4*e^3 - 125*A*b^6*d^4*e^3 + 150*B*a^2*b^4*
d^3*e^4 + 60*A*a*b^5*d^3*e^4 + 40*B*a^3*b^3*d^2*e^5 + 30*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 + 6*B*a^5*b*e^7 + 15*A*a^4*b^2*e^7)*x^2 + 6*(609*B*b^6*d^6*e - 822*B*a*b^5*d^5*e^2 - 137*A*b^6*d^5*e^2
+ 150*B*a^2*b^4*d^4*e^3 + 60*A*a*b^5*d^4*e^3 + 40*B*a^3*b^3*d^3*e^4 + 30*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 + 6*B*a^5*b*d*e^6 + 15*A*a^4*b^2*d*e^6 + 2*B*a^6*e^7 + 12*A*a^5*b*e^7)*x)*e^(-8)/(x
*e + d)^6

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Mupad [B]
time = 1.35, size = 875, normalized size = 3.15 \begin {gather*} \frac {\ln \left (d+e\,x\right )\,\left (A\,b^6\,e-7\,B\,b^6\,d+6\,B\,a\,b^5\,e\right )}{e^8}-\frac {x^3\,\left (5\,B\,a^4\,b^2\,e^6+\frac {40\,B\,a^3\,b^3\,d\,e^5}{3}+\frac {20\,A\,a^3\,b^3\,e^6}{3}+50\,B\,a^2\,b^4\,d^2\,e^4+10\,A\,a^2\,b^4\,d\,e^5-220\,B\,a\,b^5\,d^3\,e^3+20\,A\,a\,b^5\,d^2\,e^4+\frac {455\,B\,b^6\,d^4\,e^2}{3}-\frac {110\,A\,b^6\,d^3\,e^3}{3}\right )+\frac {2\,B\,a^6\,d\,e^6+10\,A\,a^6\,e^7+6\,B\,a^5\,b\,d^2\,e^5+12\,A\,a^5\,b\,d\,e^6+15\,B\,a^4\,b^2\,d^3\,e^4+15\,A\,a^4\,b^2\,d^2\,e^5+40\,B\,a^3\,b^3\,d^4\,e^3+20\,A\,a^3\,b^3\,d^3\,e^4+150\,B\,a^2\,b^4\,d^5\,e^2+30\,A\,a^2\,b^4\,d^4\,e^3-882\,B\,a\,b^5\,d^6\,e+60\,A\,a\,b^5\,d^5\,e^2+669\,B\,b^6\,d^7-147\,A\,b^6\,d^6\,e}{60\,e}+x\,\left (\frac {B\,a^6\,e^6}{5}+\frac {3\,B\,a^5\,b\,d\,e^5}{5}+\frac {6\,A\,a^5\,b\,e^6}{5}+\frac {3\,B\,a^4\,b^2\,d^2\,e^4}{2}+\frac {3\,A\,a^4\,b^2\,d\,e^5}{2}+4\,B\,a^3\,b^3\,d^3\,e^3+2\,A\,a^3\,b^3\,d^2\,e^4+15\,B\,a^2\,b^4\,d^4\,e^2+3\,A\,a^2\,b^4\,d^3\,e^3-\frac {411\,B\,a\,b^5\,d^5\,e}{5}+6\,A\,a\,b^5\,d^4\,e^2+\frac {609\,B\,b^6\,d^6}{10}-\frac {137\,A\,b^6\,d^5\,e}{10}\right )+x^5\,\left (15\,B\,a^2\,b^4\,e^6-36\,B\,a\,b^5\,d\,e^5+6\,A\,a\,b^5\,e^6+21\,B\,b^6\,d^2\,e^4-6\,A\,b^6\,d\,e^5\right )+x^2\,\left (\frac {3\,B\,a^5\,b\,e^6}{2}+\frac {15\,B\,a^4\,b^2\,d\,e^5}{4}+\frac {15\,A\,a^4\,b^2\,e^6}{4}+10\,B\,a^3\,b^3\,d^2\,e^4+5\,A\,a^3\,b^3\,d\,e^5+\frac {75\,B\,a^2\,b^4\,d^3\,e^3}{2}+\frac {15\,A\,a^2\,b^4\,d^2\,e^4}{2}-\frac {375\,B\,a\,b^5\,d^4\,e^2}{2}+15\,A\,a\,b^5\,d^3\,e^3+\frac {539\,B\,b^6\,d^5\,e}{4}-\frac {125\,A\,b^6\,d^4\,e^2}{4}\right )+x^4\,\left (10\,B\,a^3\,b^3\,e^6+\frac {75\,B\,a^2\,b^4\,d\,e^5}{2}+\frac {15\,A\,a^2\,b^4\,e^6}{2}-135\,B\,a\,b^5\,d^2\,e^4+15\,A\,a\,b^5\,d\,e^5+\frac {175\,B\,b^6\,d^3\,e^3}{2}-\frac {45\,A\,b^6\,d^2\,e^4}{2}\right )}{d^6\,e^7+6\,d^5\,e^8\,x+15\,d^4\,e^9\,x^2+20\,d^3\,e^{10}\,x^3+15\,d^2\,e^{11}\,x^4+6\,d\,e^{12}\,x^5+e^{13}\,x^6}+\frac {B\,b^6\,x}{e^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^6)/(d + e*x)^7,x)

[Out]

(log(d + e*x)*(A*b^6*e - 7*B*b^6*d + 6*B*a*b^5*e))/e^8 - (x^3*((20*A*a^3*b^3*e^6)/3 + 5*B*a^4*b^2*e^6 - (110*A
*b^6*d^3*e^3)/3 + (455*B*b^6*d^4*e^2)/3 + 20*A*a*b^5*d^2*e^4 + 10*A*a^2*b^4*d*e^5 - 220*B*a*b^5*d^3*e^3 + (40*
B*a^3*b^3*d*e^5)/3 + 50*B*a^2*b^4*d^2*e^4) + (10*A*a^6*e^7 + 669*B*b^6*d^7 - 147*A*b^6*d^6*e + 2*B*a^6*d*e^6 +
60*A*a*b^5*d^5*e^2 + 6*B*a^5*b*d^2*e^5 + 30*A*a^2*b^4*d^4*e^3 + 20*A*a^3*b^3*d^3*e^4 + 15*A*a^4*b^2*d^2*e^5 +
150*B*a^2*b^4*d^5*e^2 + 40*B*a^3*b^3*d^4*e^3 + 15*B*a^4*b^2*d^3*e^4 + 12*A*a^5*b*d*e^6 - 882*B*a*b^5*d^6*e)/(
60*e) + x*((B*a^6*e^6)/5 + (609*B*b^6*d^6)/10 + (6*A*a^5*b*e^6)/5 - (137*A*b^6*d^5*e)/10 + 6*A*a*b^5*d^4*e^2 +
(3*A*a^4*b^2*d*e^5)/2 + 3*A*a^2*b^4*d^3*e^3 + 2*A*a^3*b^3*d^2*e^4 + 15*B*a^2*b^4*d^4*e^2 + 4*B*a^3*b^3*d^3*e^
3 + (3*B*a^4*b^2*d^2*e^4)/2 - (411*B*a*b^5*d^5*e)/5 + (3*B*a^5*b*d*e^5)/5) + x^5*(6*A*a*b^5*e^6 - 6*A*b^6*d*e^
5 + 15*B*a^2*b^4*e^6 + 21*B*b^6*d^2*e^4 - 36*B*a*b^5*d*e^5) + x^2*((3*B*a^5*b*e^6)/2 + (539*B*b^6*d^5*e)/4 + (
15*A*a^4*b^2*e^6)/4 - (125*A*b^6*d^4*e^2)/4 + 15*A*a*b^5*d^3*e^3 + 5*A*a^3*b^3*d*e^5 - (375*B*a*b^5*d^4*e^2)/2
+ (15*B*a^4*b^2*d*e^5)/4 + (15*A*a^2*b^4*d^2*e^4)/2 + (75*B*a^2*b^4*d^3*e^3)/2 + 10*B*a^3*b^3*d^2*e^4) + x^4*
((15*A*a^2*b^4*e^6)/2 + 10*B*a^3*b^3*e^6 - (45*A*b^6*d^2*e^4)/2 + (175*B*b^6*d^3*e^3)/2 - 135*B*a*b^5*d^2*e^4
+ (75*B*a^2*b^4*d*e^5)/2 + 15*A*a*b^5*d*e^5))/(d^6*e^7 + e^13*x^6 + 6*d^5*e^8*x + 6*d*e^12*x^5 + 15*d^4*e^9*x^
2 + 20*d^3*e^10*x^3 + 15*d^2*e^11*x^4) + (B*b^6*x)/e^7

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