3.11.63 \(\int \frac {(a+b x)^6 (A+B x)}{(d+e x)^4} \, dx\) [1063]

Optimal. Leaf size=279 \[ -\frac {5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e) x}{e^7}+\frac {(b d-a e)^6 (B d-A e)}{3 e^8 (d+e x)^3}-\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e)}{2 e^8 (d+e x)^2}+\frac {3 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{e^8 (d+e x)}+\frac {3 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) (d+e x)^2}{2 e^8}-\frac {b^5 (7 b B d-A b e-6 a B e) (d+e x)^3}{3 e^8}+\frac {b^6 B (d+e x)^4}{4 e^8}+\frac {5 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e) \log (d+e x)}{e^8} \]

[Out]

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

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Rubi [A]
time = 0.30, antiderivative size = 279, 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 (d+e x)^3 (-6 a B e-A b e+7 b B d)}{3 e^8}+\frac {3 b^4 (d+e x)^2 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{2 e^8}-\frac {5 b^3 x (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{e^7}+\frac {5 b^2 (b d-a e)^3 \log (d+e x) (-3 a B e-4 A b e+7 b B d)}{e^8}+\frac {3 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8 (d+e x)}-\frac {(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{2 e^8 (d+e x)^2}+\frac {(b d-a e)^6 (B d-A e)}{3 e^8 (d+e x)^3}+\frac {b^6 B (d+e x)^4}{4 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^4,x]

[Out]

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

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Mathematica [A]
time = 0.13, size = 297, normalized size = 1.06 \begin {gather*} \frac {12 b^3 e \left (20 a^3 B e^3+12 a b^2 d e (5 B d-2 A e)+15 a^2 b e^2 (-4 B d+A e)+10 b^3 d^2 (-2 B d+A e)\right ) x-6 b^4 e^2 \left (-15 a^2 B e^2-6 a b e (-4 B d+A e)+2 b^2 d (-5 B d+2 A e)\right ) x^2+4 b^5 e^3 (-4 b B d+A b e+6 a B e) x^3+3 b^6 B e^4 x^4+\frac {4 (b d-a e)^6 (B d-A e)}{(d+e x)^3}-\frac {6 (b d-a e)^5 (7 b B d-6 A b e-a B e)}{(d+e x)^2}+\frac {36 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{d+e x}+60 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e) \log (d+e x)}{12 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^4,x]

[Out]

(12*b^3*e*(20*a^3*B*e^3 + 12*a*b^2*d*e*(5*B*d - 2*A*e) + 15*a^2*b*e^2*(-4*B*d + A*e) + 10*b^3*d^2*(-2*B*d + A*
e))*x - 6*b^4*e^2*(-15*a^2*B*e^2 - 6*a*b*e*(-4*B*d + A*e) + 2*b^2*d*(-5*B*d + 2*A*e))*x^2 + 4*b^5*e^3*(-4*b*B*
d + A*b*e + 6*a*B*e)*x^3 + 3*b^6*B*e^4*x^4 + (4*(b*d - a*e)^6*(B*d - A*e))/(d + e*x)^3 - (6*(b*d - a*e)^5*(7*b
*B*d - 6*A*b*e - a*B*e))/(d + e*x)^2 + (36*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e))/(d + e*x) + 60*b^2*(
b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e)*Log[d + e*x])/(12*e^8)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(824\) vs. \(2(269)=538\).
time = 0.10, size = 825, normalized size = 2.96 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)^4,x,method=_RETURNVERBOSE)

[Out]

b^3/e^7*(1/4*b^3*B*x^4*e^3+1/3*A*b^3*e^3*x^3+2*B*a*b^2*e^3*x^3-4/3*B*b^3*d*e^2*x^3+3*A*a*b^2*e^3*x^2-2*A*b^3*d
*e^2*x^2+15/2*B*a^2*b*e^3*x^2-12*B*a*b^2*d*e^2*x^2+5*B*b^3*d^2*e*x^2+15*A*a^2*b*e^3*x-24*A*a*b^2*d*e^2*x+10*A*
b^3*d^2*e*x+20*B*a^3*e^3*x-60*B*a^2*b*d*e^2*x+60*B*a*b^2*d^2*e*x-20*B*b^3*d^3*x)-1/2/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)^2-3*
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)-1/3*(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)^3+5*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)*ln(e*x+d)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 804 vs. \(2 (287) = 574\).
time = 0.34, size = 804, normalized size = 2.88 \begin {gather*} 5 \, {\left (7 \, B b^{6} d^{4} + 3 \, B a^{4} b^{2} e^{4} + 4 \, A a^{3} b^{3} e^{4} - 4 \, {\left (6 \, B a b^{5} e + A b^{6} e\right )} d^{3} + 6 \, {\left (5 \, B a^{2} b^{4} e^{2} + 2 \, A a b^{5} e^{2}\right )} d^{2} - 4 \, {\left (4 \, B a^{3} b^{3} e^{3} + 3 \, A a^{2} b^{4} e^{3}\right )} d\right )} e^{\left (-8\right )} \log \left (x e + d\right ) + \frac {1}{12} \, {\left (3 \, B b^{6} x^{4} e^{3} - 4 \, {\left (4 \, B b^{6} d e^{2} - 6 \, B a b^{5} e^{3} - A b^{6} e^{3}\right )} x^{3} + 6 \, {\left (10 \, B b^{6} d^{2} e + 15 \, B a^{2} b^{4} e^{3} + 6 \, A a b^{5} e^{3} - 4 \, {\left (6 \, B a b^{5} e^{2} + A b^{6} e^{2}\right )} d\right )} x^{2} - 12 \, {\left (20 \, B b^{6} d^{3} - 20 \, B a^{3} b^{3} e^{3} - 15 \, A a^{2} b^{4} e^{3} - 10 \, {\left (6 \, B a b^{5} e + A b^{6} e\right )} d^{2} + 12 \, {\left (5 \, B a^{2} b^{4} e^{2} + 2 \, A a b^{5} e^{2}\right )} d\right )} x\right )} e^{\left (-7\right )} + \frac {107 \, B b^{6} d^{7} - 2 \, A a^{6} e^{7} - 74 \, {\left (6 \, B a b^{5} e + A b^{6} e\right )} d^{6} + 141 \, {\left (5 \, B a^{2} b^{4} e^{2} + 2 \, A a b^{5} e^{2}\right )} d^{5} - 130 \, {\left (4 \, B a^{3} b^{3} e^{3} + 3 \, A a^{2} b^{4} e^{3}\right )} d^{4} + 55 \, {\left (3 \, B a^{4} b^{2} e^{4} + 4 \, A a^{3} b^{3} e^{4}\right )} d^{3} - 6 \, {\left (2 \, B a^{5} b e^{5} + 5 \, A a^{4} b^{2} e^{5}\right )} d^{2} + 18 \, {\left (7 \, B b^{6} d^{5} e^{2} - 2 \, B a^{5} b e^{7} - 5 \, A a^{4} b^{2} e^{7} - 5 \, {\left (6 \, B a b^{5} e^{3} + A b^{6} e^{3}\right )} d^{4} + 10 \, {\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} - {\left (B a^{6} e^{6} + 6 \, A a^{5} b e^{6}\right )} d + 3 \, {\left (77 \, B b^{6} d^{6} e - B a^{6} e^{7} - 6 \, A a^{5} b e^{7} - 54 \, {\left (6 \, B a b^{5} e^{2} + A b^{6} e^{2}\right )} d^{5} + 105 \, {\left (5 \, B a^{2} b^{4} e^{3} + 2 \, A a b^{5} e^{3}\right )} d^{4} - 100 \, {\left (4 \, B a^{3} b^{3} e^{4} + 3 \, A a^{2} b^{4} e^{4}\right )} d^{3} + 45 \, {\left (3 \, B a^{4} b^{2} e^{5} + 4 \, A a^{3} b^{3} e^{5}\right )} d^{2} - 6 \, {\left (2 \, B a^{5} b e^{6} + 5 \, A a^{4} b^{2} e^{6}\right )} d\right )} x}{6 \, {\left (x^{3} e^{11} + 3 \, d x^{2} e^{10} + 3 \, d^{2} x e^{9} + d^{3} 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)^4,x, algorithm="maxima")

[Out]

5*(7*B*b^6*d^4 + 3*B*a^4*b^2*e^4 + 4*A*a^3*b^3*e^4 - 4*(6*B*a*b^5*e + A*b^6*e)*d^3 + 6*(5*B*a^2*b^4*e^2 + 2*A*
a*b^5*e^2)*d^2 - 4*(4*B*a^3*b^3*e^3 + 3*A*a^2*b^4*e^3)*d)*e^(-8)*log(x*e + d) + 1/12*(3*B*b^6*x^4*e^3 - 4*(4*B
*b^6*d*e^2 - 6*B*a*b^5*e^3 - A*b^6*e^3)*x^3 + 6*(10*B*b^6*d^2*e + 15*B*a^2*b^4*e^3 + 6*A*a*b^5*e^3 - 4*(6*B*a*
b^5*e^2 + A*b^6*e^2)*d)*x^2 - 12*(20*B*b^6*d^3 - 20*B*a^3*b^3*e^3 - 15*A*a^2*b^4*e^3 - 10*(6*B*a*b^5*e + A*b^6
*e)*d^2 + 12*(5*B*a^2*b^4*e^2 + 2*A*a*b^5*e^2)*d)*x)*e^(-7) + 1/6*(107*B*b^6*d^7 - 2*A*a^6*e^7 - 74*(6*B*a*b^5
*e + A*b^6*e)*d^6 + 141*(5*B*a^2*b^4*e^2 + 2*A*a*b^5*e^2)*d^5 - 130*(4*B*a^3*b^3*e^3 + 3*A*a^2*b^4*e^3)*d^4 +
55*(3*B*a^4*b^2*e^4 + 4*A*a^3*b^3*e^4)*d^3 - 6*(2*B*a^5*b*e^5 + 5*A*a^4*b^2*e^5)*d^2 + 18*(7*B*b^6*d^5*e^2 - 2
*B*a^5*b*e^7 - 5*A*a^4*b^2*e^7 - 5*(6*B*a*b^5*e^3 + A*b^6*e^3)*d^4 + 10*(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 - (B*a^6*e^6 + 6*A
*a^5*b*e^6)*d + 3*(77*B*b^6*d^6*e - B*a^6*e^7 - 6*A*a^5*b*e^7 - 54*(6*B*a*b^5*e^2 + A*b^6*e^2)*d^5 + 105*(5*B*
a^2*b^4*e^3 + 2*A*a*b^5*e^3)*d^4 - 100*(4*B*a^3*b^3*e^4 + 3*A*a^2*b^4*e^4)*d^3 + 45*(3*B*a^4*b^2*e^5 + 4*A*a^3
*b^3*e^5)*d^2 - 6*(2*B*a^5*b*e^6 + 5*A*a^4*b^2*e^6)*d)*x)/(x^3*e^11 + 3*d*x^2*e^10 + 3*d^2*x*e^9 + d^3*e^8)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1187 vs. \(2 (287) = 574\).
time = 0.63, size = 1187, normalized size = 4.25 \begin {gather*} \frac {214 \, B b^{6} d^{7} + {\left (3 \, B b^{6} x^{7} - 4 \, A a^{6} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 18 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 60 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} - 36 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 6 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )} e^{7} - {\left (7 \, B b^{6} d x^{6} + 12 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d x^{5} + 90 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d x^{4} - 180 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d x^{3} - 180 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d x^{2} + 36 \, {\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 (7 \, B b^{6} d^{2} x^{5} + 20 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} x^{4} - 126 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} x^{3} - 60 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} x^{2} + 90 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} x - 4 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2}\right )} e^{5} - {\left (105 \, B b^{6} d^{3} x^{4} - 292 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} x^{3} + 54 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} x^{2} + 540 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} x - 110 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3}\right )} e^{4} - 2 \, {\left (278 \, B b^{6} d^{4} x^{3} - 78 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} x^{2} - 243 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} x + 130 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4}\right )} e^{3} - 6 \, {\left (68 \, B b^{6} d^{5} x^{2} + 34 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} x - 47 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5}\right )} e^{2} + 74 \, {\left (3 \, B b^{6} d^{6} x - 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6}\right )} e + 60 \, {\left (7 \, B b^{6} d^{7} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} e^{7} - {\left (4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d x^{3} - 3 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d x^{2}\right )} e^{6} + 3 \, {\left (2 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} x^{3} - 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} x^{2} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} x\right )} e^{5} - {\left (4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} x^{3} - 18 \, {\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} + {\left (7 \, B b^{6} d^{4} x^{3} - 12 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} x^{2} + 18 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} x - 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4}\right )} e^{3} + 3 \, {\left (7 \, B b^{6} d^{5} x^{2} - 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} x + 2 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5}\right )} e^{2} + {\left (21 \, B b^{6} d^{6} x - 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6}\right )} e\right )} \log \left (x e + d\right )}{12 \, {\left (x^{3} e^{11} + 3 \, d x^{2} e^{10} + 3 \, d^{2} x e^{9} + d^{3} 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)^4,x, algorithm="fricas")

[Out]

1/12*(214*B*b^6*d^7 + (3*B*b^6*x^7 - 4*A*a^6 + 4*(6*B*a*b^5 + A*b^6)*x^6 + 18*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 +
60*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 - 36*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 - 6*(B*a^6 + 6*A*a^5*b)*x)*e^7 - (7*B*b^
6*d*x^6 + 12*(6*B*a*b^5 + A*b^6)*d*x^5 + 90*(5*B*a^2*b^4 + 2*A*a*b^5)*d*x^4 - 180*(4*B*a^3*b^3 + 3*A*a^2*b^4)*
d*x^3 - 180*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*x^2 + 36*(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*(7*B*b^6*d^2*x^5 + 20*(6*B*a*b^5 + A*b^6)*d^2*x^4 - 126*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*x^3 - 60*(4*B*a^3*
b^3 + 3*A*a^2*b^4)*d^2*x^2 + 90*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*x - 4*(2*B*a^5*b + 5*A*a^4*b^2)*d^2)*e^5 - (10
5*B*b^6*d^3*x^4 - 292*(6*B*a*b^5 + A*b^6)*d^3*x^3 + 54*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*x^2 + 540*(4*B*a^3*b^3 +
3*A*a^2*b^4)*d^3*x - 110*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3)*e^4 - 2*(278*B*b^6*d^4*x^3 - 78*(6*B*a*b^5 + A*b^6)*
d^4*x^2 - 243*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*x + 130*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4)*e^3 - 6*(68*B*b^6*d^5*x^2
+ 34*(6*B*a*b^5 + A*b^6)*d^5*x - 47*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5)*e^2 + 74*(3*B*b^6*d^6*x - 2*(6*B*a*b^5 + A
*b^6)*d^6)*e + 60*(7*B*b^6*d^7 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3*e^7 - (4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*x^3 -
3*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*x^2)*e^6 + 3*(2*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*x^3 - 4*(4*B*a^3*b^3 + 3*A*a^2*b
^4)*d^2*x^2 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*x)*e^5 - (4*(6*B*a*b^5 + A*b^6)*d^3*x^3 - 18*(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 + (7*B*b^6*d^4*x^
3 - 12*(6*B*a*b^5 + A*b^6)*d^4*x^2 + 18*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*x - 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4)*e
^3 + 3*(7*B*b^6*d^5*x^2 - 4*(6*B*a*b^5 + A*b^6)*d^5*x + 2*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5)*e^2 + (21*B*b^6*d^6*x
- 4*(6*B*a*b^5 + A*b^6)*d^6)*e)*log(x*e + d))/(x^3*e^11 + 3*d*x^2*e^10 + 3*d^2*x*e^9 + d^3*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)**4,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 796 vs. \(2 (287) = 574\).
time = 2.95, size = 796, normalized size = 2.85 \begin {gather*} 5 \, {\left (7 \, B b^{6} d^{4} - 24 \, B a b^{5} d^{3} e - 4 \, A b^{6} d^{3} e + 30 \, B a^{2} b^{4} d^{2} e^{2} + 12 \, A a b^{5} d^{2} e^{2} - 16 \, B a^{3} b^{3} d e^{3} - 12 \, A a^{2} b^{4} d e^{3} + 3 \, B a^{4} b^{2} e^{4} + 4 \, A a^{3} b^{3} e^{4}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{12} \, {\left (3 \, B b^{6} x^{4} e^{12} - 16 \, B b^{6} d x^{3} e^{11} + 60 \, B b^{6} d^{2} x^{2} e^{10} - 240 \, B b^{6} d^{3} x e^{9} + 24 \, B a b^{5} x^{3} e^{12} + 4 \, A b^{6} x^{3} e^{12} - 144 \, B a b^{5} d x^{2} e^{11} - 24 \, A b^{6} d x^{2} e^{11} + 720 \, B a b^{5} d^{2} x e^{10} + 120 \, A b^{6} d^{2} x e^{10} + 90 \, B a^{2} b^{4} x^{2} e^{12} + 36 \, A a b^{5} x^{2} e^{12} - 720 \, B a^{2} b^{4} d x e^{11} - 288 \, A a b^{5} d x e^{11} + 240 \, B a^{3} b^{3} x e^{12} + 180 \, A a^{2} b^{4} x e^{12}\right )} e^{\left (-16\right )} + \frac {{\left (107 \, B b^{6} d^{7} - 444 \, B a b^{5} d^{6} e - 74 \, A b^{6} d^{6} e + 705 \, B a^{2} b^{4} d^{5} e^{2} + 282 \, A a b^{5} d^{5} e^{2} - 520 \, B a^{3} b^{3} d^{4} e^{3} - 390 \, A a^{2} b^{4} d^{4} e^{3} + 165 \, B a^{4} b^{2} d^{3} e^{4} + 220 \, A a^{3} b^{3} d^{3} e^{4} - 12 \, B a^{5} b d^{2} e^{5} - 30 \, A a^{4} b^{2} d^{2} e^{5} - B a^{6} d e^{6} - 6 \, A a^{5} b d e^{6} - 2 \, A a^{6} e^{7} + 18 \, {\left (7 \, B b^{6} d^{5} e^{2} - 30 \, B a b^{5} d^{4} e^{3} - 5 \, A b^{6} d^{4} e^{3} + 50 \, B a^{2} b^{4} d^{3} e^{4} + 20 \, 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} - 2 \, B a^{5} b e^{7} - 5 \, A a^{4} b^{2} e^{7}\right )} x^{2} + 3 \, {\left (77 \, B b^{6} d^{6} e - 324 \, B a b^{5} d^{5} e^{2} - 54 \, A b^{6} d^{5} e^{2} + 525 \, B a^{2} b^{4} d^{4} e^{3} + 210 \, A a b^{5} d^{4} e^{3} - 400 \, B a^{3} b^{3} d^{3} e^{4} - 300 \, A a^{2} b^{4} d^{3} e^{4} + 135 \, B a^{4} b^{2} d^{2} e^{5} + 180 \, A a^{3} b^{3} d^{2} e^{5} - 12 \, B a^{5} b d e^{6} - 30 \, 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 )}}{6 \, {\left (x e + d\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^4,x, algorithm="giac")

[Out]

5*(7*B*b^6*d^4 - 24*B*a*b^5*d^3*e - 4*A*b^6*d^3*e + 30*B*a^2*b^4*d^2*e^2 + 12*A*a*b^5*d^2*e^2 - 16*B*a^3*b^3*d
*e^3 - 12*A*a^2*b^4*d*e^3 + 3*B*a^4*b^2*e^4 + 4*A*a^3*b^3*e^4)*e^(-8)*log(abs(x*e + d)) + 1/12*(3*B*b^6*x^4*e^
12 - 16*B*b^6*d*x^3*e^11 + 60*B*b^6*d^2*x^2*e^10 - 240*B*b^6*d^3*x*e^9 + 24*B*a*b^5*x^3*e^12 + 4*A*b^6*x^3*e^1
2 - 144*B*a*b^5*d*x^2*e^11 - 24*A*b^6*d*x^2*e^11 + 720*B*a*b^5*d^2*x*e^10 + 120*A*b^6*d^2*x*e^10 + 90*B*a^2*b^
4*x^2*e^12 + 36*A*a*b^5*x^2*e^12 - 720*B*a^2*b^4*d*x*e^11 - 288*A*a*b^5*d*x*e^11 + 240*B*a^3*b^3*x*e^12 + 180*
A*a^2*b^4*x*e^12)*e^(-16) + 1/6*(107*B*b^6*d^7 - 444*B*a*b^5*d^6*e - 74*A*b^6*d^6*e + 705*B*a^2*b^4*d^5*e^2 +
282*A*a*b^5*d^5*e^2 - 520*B*a^3*b^3*d^4*e^3 - 390*A*a^2*b^4*d^4*e^3 + 165*B*a^4*b^2*d^3*e^4 + 220*A*a^3*b^3*d^
3*e^4 - 12*B*a^5*b*d^2*e^5 - 30*A*a^4*b^2*d^2*e^5 - B*a^6*d*e^6 - 6*A*a^5*b*d*e^6 - 2*A*a^6*e^7 + 18*(7*B*b^6*
d^5*e^2 - 30*B*a*b^5*d^4*e^3 - 5*A*b^6*d^4*e^3 + 50*B*a^2*b^4*d^3*e^4 + 20*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 - 2*B*a^5*b*e^7 - 5*A*a^4*b^2*e^7)*x^2 +
3*(77*B*b^6*d^6*e - 324*B*a*b^5*d^5*e^2 - 54*A*b^6*d^5*e^2 + 525*B*a^2*b^4*d^4*e^3 + 210*A*a*b^5*d^4*e^3 - 400
*B*a^3*b^3*d^3*e^4 - 300*A*a^2*b^4*d^3*e^4 + 135*B*a^4*b^2*d^2*e^5 + 180*A*a^3*b^3*d^2*e^5 - 12*B*a^5*b*d*e^6
- 30*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)^3

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Mupad [B]
time = 1.22, size = 907, normalized size = 3.25 \begin {gather*} x^3\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{3\,e^4}-\frac {4\,B\,b^6\,d}{3\,e^5}\right )-x^2\,\left (\frac {2\,d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^4}-\frac {4\,B\,b^6\,d}{e^5}\right )}{e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{2\,e^4}+\frac {3\,B\,b^6\,d^2}{e^6}\right )-\frac {\frac {B\,a^6\,d\,e^6+2\,A\,a^6\,e^7+12\,B\,a^5\,b\,d^2\,e^5+6\,A\,a^5\,b\,d\,e^6-165\,B\,a^4\,b^2\,d^3\,e^4+30\,A\,a^4\,b^2\,d^2\,e^5+520\,B\,a^3\,b^3\,d^4\,e^3-220\,A\,a^3\,b^3\,d^3\,e^4-705\,B\,a^2\,b^4\,d^5\,e^2+390\,A\,a^2\,b^4\,d^4\,e^3+444\,B\,a\,b^5\,d^6\,e-282\,A\,a\,b^5\,d^5\,e^2-107\,B\,b^6\,d^7+74\,A\,b^6\,d^6\,e}{6\,e}+x\,\left (\frac {B\,a^6\,e^6}{2}+6\,B\,a^5\,b\,d\,e^5+3\,A\,a^5\,b\,e^6-\frac {135\,B\,a^4\,b^2\,d^2\,e^4}{2}+15\,A\,a^4\,b^2\,d\,e^5+200\,B\,a^3\,b^3\,d^3\,e^3-90\,A\,a^3\,b^3\,d^2\,e^4-\frac {525\,B\,a^2\,b^4\,d^4\,e^2}{2}+150\,A\,a^2\,b^4\,d^3\,e^3+162\,B\,a\,b^5\,d^5\,e-105\,A\,a\,b^5\,d^4\,e^2-\frac {77\,B\,b^6\,d^6}{2}+27\,A\,b^6\,d^5\,e\right )+x^2\,\left (6\,B\,a^5\,b\,e^6-45\,B\,a^4\,b^2\,d\,e^5+15\,A\,a^4\,b^2\,e^6+120\,B\,a^3\,b^3\,d^2\,e^4-60\,A\,a^3\,b^3\,d\,e^5-150\,B\,a^2\,b^4\,d^3\,e^3+90\,A\,a^2\,b^4\,d^2\,e^4+90\,B\,a\,b^5\,d^4\,e^2-60\,A\,a\,b^5\,d^3\,e^3-21\,B\,b^6\,d^5\,e+15\,A\,b^6\,d^4\,e^2\right )}{d^3\,e^7+3\,d^2\,e^8\,x+3\,d\,e^9\,x^2+e^{10}\,x^3}-x\,\left (\frac {6\,d^2\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^4}-\frac {4\,B\,b^6\,d}{e^5}\right )}{e^2}-\frac {4\,d\,\left (\frac {4\,d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^4}-\frac {4\,B\,b^6\,d}{e^5}\right )}{e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{e^4}+\frac {6\,B\,b^6\,d^2}{e^6}\right )}{e}-\frac {5\,a^2\,b^3\,\left (3\,A\,b+4\,B\,a\right )}{e^4}+\frac {4\,B\,b^6\,d^3}{e^7}\right )+\frac {\ln \left (d+e\,x\right )\,\left (15\,B\,a^4\,b^2\,e^4-80\,B\,a^3\,b^3\,d\,e^3+20\,A\,a^3\,b^3\,e^4+150\,B\,a^2\,b^4\,d^2\,e^2-60\,A\,a^2\,b^4\,d\,e^3-120\,B\,a\,b^5\,d^3\,e+60\,A\,a\,b^5\,d^2\,e^2+35\,B\,b^6\,d^4-20\,A\,b^6\,d^3\,e\right )}{e^8}+\frac {B\,b^6\,x^4}{4\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^6)/(d + e*x)^4,x)

[Out]

x^3*((A*b^6 + 6*B*a*b^5)/(3*e^4) - (4*B*b^6*d)/(3*e^5)) - x^2*((2*d*((A*b^6 + 6*B*a*b^5)/e^4 - (4*B*b^6*d)/e^5
))/e - (3*a*b^4*(2*A*b + 5*B*a))/(2*e^4) + (3*B*b^6*d^2)/e^6) - ((2*A*a^6*e^7 - 107*B*b^6*d^7 + 74*A*b^6*d^6*e
+ B*a^6*d*e^6 - 282*A*a*b^5*d^5*e^2 + 12*B*a^5*b*d^2*e^5 + 390*A*a^2*b^4*d^4*e^3 - 220*A*a^3*b^3*d^3*e^4 + 30
*A*a^4*b^2*d^2*e^5 - 705*B*a^2*b^4*d^5*e^2 + 520*B*a^3*b^3*d^4*e^3 - 165*B*a^4*b^2*d^3*e^4 + 6*A*a^5*b*d*e^6 +
444*B*a*b^5*d^6*e)/(6*e) + x*((B*a^6*e^6)/2 - (77*B*b^6*d^6)/2 + 3*A*a^5*b*e^6 + 27*A*b^6*d^5*e - 105*A*a*b^5
*d^4*e^2 + 15*A*a^4*b^2*d*e^5 + 150*A*a^2*b^4*d^3*e^3 - 90*A*a^3*b^3*d^2*e^4 - (525*B*a^2*b^4*d^4*e^2)/2 + 200
*B*a^3*b^3*d^3*e^3 - (135*B*a^4*b^2*d^2*e^4)/2 + 162*B*a*b^5*d^5*e + 6*B*a^5*b*d*e^5) + x^2*(6*B*a^5*b*e^6 - 2
1*B*b^6*d^5*e + 15*A*a^4*b^2*e^6 + 15*A*b^6*d^4*e^2 - 60*A*a*b^5*d^3*e^3 - 60*A*a^3*b^3*d*e^5 + 90*B*a*b^5*d^4
*e^2 - 45*B*a^4*b^2*d*e^5 + 90*A*a^2*b^4*d^2*e^4 - 150*B*a^2*b^4*d^3*e^3 + 120*B*a^3*b^3*d^2*e^4))/(d^3*e^7 +
e^10*x^3 + 3*d^2*e^8*x + 3*d*e^9*x^2) - x*((6*d^2*((A*b^6 + 6*B*a*b^5)/e^4 - (4*B*b^6*d)/e^5))/e^2 - (4*d*((4*
d*((A*b^6 + 6*B*a*b^5)/e^4 - (4*B*b^6*d)/e^5))/e - (3*a*b^4*(2*A*b + 5*B*a))/e^4 + (6*B*b^6*d^2)/e^6))/e - (5*
a^2*b^3*(3*A*b + 4*B*a))/e^4 + (4*B*b^6*d^3)/e^7) + (log(d + e*x)*(35*B*b^6*d^4 - 20*A*b^6*d^3*e + 20*A*a^3*b^
3*e^4 + 15*B*a^4*b^2*e^4 + 60*A*a*b^5*d^2*e^2 - 60*A*a^2*b^4*d*e^3 - 80*B*a^3*b^3*d*e^3 + 150*B*a^2*b^4*d^2*e^
2 - 120*B*a*b^5*d^3*e))/e^8 + (B*b^6*x^4)/(4*e^4)

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