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

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

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Rubi [A]  time = 0.45, 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 = {77} \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 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)^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 = 0.19, size = 297, normalized size = 1.06 \begin {gather*} \frac {-6 b^4 e^2 x^2 \left (-15 a^2 B e^2-6 a b e (A e-4 B d)+2 b^2 d (2 A e-5 B d)\right )+12 b^3 e x \left (20 a^3 B e^3+15 a^2 b e^2 (A e-4 B d)+12 a b^2 d e (5 B d-2 A e)+10 b^3 d^2 (A e-2 B d)\right )+4 b^5 e^3 x^3 (6 a B e+A b e-4 b B d)+60 b^2 (b d-a e)^3 \log (d+e x) (-3 a B e-4 A b e+7 b B d)+\frac {36 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{d+e x}-\frac {6 (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{(d+e x)^2}+\frac {4 (b d-a e)^6 (B d-A e)}{(d+e x)^3}+3 b^6 B e^4 x^4}{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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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)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.89, size = 1225, normalized size = 4.39

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)^4,x, algorithm="fricas")

[Out]

1/12*(3*B*b^6*e^7*x^7 + 214*B*b^6*d^7 - 4*A*a^6*e^7 - 148*(6*B*a*b^5 + A*b^6)*d^6*e + 282*(5*B*a^2*b^4 + 2*A*a
*b^5)*d^5*e^2 - 260*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 110*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 12*(2*B*a^
5*b + 5*A*a^4*b^2)*d^2*e^5 - 2*(B*a^6 + 6*A*a^5*b)*d*e^6 - (7*B*b^6*d*e^6 - 4*(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 3
*(7*B*b^6*d^2*e^5 - 4*(6*B*a*b^5 + A*b^6)*d*e^6 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 - 15*(7*B*b^6*d^3*e^4 -
4*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 - 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 -
2*(278*B*b^6*d^4*e^3 - 146*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 189*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 - 90*(4*B*a^3*b
^3 + 3*A*a^2*b^4)*d*e^6)*x^3 - 6*(68*B*b^6*d^5*e^2 - 26*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 9*(5*B*a^2*b^4 + 2*A*a*b
^5)*d^3*e^4 + 30*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 - 30*(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 + 6*(37*B*b^6*d^6*e - 34*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 81*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e
^3 - 90*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 45*(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 + 60*(7*B*b^6*d^7 - 4*(6*B*a*b^5 + A*b^6)*d^6*e + 6*(5*B*a^2*b^4 + 2*A
*a*b^5)*d^5*e^2 - 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + (7*B*b^6*d^4*e
^3 - 4*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 6*(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 + 3*(7*B*b^6*d^5*e^2 - 4*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 6*(5*B*a^2*b^4
+ 2*A*a*b^5)*d^3*e^4 - 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6)*x^2 + 3*(7*B
*b^6*d^6*e - 4*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 4*(4*B*a^3*b^3 + 3*A*a^2*b^
4)*d^3*e^4 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5)*x)*log(e*x + d))/(e^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9*x + d^
3*e^8)

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giac [B]  time = 1.21, 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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maple [B]  time = 0.02, size = 1143, normalized size = 4.10

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)^4,x)

[Out]

-1/2/e^2/(e*x+d)^2*B*a^6+1/4*b^6/e^4*B*x^4+1/3*b^6/e^4*A*x^3-1/3/e/(e*x+d)^3*A*a^6+2*b^5/e^4*B*x^3*a-4/3*b^6/e
^5*B*x^3*d-1/3/e^7/(e*x+d)^3*A*b^6*d^6+1/3/e^2/(e*x+d)^3*B*d*a^6+1/3/e^8/(e*x+d)^3*B*b^6*d^7+20*b^3/e^4*ln(e*x
+d)*A*a^3-20*b^6/e^7*ln(e*x+d)*A*d^3+2/e^2/(e*x+d)^3*A*d*a^5*b-5/e^3/(e*x+d)^3*A*d^2*a^4*b^2+60*b^3/e^4/(e*x+d
)*A*a^3*d-90*b^4/e^5/(e*x+d)*A*a^2*d^2+60*b^5/e^6/(e*x+d)*A*a*d^3+45*b^2/e^4/(e*x+d)*B*a^4*d-120*b^3/e^5/(e*x+
d)*B*a^3*d^2+150*b^4/e^6/(e*x+d)*B*a^2*d^3-90*b^5/e^7/(e*x+d)*B*a*d^4-12*b^5/e^5*B*x^2*a*d-24*b^5/e^5*A*x*a*d-
6*b/e^3/(e*x+d)*B*a^5+21*b^6/e^8/(e*x+d)*B*d^5-60*b^4/e^5*B*x*a^2*d+60*b^5/e^6*B*x*a*d^2+20/3/e^4/(e*x+d)^3*A*
d^3*a^3*b^3-5/e^5/(e*x+d)^3*A*a^2*b^4*d^4+2/e^6/(e*x+d)^3*A*a*b^5*d^5-2/e^3/(e*x+d)^3*B*d^2*a^5*b+5/e^4/(e*x+d
)^3*B*d^3*a^4*b^2-20/3/e^5/(e*x+d)^3*B*a^3*b^3*d^4+5/e^6/(e*x+d)^3*B*a^2*b^4*d^5-2/e^7/(e*x+d)^3*B*a*b^5*d^6-6
0*b^4/e^5*ln(e*x+d)*A*a^2*d+60*b^5/e^6*ln(e*x+d)*A*a*d^2-80*b^3/e^5*ln(e*x+d)*B*a^3*d+150*b^4/e^6*ln(e*x+d)*B*
a^2*d^2-120*b^5/e^7*ln(e*x+d)*B*a*d^3+15/e^3/(e*x+d)^2*A*a^4*b^2*d-30/e^4/(e*x+d)^2*A*a^3*b^3*d^2+30/e^5/(e*x+
d)^2*A*a^2*b^4*d^3-15/e^6/(e*x+d)^2*A*a*b^5*d^4+6/e^3/(e*x+d)^2*B*a^5*b*d-45/2/e^4/(e*x+d)^2*B*a^4*b^2*d^2+40/
e^5/(e*x+d)^2*B*a^3*b^3*d^3-75/2/e^6/(e*x+d)^2*B*a^2*b^4*d^4+18/e^7/(e*x+d)^2*B*a*b^5*d^5+15*b^2/e^4*ln(e*x+d)
*B*a^4+35*b^6/e^8*ln(e*x+d)*B*d^4-3/e^2/(e*x+d)^2*A*a^5*b+3/e^7/(e*x+d)^2*A*b^6*d^5-7/2/e^8/(e*x+d)^2*B*b^6*d^
6+3*b^5/e^4*A*x^2*a-2*b^6/e^5*A*x^2*d+15/2*b^4/e^4*B*x^2*a^2+5*b^6/e^6*B*x^2*d^2+15*b^4/e^4*A*x*a^2+10*b^6/e^6
*A*x*d^2+20*b^3/e^4*B*x*a^3-20*b^6/e^7*B*x*d^3-15*b^2/e^3/(e*x+d)*A*a^4-15*b^6/e^7/(e*x+d)*A*d^4

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maxima [B]  time = 0.85, size = 793, normalized size = 2.84 \begin {gather*} \frac {107 \, B b^{6} d^{7} - 2 \, A a^{6} e^{7} - 74 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 141 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} - 130 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 55 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} - 6 \, {\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} + 18 \, {\left (7 \, B b^{6} d^{5} e^{2} - 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} - 10 \, {\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} - {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 3 \, {\left (77 \, B b^{6} d^{6} e - 54 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 105 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} - 100 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 45 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} - 6 \, {\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}{6 \, {\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} + \frac {3 \, B b^{6} e^{3} x^{4} - 4 \, {\left (4 \, B b^{6} d e^{2} - {\left (6 \, B a b^{5} + A b^{6}\right )} e^{3}\right )} x^{3} + 6 \, {\left (10 \, B b^{6} d^{2} e - 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{2} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{3}\right )} x^{2} - 12 \, {\left (20 \, B b^{6} d^{3} - 10 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e + 12 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{2} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{3}\right )} x}{12 \, e^{7}} + \frac {5 \, {\left (7 \, B b^{6} d^{4} - 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{2} - 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{3} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{4}\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)^4,x, algorithm="maxima")

[Out]

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

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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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sympy [B]  time = 62.59, size = 867, normalized size = 3.11 \begin {gather*} \frac {B b^{6} x^{4}}{4 e^{4}} + \frac {5 b^{2} \left (a e - b d\right )^{3} \left (4 A b e + 3 B a e - 7 B b d\right ) \log {\left (d + e x \right )}}{e^{8}} + x^{3} \left (\frac {A b^{6}}{3 e^{4}} + \frac {2 B a b^{5}}{e^{4}} - \frac {4 B b^{6} d}{3 e^{5}}\right ) + x^{2} \left (\frac {3 A a b^{5}}{e^{4}} - \frac {2 A b^{6} d}{e^{5}} + \frac {15 B a^{2} b^{4}}{2 e^{4}} - \frac {12 B a b^{5} d}{e^{5}} + \frac {5 B b^{6} d^{2}}{e^{6}}\right ) + x \left (\frac {15 A a^{2} b^{4}}{e^{4}} - \frac {24 A a b^{5} d}{e^{5}} + \frac {10 A b^{6} d^{2}}{e^{6}} + \frac {20 B a^{3} b^{3}}{e^{4}} - \frac {60 B a^{2} b^{4} d}{e^{5}} + \frac {60 B a b^{5} d^{2}}{e^{6}} - \frac {20 B b^{6} d^{3}}{e^{7}}\right ) + \frac {- 2 A a^{6} e^{7} - 6 A a^{5} b d e^{6} - 30 A a^{4} b^{2} d^{2} e^{5} + 220 A a^{3} b^{3} d^{3} e^{4} - 390 A a^{2} b^{4} d^{4} e^{3} + 282 A a b^{5} d^{5} e^{2} - 74 A b^{6} d^{6} e - B a^{6} d e^{6} - 12 B a^{5} b d^{2} e^{5} + 165 B a^{4} b^{2} d^{3} e^{4} - 520 B a^{3} b^{3} d^{4} e^{3} + 705 B a^{2} b^{4} d^{5} e^{2} - 444 B a b^{5} d^{6} e + 107 B b^{6} d^{7} + x^{2} \left (- 90 A a^{4} b^{2} e^{7} + 360 A a^{3} b^{3} d e^{6} - 540 A a^{2} b^{4} d^{2} e^{5} + 360 A a b^{5} d^{3} e^{4} - 90 A b^{6} d^{4} e^{3} - 36 B a^{5} b e^{7} + 270 B a^{4} b^{2} d e^{6} - 720 B a^{3} b^{3} d^{2} e^{5} + 900 B a^{2} b^{4} d^{3} e^{4} - 540 B a b^{5} d^{4} e^{3} + 126 B b^{6} d^{5} e^{2}\right ) + x \left (- 18 A a^{5} b e^{7} - 90 A a^{4} b^{2} d e^{6} + 540 A a^{3} b^{3} d^{2} e^{5} - 900 A a^{2} b^{4} d^{3} e^{4} + 630 A a b^{5} d^{4} e^{3} - 162 A b^{6} d^{5} e^{2} - 3 B a^{6} e^{7} - 36 B a^{5} b d e^{6} + 405 B a^{4} b^{2} d^{2} e^{5} - 1200 B a^{3} b^{3} d^{3} e^{4} + 1575 B a^{2} b^{4} d^{4} e^{3} - 972 B a b^{5} d^{5} e^{2} + 231 B b^{6} d^{6} e\right )}{6 d^{3} e^{8} + 18 d^{2} e^{9} x + 18 d e^{10} x^{2} + 6 e^{11} x^{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)

[Out]

B*b**6*x**4/(4*e**4) + 5*b**2*(a*e - b*d)**3*(4*A*b*e + 3*B*a*e - 7*B*b*d)*log(d + e*x)/e**8 + x**3*(A*b**6/(3
*e**4) + 2*B*a*b**5/e**4 - 4*B*b**6*d/(3*e**5)) + x**2*(3*A*a*b**5/e**4 - 2*A*b**6*d/e**5 + 15*B*a**2*b**4/(2*
e**4) - 12*B*a*b**5*d/e**5 + 5*B*b**6*d**2/e**6) + x*(15*A*a**2*b**4/e**4 - 24*A*a*b**5*d/e**5 + 10*A*b**6*d**
2/e**6 + 20*B*a**3*b**3/e**4 - 60*B*a**2*b**4*d/e**5 + 60*B*a*b**5*d**2/e**6 - 20*B*b**6*d**3/e**7) + (-2*A*a*
*6*e**7 - 6*A*a**5*b*d*e**6 - 30*A*a**4*b**2*d**2*e**5 + 220*A*a**3*b**3*d**3*e**4 - 390*A*a**2*b**4*d**4*e**3
+ 282*A*a*b**5*d**5*e**2 - 74*A*b**6*d**6*e - B*a**6*d*e**6 - 12*B*a**5*b*d**2*e**5 + 165*B*a**4*b**2*d**3*e*
*4 - 520*B*a**3*b**3*d**4*e**3 + 705*B*a**2*b**4*d**5*e**2 - 444*B*a*b**5*d**6*e + 107*B*b**6*d**7 + x**2*(-90
*A*a**4*b**2*e**7 + 360*A*a**3*b**3*d*e**6 - 540*A*a**2*b**4*d**2*e**5 + 360*A*a*b**5*d**3*e**4 - 90*A*b**6*d*
*4*e**3 - 36*B*a**5*b*e**7 + 270*B*a**4*b**2*d*e**6 - 720*B*a**3*b**3*d**2*e**5 + 900*B*a**2*b**4*d**3*e**4 -
540*B*a*b**5*d**4*e**3 + 126*B*b**6*d**5*e**2) + x*(-18*A*a**5*b*e**7 - 90*A*a**4*b**2*d*e**6 + 540*A*a**3*b**
3*d**2*e**5 - 900*A*a**2*b**4*d**3*e**4 + 630*A*a*b**5*d**4*e**3 - 162*A*b**6*d**5*e**2 - 3*B*a**6*e**7 - 36*B
*a**5*b*d*e**6 + 405*B*a**4*b**2*d**2*e**5 - 1200*B*a**3*b**3*d**3*e**4 + 1575*B*a**2*b**4*d**4*e**3 - 972*B*a
*b**5*d**5*e**2 + 231*B*b**6*d**6*e))/(6*d**3*e**8 + 18*d**2*e**9*x + 18*d*e**10*x**2 + 6*e**11*x**3)

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