3.12.42 \(\int \frac {A+B x}{(a+b x)^3 (d+e x)^4} \, dx\) [1142]

Optimal. Leaf size=248 \[ -\frac {b^2 (A b-a B)}{2 (b d-a e)^4 (a+b x)^2}-\frac {b^2 (b B d-4 A b e+3 a B e)}{(b d-a e)^5 (a+b x)}-\frac {e (B d-A e)}{3 (b d-a e)^3 (d+e x)^3}-\frac {e (2 b B d-3 A b e+a B e)}{2 (b d-a e)^4 (d+e x)^2}-\frac {3 b e (b B d-2 A b e+a B e)}{(b d-a e)^5 (d+e x)}-\frac {2 b^2 e (2 b B d-5 A b e+3 a B e) \log (a+b x)}{(b d-a e)^6}+\frac {2 b^2 e (2 b B d-5 A b e+3 a B e) \log (d+e x)}{(b d-a e)^6} \]

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 248, 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^2 (3 a B e-4 A b e+b B d)}{(a+b x) (b d-a e)^5}-\frac {b^2 (A b-a B)}{2 (a+b x)^2 (b d-a e)^4}-\frac {2 b^2 e \log (a+b x) (3 a B e-5 A b e+2 b B d)}{(b d-a e)^6}+\frac {2 b^2 e \log (d+e x) (3 a B e-5 A b e+2 b B d)}{(b d-a e)^6}-\frac {3 b e (a B e-2 A b e+b B d)}{(d+e x) (b d-a e)^5}-\frac {e (a B e-3 A b e+2 b B d)}{2 (d+e x)^2 (b d-a e)^4}-\frac {e (B d-A e)}{3 (d+e x)^3 (b d-a e)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((a + b*x)^3*(d + e*x)^4),x]

[Out]

-1/2*(b^2*(A*b - a*B))/((b*d - a*e)^4*(a + b*x)^2) - (b^2*(b*B*d - 4*A*b*e + 3*a*B*e))/((b*d - a*e)^5*(a + b*x
)) - (e*(B*d - A*e))/(3*(b*d - a*e)^3*(d + e*x)^3) - (e*(2*b*B*d - 3*A*b*e + a*B*e))/(2*(b*d - a*e)^4*(d + e*x
)^2) - (3*b*e*(b*B*d - 2*A*b*e + a*B*e))/((b*d - a*e)^5*(d + e*x)) - (2*b^2*e*(2*b*B*d - 5*A*b*e + 3*a*B*e)*Lo
g[a + b*x])/(b*d - a*e)^6 + (2*b^2*e*(2*b*B*d - 5*A*b*e + 3*a*B*e)*Log[d + e*x])/(b*d - a*e)^6

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

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Mathematica [A]
time = 0.16, size = 233, normalized size = 0.94 \begin {gather*} \frac {-\frac {3 b^2 (A b-a B) (b d-a e)^2}{(a+b x)^2}-\frac {6 b^2 (b d-a e) (b B d-4 A b e+3 a B e)}{a+b x}+\frac {2 e (b d-a e)^3 (-B d+A e)}{(d+e x)^3}+\frac {3 e (b d-a e)^2 (-2 b B d+3 A b e-a B e)}{(d+e x)^2}+\frac {18 b e (-b d+a e) (b B d-2 A b e+a B e)}{d+e x}+12 b^2 e (-2 b B d+5 A b e-3 a B e) \log (a+b x)+12 b^2 e (2 b B d-5 A b e+3 a B e) \log (d+e x)}{6 (b d-a e)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((a + b*x)^3*(d + e*x)^4),x]

[Out]

((-3*b^2*(A*b - a*B)*(b*d - a*e)^2)/(a + b*x)^2 - (6*b^2*(b*d - a*e)*(b*B*d - 4*A*b*e + 3*a*B*e))/(a + b*x) +
(2*e*(b*d - a*e)^3*(-(B*d) + A*e))/(d + e*x)^3 + (3*e*(b*d - a*e)^2*(-2*b*B*d + 3*A*b*e - a*B*e))/(d + e*x)^2
+ (18*b*e*(-(b*d) + a*e)*(b*B*d - 2*A*b*e + a*B*e))/(d + e*x) + 12*b^2*e*(-2*b*B*d + 5*A*b*e - 3*a*B*e)*Log[a
+ b*x] + 12*b^2*e*(2*b*B*d - 5*A*b*e + 3*a*B*e)*Log[d + e*x])/(6*(b*d - a*e)^6)

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Maple [A]
time = 0.13, size = 247, normalized size = 1.00 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(b*x+a)^3/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1092 vs. \(2 (269) = 538\).
time = 0.37, size = 1092, normalized size = 4.40 \begin {gather*} -\frac {2 \, {\left (2 \, B b^{3} d e + 3 \, B a b^{2} e^{2} - 5 \, A b^{3} e^{2}\right )} \log \left (b x + a\right )}{b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}} + \frac {2 \, {\left (2 \, B b^{3} d e + 3 \, B a b^{2} e^{2} - 5 \, A b^{3} e^{2}\right )} \log \left (x e + d\right )}{b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}} + \frac {2 \, A a^{4} e^{4} - 3 \, {\left (B a b^{3} + A b^{4}\right )} d^{4} - 12 \, {\left (2 \, B b^{4} d e^{3} + 3 \, B a b^{3} e^{4} - 5 \, A b^{4} e^{4}\right )} x^{4} - {\left (47 \, B a^{2} b^{2} e - 27 \, A a b^{3} e\right )} d^{3} - 6 \, {\left (10 \, B b^{4} d^{2} e^{2} + 9 \, B a^{2} b^{2} e^{4} - 15 \, A a b^{3} e^{4} + {\left (21 \, B a b^{3} e^{3} - 25 \, A b^{4} e^{3}\right )} d\right )} x^{3} - {\left (11 \, B a^{3} b e^{2} - 47 \, A a^{2} b^{2} e^{2}\right )} d^{2} - 2 \, {\left (22 \, B b^{4} d^{3} e + 6 \, B a^{3} b e^{4} - 10 \, A a^{2} b^{2} e^{4} + {\left (79 \, B a b^{3} e^{2} - 55 \, A b^{4} e^{2}\right )} d^{2} + {\left (73 \, B a^{2} b^{2} e^{3} - 115 \, A a b^{3} e^{3}\right )} d\right )} x^{2} + {\left (B a^{4} e^{3} - 13 \, A a^{3} b e^{3}\right )} d - {\left (6 \, B b^{4} d^{4} - 3 \, B a^{4} e^{4} + 5 \, A a^{3} b e^{4} + {\left (79 \, B a b^{3} e - 15 \, A b^{4} e\right )} d^{3} + {\left (127 \, B a^{2} b^{2} e^{2} - 175 \, A a b^{3} e^{2}\right )} d^{2} + {\left (31 \, B a^{3} b e^{3} - 55 \, A a^{2} b^{2} e^{3}\right )} d\right )} x}{6 \, {\left (a^{2} b^{5} d^{8} - 5 \, a^{3} b^{4} d^{7} e + 10 \, a^{4} b^{3} d^{6} e^{2} - 10 \, a^{5} b^{2} d^{5} e^{3} + 5 \, a^{6} b d^{4} e^{4} - a^{7} d^{3} e^{5} + {\left (b^{7} d^{5} e^{3} - 5 \, a b^{6} d^{4} e^{4} + 10 \, a^{2} b^{5} d^{3} e^{5} - 10 \, a^{3} b^{4} d^{2} e^{6} + 5 \, a^{4} b^{3} d e^{7} - a^{5} b^{2} e^{8}\right )} x^{5} + {\left (3 \, b^{7} d^{6} e^{2} - 13 \, a b^{6} d^{5} e^{3} + 20 \, a^{2} b^{5} d^{4} e^{4} - 10 \, a^{3} b^{4} d^{3} e^{5} - 5 \, a^{4} b^{3} d^{2} e^{6} + 7 \, a^{5} b^{2} d e^{7} - 2 \, a^{6} b e^{8}\right )} x^{4} + {\left (3 \, b^{7} d^{7} e - 9 \, a b^{6} d^{6} e^{2} + a^{2} b^{5} d^{5} e^{3} + 25 \, a^{3} b^{4} d^{4} e^{4} - 35 \, a^{4} b^{3} d^{3} e^{5} + 17 \, a^{5} b^{2} d^{2} e^{6} - a^{6} b d e^{7} - a^{7} e^{8}\right )} x^{3} + {\left (b^{7} d^{8} + a b^{6} d^{7} e - 17 \, a^{2} b^{5} d^{6} e^{2} + 35 \, a^{3} b^{4} d^{5} e^{3} - 25 \, a^{4} b^{3} d^{4} e^{4} - a^{5} b^{2} d^{3} e^{5} + 9 \, a^{6} b d^{2} e^{6} - 3 \, a^{7} d e^{7}\right )} x^{2} + {\left (2 \, a b^{6} d^{8} - 7 \, a^{2} b^{5} d^{7} e + 5 \, a^{3} b^{4} d^{6} e^{2} + 10 \, a^{4} b^{3} d^{5} e^{3} - 20 \, a^{5} b^{2} d^{4} e^{4} + 13 \, a^{6} b d^{3} e^{5} - 3 \, a^{7} d^{2} e^{6}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)^3/(e*x+d)^4,x, algorithm="maxima")

[Out]

-2*(2*B*b^3*d*e + 3*B*a*b^2*e^2 - 5*A*b^3*e^2)*log(b*x + a)/(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20
*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6) + 2*(2*B*b^3*d*e + 3*B*a*b^2*e^2 - 5*A*b^3*e^
2)*log(x*e + d)/(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^
5*b*d*e^5 + a^6*e^6) + 1/6*(2*A*a^4*e^4 - 3*(B*a*b^3 + A*b^4)*d^4 - 12*(2*B*b^4*d*e^3 + 3*B*a*b^3*e^4 - 5*A*b^
4*e^4)*x^4 - (47*B*a^2*b^2*e - 27*A*a*b^3*e)*d^3 - 6*(10*B*b^4*d^2*e^2 + 9*B*a^2*b^2*e^4 - 15*A*a*b^3*e^4 + (2
1*B*a*b^3*e^3 - 25*A*b^4*e^3)*d)*x^3 - (11*B*a^3*b*e^2 - 47*A*a^2*b^2*e^2)*d^2 - 2*(22*B*b^4*d^3*e + 6*B*a^3*b
*e^4 - 10*A*a^2*b^2*e^4 + (79*B*a*b^3*e^2 - 55*A*b^4*e^2)*d^2 + (73*B*a^2*b^2*e^3 - 115*A*a*b^3*e^3)*d)*x^2 +
(B*a^4*e^3 - 13*A*a^3*b*e^3)*d - (6*B*b^4*d^4 - 3*B*a^4*e^4 + 5*A*a^3*b*e^4 + (79*B*a*b^3*e - 15*A*b^4*e)*d^3
+ (127*B*a^2*b^2*e^2 - 175*A*a*b^3*e^2)*d^2 + (31*B*a^3*b*e^3 - 55*A*a^2*b^2*e^3)*d)*x)/(a^2*b^5*d^8 - 5*a^3*b
^4*d^7*e + 10*a^4*b^3*d^6*e^2 - 10*a^5*b^2*d^5*e^3 + 5*a^6*b*d^4*e^4 - a^7*d^3*e^5 + (b^7*d^5*e^3 - 5*a*b^6*d^
4*e^4 + 10*a^2*b^5*d^3*e^5 - 10*a^3*b^4*d^2*e^6 + 5*a^4*b^3*d*e^7 - a^5*b^2*e^8)*x^5 + (3*b^7*d^6*e^2 - 13*a*b
^6*d^5*e^3 + 20*a^2*b^5*d^4*e^4 - 10*a^3*b^4*d^3*e^5 - 5*a^4*b^3*d^2*e^6 + 7*a^5*b^2*d*e^7 - 2*a^6*b*e^8)*x^4
+ (3*b^7*d^7*e - 9*a*b^6*d^6*e^2 + a^2*b^5*d^5*e^3 + 25*a^3*b^4*d^4*e^4 - 35*a^4*b^3*d^3*e^5 + 17*a^5*b^2*d^2*
e^6 - a^6*b*d*e^7 - a^7*e^8)*x^3 + (b^7*d^8 + a*b^6*d^7*e - 17*a^2*b^5*d^6*e^2 + 35*a^3*b^4*d^5*e^3 - 25*a^4*b
^3*d^4*e^4 - a^5*b^2*d^3*e^5 + 9*a^6*b*d^2*e^6 - 3*a^7*d*e^7)*x^2 + (2*a*b^6*d^8 - 7*a^2*b^5*d^7*e + 5*a^3*b^4
*d^6*e^2 + 10*a^4*b^3*d^5*e^3 - 20*a^5*b^2*d^4*e^4 + 13*a^6*b*d^3*e^5 - 3*a^7*d^2*e^6)*x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1804 vs. \(2 (269) = 538\).
time = 1.36, size = 1804, normalized size = 7.27 \begin {gather*} -\frac {6 \, B b^{5} d^{5} x + 3 \, {\left (B a b^{4} + A b^{5}\right )} d^{5} + {\left (2 \, A a^{5} - 12 \, {\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} - 18 \, {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} - 4 \, {\left (3 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} + {\left (3 \, B a^{5} - 5 \, A a^{4} b\right )} x\right )} e^{5} + {\left (12 \, {\left (B a b^{4} - 5 \, A b^{5}\right )} d x^{4} - 12 \, {\left (6 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} d x^{3} - 2 \, {\left (67 \, B a^{3} b^{2} - 105 \, A a^{2} b^{3}\right )} d x^{2} - 2 \, {\left (17 \, B a^{4} b - 30 \, A a^{3} b^{2}\right )} d x + {\left (B a^{5} - 15 \, A a^{4} b\right )} d\right )} e^{4} + 6 \, {\left (4 \, B b^{5} d^{2} x^{4} + {\left (11 \, B a b^{4} - 25 \, A b^{5}\right )} d^{2} x^{3} - 2 \, {\left (B a^{2} b^{3} + 10 \, A a b^{4}\right )} d^{2} x^{2} - 4 \, {\left (4 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} d^{2} x - 2 \, {\left (B a^{4} b - 5 \, A a^{3} b^{2}\right )} d^{2}\right )} e^{3} + 2 \, {\left (30 \, B b^{5} d^{3} x^{3} + {\left (57 \, B a b^{4} - 55 \, A b^{5}\right )} d^{3} x^{2} + 8 \, {\left (3 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} d^{3} x - 2 \, {\left (9 \, B a^{3} b^{2} + 5 \, A a^{2} b^{3}\right )} d^{3}\right )} e^{2} + {\left (44 \, B b^{5} d^{4} x^{2} + {\left (73 \, B a b^{4} - 15 \, A b^{5}\right )} d^{4} x + 2 \, {\left (22 \, B a^{2} b^{3} - 15 \, A a b^{4}\right )} d^{4}\right )} e + 12 \, {\left ({\left ({\left (3 \, B a b^{4} - 5 \, A b^{5}\right )} x^{5} + 2 \, {\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} + {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3}\right )} e^{5} + {\left (2 \, B b^{5} d x^{5} + {\left (13 \, B a b^{4} - 15 \, A b^{5}\right )} d x^{4} + 10 \, {\left (2 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} d x^{3} + 3 \, {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} d x^{2}\right )} e^{4} + 3 \, {\left (2 \, B b^{5} d^{2} x^{4} + {\left (7 \, B a b^{4} - 5 \, A b^{5}\right )} d^{2} x^{3} + 2 \, {\left (4 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} d^{2} x^{2} + {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} d^{2} x\right )} e^{3} + {\left (6 \, B b^{5} d^{3} x^{3} + 5 \, {\left (3 \, B a b^{4} - A b^{5}\right )} d^{3} x^{2} + 2 \, {\left (6 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} d^{3} x + {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} d^{3}\right )} e^{2} + 2 \, {\left (B b^{5} d^{4} x^{2} + 2 \, B a b^{4} d^{4} x + B a^{2} b^{3} d^{4}\right )} e\right )} \log \left (b x + a\right ) - 12 \, {\left ({\left ({\left (3 \, B a b^{4} - 5 \, A b^{5}\right )} x^{5} + 2 \, {\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} + {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3}\right )} e^{5} + {\left (2 \, B b^{5} d x^{5} + {\left (13 \, B a b^{4} - 15 \, A b^{5}\right )} d x^{4} + 10 \, {\left (2 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} d x^{3} + 3 \, {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} d x^{2}\right )} e^{4} + 3 \, {\left (2 \, B b^{5} d^{2} x^{4} + {\left (7 \, B a b^{4} - 5 \, A b^{5}\right )} d^{2} x^{3} + 2 \, {\left (4 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} d^{2} x^{2} + {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} d^{2} x\right )} e^{3} + {\left (6 \, B b^{5} d^{3} x^{3} + 5 \, {\left (3 \, B a b^{4} - A b^{5}\right )} d^{3} x^{2} + 2 \, {\left (6 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} d^{3} x + {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} d^{3}\right )} e^{2} + 2 \, {\left (B b^{5} d^{4} x^{2} + 2 \, B a b^{4} d^{4} x + B a^{2} b^{3} d^{4}\right )} e\right )} \log \left (x e + d\right )}{6 \, {\left (b^{8} d^{9} x^{2} + 2 \, a b^{7} d^{9} x + a^{2} b^{6} d^{9} + {\left (a^{6} b^{2} x^{5} + 2 \, a^{7} b x^{4} + a^{8} x^{3}\right )} e^{9} - 3 \, {\left (2 \, a^{5} b^{3} d x^{5} + 3 \, a^{6} b^{2} d x^{4} - a^{8} d x^{2}\right )} e^{8} + 3 \, {\left (5 \, a^{4} b^{4} d^{2} x^{5} + 4 \, a^{5} b^{3} d^{2} x^{4} - 6 \, a^{6} b^{2} d^{2} x^{3} - 4 \, a^{7} b d^{2} x^{2} + a^{8} d^{2} x\right )} e^{7} - {\left (20 \, a^{3} b^{5} d^{3} x^{5} - 5 \, a^{4} b^{4} d^{3} x^{4} - 52 \, a^{5} b^{3} d^{3} x^{3} - 10 \, a^{6} b^{2} d^{3} x^{2} + 16 \, a^{7} b d^{3} x - a^{8} d^{3}\right )} e^{6} + 3 \, {\left (5 \, a^{2} b^{6} d^{4} x^{5} - 10 \, a^{3} b^{5} d^{4} x^{4} - 20 \, a^{4} b^{4} d^{4} x^{3} + 8 \, a^{5} b^{3} d^{4} x^{2} + 11 \, a^{6} b^{2} d^{4} x - 2 \, a^{7} b d^{4}\right )} e^{5} - 3 \, {\left (2 \, a b^{7} d^{5} x^{5} - 11 \, a^{2} b^{6} d^{5} x^{4} - 8 \, a^{3} b^{5} d^{5} x^{3} + 20 \, a^{4} b^{4} d^{5} x^{2} + 10 \, a^{5} b^{3} d^{5} x - 5 \, a^{6} b^{2} d^{5}\right )} e^{4} + {\left (b^{8} d^{6} x^{5} - 16 \, a b^{7} d^{6} x^{4} + 10 \, a^{2} b^{6} d^{6} x^{3} + 52 \, a^{3} b^{5} d^{6} x^{2} + 5 \, a^{4} b^{4} d^{6} x - 20 \, a^{5} b^{3} d^{6}\right )} e^{3} + 3 \, {\left (b^{8} d^{7} x^{4} - 4 \, a b^{7} d^{7} x^{3} - 6 \, a^{2} b^{6} d^{7} x^{2} + 4 \, a^{3} b^{5} d^{7} x + 5 \, a^{4} b^{4} d^{7}\right )} e^{2} + 3 \, {\left (b^{8} d^{8} x^{3} - 3 \, a^{2} b^{6} d^{8} x - 2 \, a^{3} b^{5} d^{8}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)^3/(e*x+d)^4,x, algorithm="fricas")

[Out]

-1/6*(6*B*b^5*d^5*x + 3*(B*a*b^4 + A*b^5)*d^5 + (2*A*a^5 - 12*(3*B*a^2*b^3 - 5*A*a*b^4)*x^4 - 18*(3*B*a^3*b^2
- 5*A*a^2*b^3)*x^3 - 4*(3*B*a^4*b - 5*A*a^3*b^2)*x^2 + (3*B*a^5 - 5*A*a^4*b)*x)*e^5 + (12*(B*a*b^4 - 5*A*b^5)*
d*x^4 - 12*(6*B*a^2*b^3 - 5*A*a*b^4)*d*x^3 - 2*(67*B*a^3*b^2 - 105*A*a^2*b^3)*d*x^2 - 2*(17*B*a^4*b - 30*A*a^3
*b^2)*d*x + (B*a^5 - 15*A*a^4*b)*d)*e^4 + 6*(4*B*b^5*d^2*x^4 + (11*B*a*b^4 - 25*A*b^5)*d^2*x^3 - 2*(B*a^2*b^3
+ 10*A*a*b^4)*d^2*x^2 - 4*(4*B*a^3*b^2 - 5*A*a^2*b^3)*d^2*x - 2*(B*a^4*b - 5*A*a^3*b^2)*d^2)*e^3 + 2*(30*B*b^5
*d^3*x^3 + (57*B*a*b^4 - 55*A*b^5)*d^3*x^2 + 8*(3*B*a^2*b^3 - 10*A*a*b^4)*d^3*x - 2*(9*B*a^3*b^2 + 5*A*a^2*b^3
)*d^3)*e^2 + (44*B*b^5*d^4*x^2 + (73*B*a*b^4 - 15*A*b^5)*d^4*x + 2*(22*B*a^2*b^3 - 15*A*a*b^4)*d^4)*e + 12*(((
3*B*a*b^4 - 5*A*b^5)*x^5 + 2*(3*B*a^2*b^3 - 5*A*a*b^4)*x^4 + (3*B*a^3*b^2 - 5*A*a^2*b^3)*x^3)*e^5 + (2*B*b^5*d
*x^5 + (13*B*a*b^4 - 15*A*b^5)*d*x^4 + 10*(2*B*a^2*b^3 - 3*A*a*b^4)*d*x^3 + 3*(3*B*a^3*b^2 - 5*A*a^2*b^3)*d*x^
2)*e^4 + 3*(2*B*b^5*d^2*x^4 + (7*B*a*b^4 - 5*A*b^5)*d^2*x^3 + 2*(4*B*a^2*b^3 - 5*A*a*b^4)*d^2*x^2 + (3*B*a^3*b
^2 - 5*A*a^2*b^3)*d^2*x)*e^3 + (6*B*b^5*d^3*x^3 + 5*(3*B*a*b^4 - A*b^5)*d^3*x^2 + 2*(6*B*a^2*b^3 - 5*A*a*b^4)*
d^3*x + (3*B*a^3*b^2 - 5*A*a^2*b^3)*d^3)*e^2 + 2*(B*b^5*d^4*x^2 + 2*B*a*b^4*d^4*x + B*a^2*b^3*d^4)*e)*log(b*x
+ a) - 12*(((3*B*a*b^4 - 5*A*b^5)*x^5 + 2*(3*B*a^2*b^3 - 5*A*a*b^4)*x^4 + (3*B*a^3*b^2 - 5*A*a^2*b^3)*x^3)*e^5
+ (2*B*b^5*d*x^5 + (13*B*a*b^4 - 15*A*b^5)*d*x^4 + 10*(2*B*a^2*b^3 - 3*A*a*b^4)*d*x^3 + 3*(3*B*a^3*b^2 - 5*A*
a^2*b^3)*d*x^2)*e^4 + 3*(2*B*b^5*d^2*x^4 + (7*B*a*b^4 - 5*A*b^5)*d^2*x^3 + 2*(4*B*a^2*b^3 - 5*A*a*b^4)*d^2*x^2
+ (3*B*a^3*b^2 - 5*A*a^2*b^3)*d^2*x)*e^3 + (6*B*b^5*d^3*x^3 + 5*(3*B*a*b^4 - A*b^5)*d^3*x^2 + 2*(6*B*a^2*b^3
- 5*A*a*b^4)*d^3*x + (3*B*a^3*b^2 - 5*A*a^2*b^3)*d^3)*e^2 + 2*(B*b^5*d^4*x^2 + 2*B*a*b^4*d^4*x + B*a^2*b^3*d^4
)*e)*log(x*e + d))/(b^8*d^9*x^2 + 2*a*b^7*d^9*x + a^2*b^6*d^9 + (a^6*b^2*x^5 + 2*a^7*b*x^4 + a^8*x^3)*e^9 - 3*
(2*a^5*b^3*d*x^5 + 3*a^6*b^2*d*x^4 - a^8*d*x^2)*e^8 + 3*(5*a^4*b^4*d^2*x^5 + 4*a^5*b^3*d^2*x^4 - 6*a^6*b^2*d^2
*x^3 - 4*a^7*b*d^2*x^2 + a^8*d^2*x)*e^7 - (20*a^3*b^5*d^3*x^5 - 5*a^4*b^4*d^3*x^4 - 52*a^5*b^3*d^3*x^3 - 10*a^
6*b^2*d^3*x^2 + 16*a^7*b*d^3*x - a^8*d^3)*e^6 + 3*(5*a^2*b^6*d^4*x^5 - 10*a^3*b^5*d^4*x^4 - 20*a^4*b^4*d^4*x^3
+ 8*a^5*b^3*d^4*x^2 + 11*a^6*b^2*d^4*x - 2*a^7*b*d^4)*e^5 - 3*(2*a*b^7*d^5*x^5 - 11*a^2*b^6*d^5*x^4 - 8*a^3*b
^5*d^5*x^3 + 20*a^4*b^4*d^5*x^2 + 10*a^5*b^3*d^5*x - 5*a^6*b^2*d^5)*e^4 + (b^8*d^6*x^5 - 16*a*b^7*d^6*x^4 + 10
*a^2*b^6*d^6*x^3 + 52*a^3*b^5*d^6*x^2 + 5*a^4*b^4*d^6*x - 20*a^5*b^3*d^6)*e^3 + 3*(b^8*d^7*x^4 - 4*a*b^7*d^7*x
^3 - 6*a^2*b^6*d^7*x^2 + 4*a^3*b^5*d^7*x + 5*a^4*b^4*d^7)*e^2 + 3*(b^8*d^8*x^3 - 3*a^2*b^6*d^8*x - 2*a^3*b^5*d
^8)*e)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1975 vs. \(2 (245) = 490\).
time = 7.02, size = 1975, normalized size = 7.96 \begin {gather*} \frac {2 b^{2} e \left (- 5 A b e + 3 B a e + 2 B b d\right ) \log {\left (x + \frac {- 10 A a b^{3} e^{3} - 10 A b^{4} d e^{2} + 6 B a^{2} b^{2} e^{3} + 10 B a b^{3} d e^{2} + 4 B b^{4} d^{2} e - \frac {2 a^{7} b^{2} e^{8} \left (- 5 A b e + 3 B a e + 2 B b d\right )}{\left (a e - b d\right )^{6}} + \frac {14 a^{6} b^{3} d e^{7} \left (- 5 A b e + 3 B a e + 2 B b d\right )}{\left (a e - b d\right )^{6}} - \frac {42 a^{5} b^{4} d^{2} e^{6} \left (- 5 A b e + 3 B a e + 2 B b d\right )}{\left (a e - b d\right )^{6}} + \frac {70 a^{4} b^{5} d^{3} e^{5} \left (- 5 A b e + 3 B a e + 2 B b d\right )}{\left (a e - b d\right )^{6}} - \frac {70 a^{3} b^{6} d^{4} e^{4} \left (- 5 A b e + 3 B a e + 2 B b d\right )}{\left (a e - b d\right )^{6}} + \frac {42 a^{2} b^{7} d^{5} e^{3} \left (- 5 A b e + 3 B a e + 2 B b d\right )}{\left (a e - b d\right )^{6}} - \frac {14 a b^{8} d^{6} e^{2} \left (- 5 A b e + 3 B a e + 2 B b d\right )}{\left (a e - b d\right )^{6}} + \frac {2 b^{9} d^{7} e \left (- 5 A b e + 3 B a e + 2 B b d\right )}{\left (a e - b d\right )^{6}}}{- 20 A b^{4} e^{3} + 12 B a b^{3} e^{3} + 8 B b^{4} d e^{2}} \right )}}{\left (a e - b d\right )^{6}} - \frac {2 b^{2} e \left (- 5 A b e + 3 B a e + 2 B b d\right ) \log {\left (x + \frac {- 10 A a b^{3} e^{3} - 10 A b^{4} d e^{2} + 6 B a^{2} b^{2} e^{3} + 10 B a b^{3} d e^{2} + 4 B b^{4} d^{2} e + \frac {2 a^{7} b^{2} e^{8} \left (- 5 A b e + 3 B a e + 2 B b d\right )}{\left (a e - b d\right )^{6}} - \frac {14 a^{6} b^{3} d e^{7} \left (- 5 A b e + 3 B a e + 2 B b d\right )}{\left (a e - b d\right )^{6}} + \frac {42 a^{5} b^{4} d^{2} e^{6} \left (- 5 A b e + 3 B a e + 2 B b d\right )}{\left (a e - b d\right )^{6}} - \frac {70 a^{4} b^{5} d^{3} e^{5} \left (- 5 A b e + 3 B a e + 2 B b d\right )}{\left (a e - b d\right )^{6}} + \frac {70 a^{3} b^{6} d^{4} e^{4} \left (- 5 A b e + 3 B a e + 2 B b d\right )}{\left (a e - b d\right )^{6}} - \frac {42 a^{2} b^{7} d^{5} e^{3} \left (- 5 A b e + 3 B a e + 2 B b d\right )}{\left (a e - b d\right )^{6}} + \frac {14 a b^{8} d^{6} e^{2} \left (- 5 A b e + 3 B a e + 2 B b d\right )}{\left (a e - b d\right )^{6}} - \frac {2 b^{9} d^{7} e \left (- 5 A b e + 3 B a e + 2 B b d\right )}{\left (a e - b d\right )^{6}}}{- 20 A b^{4} e^{3} + 12 B a b^{3} e^{3} + 8 B b^{4} d e^{2}} \right )}}{\left (a e - b d\right )^{6}} + \frac {- 2 A a^{4} e^{4} + 13 A a^{3} b d e^{3} - 47 A a^{2} b^{2} d^{2} e^{2} - 27 A a b^{3} d^{3} e + 3 A b^{4} d^{4} - B a^{4} d e^{3} + 11 B a^{3} b d^{2} e^{2} + 47 B a^{2} b^{2} d^{3} e + 3 B a b^{3} d^{4} + x^{4} \left (- 60 A b^{4} e^{4} + 36 B a b^{3} e^{4} + 24 B b^{4} d e^{3}\right ) + x^{3} \left (- 90 A a b^{3} e^{4} - 150 A b^{4} d e^{3} + 54 B a^{2} b^{2} e^{4} + 126 B a b^{3} d e^{3} + 60 B b^{4} d^{2} e^{2}\right ) + x^{2} \left (- 20 A a^{2} b^{2} e^{4} - 230 A a b^{3} d e^{3} - 110 A b^{4} d^{2} e^{2} + 12 B a^{3} b e^{4} + 146 B a^{2} b^{2} d e^{3} + 158 B a b^{3} d^{2} e^{2} + 44 B b^{4} d^{3} e\right ) + x \left (5 A a^{3} b e^{4} - 55 A a^{2} b^{2} d e^{3} - 175 A a b^{3} d^{2} e^{2} - 15 A b^{4} d^{3} e - 3 B a^{4} e^{4} + 31 B a^{3} b d e^{3} + 127 B a^{2} b^{2} d^{2} e^{2} + 79 B a b^{3} d^{3} e + 6 B b^{4} d^{4}\right )}{6 a^{7} d^{3} e^{5} - 30 a^{6} b d^{4} e^{4} + 60 a^{5} b^{2} d^{5} e^{3} - 60 a^{4} b^{3} d^{6} e^{2} + 30 a^{3} b^{4} d^{7} e - 6 a^{2} b^{5} d^{8} + x^{5} \cdot \left (6 a^{5} b^{2} e^{8} - 30 a^{4} b^{3} d e^{7} + 60 a^{3} b^{4} d^{2} e^{6} - 60 a^{2} b^{5} d^{3} e^{5} + 30 a b^{6} d^{4} e^{4} - 6 b^{7} d^{5} e^{3}\right ) + x^{4} \cdot \left (12 a^{6} b e^{8} - 42 a^{5} b^{2} d e^{7} + 30 a^{4} b^{3} d^{2} e^{6} + 60 a^{3} b^{4} d^{3} e^{5} - 120 a^{2} b^{5} d^{4} e^{4} + 78 a b^{6} d^{5} e^{3} - 18 b^{7} d^{6} e^{2}\right ) + x^{3} \cdot \left (6 a^{7} e^{8} + 6 a^{6} b d e^{7} - 102 a^{5} b^{2} d^{2} e^{6} + 210 a^{4} b^{3} d^{3} e^{5} - 150 a^{3} b^{4} d^{4} e^{4} - 6 a^{2} b^{5} d^{5} e^{3} + 54 a b^{6} d^{6} e^{2} - 18 b^{7} d^{7} e\right ) + x^{2} \cdot \left (18 a^{7} d e^{7} - 54 a^{6} b d^{2} e^{6} + 6 a^{5} b^{2} d^{3} e^{5} + 150 a^{4} b^{3} d^{4} e^{4} - 210 a^{3} b^{4} d^{5} e^{3} + 102 a^{2} b^{5} d^{6} e^{2} - 6 a b^{6} d^{7} e - 6 b^{7} d^{8}\right ) + x \left (18 a^{7} d^{2} e^{6} - 78 a^{6} b d^{3} e^{5} + 120 a^{5} b^{2} d^{4} e^{4} - 60 a^{4} b^{3} d^{5} e^{3} - 30 a^{3} b^{4} d^{6} e^{2} + 42 a^{2} b^{5} d^{7} e - 12 a b^{6} d^{8}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)**3/(e*x+d)**4,x)

[Out]

2*b**2*e*(-5*A*b*e + 3*B*a*e + 2*B*b*d)*log(x + (-10*A*a*b**3*e**3 - 10*A*b**4*d*e**2 + 6*B*a**2*b**2*e**3 + 1
0*B*a*b**3*d*e**2 + 4*B*b**4*d**2*e - 2*a**7*b**2*e**8*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 + 14*a**6
*b**3*d*e**7*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 - 42*a**5*b**4*d**2*e**6*(-5*A*b*e + 3*B*a*e + 2*B*
b*d)/(a*e - b*d)**6 + 70*a**4*b**5*d**3*e**5*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 - 70*a**3*b**6*d**4
*e**4*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 + 42*a**2*b**7*d**5*e**3*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a
*e - b*d)**6 - 14*a*b**8*d**6*e**2*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 + 2*b**9*d**7*e*(-5*A*b*e + 3
*B*a*e + 2*B*b*d)/(a*e - b*d)**6)/(-20*A*b**4*e**3 + 12*B*a*b**3*e**3 + 8*B*b**4*d*e**2))/(a*e - b*d)**6 - 2*b
**2*e*(-5*A*b*e + 3*B*a*e + 2*B*b*d)*log(x + (-10*A*a*b**3*e**3 - 10*A*b**4*d*e**2 + 6*B*a**2*b**2*e**3 + 10*B
*a*b**3*d*e**2 + 4*B*b**4*d**2*e + 2*a**7*b**2*e**8*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 - 14*a**6*b*
*3*d*e**7*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 + 42*a**5*b**4*d**2*e**6*(-5*A*b*e + 3*B*a*e + 2*B*b*d
)/(a*e - b*d)**6 - 70*a**4*b**5*d**3*e**5*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 + 70*a**3*b**6*d**4*e*
*4*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 - 42*a**2*b**7*d**5*e**3*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e
- b*d)**6 + 14*a*b**8*d**6*e**2*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 - 2*b**9*d**7*e*(-5*A*b*e + 3*B*
a*e + 2*B*b*d)/(a*e - b*d)**6)/(-20*A*b**4*e**3 + 12*B*a*b**3*e**3 + 8*B*b**4*d*e**2))/(a*e - b*d)**6 + (-2*A*
a**4*e**4 + 13*A*a**3*b*d*e**3 - 47*A*a**2*b**2*d**2*e**2 - 27*A*a*b**3*d**3*e + 3*A*b**4*d**4 - B*a**4*d*e**3
+ 11*B*a**3*b*d**2*e**2 + 47*B*a**2*b**2*d**3*e + 3*B*a*b**3*d**4 + x**4*(-60*A*b**4*e**4 + 36*B*a*b**3*e**4
+ 24*B*b**4*d*e**3) + x**3*(-90*A*a*b**3*e**4 - 150*A*b**4*d*e**3 + 54*B*a**2*b**2*e**4 + 126*B*a*b**3*d*e**3
+ 60*B*b**4*d**2*e**2) + x**2*(-20*A*a**2*b**2*e**4 - 230*A*a*b**3*d*e**3 - 110*A*b**4*d**2*e**2 + 12*B*a**3*b
*e**4 + 146*B*a**2*b**2*d*e**3 + 158*B*a*b**3*d**2*e**2 + 44*B*b**4*d**3*e) + x*(5*A*a**3*b*e**4 - 55*A*a**2*b
**2*d*e**3 - 175*A*a*b**3*d**2*e**2 - 15*A*b**4*d**3*e - 3*B*a**4*e**4 + 31*B*a**3*b*d*e**3 + 127*B*a**2*b**2*
d**2*e**2 + 79*B*a*b**3*d**3*e + 6*B*b**4*d**4))/(6*a**7*d**3*e**5 - 30*a**6*b*d**4*e**4 + 60*a**5*b**2*d**5*e
**3 - 60*a**4*b**3*d**6*e**2 + 30*a**3*b**4*d**7*e - 6*a**2*b**5*d**8 + x**5*(6*a**5*b**2*e**8 - 30*a**4*b**3*
d*e**7 + 60*a**3*b**4*d**2*e**6 - 60*a**2*b**5*d**3*e**5 + 30*a*b**6*d**4*e**4 - 6*b**7*d**5*e**3) + x**4*(12*
a**6*b*e**8 - 42*a**5*b**2*d*e**7 + 30*a**4*b**3*d**2*e**6 + 60*a**3*b**4*d**3*e**5 - 120*a**2*b**5*d**4*e**4
+ 78*a*b**6*d**5*e**3 - 18*b**7*d**6*e**2) + x**3*(6*a**7*e**8 + 6*a**6*b*d*e**7 - 102*a**5*b**2*d**2*e**6 + 2
10*a**4*b**3*d**3*e**5 - 150*a**3*b**4*d**4*e**4 - 6*a**2*b**5*d**5*e**3 + 54*a*b**6*d**6*e**2 - 18*b**7*d**7*
e) + x**2*(18*a**7*d*e**7 - 54*a**6*b*d**2*e**6 + 6*a**5*b**2*d**3*e**5 + 150*a**4*b**3*d**4*e**4 - 210*a**3*b
**4*d**5*e**3 + 102*a**2*b**5*d**6*e**2 - 6*a*b**6*d**7*e - 6*b**7*d**8) + x*(18*a**7*d**2*e**6 - 78*a**6*b*d*
*3*e**5 + 120*a**5*b**2*d**4*e**4 - 60*a**4*b**3*d**5*e**3 - 30*a**3*b**4*d**6*e**2 + 42*a**2*b**5*d**7*e - 12
*a*b**6*d**8))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 760 vs. \(2 (269) = 538\).
time = 1.00, size = 760, normalized size = 3.06 \begin {gather*} -\frac {2 \, {\left (2 \, B b^{4} d e + 3 \, B a b^{3} e^{2} - 5 \, A b^{4} e^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}} + \frac {2 \, {\left (2 \, B b^{3} d e^{2} + 3 \, B a b^{2} e^{3} - 5 \, A b^{3} e^{3}\right )} \log \left ({\left | x e + d \right |}\right )}{b^{6} d^{6} e - 6 \, a b^{5} d^{5} e^{2} + 15 \, a^{2} b^{4} d^{4} e^{3} - 20 \, a^{3} b^{3} d^{3} e^{4} + 15 \, a^{4} b^{2} d^{2} e^{5} - 6 \, a^{5} b d e^{6} + a^{6} e^{7}} - \frac {3 \, B a b^{4} d^{5} + 3 \, A b^{5} d^{5} + 44 \, B a^{2} b^{3} d^{4} e - 30 \, A a b^{4} d^{4} e - 36 \, B a^{3} b^{2} d^{3} e^{2} - 20 \, A a^{2} b^{3} d^{3} e^{2} - 12 \, B a^{4} b d^{2} e^{3} + 60 \, A a^{3} b^{2} d^{2} e^{3} + B a^{5} d e^{4} - 15 \, A a^{4} b d e^{4} + 2 \, A a^{5} e^{5} + 12 \, {\left (2 \, B b^{5} d^{2} e^{3} + B a b^{4} d e^{4} - 5 \, A b^{5} d e^{4} - 3 \, B a^{2} b^{3} e^{5} + 5 \, A a b^{4} e^{5}\right )} x^{4} + 6 \, {\left (10 \, B b^{5} d^{3} e^{2} + 11 \, B a b^{4} d^{2} e^{3} - 25 \, A b^{5} d^{2} e^{3} - 12 \, B a^{2} b^{3} d e^{4} + 10 \, A a b^{4} d e^{4} - 9 \, B a^{3} b^{2} e^{5} + 15 \, A a^{2} b^{3} e^{5}\right )} x^{3} + 2 \, {\left (22 \, B b^{5} d^{4} e + 57 \, B a b^{4} d^{3} e^{2} - 55 \, A b^{5} d^{3} e^{2} - 6 \, B a^{2} b^{3} d^{2} e^{3} - 60 \, A a b^{4} d^{2} e^{3} - 67 \, B a^{3} b^{2} d e^{4} + 105 \, A a^{2} b^{3} d e^{4} - 6 \, B a^{4} b e^{5} + 10 \, A a^{3} b^{2} e^{5}\right )} x^{2} + {\left (6 \, B b^{5} d^{5} + 73 \, B a b^{4} d^{4} e - 15 \, A b^{5} d^{4} e + 48 \, B a^{2} b^{3} d^{3} e^{2} - 160 \, A a b^{4} d^{3} e^{2} - 96 \, B a^{3} b^{2} d^{2} e^{3} + 120 \, A a^{2} b^{3} d^{2} e^{3} - 34 \, B a^{4} b d e^{4} + 60 \, A a^{3} b^{2} d e^{4} + 3 \, B a^{5} e^{5} - 5 \, A a^{4} b e^{5}\right )} x}{6 \, {\left (b d - a e\right )}^{6} {\left (b x + a\right )}^{2} {\left (x e + d\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)^3/(e*x+d)^4,x, algorithm="giac")

[Out]

-2*(2*B*b^4*d*e + 3*B*a*b^3*e^2 - 5*A*b^4*e^2)*log(abs(b*x + a))/(b^7*d^6 - 6*a*b^6*d^5*e + 15*a^2*b^5*d^4*e^2
- 20*a^3*b^4*d^3*e^3 + 15*a^4*b^3*d^2*e^4 - 6*a^5*b^2*d*e^5 + a^6*b*e^6) + 2*(2*B*b^3*d*e^2 + 3*B*a*b^2*e^3 -
5*A*b^3*e^3)*log(abs(x*e + d))/(b^6*d^6*e - 6*a*b^5*d^5*e^2 + 15*a^2*b^4*d^4*e^3 - 20*a^3*b^3*d^3*e^4 + 15*a^
4*b^2*d^2*e^5 - 6*a^5*b*d*e^6 + a^6*e^7) - 1/6*(3*B*a*b^4*d^5 + 3*A*b^5*d^5 + 44*B*a^2*b^3*d^4*e - 30*A*a*b^4*
d^4*e - 36*B*a^3*b^2*d^3*e^2 - 20*A*a^2*b^3*d^3*e^2 - 12*B*a^4*b*d^2*e^3 + 60*A*a^3*b^2*d^2*e^3 + B*a^5*d*e^4
- 15*A*a^4*b*d*e^4 + 2*A*a^5*e^5 + 12*(2*B*b^5*d^2*e^3 + B*a*b^4*d*e^4 - 5*A*b^5*d*e^4 - 3*B*a^2*b^3*e^5 + 5*A
*a*b^4*e^5)*x^4 + 6*(10*B*b^5*d^3*e^2 + 11*B*a*b^4*d^2*e^3 - 25*A*b^5*d^2*e^3 - 12*B*a^2*b^3*d*e^4 + 10*A*a*b^
4*d*e^4 - 9*B*a^3*b^2*e^5 + 15*A*a^2*b^3*e^5)*x^3 + 2*(22*B*b^5*d^4*e + 57*B*a*b^4*d^3*e^2 - 55*A*b^5*d^3*e^2
- 6*B*a^2*b^3*d^2*e^3 - 60*A*a*b^4*d^2*e^3 - 67*B*a^3*b^2*d*e^4 + 105*A*a^2*b^3*d*e^4 - 6*B*a^4*b*e^5 + 10*A*a
^3*b^2*e^5)*x^2 + (6*B*b^5*d^5 + 73*B*a*b^4*d^4*e - 15*A*b^5*d^4*e + 48*B*a^2*b^3*d^3*e^2 - 160*A*a*b^4*d^3*e^
2 - 96*B*a^3*b^2*d^2*e^3 + 120*A*a^2*b^3*d^2*e^3 - 34*B*a^4*b*d*e^4 + 60*A*a^3*b^2*d*e^4 + 3*B*a^5*e^5 - 5*A*a
^4*b*e^5)*x)/((b*d - a*e)^6*(b*x + a)^2*(x*e + d)^3)

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Mupad [B]
time = 1.91, size = 1030, normalized size = 4.15 \begin {gather*} \frac {\frac {-B\,a^4\,d\,e^3-2\,A\,a^4\,e^4+11\,B\,a^3\,b\,d^2\,e^2+13\,A\,a^3\,b\,d\,e^3+47\,B\,a^2\,b^2\,d^3\,e-47\,A\,a^2\,b^2\,d^2\,e^2+3\,B\,a\,b^3\,d^4-27\,A\,a\,b^3\,d^3\,e+3\,A\,b^4\,d^4}{6\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {x\,\left (3\,B\,a\,e-5\,A\,b\,e+2\,B\,b\,d\right )\,\left (-a^3\,e^3+11\,a^2\,b\,d\,e^2+35\,a\,b^2\,d^2\,e+3\,b^3\,d^3\right )}{6\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {2\,b^3\,e^3\,x^4\,\left (3\,B\,a\,e-5\,A\,b\,e+2\,B\,b\,d\right )}{a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5}+\frac {b\,x^3\,\left (5\,d\,b^2\,e^2+3\,a\,b\,e^3\right )\,\left (3\,B\,a\,e-5\,A\,b\,e+2\,B\,b\,d\right )}{a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5}+\frac {b\,x^2\,\left (3\,B\,a\,e-5\,A\,b\,e+2\,B\,b\,d\right )\,\left (2\,a^2\,e^3+23\,a\,b\,d\,e^2+11\,b^2\,d^2\,e\right )}{3\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}}{x^2\,\left (3\,a^2\,d\,e^2+6\,a\,b\,d^2\,e+b^2\,d^3\right )+x^3\,\left (a^2\,e^3+6\,a\,b\,d\,e^2+3\,b^2\,d^2\,e\right )+x\,\left (3\,e\,a^2\,d^2+2\,b\,a\,d^3\right )+x^4\,\left (3\,d\,b^2\,e^2+2\,a\,b\,e^3\right )+a^2\,d^3+b^2\,e^3\,x^5}-\frac {2\,\mathrm {atanh}\left (\frac {\left (2\,b^2\,e^2\,\left (5\,A\,b-3\,B\,a\right )-4\,B\,b^3\,d\,e\right )\,\left (a^6\,e^6-4\,a^5\,b\,d\,e^5+5\,a^4\,b^2\,d^2\,e^4-5\,a^2\,b^4\,d^4\,e^2+4\,a\,b^5\,d^5\,e-b^6\,d^6\right )}{{\left (a\,e-b\,d\right )}^6\,\left (-10\,A\,b^3\,e^2+4\,B\,d\,b^3\,e+6\,B\,a\,b^2\,e^2\right )}+\frac {2\,b\,e\,x\,\left (2\,b^2\,e^2\,\left (5\,A\,b-3\,B\,a\right )-4\,B\,b^3\,d\,e\right )\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}{{\left (a\,e-b\,d\right )}^6\,\left (-10\,A\,b^3\,e^2+4\,B\,d\,b^3\,e+6\,B\,a\,b^2\,e^2\right )}\right )\,\left (2\,b^2\,e^2\,\left (5\,A\,b-3\,B\,a\right )-4\,B\,b^3\,d\,e\right )}{{\left (a\,e-b\,d\right )}^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/((a + b*x)^3*(d + e*x)^4),x)

[Out]

((3*A*b^4*d^4 - 2*A*a^4*e^4 + 3*B*a*b^3*d^4 - B*a^4*d*e^3 + 47*B*a^2*b^2*d^3*e + 11*B*a^3*b*d^2*e^2 - 47*A*a^2
*b^2*d^2*e^2 - 27*A*a*b^3*d^3*e + 13*A*a^3*b*d*e^3)/(6*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^
2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4)) + (x*(3*B*a*e - 5*A*b*e + 2*B*b*d)*(3*b^3*d^3 - a^3*e^3 + 35*a*b^2*d^2
*e + 11*a^2*b*d*e^2))/(6*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*
b*d*e^4)) + (2*b^3*e^3*x^4*(3*B*a*e - 5*A*b*e + 2*B*b*d))/(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2
*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4) + (b*x^3*(5*b^2*d*e^2 + 3*a*b*e^3)*(3*B*a*e - 5*A*b*e + 2*B*b*d))/(a
^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4) + (b*x^2*(3*B*a*e
- 5*A*b*e + 2*B*b*d)*(2*a^2*e^3 + 11*b^2*d^2*e + 23*a*b*d*e^2))/(3*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 1
0*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4)))/(x^2*(b^2*d^3 + 3*a^2*d*e^2 + 6*a*b*d^2*e) + x^3*(a^2*e^3
+ 3*b^2*d^2*e + 6*a*b*d*e^2) + x*(3*a^2*d^2*e + 2*a*b*d^3) + x^4*(3*b^2*d*e^2 + 2*a*b*e^3) + a^2*d^3 + b^2*e^
3*x^5) - (2*atanh(((2*b^2*e^2*(5*A*b - 3*B*a) - 4*B*b^3*d*e)*(a^6*e^6 - b^6*d^6 - 5*a^2*b^4*d^4*e^2 + 5*a^4*b^
2*d^2*e^4 + 4*a*b^5*d^5*e - 4*a^5*b*d*e^5))/((a*e - b*d)^6*(4*B*b^3*d*e - 10*A*b^3*e^2 + 6*B*a*b^2*e^2)) + (2*
b*e*x*(2*b^2*e^2*(5*A*b - 3*B*a) - 4*B*b^3*d*e)*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 +
5*a*b^4*d^4*e - 5*a^4*b*d*e^4))/((a*e - b*d)^6*(4*B*b^3*d*e - 10*A*b^3*e^2 + 6*B*a*b^2*e^2)))*(2*b^2*e^2*(5*A
*b - 3*B*a) - 4*B*b^3*d*e))/(a*e - b*d)^6

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