3.11.2 \(\int \frac {A+B x}{(a+b x)^3 (d+e x)^3} \, dx\)

Optimal. Leaf size=199 \[ -\frac {b (A b-a B)}{2 (a+b x)^2 (b d-a e)^3}-\frac {b (2 a B e-3 A b e+b B d)}{(a+b x) (b d-a e)^4}-\frac {e (a B e-3 A b e+2 b B d)}{(d+e x) (b d-a e)^4}-\frac {e (B d-A e)}{2 (d+e x)^2 (b d-a e)^3}-\frac {3 b e \log (a+b x) (a B e-2 A b e+b B d)}{(b d-a e)^5}+\frac {3 b e \log (d+e x) (a B e-2 A b e+b B d)}{(b d-a e)^5} \]

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Rubi [A]  time = 0.22, antiderivative size = 199, 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 (A b-a B)}{2 (a+b x)^2 (b d-a e)^3}-\frac {b (2 a B e-3 A b e+b B d)}{(a+b x) (b d-a e)^4}-\frac {e (a B e-3 A b e+2 b B d)}{(d+e x) (b d-a e)^4}-\frac {e (B d-A e)}{2 (d+e x)^2 (b d-a e)^3}-\frac {3 b e \log (a+b x) (a B e-2 A b e+b B d)}{(b d-a e)^5}+\frac {3 b e \log (d+e x) (a B e-2 A b e+b B d)}{(b d-a e)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.22, size = 185, normalized size = 0.93 \begin {gather*} \frac {-\frac {b (A b-a B) (b d-a e)^2}{(a+b x)^2}+\frac {e (b d-a e)^2 (A e-B d)}{(d+e x)^2}-\frac {2 b (b d-a e) (2 a B e-3 A b e+b B d)}{a+b x}+\frac {2 e (b d-a e) (-a B e+3 A b e-2 b B d)}{d+e x}-6 b e \log (a+b x) (a B e-2 A b e+b B d)+6 b e \log (d+e x) (a B e-2 A b e+b B d)}{2 (b d-a e)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x}{(a+b x)^3 (d+e x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.53, size = 1215, normalized size = 6.11

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(9*B*a^3*b*d^2*e^2 + A*a^4*e^4 - (B*a*b^3 + A*b^4)*d^4 - (9*B*a^2*b^2 - 8*A*a*b^3)*d^3*e + (B*a^4 - 8*A*a^
3*b)*d*e^3 - 6*(B*b^4*d^2*e^2 - 2*A*b^4*d*e^3 - (B*a^2*b^2 - 2*A*a*b^3)*e^4)*x^3 - 9*(B*b^4*d^3*e - B*a^2*b^2*
d*e^3 + (B*a*b^3 - 2*A*b^4)*d^2*e^2 - (B*a^3*b - 2*A*a^2*b^2)*e^4)*x^2 - 2*(B*b^4*d^4 - 12*A*a*b^3*d^2*e^2 + (
7*B*a*b^3 - 2*A*b^4)*d^3*e - (7*B*a^3*b - 12*A*a^2*b^2)*d*e^3 - (B*a^4 - 2*A*a^3*b)*e^4)*x - 6*(B*a^2*b^2*d^3*
e + (B*a^3*b - 2*A*a^2*b^2)*d^2*e^2 + (B*b^4*d*e^3 + (B*a*b^3 - 2*A*b^4)*e^4)*x^4 + 2*(B*b^4*d^2*e^2 + 2*(B*a*
b^3 - A*b^4)*d*e^3 + (B*a^2*b^2 - 2*A*a*b^3)*e^4)*x^3 + (B*b^4*d^3*e + (5*B*a*b^3 - 2*A*b^4)*d^2*e^2 + (5*B*a^
2*b^2 - 8*A*a*b^3)*d*e^3 + (B*a^3*b - 2*A*a^2*b^2)*e^4)*x^2 + 2*(B*a*b^3*d^3*e + 2*(B*a^2*b^2 - A*a*b^3)*d^2*e
^2 + (B*a^3*b - 2*A*a^2*b^2)*d*e^3)*x)*log(b*x + a) + 6*(B*a^2*b^2*d^3*e + (B*a^3*b - 2*A*a^2*b^2)*d^2*e^2 + (
B*b^4*d*e^3 + (B*a*b^3 - 2*A*b^4)*e^4)*x^4 + 2*(B*b^4*d^2*e^2 + 2*(B*a*b^3 - A*b^4)*d*e^3 + (B*a^2*b^2 - 2*A*a
*b^3)*e^4)*x^3 + (B*b^4*d^3*e + (5*B*a*b^3 - 2*A*b^4)*d^2*e^2 + (5*B*a^2*b^2 - 8*A*a*b^3)*d*e^3 + (B*a^3*b - 2
*A*a^2*b^2)*e^4)*x^2 + 2*(B*a*b^3*d^3*e + 2*(B*a^2*b^2 - A*a*b^3)*d^2*e^2 + (B*a^3*b - 2*A*a^2*b^2)*d*e^3)*x)*
log(e*x + d))/(a^2*b^5*d^7 - 5*a^3*b^4*d^6*e + 10*a^4*b^3*d^5*e^2 - 10*a^5*b^2*d^4*e^3 + 5*a^6*b*d^3*e^4 - a^7
*d^2*e^5 + (b^7*d^5*e^2 - 5*a*b^6*d^4*e^3 + 10*a^2*b^5*d^3*e^4 - 10*a^3*b^4*d^2*e^5 + 5*a^4*b^3*d*e^6 - a^5*b^
2*e^7)*x^4 + 2*(b^7*d^6*e - 4*a*b^6*d^5*e^2 + 5*a^2*b^5*d^4*e^3 - 5*a^4*b^3*d^2*e^5 + 4*a^5*b^2*d*e^6 - a^6*b*
e^7)*x^3 + (b^7*d^7 - a*b^6*d^6*e - 9*a^2*b^5*d^5*e^2 + 25*a^3*b^4*d^4*e^3 - 25*a^4*b^3*d^3*e^4 + 9*a^5*b^2*d^
2*e^5 + a^6*b*d*e^6 - a^7*e^7)*x^2 + 2*(a*b^6*d^7 - 4*a^2*b^5*d^6*e + 5*a^3*b^4*d^5*e^2 - 5*a^5*b^2*d^3*e^4 +
4*a^6*b*d^2*e^5 - a^7*d*e^6)*x)

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giac [B]  time = 1.25, size = 513, normalized size = 2.58 \begin {gather*} -\frac {3 \, {\left (B b^{3} d e + B a b^{2} e^{2} - 2 \, A b^{3} e^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}} + \frac {3 \, {\left (B b^{2} d e^{2} + B a b e^{3} - 2 \, A b^{2} e^{3}\right )} \log \left ({\left | x e + d \right |}\right )}{b^{5} d^{5} e - 5 \, a b^{4} d^{4} e^{2} + 10 \, a^{2} b^{3} d^{3} e^{3} - 10 \, a^{3} b^{2} d^{2} e^{4} + 5 \, a^{4} b d e^{5} - a^{5} e^{6}} - \frac {6 \, B b^{3} d x^{3} e^{2} + 9 \, B b^{3} d^{2} x^{2} e + 2 \, B b^{3} d^{3} x + 6 \, B a b^{2} x^{3} e^{3} - 12 \, A b^{3} x^{3} e^{3} + 18 \, B a b^{2} d x^{2} e^{2} - 18 \, A b^{3} d x^{2} e^{2} + 16 \, B a b^{2} d^{2} x e - 4 \, A b^{3} d^{2} x e + B a b^{2} d^{3} + A b^{3} d^{3} + 9 \, B a^{2} b x^{2} e^{3} - 18 \, A a b^{2} x^{2} e^{3} + 16 \, B a^{2} b d x e^{2} - 28 \, A a b^{2} d x e^{2} + 10 \, B a^{2} b d^{2} e - 7 \, A a b^{2} d^{2} e + 2 \, B a^{3} x e^{3} - 4 \, A a^{2} b x e^{3} + B a^{3} d e^{2} - 7 \, A a^{2} b d e^{2} + A a^{3} e^{3}}{2 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left (b x^{2} e + b d x + a x e + a d\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*(B*b^3*d*e + B*a*b^2*e^2 - 2*A*b^3*e^2)*log(abs(b*x + a))/(b^6*d^5 - 5*a*b^5*d^4*e + 10*a^2*b^4*d^3*e^2 - 1
0*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^4 - a^5*b*e^5) + 3*(B*b^2*d*e^2 + B*a*b*e^3 - 2*A*b^2*e^3)*log(abs(x*e + d))
/(b^5*d^5*e - 5*a*b^4*d^4*e^2 + 10*a^2*b^3*d^3*e^3 - 10*a^3*b^2*d^2*e^4 + 5*a^4*b*d*e^5 - a^5*e^6) - 1/2*(6*B*
b^3*d*x^3*e^2 + 9*B*b^3*d^2*x^2*e + 2*B*b^3*d^3*x + 6*B*a*b^2*x^3*e^3 - 12*A*b^3*x^3*e^3 + 18*B*a*b^2*d*x^2*e^
2 - 18*A*b^3*d*x^2*e^2 + 16*B*a*b^2*d^2*x*e - 4*A*b^3*d^2*x*e + B*a*b^2*d^3 + A*b^3*d^3 + 9*B*a^2*b*x^2*e^3 -
18*A*a*b^2*x^2*e^3 + 16*B*a^2*b*d*x*e^2 - 28*A*a*b^2*d*x*e^2 + 10*B*a^2*b*d^2*e - 7*A*a*b^2*d^2*e + 2*B*a^3*x*
e^3 - 4*A*a^2*b*x*e^3 + B*a^3*d*e^2 - 7*A*a^2*b*d*e^2 + A*a^3*e^3)/((b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e
^2 - 4*a^3*b*d*e^3 + a^4*e^4)*(b*x^2*e + b*d*x + a*x*e + a*d)^2)

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maple [A]  time = 0.02, size = 380, normalized size = 1.91 \begin {gather*} -\frac {6 A \,b^{2} e^{2} \ln \left (b x +a \right )}{\left (a e -b d \right )^{5}}+\frac {6 A \,b^{2} e^{2} \ln \left (e x +d \right )}{\left (a e -b d \right )^{5}}+\frac {3 B a b \,e^{2} \ln \left (b x +a \right )}{\left (a e -b d \right )^{5}}-\frac {3 B a b \,e^{2} \ln \left (e x +d \right )}{\left (a e -b d \right )^{5}}+\frac {3 B \,b^{2} d e \ln \left (b x +a \right )}{\left (a e -b d \right )^{5}}-\frac {3 B \,b^{2} d e \ln \left (e x +d \right )}{\left (a e -b d \right )^{5}}+\frac {3 A \,b^{2} e}{\left (a e -b d \right )^{4} \left (b x +a \right )}+\frac {3 A b \,e^{2}}{\left (a e -b d \right )^{4} \left (e x +d \right )}-\frac {2 B a b e}{\left (a e -b d \right )^{4} \left (b x +a \right )}-\frac {B a \,e^{2}}{\left (a e -b d \right )^{4} \left (e x +d \right )}-\frac {B \,b^{2} d}{\left (a e -b d \right )^{4} \left (b x +a \right )}-\frac {2 B b d e}{\left (a e -b d \right )^{4} \left (e x +d \right )}+\frac {A \,b^{2}}{2 \left (a e -b d \right )^{3} \left (b x +a \right )^{2}}-\frac {A \,e^{2}}{2 \left (a e -b d \right )^{3} \left (e x +d \right )^{2}}-\frac {B a b}{2 \left (a e -b d \right )^{3} \left (b x +a \right )^{2}}+\frac {B d e}{2 \left (a e -b d \right )^{3} \left (e x +d \right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.74, size = 745, normalized size = 3.74 \begin {gather*} -\frac {3 \, {\left (B b^{2} d e + {\left (B a b - 2 \, A b^{2}\right )} e^{2}\right )} \log \left (b x + a\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} + \frac {3 \, {\left (B b^{2} d e + {\left (B a b - 2 \, A b^{2}\right )} e^{2}\right )} \log \left (e x + d\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac {A a^{3} e^{3} + {\left (B a b^{2} + A b^{3}\right )} d^{3} + {\left (10 \, B a^{2} b - 7 \, A a b^{2}\right )} d^{2} e + {\left (B a^{3} - 7 \, A a^{2} b\right )} d e^{2} + 6 \, {\left (B b^{3} d e^{2} + {\left (B a b^{2} - 2 \, A b^{3}\right )} e^{3}\right )} x^{3} + 9 \, {\left (B b^{3} d^{2} e + 2 \, {\left (B a b^{2} - A b^{3}\right )} d e^{2} + {\left (B a^{2} b - 2 \, A a b^{2}\right )} e^{3}\right )} x^{2} + 2 \, {\left (B b^{3} d^{3} + 2 \, {\left (4 \, B a b^{2} - A b^{3}\right )} d^{2} e + 2 \, {\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} d e^{2} + {\left (B a^{3} - 2 \, A a^{2} b\right )} e^{3}\right )} x}{2 \, {\left (a^{2} b^{4} d^{6} - 4 \, a^{3} b^{3} d^{5} e + 6 \, a^{4} b^{2} d^{4} e^{2} - 4 \, a^{5} b d^{3} e^{3} + a^{6} d^{2} e^{4} + {\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{4} + 2 \, {\left (b^{6} d^{5} e - 3 \, a b^{5} d^{4} e^{2} + 2 \, a^{2} b^{4} d^{3} e^{3} + 2 \, a^{3} b^{3} d^{2} e^{4} - 3 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x^{3} + {\left (b^{6} d^{6} - 9 \, a^{2} b^{4} d^{4} e^{2} + 16 \, a^{3} b^{3} d^{3} e^{3} - 9 \, a^{4} b^{2} d^{2} e^{4} + a^{6} e^{6}\right )} x^{2} + 2 \, {\left (a b^{5} d^{6} - 3 \, a^{2} b^{4} d^{5} e + 2 \, a^{3} b^{3} d^{4} e^{2} + 2 \, a^{4} b^{2} d^{3} e^{3} - 3 \, a^{5} b d^{2} e^{4} + a^{6} d e^{5}\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)^3,x, algorithm="maxima")

[Out]

-3*(B*b^2*d*e + (B*a*b - 2*A*b^2)*e^2)*log(b*x + a)/(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2
*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5) + 3*(B*b^2*d*e + (B*a*b - 2*A*b^2)*e^2)*log(e*x + d)/(b^5*d^5 - 5*a*b^4*d^
4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5) - 1/2*(A*a^3*e^3 + (B*a*b^2 + A*b^3)*
d^3 + (10*B*a^2*b - 7*A*a*b^2)*d^2*e + (B*a^3 - 7*A*a^2*b)*d*e^2 + 6*(B*b^3*d*e^2 + (B*a*b^2 - 2*A*b^3)*e^3)*x
^3 + 9*(B*b^3*d^2*e + 2*(B*a*b^2 - A*b^3)*d*e^2 + (B*a^2*b - 2*A*a*b^2)*e^3)*x^2 + 2*(B*b^3*d^3 + 2*(4*B*a*b^2
- A*b^3)*d^2*e + 2*(4*B*a^2*b - 7*A*a*b^2)*d*e^2 + (B*a^3 - 2*A*a^2*b)*e^3)*x)/(a^2*b^4*d^6 - 4*a^3*b^3*d^5*e
+ 6*a^4*b^2*d^4*e^2 - 4*a^5*b*d^3*e^3 + a^6*d^2*e^4 + (b^6*d^4*e^2 - 4*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 - 4*
a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^4 + 2*(b^6*d^5*e - 3*a*b^5*d^4*e^2 + 2*a^2*b^4*d^3*e^3 + 2*a^3*b^3*d^2*e^4 - 3*
a^4*b^2*d*e^5 + a^5*b*e^6)*x^3 + (b^6*d^6 - 9*a^2*b^4*d^4*e^2 + 16*a^3*b^3*d^3*e^3 - 9*a^4*b^2*d^2*e^4 + a^6*e
^6)*x^2 + 2*(a*b^5*d^6 - 3*a^2*b^4*d^5*e + 2*a^3*b^3*d^4*e^2 + 2*a^4*b^2*d^3*e^3 - 3*a^5*b*d^2*e^4 + a^6*d*e^5
)*x)

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mupad [B]  time = 1.67, size = 726, normalized size = 3.65 \begin {gather*} \frac {2\,\mathrm {atanh}\left (\frac {\left (3\,b\,e^2\,\left (2\,A\,b-B\,a\right )-3\,B\,b^2\,d\,e\right )\,\left (\frac {a^5\,e^5-3\,a^4\,b\,d\,e^4+2\,a^3\,b^2\,d^2\,e^3+2\,a^2\,b^3\,d^3\,e^2-3\,a\,b^4\,d^4\,e+b^5\,d^5}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}+2\,b\,e\,x\right )\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{{\left (a\,e-b\,d\right )}^5\,\left (-6\,A\,b^2\,e^2+3\,B\,d\,b^2\,e+3\,B\,a\,b\,e^2\right )}\right )\,\left (3\,b\,e^2\,\left (2\,A\,b-B\,a\right )-3\,B\,b^2\,d\,e\right )}{{\left (a\,e-b\,d\right )}^5}-\frac {\frac {B\,a^3\,d\,e^2+A\,a^3\,e^3+10\,B\,a^2\,b\,d^2\,e-7\,A\,a^2\,b\,d\,e^2+B\,a\,b^2\,d^3-7\,A\,a\,b^2\,d^2\,e+A\,b^3\,d^3}{2\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}+\frac {9\,x^2\,\left (d\,b^2\,e+a\,b\,e^2\right )\,\left (B\,a\,e-2\,A\,b\,e+B\,b\,d\right )}{2\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}+\frac {x\,\left (a^2\,e^2+7\,a\,b\,d\,e+b^2\,d^2\right )\,\left (B\,a\,e-2\,A\,b\,e+B\,b\,d\right )}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}+\frac {3\,b^2\,e^2\,x^3\,\left (B\,a\,e-2\,A\,b\,e+B\,b\,d\right )}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}}{x\,\left (2\,e\,a^2\,d+2\,b\,a\,d^2\right )+x^2\,\left (a^2\,e^2+4\,a\,b\,d\,e+b^2\,d^2\right )+x^3\,\left (2\,d\,b^2\,e+2\,a\,b\,e^2\right )+a^2\,d^2+b^2\,e^2\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*atanh(((3*b*e^2*(2*A*b - B*a) - 3*B*b^2*d*e)*((a^5*e^5 + b^5*d^5 + 2*a^2*b^3*d^3*e^2 + 2*a^3*b^2*d^2*e^3 -
3*a*b^4*d^4*e - 3*a^4*b*d*e^4)/(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3) + 2*b*e
*x)*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3))/((a*e - b*d)^5*(3*B*a*b*e^2 - 6*A
*b^2*e^2 + 3*B*b^2*d*e)))*(3*b*e^2*(2*A*b - B*a) - 3*B*b^2*d*e))/(a*e - b*d)^5 - ((A*a^3*e^3 + A*b^3*d^3 + B*a
*b^2*d^3 + B*a^3*d*e^2 - 7*A*a*b^2*d^2*e - 7*A*a^2*b*d*e^2 + 10*B*a^2*b*d^2*e)/(2*(a^4*e^4 + b^4*d^4 + 6*a^2*b
^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3)) + (9*x^2*(a*b*e^2 + b^2*d*e)*(B*a*e - 2*A*b*e + B*b*d))/(2*(a^4*e
^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3)) + (x*(a^2*e^2 + b^2*d^2 + 7*a*b*d*e)*(B*a*e
- 2*A*b*e + B*b*d))/(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3) + (3*b^2*e^2*x^3*
(B*a*e - 2*A*b*e + B*b*d))/(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3))/(x*(2*a*b*
d^2 + 2*a^2*d*e) + x^2*(a^2*e^2 + b^2*d^2 + 4*a*b*d*e) + x^3*(2*a*b*e^2 + 2*b^2*d*e) + a^2*d^2 + b^2*e^2*x^4)

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sympy [B]  time = 5.67, size = 1431, normalized size = 7.19

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*b*e*(-2*A*b*e + B*a*e + B*b*d)*log(x + (-6*A*a*b**2*e**3 - 6*A*b**3*d*e**2 + 3*B*a**2*b*e**3 + 6*B*a*b**2*d
*e**2 + 3*B*b**3*d**2*e - 3*a**6*b*e**7*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**5 + 18*a**5*b**2*d*e**6*(-2*A*
b*e + B*a*e + B*b*d)/(a*e - b*d)**5 - 45*a**4*b**3*d**2*e**5*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**5 + 60*a*
*3*b**4*d**3*e**4*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**5 - 45*a**2*b**5*d**4*e**3*(-2*A*b*e + B*a*e + B*b*d
)/(a*e - b*d)**5 + 18*a*b**6*d**5*e**2*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**5 - 3*b**7*d**6*e*(-2*A*b*e + B
*a*e + B*b*d)/(a*e - b*d)**5)/(-12*A*b**3*e**3 + 6*B*a*b**2*e**3 + 6*B*b**3*d*e**2))/(a*e - b*d)**5 + 3*b*e*(-
2*A*b*e + B*a*e + B*b*d)*log(x + (-6*A*a*b**2*e**3 - 6*A*b**3*d*e**2 + 3*B*a**2*b*e**3 + 6*B*a*b**2*d*e**2 + 3
*B*b**3*d**2*e + 3*a**6*b*e**7*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**5 - 18*a**5*b**2*d*e**6*(-2*A*b*e + B*a
*e + B*b*d)/(a*e - b*d)**5 + 45*a**4*b**3*d**2*e**5*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**5 - 60*a**3*b**4*d
**3*e**4*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**5 + 45*a**2*b**5*d**4*e**3*(-2*A*b*e + B*a*e + B*b*d)/(a*e -
b*d)**5 - 18*a*b**6*d**5*e**2*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**5 + 3*b**7*d**6*e*(-2*A*b*e + B*a*e + B*
b*d)/(a*e - b*d)**5)/(-12*A*b**3*e**3 + 6*B*a*b**2*e**3 + 6*B*b**3*d*e**2))/(a*e - b*d)**5 + (-A*a**3*e**3 + 7
*A*a**2*b*d*e**2 + 7*A*a*b**2*d**2*e - A*b**3*d**3 - B*a**3*d*e**2 - 10*B*a**2*b*d**2*e - B*a*b**2*d**3 + x**3
*(12*A*b**3*e**3 - 6*B*a*b**2*e**3 - 6*B*b**3*d*e**2) + x**2*(18*A*a*b**2*e**3 + 18*A*b**3*d*e**2 - 9*B*a**2*b
*e**3 - 18*B*a*b**2*d*e**2 - 9*B*b**3*d**2*e) + x*(4*A*a**2*b*e**3 + 28*A*a*b**2*d*e**2 + 4*A*b**3*d**2*e - 2*
B*a**3*e**3 - 16*B*a**2*b*d*e**2 - 16*B*a*b**2*d**2*e - 2*B*b**3*d**3))/(2*a**6*d**2*e**4 - 8*a**5*b*d**3*e**3
+ 12*a**4*b**2*d**4*e**2 - 8*a**3*b**3*d**5*e + 2*a**2*b**4*d**6 + x**4*(2*a**4*b**2*e**6 - 8*a**3*b**3*d*e**
5 + 12*a**2*b**4*d**2*e**4 - 8*a*b**5*d**3*e**3 + 2*b**6*d**4*e**2) + x**3*(4*a**5*b*e**6 - 12*a**4*b**2*d*e**
5 + 8*a**3*b**3*d**2*e**4 + 8*a**2*b**4*d**3*e**3 - 12*a*b**5*d**4*e**2 + 4*b**6*d**5*e) + x**2*(2*a**6*e**6 -
18*a**4*b**2*d**2*e**4 + 32*a**3*b**3*d**3*e**3 - 18*a**2*b**4*d**4*e**2 + 2*b**6*d**6) + x*(4*a**6*d*e**5 -
12*a**5*b*d**2*e**4 + 8*a**4*b**2*d**3*e**3 + 8*a**3*b**3*d**4*e**2 - 12*a**2*b**4*d**5*e + 4*a*b**5*d**6))

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