3.10.93 \(\int \frac {A+B x}{(a+b x)^2 (d+e x)^5} \, dx\)

Optimal. Leaf size=239 \[ -\frac {b^3 (A b-a B)}{(a+b x) (b d-a e)^5}+\frac {b^3 \log (a+b x) (4 a B e-5 A b e+b B d)}{(b d-a e)^6}-\frac {b^3 \log (d+e x) (4 a B e-5 A b e+b B d)}{(b d-a e)^6}+\frac {b^2 (3 a B e-4 A b e+b B d)}{(d+e x) (b d-a e)^5}+\frac {b (2 a B e-3 A b e+b B d)}{2 (d+e x)^2 (b d-a e)^4}+\frac {a B e-2 A b e+b B d}{3 (d+e x)^3 (b d-a e)^3}+\frac {B d-A e}{4 (d+e x)^4 (b d-a e)^2} \]

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Rubi [A]  time = 0.28, antiderivative size = 239, 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^3 (A b-a B)}{(a+b x) (b d-a e)^5}+\frac {b^2 (3 a B e-4 A b e+b B d)}{(d+e x) (b d-a e)^5}+\frac {b^3 \log (a+b x) (4 a B e-5 A b e+b B d)}{(b d-a e)^6}-\frac {b^3 \log (d+e x) (4 a B e-5 A b e+b B d)}{(b d-a e)^6}+\frac {b (2 a B e-3 A b e+b B d)}{2 (d+e x)^2 (b d-a e)^4}+\frac {a B e-2 A b e+b B d}{3 (d+e x)^3 (b d-a e)^3}+\frac {B d-A e}{4 (d+e x)^4 (b d-a e)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((a + b*x)^2*(d + e*x)^5),x]

[Out]

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

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

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Mathematica [A]  time = 0.28, size = 225, normalized size = 0.94 \begin {gather*} \frac {-\frac {12 b^3 (A b-a B) (b d-a e)}{a+b x}+12 b^3 \log (a+b x) (4 a B e-5 A b e+b B d)-12 b^3 \log (d+e x) (4 a B e-5 A b e+b B d)+\frac {12 b^2 (b d-a e) (3 a B e-4 A b e+b B d)}{d+e x}+\frac {3 (b d-a e)^4 (B d-A e)}{(d+e x)^4}+\frac {4 (b d-a e)^3 (a B e-2 A b e+b B d)}{(d+e x)^3}+\frac {6 b (b d-a e)^2 (2 a B e-3 A b e+b B d)}{(d+e x)^2}}{12 (b d-a e)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((a + b*x)^2*(d + e*x)^5),x]

[Out]

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

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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)^2 (d+e x)^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(A + B*x)/((a + b*x)^2*(d + e*x)^5),x]

[Out]

IntegrateAlgebraic[(A + B*x)/((a + b*x)^2*(d + e*x)^5), x]

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fricas [B]  time = 0.65, size = 1702, normalized size = 7.12

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)^2/(e*x+d)^5,x, algorithm="fricas")

[Out]

-1/12*(3*A*a^5*e^5 - (37*B*a*b^4 - 12*A*b^5)*d^5 + (8*B*a^2*b^3 + 65*A*a*b^4)*d^4*e + 12*(3*B*a^3*b^2 - 10*A*a
^2*b^3)*d^3*e^2 - 4*(2*B*a^4*b - 15*A*a^3*b^2)*d^2*e^3 + (B*a^5 - 20*A*a^4*b)*d*e^4 - 12*(B*b^5*d^2*e^3 + (3*B
*a*b^4 - 5*A*b^5)*d*e^4 - (4*B*a^2*b^3 - 5*A*a*b^4)*e^5)*x^4 - 6*(7*B*b^5*d^3*e^2 + (22*B*a*b^4 - 35*A*b^5)*d^
2*e^3 - 5*(5*B*a^2*b^3 - 6*A*a*b^4)*d*e^4 - (4*B*a^3*b^2 - 5*A*a^2*b^3)*e^5)*x^3 - 2*(26*B*b^5*d^4*e + (89*B*a
*b^4 - 130*A*b^5)*d^3*e^2 - 3*(24*B*a^2*b^3 - 25*A*a*b^4)*d^2*e^3 - (47*B*a^3*b^2 - 60*A*a^2*b^3)*d*e^4 + (4*B
*a^4*b - 5*A*a^3*b^2)*e^5)*x^2 - (25*B*b^5*d^5 + (104*B*a*b^4 - 125*A*b^5)*d^4*e - 20*(B*a^2*b^3 + A*a*b^4)*d^
3*e^2 - 4*(34*B*a^3*b^2 - 45*A*a^2*b^3)*d^2*e^3 + (31*B*a^4*b - 40*A*a^3*b^2)*d*e^4 - (4*B*a^5 - 5*A*a^4*b)*e^
5)*x - 12*(B*a*b^4*d^5 + (4*B*a^2*b^3 - 5*A*a*b^4)*d^4*e + (B*b^5*d*e^4 + (4*B*a*b^4 - 5*A*b^5)*e^5)*x^5 + (4*
B*b^5*d^2*e^3 + (17*B*a*b^4 - 20*A*b^5)*d*e^4 + (4*B*a^2*b^3 - 5*A*a*b^4)*e^5)*x^4 + 2*(3*B*b^5*d^3*e^2 + (14*
B*a*b^4 - 15*A*b^5)*d^2*e^3 + 2*(4*B*a^2*b^3 - 5*A*a*b^4)*d*e^4)*x^3 + 2*(2*B*b^5*d^4*e + (11*B*a*b^4 - 10*A*b
^5)*d^3*e^2 + 3*(4*B*a^2*b^3 - 5*A*a*b^4)*d^2*e^3)*x^2 + (B*b^5*d^5 + (8*B*a*b^4 - 5*A*b^5)*d^4*e + 4*(4*B*a^2
*b^3 - 5*A*a*b^4)*d^3*e^2)*x)*log(b*x + a) + 12*(B*a*b^4*d^5 + (4*B*a^2*b^3 - 5*A*a*b^4)*d^4*e + (B*b^5*d*e^4
+ (4*B*a*b^4 - 5*A*b^5)*e^5)*x^5 + (4*B*b^5*d^2*e^3 + (17*B*a*b^4 - 20*A*b^5)*d*e^4 + (4*B*a^2*b^3 - 5*A*a*b^4
)*e^5)*x^4 + 2*(3*B*b^5*d^3*e^2 + (14*B*a*b^4 - 15*A*b^5)*d^2*e^3 + 2*(4*B*a^2*b^3 - 5*A*a*b^4)*d*e^4)*x^3 + 2
*(2*B*b^5*d^4*e + (11*B*a*b^4 - 10*A*b^5)*d^3*e^2 + 3*(4*B*a^2*b^3 - 5*A*a*b^4)*d^2*e^3)*x^2 + (B*b^5*d^5 + (8
*B*a*b^4 - 5*A*b^5)*d^4*e + 4*(4*B*a^2*b^3 - 5*A*a*b^4)*d^3*e^2)*x)*log(e*x + d))/(a*b^6*d^10 - 6*a^2*b^5*d^9*
e + 15*a^3*b^4*d^8*e^2 - 20*a^4*b^3*d^7*e^3 + 15*a^5*b^2*d^6*e^4 - 6*a^6*b*d^5*e^5 + a^7*d^4*e^6 + (b^7*d^6*e^
4 - 6*a*b^6*d^5*e^5 + 15*a^2*b^5*d^4*e^6 - 20*a^3*b^4*d^3*e^7 + 15*a^4*b^3*d^2*e^8 - 6*a^5*b^2*d*e^9 + a^6*b*e
^10)*x^5 + (4*b^7*d^7*e^3 - 23*a*b^6*d^6*e^4 + 54*a^2*b^5*d^5*e^5 - 65*a^3*b^4*d^4*e^6 + 40*a^4*b^3*d^3*e^7 -
9*a^5*b^2*d^2*e^8 - 2*a^6*b*d*e^9 + a^7*e^10)*x^4 + 2*(3*b^7*d^8*e^2 - 16*a*b^6*d^7*e^3 + 33*a^2*b^5*d^6*e^4 -
30*a^3*b^4*d^5*e^5 + 5*a^4*b^3*d^4*e^6 + 12*a^5*b^2*d^3*e^7 - 9*a^6*b*d^2*e^8 + 2*a^7*d*e^9)*x^3 + 2*(2*b^7*d
^9*e - 9*a*b^6*d^8*e^2 + 12*a^2*b^5*d^7*e^3 + 5*a^3*b^4*d^6*e^4 - 30*a^4*b^3*d^5*e^5 + 33*a^5*b^2*d^4*e^6 - 16
*a^6*b*d^3*e^7 + 3*a^7*d^2*e^8)*x^2 + (b^7*d^10 - 2*a*b^6*d^9*e - 9*a^2*b^5*d^8*e^2 + 40*a^3*b^4*d^7*e^3 - 65*
a^4*b^3*d^6*e^4 + 54*a^5*b^2*d^5*e^5 - 23*a^6*b*d^4*e^6 + 4*a^7*d^3*e^7)*x)

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giac [B]  time = 1.43, size = 568, normalized size = 2.38 \begin {gather*} -\frac {{\left (B b^{5} d + 4 \, B a b^{4} e - 5 \, A b^{5} e\right )} \log \left ({\left | \frac {b d}{b x + a} - \frac {a e}{b x + a} + e \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 {\frac {B a b^{8}}{b x + a} - \frac {A b^{9}}{b x + a}}{b^{10} d^{5} - 5 \, a b^{9} d^{4} e + 10 \, a^{2} b^{8} d^{3} e^{2} - 10 \, a^{3} b^{7} d^{2} e^{3} + 5 \, a^{4} b^{6} d e^{4} - a^{5} b^{5} e^{5}} - \frac {25 \, B b^{4} d e^{4} + 52 \, B a b^{3} e^{5} - 77 \, A b^{4} e^{5} + \frac {4 \, {\left (22 \, B b^{6} d^{2} e^{3} + 21 \, B a b^{5} d e^{4} - 65 \, A b^{6} d e^{4} - 43 \, B a^{2} b^{4} e^{5} + 65 \, A a b^{5} e^{5}\right )}}{{\left (b x + a\right )} b} + \frac {12 \, {\left (9 \, B b^{8} d^{3} e^{2} - 2 \, B a b^{7} d^{2} e^{3} - 25 \, A b^{8} d^{2} e^{3} - 23 \, B a^{2} b^{6} d e^{4} + 50 \, A a b^{7} d e^{4} + 16 \, B a^{3} b^{5} e^{5} - 25 \, A a^{2} b^{6} e^{5}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac {24 \, {\left (2 \, B b^{10} d^{4} e - 3 \, B a b^{9} d^{3} e^{2} - 5 \, A b^{10} d^{3} e^{2} - 3 \, B a^{2} b^{8} d^{2} e^{3} + 15 \, A a b^{9} d^{2} e^{3} + 7 \, B a^{3} b^{7} d e^{4} - 15 \, A a^{2} b^{8} d e^{4} - 3 \, B a^{4} b^{6} e^{5} + 5 \, A a^{3} b^{7} e^{5}\right )}}{{\left (b x + a\right )}^{3} b^{3}}}{12 \, {\left (b d - a e\right )}^{6} {\left (\frac {b d}{b x + a} - \frac {a e}{b x + a} + e\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)^2/(e*x+d)^5,x, algorithm="giac")

[Out]

-(B*b^5*d + 4*B*a*b^4*e - 5*A*b^5*e)*log(abs(b*d/(b*x + a) - a*e/(b*x + a) + e))/(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) + (B*a*b^8/(b*x + a)
- A*b^9/(b*x + a))/(b^10*d^5 - 5*a*b^9*d^4*e + 10*a^2*b^8*d^3*e^2 - 10*a^3*b^7*d^2*e^3 + 5*a^4*b^6*d*e^4 - a^
5*b^5*e^5) - 1/12*(25*B*b^4*d*e^4 + 52*B*a*b^3*e^5 - 77*A*b^4*e^5 + 4*(22*B*b^6*d^2*e^3 + 21*B*a*b^5*d*e^4 - 6
5*A*b^6*d*e^4 - 43*B*a^2*b^4*e^5 + 65*A*a*b^5*e^5)/((b*x + a)*b) + 12*(9*B*b^8*d^3*e^2 - 2*B*a*b^7*d^2*e^3 - 2
5*A*b^8*d^2*e^3 - 23*B*a^2*b^6*d*e^4 + 50*A*a*b^7*d*e^4 + 16*B*a^3*b^5*e^5 - 25*A*a^2*b^6*e^5)/((b*x + a)^2*b^
2) + 24*(2*B*b^10*d^4*e - 3*B*a*b^9*d^3*e^2 - 5*A*b^10*d^3*e^2 - 3*B*a^2*b^8*d^2*e^3 + 15*A*a*b^9*d^2*e^3 + 7*
B*a^3*b^7*d*e^4 - 15*A*a^2*b^8*d*e^4 - 3*B*a^4*b^6*e^5 + 5*A*a^3*b^7*e^5)/((b*x + a)^3*b^3))/((b*d - a*e)^6*(b
*d/(b*x + a) - a*e/(b*x + a) + e)^4)

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maple [A]  time = 0.02, size = 438, normalized size = 1.83 \begin {gather*} -\frac {5 A \,b^{4} e \ln \left (b x +a \right )}{\left (a e -b d \right )^{6}}+\frac {5 A \,b^{4} e \ln \left (e x +d \right )}{\left (a e -b d \right )^{6}}+\frac {4 B a \,b^{3} e \ln \left (b x +a \right )}{\left (a e -b d \right )^{6}}-\frac {4 B a \,b^{3} e \ln \left (e x +d \right )}{\left (a e -b d \right )^{6}}+\frac {B \,b^{4} d \ln \left (b x +a \right )}{\left (a e -b d \right )^{6}}-\frac {B \,b^{4} d \ln \left (e x +d \right )}{\left (a e -b d \right )^{6}}+\frac {A \,b^{4}}{\left (a e -b d \right )^{5} \left (b x +a \right )}+\frac {4 A \,b^{3} e}{\left (a e -b d \right )^{5} \left (e x +d \right )}-\frac {B a \,b^{3}}{\left (a e -b d \right )^{5} \left (b x +a \right )}-\frac {3 B a \,b^{2} e}{\left (a e -b d \right )^{5} \left (e x +d \right )}-\frac {B \,b^{3} d}{\left (a e -b d \right )^{5} \left (e x +d \right )}-\frac {3 A \,b^{2} e}{2 \left (a e -b d \right )^{4} \left (e x +d \right )^{2}}+\frac {B a b e}{\left (a e -b d \right )^{4} \left (e x +d \right )^{2}}+\frac {B \,b^{2} d}{2 \left (a e -b d \right )^{4} \left (e x +d \right )^{2}}+\frac {2 A b e}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{3}}-\frac {B a e}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{3}}-\frac {B b d}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{3}}-\frac {A e}{4 \left (a e -b d \right )^{2} \left (e x +d \right )^{4}}+\frac {B d}{4 \left (a e -b d \right )^{2} \left (e x +d \right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(b*x+a)^2/(e*x+d)^5,x)

[Out]

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

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maxima [B]  time = 0.88, size = 1090, normalized size = 4.56 \begin {gather*} \frac {{\left (B b^{4} d + {\left (4 \, B a b^{3} - 5 \, A b^{4}\right )} e\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 {{\left (B b^{4} d + {\left (4 \, B a b^{3} - 5 \, A b^{4}\right )} e\right )} \log \left (e x + 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 {3 \, A a^{4} e^{4} + {\left (37 \, B a b^{3} - 12 \, A b^{4}\right )} d^{4} + {\left (29 \, B a^{2} b^{2} - 77 \, A a b^{3}\right )} d^{3} e - {\left (7 \, B a^{3} b - 43 \, A a^{2} b^{2}\right )} d^{2} e^{2} + {\left (B a^{4} - 17 \, A a^{3} b\right )} d e^{3} + 12 \, {\left (B b^{4} d e^{3} + {\left (4 \, B a b^{3} - 5 \, A b^{4}\right )} e^{4}\right )} x^{4} + 6 \, {\left (7 \, B b^{4} d^{2} e^{2} + {\left (29 \, B a b^{3} - 35 \, A b^{4}\right )} d e^{3} + {\left (4 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} e^{4}\right )} x^{3} + 2 \, {\left (26 \, B b^{4} d^{3} e + 5 \, {\left (23 \, B a b^{3} - 26 \, A b^{4}\right )} d^{2} e^{2} + {\left (43 \, B a^{2} b^{2} - 55 \, A a b^{3}\right )} d e^{3} - {\left (4 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} e^{4}\right )} x^{2} + {\left (25 \, B b^{4} d^{4} + {\left (129 \, B a b^{3} - 125 \, A b^{4}\right )} d^{3} e + {\left (109 \, B a^{2} b^{2} - 145 \, A a b^{3}\right )} d^{2} e^{2} - {\left (27 \, B a^{3} b - 35 \, A a^{2} b^{2}\right )} d e^{3} + {\left (4 \, B a^{4} - 5 \, A a^{3} b\right )} e^{4}\right )} x}{12 \, {\left (a b^{5} d^{9} - 5 \, a^{2} b^{4} d^{8} e + 10 \, a^{3} b^{3} d^{7} e^{2} - 10 \, a^{4} b^{2} d^{6} e^{3} + 5 \, a^{5} b d^{5} e^{4} - a^{6} d^{4} e^{5} + {\left (b^{6} d^{5} e^{4} - 5 \, a b^{5} d^{4} e^{5} + 10 \, a^{2} b^{4} d^{3} e^{6} - 10 \, a^{3} b^{3} d^{2} e^{7} + 5 \, a^{4} b^{2} d e^{8} - a^{5} b e^{9}\right )} x^{5} + {\left (4 \, b^{6} d^{6} e^{3} - 19 \, a b^{5} d^{5} e^{4} + 35 \, a^{2} b^{4} d^{4} e^{5} - 30 \, a^{3} b^{3} d^{3} e^{6} + 10 \, a^{4} b^{2} d^{2} e^{7} + a^{5} b d e^{8} - a^{6} e^{9}\right )} x^{4} + 2 \, {\left (3 \, b^{6} d^{7} e^{2} - 13 \, a b^{5} d^{6} e^{3} + 20 \, a^{2} b^{4} d^{5} e^{4} - 10 \, a^{3} b^{3} d^{4} e^{5} - 5 \, a^{4} b^{2} d^{3} e^{6} + 7 \, a^{5} b d^{2} e^{7} - 2 \, a^{6} d e^{8}\right )} x^{3} + 2 \, {\left (2 \, b^{6} d^{8} e - 7 \, a b^{5} d^{7} e^{2} + 5 \, a^{2} b^{4} d^{6} e^{3} + 10 \, a^{3} b^{3} d^{5} e^{4} - 20 \, a^{4} b^{2} d^{4} e^{5} + 13 \, a^{5} b d^{3} e^{6} - 3 \, a^{6} d^{2} e^{7}\right )} x^{2} + {\left (b^{6} d^{9} - a b^{5} d^{8} e - 10 \, a^{2} b^{4} d^{7} e^{2} + 30 \, a^{3} b^{3} d^{6} e^{3} - 35 \, a^{4} b^{2} d^{5} e^{4} + 19 \, a^{5} b d^{4} e^{5} - 4 \, a^{6} d^{3} 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)^2/(e*x+d)^5,x, algorithm="maxima")

[Out]

(B*b^4*d + (4*B*a*b^3 - 5*A*b^4)*e)*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) - (B*b^4*d + (4*B*a*b^3 - 5*A*b^4)*e)*log(e*x + 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)
+ 1/12*(3*A*a^4*e^4 + (37*B*a*b^3 - 12*A*b^4)*d^4 + (29*B*a^2*b^2 - 77*A*a*b^3)*d^3*e - (7*B*a^3*b - 43*A*a^2*
b^2)*d^2*e^2 + (B*a^4 - 17*A*a^3*b)*d*e^3 + 12*(B*b^4*d*e^3 + (4*B*a*b^3 - 5*A*b^4)*e^4)*x^4 + 6*(7*B*b^4*d^2*
e^2 + (29*B*a*b^3 - 35*A*b^4)*d*e^3 + (4*B*a^2*b^2 - 5*A*a*b^3)*e^4)*x^3 + 2*(26*B*b^4*d^3*e + 5*(23*B*a*b^3 -
26*A*b^4)*d^2*e^2 + (43*B*a^2*b^2 - 55*A*a*b^3)*d*e^3 - (4*B*a^3*b - 5*A*a^2*b^2)*e^4)*x^2 + (25*B*b^4*d^4 +
(129*B*a*b^3 - 125*A*b^4)*d^3*e + (109*B*a^2*b^2 - 145*A*a*b^3)*d^2*e^2 - (27*B*a^3*b - 35*A*a^2*b^2)*d*e^3 +
(4*B*a^4 - 5*A*a^3*b)*e^4)*x)/(a*b^5*d^9 - 5*a^2*b^4*d^8*e + 10*a^3*b^3*d^7*e^2 - 10*a^4*b^2*d^6*e^3 + 5*a^5*b
*d^5*e^4 - a^6*d^4*e^5 + (b^6*d^5*e^4 - 5*a*b^5*d^4*e^5 + 10*a^2*b^4*d^3*e^6 - 10*a^3*b^3*d^2*e^7 + 5*a^4*b^2*
d*e^8 - a^5*b*e^9)*x^5 + (4*b^6*d^6*e^3 - 19*a*b^5*d^5*e^4 + 35*a^2*b^4*d^4*e^5 - 30*a^3*b^3*d^3*e^6 + 10*a^4*
b^2*d^2*e^7 + a^5*b*d*e^8 - a^6*e^9)*x^4 + 2*(3*b^6*d^7*e^2 - 13*a*b^5*d^6*e^3 + 20*a^2*b^4*d^5*e^4 - 10*a^3*b
^3*d^4*e^5 - 5*a^4*b^2*d^3*e^6 + 7*a^5*b*d^2*e^7 - 2*a^6*d*e^8)*x^3 + 2*(2*b^6*d^8*e - 7*a*b^5*d^7*e^2 + 5*a^2
*b^4*d^6*e^3 + 10*a^3*b^3*d^5*e^4 - 20*a^4*b^2*d^4*e^5 + 13*a^5*b*d^3*e^6 - 3*a^6*d^2*e^7)*x^2 + (b^6*d^9 - a*
b^5*d^8*e - 10*a^2*b^4*d^7*e^2 + 30*a^3*b^3*d^6*e^3 - 35*a^4*b^2*d^5*e^4 + 19*a^5*b*d^4*e^5 - 4*a^6*d^3*e^6)*x
)

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mupad [B]  time = 1.84, size = 767, normalized size = 3.21 \begin {gather*} \frac {\ln \left (a+b\,x\right )\,\left (B\,b^4\,d-b^3\,e\,\left (5\,A\,b-4\,B\,a\right )\right )}{{\left (a\,e-b\,d\right )}^6}-\frac {\frac {B\,a^4\,d\,e^3+3\,A\,a^4\,e^4-7\,B\,a^3\,b\,d^2\,e^2-17\,A\,a^3\,b\,d\,e^3+29\,B\,a^2\,b^2\,d^3\,e+43\,A\,a^2\,b^2\,d^2\,e^2+37\,B\,a\,b^3\,d^4-77\,A\,a\,b^3\,d^3\,e-12\,A\,b^4\,d^4}{12\,\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 (4\,B\,a\,e-5\,A\,b\,e+B\,b\,d\right )\,\left (a^3\,e^3-7\,a^2\,b\,d\,e^2+29\,a\,b^2\,d^2\,e+25\,b^3\,d^3\right )}{12\,\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 {b^3\,e^3\,x^4\,\left (4\,B\,a\,e-5\,A\,b\,e+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^2\,x^3\,\left (a\,e^3+7\,b\,d\,e^2\right )\,\left (4\,B\,a\,e-5\,A\,b\,e+B\,b\,d\right )}{2\,\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 {b\,x^2\,\left (4\,B\,a\,e-5\,A\,b\,e+B\,b\,d\right )\,\left (-a^2\,e^3+11\,a\,b\,d\,e^2+26\,b^2\,d^2\,e\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 )}}{x^4\,\left (a\,e^4+4\,b\,d\,e^3\right )+a\,d^4+x\,\left (b\,d^4+4\,a\,e\,d^3\right )+x^2\,\left (4\,b\,d^3\,e+6\,a\,d^2\,e^2\right )+x^3\,\left (6\,b\,d^2\,e^2+4\,a\,d\,e^3\right )+b\,e^4\,x^5}+\frac {\ln \left (d+e\,x\right )\,\left (b^4\,\left (5\,A\,e-B\,d\right )-4\,B\,a\,b^3\,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)^2*(d + e*x)^5),x)

[Out]

(log(a + b*x)*(B*b^4*d - b^3*e*(5*A*b - 4*B*a)))/(a*e - b*d)^6 - ((3*A*a^4*e^4 - 12*A*b^4*d^4 + 37*B*a*b^3*d^4
+ B*a^4*d*e^3 + 29*B*a^2*b^2*d^3*e - 7*B*a^3*b*d^2*e^2 + 43*A*a^2*b^2*d^2*e^2 - 77*A*a*b^3*d^3*e - 17*A*a^3*b
*d*e^3)/(12*(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*(4*B*a*e - 5*A*b*e + B*b*d)*(a^3*e^3 + 25*b^3*d^3 + 29*a*b^2*d^2*e - 7*a^2*b*d*e^2))/(12*(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^3*e^3*x^4*(4*B*a*e - 5*A*b*e
+ 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^2
*x^3*(a*e^3 + 7*b*d*e^2)*(4*B*a*e - 5*A*b*e + B*b*d))/(2*(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*(4*B*a*e - 5*A*b*e + B*b*d)*(26*b^2*d^2*e - a^2*e^3 + 11*a*
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)))/(
x^4*(a*e^4 + 4*b*d*e^3) + a*d^4 + x*(b*d^4 + 4*a*d^3*e) + x^2*(6*a*d^2*e^2 + 4*b*d^3*e) + x^3*(6*b*d^2*e^2 + 4
*a*d*e^3) + b*e^4*x^5) + (log(d + e*x)*(b^4*(5*A*e - B*d) - 4*B*a*b^3*e))/(a*e - b*d)^6

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sympy [B]  time = 7.88, size = 1877, normalized size = 7.85

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)**2/(e*x+d)**5,x)

[Out]

-b**3*(-5*A*b*e + 4*B*a*e + B*b*d)*log(x + (-5*A*a*b**4*e**2 - 5*A*b**5*d*e + 4*B*a**2*b**3*e**2 + 5*B*a*b**4*
d*e + B*b**5*d**2 - a**7*b**3*e**7*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6 + 7*a**6*b**4*d*e**6*(-5*A*b*e
+ 4*B*a*e + B*b*d)/(a*e - b*d)**6 - 21*a**5*b**5*d**2*e**5*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6 + 35*a*
*4*b**6*d**3*e**4*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6 - 35*a**3*b**7*d**4*e**3*(-5*A*b*e + 4*B*a*e + B
*b*d)/(a*e - b*d)**6 + 21*a**2*b**8*d**5*e**2*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6 - 7*a*b**9*d**6*e*(-
5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6 + b**10*d**7*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6)/(-10*A*b**
5*e**2 + 8*B*a*b**4*e**2 + 2*B*b**5*d*e))/(a*e - b*d)**6 + b**3*(-5*A*b*e + 4*B*a*e + B*b*d)*log(x + (-5*A*a*b
**4*e**2 - 5*A*b**5*d*e + 4*B*a**2*b**3*e**2 + 5*B*a*b**4*d*e + B*b**5*d**2 + a**7*b**3*e**7*(-5*A*b*e + 4*B*a
*e + B*b*d)/(a*e - b*d)**6 - 7*a**6*b**4*d*e**6*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6 + 21*a**5*b**5*d**
2*e**5*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6 - 35*a**4*b**6*d**3*e**4*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e
- b*d)**6 + 35*a**3*b**7*d**4*e**3*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6 - 21*a**2*b**8*d**5*e**2*(-5*A*
b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6 + 7*a*b**9*d**6*e*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6 - b**10*d*
*7*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6)/(-10*A*b**5*e**2 + 8*B*a*b**4*e**2 + 2*B*b**5*d*e))/(a*e - b*d
)**6 + (-3*A*a**4*e**4 + 17*A*a**3*b*d*e**3 - 43*A*a**2*b**2*d**2*e**2 + 77*A*a*b**3*d**3*e + 12*A*b**4*d**4 -
B*a**4*d*e**3 + 7*B*a**3*b*d**2*e**2 - 29*B*a**2*b**2*d**3*e - 37*B*a*b**3*d**4 + x**4*(60*A*b**4*e**4 - 48*B
*a*b**3*e**4 - 12*B*b**4*d*e**3) + x**3*(30*A*a*b**3*e**4 + 210*A*b**4*d*e**3 - 24*B*a**2*b**2*e**4 - 174*B*a*
b**3*d*e**3 - 42*B*b**4*d**2*e**2) + x**2*(-10*A*a**2*b**2*e**4 + 110*A*a*b**3*d*e**3 + 260*A*b**4*d**2*e**2 +
8*B*a**3*b*e**4 - 86*B*a**2*b**2*d*e**3 - 230*B*a*b**3*d**2*e**2 - 52*B*b**4*d**3*e) + x*(5*A*a**3*b*e**4 - 3
5*A*a**2*b**2*d*e**3 + 145*A*a*b**3*d**2*e**2 + 125*A*b**4*d**3*e - 4*B*a**4*e**4 + 27*B*a**3*b*d*e**3 - 109*B
*a**2*b**2*d**2*e**2 - 129*B*a*b**3*d**3*e - 25*B*b**4*d**4))/(12*a**6*d**4*e**5 - 60*a**5*b*d**5*e**4 + 120*a
**4*b**2*d**6*e**3 - 120*a**3*b**3*d**7*e**2 + 60*a**2*b**4*d**8*e - 12*a*b**5*d**9 + x**5*(12*a**5*b*e**9 - 6
0*a**4*b**2*d*e**8 + 120*a**3*b**3*d**2*e**7 - 120*a**2*b**4*d**3*e**6 + 60*a*b**5*d**4*e**5 - 12*b**6*d**5*e*
*4) + x**4*(12*a**6*e**9 - 12*a**5*b*d*e**8 - 120*a**4*b**2*d**2*e**7 + 360*a**3*b**3*d**3*e**6 - 420*a**2*b**
4*d**4*e**5 + 228*a*b**5*d**5*e**4 - 48*b**6*d**6*e**3) + x**3*(48*a**6*d*e**8 - 168*a**5*b*d**2*e**7 + 120*a*
*4*b**2*d**3*e**6 + 240*a**3*b**3*d**4*e**5 - 480*a**2*b**4*d**5*e**4 + 312*a*b**5*d**6*e**3 - 72*b**6*d**7*e*
*2) + x**2*(72*a**6*d**2*e**7 - 312*a**5*b*d**3*e**6 + 480*a**4*b**2*d**4*e**5 - 240*a**3*b**3*d**5*e**4 - 120
*a**2*b**4*d**6*e**3 + 168*a*b**5*d**7*e**2 - 48*b**6*d**8*e) + x*(48*a**6*d**3*e**6 - 228*a**5*b*d**4*e**5 +
420*a**4*b**2*d**5*e**4 - 360*a**3*b**3*d**6*e**3 + 120*a**2*b**4*d**7*e**2 + 12*a*b**5*d**8*e - 12*b**6*d**9)
)

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