3.1685 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^2}{(d+e x)^5} \, dx\)

Optimal. Leaf size=189 \[ -\frac {b^3 \log (d+e x) (-4 a B e-A b e+5 b B d)}{e^6}-\frac {2 b^2 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{e^6 (d+e x)}+\frac {b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{e^6 (d+e x)^2}-\frac {(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{3 e^6 (d+e x)^3}+\frac {(b d-a e)^4 (B d-A e)}{4 e^6 (d+e x)^4}+\frac {b^4 B x}{e^5} \]

[Out]

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

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Rubi [A]  time = 0.20, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 77} \[ -\frac {2 b^2 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{e^6 (d+e x)}-\frac {b^3 \log (d+e x) (-4 a B e-A b e+5 b B d)}{e^6}+\frac {b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{e^6 (d+e x)^2}-\frac {(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{3 e^6 (d+e x)^3}+\frac {(b d-a e)^4 (B d-A e)}{4 e^6 (d+e x)^4}+\frac {b^4 B x}{e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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

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Mathematica [A]  time = 0.17, size = 338, normalized size = 1.79 \[ -\frac {a^4 e^4 (3 A e+B (d+4 e x))+4 a^3 b e^3 \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )+6 a^2 b^2 e^2 \left (A e \left (d^2+4 d e x+6 e^2 x^2\right )+3 B \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )-4 a b^3 e \left (B d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )-3 A e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+12 b^3 (d+e x)^4 \log (d+e x) (-4 a B e-A b e+5 b B d)-\left (b^4 \left (A d e \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )-B \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )\right )\right )}{12 e^6 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(a^4*e^4*(3*A*e + B*(d + 4*e*x)) + 4*a^3*b*e^3*(A*e*(d + 4*e*x) + B*(d^2 + 4*d*e*x + 6*e^2*x^2)) + 6*a^2
*b^2*e^2*(A*e*(d^2 + 4*d*e*x + 6*e^2*x^2) + 3*B*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3)) - 4*a*b^3*e*(-3*A
*e*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) + B*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3)) - b^4
*(A*d*e*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3) - B*(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2 + 48*d^
2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5)) + 12*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x)^4*Log[d + e*x])/(e^6*(d
+ e*x)^4)

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fricas [B]  time = 0.87, size = 602, normalized size = 3.19 \[ \frac {12 \, B b^{4} e^{5} x^{5} + 48 \, B b^{4} d e^{4} x^{4} - 77 \, B b^{4} d^{5} - 3 \, A a^{4} e^{5} + 25 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e - 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} - {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} - 24 \, {\left (2 \, B b^{4} d^{2} e^{3} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} - 12 \, {\left (21 \, B b^{4} d^{3} e^{2} - 9 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} - 4 \, {\left (62 \, B b^{4} d^{4} e - 22 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x - 12 \, {\left (5 \, B b^{4} d^{5} - {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + {\left (5 \, B b^{4} d e^{4} - {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 4 \, {\left (5 \, B b^{4} d^{2} e^{3} - {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4}\right )} x^{3} + 6 \, {\left (5 \, B b^{4} d^{3} e^{2} - {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3}\right )} x^{2} + 4 \, {\left (5 \, B b^{4} d^{4} e - {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(12*B*b^4*e^5*x^5 + 48*B*b^4*d*e^4*x^4 - 77*B*b^4*d^5 - 3*A*a^4*e^5 + 25*(4*B*a*b^3 + A*b^4)*d^4*e - 6*(3
*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 - 2*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 - (B*a^4 + 4*A*a^3*b)*d*e^4 - 24*(2*B*b^
4*d^2*e^3 - 2*(4*B*a*b^3 + A*b^4)*d*e^4 + (3*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 - 12*(21*B*b^4*d^3*e^2 - 9*(4*B*a
*b^3 + A*b^4)*d^2*e^3 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + (2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 - 4*(62*B*b^4*d
^4*e - 22*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 2*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^
4 + (B*a^4 + 4*A*a^3*b)*e^5)*x - 12*(5*B*b^4*d^5 - (4*B*a*b^3 + A*b^4)*d^4*e + (5*B*b^4*d*e^4 - (4*B*a*b^3 + A
*b^4)*e^5)*x^4 + 4*(5*B*b^4*d^2*e^3 - (4*B*a*b^3 + A*b^4)*d*e^4)*x^3 + 6*(5*B*b^4*d^3*e^2 - (4*B*a*b^3 + A*b^4
)*d^2*e^3)*x^2 + 4*(5*B*b^4*d^4*e - (4*B*a*b^3 + A*b^4)*d^3*e^2)*x)*log(e*x + d))/(e^10*x^4 + 4*d*e^9*x^3 + 6*
d^2*e^8*x^2 + 4*d^3*e^7*x + d^4*e^6)

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giac [B]  time = 0.18, size = 652, normalized size = 3.45 \[ {\left (x e + d\right )} B b^{4} e^{\left (-6\right )} + {\left (5 \, B b^{4} d - 4 \, B a b^{3} e - A b^{4} e\right )} e^{\left (-6\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac {1}{12} \, {\left (\frac {120 \, B b^{4} d^{2} e^{22}}{x e + d} - \frac {60 \, B b^{4} d^{3} e^{22}}{{\left (x e + d\right )}^{2}} + \frac {20 \, B b^{4} d^{4} e^{22}}{{\left (x e + d\right )}^{3}} - \frac {3 \, B b^{4} d^{5} e^{22}}{{\left (x e + d\right )}^{4}} - \frac {192 \, B a b^{3} d e^{23}}{x e + d} - \frac {48 \, A b^{4} d e^{23}}{x e + d} + \frac {144 \, B a b^{3} d^{2} e^{23}}{{\left (x e + d\right )}^{2}} + \frac {36 \, A b^{4} d^{2} e^{23}}{{\left (x e + d\right )}^{2}} - \frac {64 \, B a b^{3} d^{3} e^{23}}{{\left (x e + d\right )}^{3}} - \frac {16 \, A b^{4} d^{3} e^{23}}{{\left (x e + d\right )}^{3}} + \frac {12 \, B a b^{3} d^{4} e^{23}}{{\left (x e + d\right )}^{4}} + \frac {3 \, A b^{4} d^{4} e^{23}}{{\left (x e + d\right )}^{4}} + \frac {72 \, B a^{2} b^{2} e^{24}}{x e + d} + \frac {48 \, A a b^{3} e^{24}}{x e + d} - \frac {108 \, B a^{2} b^{2} d e^{24}}{{\left (x e + d\right )}^{2}} - \frac {72 \, A a b^{3} d e^{24}}{{\left (x e + d\right )}^{2}} + \frac {72 \, B a^{2} b^{2} d^{2} e^{24}}{{\left (x e + d\right )}^{3}} + \frac {48 \, A a b^{3} d^{2} e^{24}}{{\left (x e + d\right )}^{3}} - \frac {18 \, B a^{2} b^{2} d^{3} e^{24}}{{\left (x e + d\right )}^{4}} - \frac {12 \, A a b^{3} d^{3} e^{24}}{{\left (x e + d\right )}^{4}} + \frac {24 \, B a^{3} b e^{25}}{{\left (x e + d\right )}^{2}} + \frac {36 \, A a^{2} b^{2} e^{25}}{{\left (x e + d\right )}^{2}} - \frac {32 \, B a^{3} b d e^{25}}{{\left (x e + d\right )}^{3}} - \frac {48 \, A a^{2} b^{2} d e^{25}}{{\left (x e + d\right )}^{3}} + \frac {12 \, B a^{3} b d^{2} e^{25}}{{\left (x e + d\right )}^{4}} + \frac {18 \, A a^{2} b^{2} d^{2} e^{25}}{{\left (x e + d\right )}^{4}} + \frac {4 \, B a^{4} e^{26}}{{\left (x e + d\right )}^{3}} + \frac {16 \, A a^{3} b e^{26}}{{\left (x e + d\right )}^{3}} - \frac {3 \, B a^{4} d e^{26}}{{\left (x e + d\right )}^{4}} - \frac {12 \, A a^{3} b d e^{26}}{{\left (x e + d\right )}^{4}} + \frac {3 \, A a^{4} e^{27}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-28\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*e + d)*B*b^4*e^(-6) + (5*B*b^4*d - 4*B*a*b^3*e - A*b^4*e)*e^(-6)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - 1/1
2*(120*B*b^4*d^2*e^22/(x*e + d) - 60*B*b^4*d^3*e^22/(x*e + d)^2 + 20*B*b^4*d^4*e^22/(x*e + d)^3 - 3*B*b^4*d^5*
e^22/(x*e + d)^4 - 192*B*a*b^3*d*e^23/(x*e + d) - 48*A*b^4*d*e^23/(x*e + d) + 144*B*a*b^3*d^2*e^23/(x*e + d)^2
+ 36*A*b^4*d^2*e^23/(x*e + d)^2 - 64*B*a*b^3*d^3*e^23/(x*e + d)^3 - 16*A*b^4*d^3*e^23/(x*e + d)^3 + 12*B*a*b^
3*d^4*e^23/(x*e + d)^4 + 3*A*b^4*d^4*e^23/(x*e + d)^4 + 72*B*a^2*b^2*e^24/(x*e + d) + 48*A*a*b^3*e^24/(x*e + d
) - 108*B*a^2*b^2*d*e^24/(x*e + d)^2 - 72*A*a*b^3*d*e^24/(x*e + d)^2 + 72*B*a^2*b^2*d^2*e^24/(x*e + d)^3 + 48*
A*a*b^3*d^2*e^24/(x*e + d)^3 - 18*B*a^2*b^2*d^3*e^24/(x*e + d)^4 - 12*A*a*b^3*d^3*e^24/(x*e + d)^4 + 24*B*a^3*
b*e^25/(x*e + d)^2 + 36*A*a^2*b^2*e^25/(x*e + d)^2 - 32*B*a^3*b*d*e^25/(x*e + d)^3 - 48*A*a^2*b^2*d*e^25/(x*e
+ d)^3 + 12*B*a^3*b*d^2*e^25/(x*e + d)^4 + 18*A*a^2*b^2*d^2*e^25/(x*e + d)^4 + 4*B*a^4*e^26/(x*e + d)^3 + 16*A
*a^3*b*e^26/(x*e + d)^3 - 3*B*a^4*d*e^26/(x*e + d)^4 - 12*A*a^3*b*d*e^26/(x*e + d)^4 + 3*A*a^4*e^27/(x*e + d)^
4)*e^(-28)

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maple [B]  time = 0.06, size = 641, normalized size = 3.39 \[ -\frac {A \,a^{4}}{4 \left (e x +d \right )^{4} e}+\frac {A \,a^{3} b d}{\left (e x +d \right )^{4} e^{2}}-\frac {3 A \,a^{2} b^{2} d^{2}}{2 \left (e x +d \right )^{4} e^{3}}+\frac {A a \,b^{3} d^{3}}{\left (e x +d \right )^{4} e^{4}}-\frac {A \,b^{4} d^{4}}{4 \left (e x +d \right )^{4} e^{5}}+\frac {B \,a^{4} d}{4 \left (e x +d \right )^{4} e^{2}}-\frac {B \,a^{3} b \,d^{2}}{\left (e x +d \right )^{4} e^{3}}+\frac {3 B \,a^{2} b^{2} d^{3}}{2 \left (e x +d \right )^{4} e^{4}}-\frac {B a \,b^{3} d^{4}}{\left (e x +d \right )^{4} e^{5}}+\frac {B \,b^{4} d^{5}}{4 \left (e x +d \right )^{4} e^{6}}-\frac {4 A \,a^{3} b}{3 \left (e x +d \right )^{3} e^{2}}+\frac {4 A \,a^{2} b^{2} d}{\left (e x +d \right )^{3} e^{3}}-\frac {4 A a \,b^{3} d^{2}}{\left (e x +d \right )^{3} e^{4}}+\frac {4 A \,b^{4} d^{3}}{3 \left (e x +d \right )^{3} e^{5}}-\frac {B \,a^{4}}{3 \left (e x +d \right )^{3} e^{2}}+\frac {8 B \,a^{3} b d}{3 \left (e x +d \right )^{3} e^{3}}-\frac {6 B \,a^{2} b^{2} d^{2}}{\left (e x +d \right )^{3} e^{4}}+\frac {16 B a \,b^{3} d^{3}}{3 \left (e x +d \right )^{3} e^{5}}-\frac {5 B \,b^{4} d^{4}}{3 \left (e x +d \right )^{3} e^{6}}-\frac {3 A \,a^{2} b^{2}}{\left (e x +d \right )^{2} e^{3}}+\frac {6 A a \,b^{3} d}{\left (e x +d \right )^{2} e^{4}}-\frac {3 A \,b^{4} d^{2}}{\left (e x +d \right )^{2} e^{5}}-\frac {2 B \,a^{3} b}{\left (e x +d \right )^{2} e^{3}}+\frac {9 B \,a^{2} b^{2} d}{\left (e x +d \right )^{2} e^{4}}-\frac {12 B a \,b^{3} d^{2}}{\left (e x +d \right )^{2} e^{5}}+\frac {5 B \,b^{4} d^{3}}{\left (e x +d \right )^{2} e^{6}}-\frac {4 A a \,b^{3}}{\left (e x +d \right ) e^{4}}+\frac {4 A \,b^{4} d}{\left (e x +d \right ) e^{5}}+\frac {A \,b^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {6 B \,a^{2} b^{2}}{\left (e x +d \right ) e^{4}}+\frac {16 B a \,b^{3} d}{\left (e x +d \right ) e^{5}}+\frac {4 B a \,b^{3} \ln \left (e x +d \right )}{e^{5}}-\frac {10 B \,b^{4} d^{2}}{\left (e x +d \right ) e^{6}}-\frac {5 B \,b^{4} d \ln \left (e x +d \right )}{e^{6}}+\frac {B \,b^{4} x}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/3/e^3/(e*x+d)^3*B*a^3*b*d+9*b^2/e^4/(e*x+d)^2*B*a^2*d+6*b^3/e^4/(e*x+d)^2*A*a*d-12*b^3/e^5/(e*x+d)^2*B*a*d^2
+4*b^3/e^5*ln(e*x+d)*a*B-5*b^4/e^6*ln(e*x+d)*B*d-3*b^2/e^3/(e*x+d)^2*A*a^2-3/2/e^3/(e*x+d)^4*A*d^2*a^2*b^2+1/e
^4/(e*x+d)^4*A*d^3*a*b^3-1/e^3/(e*x+d)^4*B*d^2*a^3*b+16*b^3/e^5/(e*x+d)*B*d*a+1/e^2/(e*x+d)^4*A*d*a^3*b-1/4/e/
(e*x+d)^4*A*a^4+b^4/e^5*ln(e*x+d)*A-1/3/e^2/(e*x+d)^3*B*a^4+16/3/e^5/(e*x+d)^3*B*a*b^3*d^3+4/e^3/(e*x+d)^3*A*a
^2*b^2*d-4/e^4/(e*x+d)^3*A*a*b^3*d^2+3/2/e^4/(e*x+d)^4*B*d^3*a^2*b^2-1/e^5/(e*x+d)^4*B*d^4*a*b^3+5*b^4/e^6/(e*
x+d)^2*B*d^3-6/e^4/(e*x+d)^3*B*a^2*b^2*d^2-5/3/e^6/(e*x+d)^3*B*b^4*d^4-3*b^4/e^5/(e*x+d)^2*A*d^2-2*b/e^3/(e*x+
d)^2*B*a^3-1/4/e^5/(e*x+d)^4*A*d^4*b^4+1/4/e^2/(e*x+d)^4*B*d*a^4+1/4/e^6/(e*x+d)^4*B*b^4*d^5-4/3/e^2/(e*x+d)^3
*A*a^3*b+4/3/e^5/(e*x+d)^3*A*b^4*d^3-6*b^2/e^4/(e*x+d)*B*a^2-4*b^3/e^4/(e*x+d)*A*a+4*b^4/e^5/(e*x+d)*A*d-10*b^
4/e^6/(e*x+d)*B*d^2+b^4*B*x/e^5

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maxima [B]  time = 0.62, size = 440, normalized size = 2.33 \[ \frac {B b^{4} x}{e^{5}} - \frac {77 \, B b^{4} d^{5} + 3 \, A a^{4} e^{5} - 25 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 24 \, {\left (5 \, B b^{4} d^{2} e^{3} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 12 \, {\left (25 \, B b^{4} d^{3} e^{2} - 9 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 4 \, {\left (65 \, B b^{4} d^{4} e - 22 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{12 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} - \frac {{\left (5 \, B b^{4} d - {\left (4 \, B a b^{3} + A b^{4}\right )} e\right )} \log \left (e x + d\right )}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*b^4*x/e^5 - 1/12*(77*B*b^4*d^5 + 3*A*a^4*e^5 - 25*(4*B*a*b^3 + A*b^4)*d^4*e + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^
3*e^2 + 2*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + (B*a^4 + 4*A*a^3*b)*d*e^4 + 24*(5*B*b^4*d^2*e^3 - 2*(4*B*a*b^3 +
A*b^4)*d*e^4 + (3*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 + 12*(25*B*b^4*d^3*e^2 - 9*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 3*
(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + (2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 4*(65*B*b^4*d^4*e - 22*(4*B*a*b^3 + A*b
^4)*d^3*e^2 + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 2*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + (B*a^4 + 4*A*a^3*b)*e^
5)*x)/(e^10*x^4 + 4*d*e^9*x^3 + 6*d^2*e^8*x^2 + 4*d^3*e^7*x + d^4*e^6) - (5*B*b^4*d - (4*B*a*b^3 + A*b^4)*e)*l
og(e*x + d)/e^6

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mupad [B]  time = 2.40, size = 462, normalized size = 2.44 \[ \frac {\ln \left (d+e\,x\right )\,\left (A\,b^4\,e-5\,B\,b^4\,d+4\,B\,a\,b^3\,e\right )}{e^6}-\frac {x^3\,\left (6\,B\,a^2\,b^2\,e^4-16\,B\,a\,b^3\,d\,e^3+4\,A\,a\,b^3\,e^4+10\,B\,b^4\,d^2\,e^2-4\,A\,b^4\,d\,e^3\right )+\frac {B\,a^4\,d\,e^4+3\,A\,a^4\,e^5+4\,B\,a^3\,b\,d^2\,e^3+4\,A\,a^3\,b\,d\,e^4+18\,B\,a^2\,b^2\,d^3\,e^2+6\,A\,a^2\,b^2\,d^2\,e^3-100\,B\,a\,b^3\,d^4\,e+12\,A\,a\,b^3\,d^3\,e^2+77\,B\,b^4\,d^5-25\,A\,b^4\,d^4\,e}{12\,e}+x\,\left (\frac {B\,a^4\,e^4}{3}+\frac {4\,B\,a^3\,b\,d\,e^3}{3}+\frac {4\,A\,a^3\,b\,e^4}{3}+6\,B\,a^2\,b^2\,d^2\,e^2+2\,A\,a^2\,b^2\,d\,e^3-\frac {88\,B\,a\,b^3\,d^3\,e}{3}+4\,A\,a\,b^3\,d^2\,e^2+\frac {65\,B\,b^4\,d^4}{3}-\frac {22\,A\,b^4\,d^3\,e}{3}\right )+x^2\,\left (2\,B\,a^3\,b\,e^4+9\,B\,a^2\,b^2\,d\,e^3+3\,A\,a^2\,b^2\,e^4-36\,B\,a\,b^3\,d^2\,e^2+6\,A\,a\,b^3\,d\,e^3+25\,B\,b^4\,d^3\,e-9\,A\,b^4\,d^2\,e^2\right )}{d^4\,e^5+4\,d^3\,e^6\,x+6\,d^2\,e^7\,x^2+4\,d\,e^8\,x^3+e^9\,x^4}+\frac {B\,b^4\,x}{e^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(d + e*x)*(A*b^4*e - 5*B*b^4*d + 4*B*a*b^3*e))/e^6 - (x^3*(4*A*a*b^3*e^4 - 4*A*b^4*d*e^3 + 6*B*a^2*b^2*e^4
+ 10*B*b^4*d^2*e^2 - 16*B*a*b^3*d*e^3) + (3*A*a^4*e^5 + 77*B*b^4*d^5 - 25*A*b^4*d^4*e + B*a^4*d*e^4 + 12*A*a*
b^3*d^3*e^2 + 4*B*a^3*b*d^2*e^3 + 6*A*a^2*b^2*d^2*e^3 + 18*B*a^2*b^2*d^3*e^2 + 4*A*a^3*b*d*e^4 - 100*B*a*b^3*d
^4*e)/(12*e) + x*((B*a^4*e^4)/3 + (65*B*b^4*d^4)/3 + (4*A*a^3*b*e^4)/3 - (22*A*b^4*d^3*e)/3 + 4*A*a*b^3*d^2*e^
2 + 2*A*a^2*b^2*d*e^3 + 6*B*a^2*b^2*d^2*e^2 - (88*B*a*b^3*d^3*e)/3 + (4*B*a^3*b*d*e^3)/3) + x^2*(2*B*a^3*b*e^4
+ 25*B*b^4*d^3*e + 3*A*a^2*b^2*e^4 - 9*A*b^4*d^2*e^2 - 36*B*a*b^3*d^2*e^2 + 9*B*a^2*b^2*d*e^3 + 6*A*a*b^3*d*e
^3))/(d^4*e^5 + e^9*x^4 + 4*d^3*e^6*x + 4*d*e^8*x^3 + 6*d^2*e^7*x^2) + (B*b^4*x)/e^5

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sympy [B]  time = 62.24, size = 518, normalized size = 2.74 \[ \frac {B b^{4} x}{e^{5}} + \frac {b^{3} \left (A b e + 4 B a e - 5 B b d\right ) \log {\left (d + e x \right )}}{e^{6}} + \frac {- 3 A a^{4} e^{5} - 4 A a^{3} b d e^{4} - 6 A a^{2} b^{2} d^{2} e^{3} - 12 A a b^{3} d^{3} e^{2} + 25 A b^{4} d^{4} e - B a^{4} d e^{4} - 4 B a^{3} b d^{2} e^{3} - 18 B a^{2} b^{2} d^{3} e^{2} + 100 B a b^{3} d^{4} e - 77 B b^{4} d^{5} + x^{3} \left (- 48 A a b^{3} e^{5} + 48 A b^{4} d e^{4} - 72 B a^{2} b^{2} e^{5} + 192 B a b^{3} d e^{4} - 120 B b^{4} d^{2} e^{3}\right ) + x^{2} \left (- 36 A a^{2} b^{2} e^{5} - 72 A a b^{3} d e^{4} + 108 A b^{4} d^{2} e^{3} - 24 B a^{3} b e^{5} - 108 B a^{2} b^{2} d e^{4} + 432 B a b^{3} d^{2} e^{3} - 300 B b^{4} d^{3} e^{2}\right ) + x \left (- 16 A a^{3} b e^{5} - 24 A a^{2} b^{2} d e^{4} - 48 A a b^{3} d^{2} e^{3} + 88 A b^{4} d^{3} e^{2} - 4 B a^{4} e^{5} - 16 B a^{3} b d e^{4} - 72 B a^{2} b^{2} d^{2} e^{3} + 352 B a b^{3} d^{3} e^{2} - 260 B b^{4} d^{4} e\right )}{12 d^{4} e^{6} + 48 d^{3} e^{7} x + 72 d^{2} e^{8} x^{2} + 48 d e^{9} x^{3} + 12 e^{10} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*b**4*x/e**5 + b**3*(A*b*e + 4*B*a*e - 5*B*b*d)*log(d + e*x)/e**6 + (-3*A*a**4*e**5 - 4*A*a**3*b*d*e**4 - 6*A
*a**2*b**2*d**2*e**3 - 12*A*a*b**3*d**3*e**2 + 25*A*b**4*d**4*e - B*a**4*d*e**4 - 4*B*a**3*b*d**2*e**3 - 18*B*
a**2*b**2*d**3*e**2 + 100*B*a*b**3*d**4*e - 77*B*b**4*d**5 + x**3*(-48*A*a*b**3*e**5 + 48*A*b**4*d*e**4 - 72*B
*a**2*b**2*e**5 + 192*B*a*b**3*d*e**4 - 120*B*b**4*d**2*e**3) + x**2*(-36*A*a**2*b**2*e**5 - 72*A*a*b**3*d*e**
4 + 108*A*b**4*d**2*e**3 - 24*B*a**3*b*e**5 - 108*B*a**2*b**2*d*e**4 + 432*B*a*b**3*d**2*e**3 - 300*B*b**4*d**
3*e**2) + x*(-16*A*a**3*b*e**5 - 24*A*a**2*b**2*d*e**4 - 48*A*a*b**3*d**2*e**3 + 88*A*b**4*d**3*e**2 - 4*B*a**
4*e**5 - 16*B*a**3*b*d*e**4 - 72*B*a**2*b**2*d**2*e**3 + 352*B*a*b**3*d**3*e**2 - 260*B*b**4*d**4*e))/(12*d**4
*e**6 + 48*d**3*e**7*x + 72*d**2*e**8*x**2 + 48*d*e**9*x**3 + 12*e**10*x**4)

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