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

[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

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Rubi [A]  time = 0.197728, antiderivative size = 189, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.176529, size = 338, normalized size = 1.79 \[ -\frac{6 a^2 b^2 e^2 \left (A e \left (d^2+4 d e x+6 e^2 x^2\right )+3 B \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )\right )+4 a^3 b e^3 \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )+a^4 e^4 (3 A e+B (d+4 e x))-4 a b^3 e \left (B d \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )-3 A e \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )\right )+12 b^3 (d+e x)^4 \log (d+e x) (-4 a B e-A b e+5 b B d)+b^4 \left (-\left (A d e \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )-B \left (252 d^3 e^2 x^2+48 d^2 e^3 x^3+248 d^4 e x+77 d^5-48 d e^4 x^4-12 e^5 x^5\right )\right )\right )}{12 e^6 (d+e x)^4} \]

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]

-(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])/(12*e^6*(d +
e*x)^4)

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Maple [B]  time = 0.013, size = 641, normalized size = 3.4 \begin{align*} \text{result too large to display} \end{align*}

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]

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

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Maxima [B]  time = 1.08343, size = 594, normalized size = 3.14 \begin{align*} \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}} \end{align*}

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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Fricas [B]  time = 1.58045, size = 1239, normalized size = 6.56 \begin{align*} \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 )}} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [B]  time = 1.15998, size = 880, normalized size = 4.66 \begin{align*} \text{result too large to display} \end{align*}

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)