3.2315 \(\int \frac{(A+B x) (a+b x+c x^2)}{(d+e x)^5} \, dx\)

Optimal. Leaf size=132 \[ \frac{A e (2 c d-b e)-B \left (3 c d^2-e (2 b d-a e)\right )}{3 e^4 (d+e x)^3}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )}{4 e^4 (d+e x)^4}+\frac{-A c e-b B e+3 B c d}{2 e^4 (d+e x)^2}-\frac{B c}{e^4 (d+e x)} \]

[Out]

((B*d - A*e)*(c*d^2 - b*d*e + a*e^2))/(4*e^4*(d + e*x)^4) + (A*e*(2*c*d - b*e) - B*(3*c*d^2 - e*(2*b*d - a*e))
)/(3*e^4*(d + e*x)^3) + (3*B*c*d - b*B*e - A*c*e)/(2*e^4*(d + e*x)^2) - (B*c)/(e^4*(d + e*x))

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Rubi [A]  time = 0.120943, antiderivative size = 131, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {771} \[ -\frac{-B e (2 b d-a e)-A e (2 c d-b e)+3 B c d^2}{3 e^4 (d+e x)^3}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )}{4 e^4 (d+e x)^4}+\frac{-A c e-b B e+3 B c d}{2 e^4 (d+e x)^2}-\frac{B c}{e^4 (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((B*d - A*e)*(c*d^2 - b*d*e + a*e^2))/(4*e^4*(d + e*x)^4) - (3*B*c*d^2 - B*e*(2*b*d - a*e) - A*e*(2*c*d - b*e)
)/(3*e^4*(d + e*x)^3) + (3*B*c*d - b*B*e - A*c*e)/(2*e^4*(d + e*x)^2) - (B*c)/(e^4*(d + e*x))

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.0725328, size = 118, normalized size = 0.89 \[ -\frac{A e \left (e (3 a e+b d+4 b e x)+c \left (d^2+4 d e x+6 e^2 x^2\right )\right )+B \left (e \left (a e (d+4 e x)+b \left (d^2+4 d e x+6 e^2 x^2\right )\right )+3 c \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )\right )}{12 e^4 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(A*e*(e*(b*d + 3*a*e + 4*b*e*x) + c*(d^2 + 4*d*e*x + 6*e^2*x^2)) + B*(3*c*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*
e^3*x^3) + e*(a*e*(d + 4*e*x) + b*(d^2 + 4*d*e*x + 6*e^2*x^2))))/(12*e^4*(d + e*x)^4)

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Maple [A]  time = 0.006, size = 142, normalized size = 1.1 \begin{align*} -{\frac{Ab{e}^{2}-2\,Acde+aB{e}^{2}-2\,Bbde+3\,Bc{d}^{2}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{Ace+bBe-3\,Bcd}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{Bc}{{e}^{4} \left ( ex+d \right ) }}-{\frac{aA{e}^{3}-Abd{e}^{2}+Ac{d}^{2}e-aBd{e}^{2}+B{d}^{2}be-Bc{d}^{3}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.07634, size = 212, normalized size = 1.61 \begin{align*} -\frac{12 \, B c e^{3} x^{3} + 3 \, B c d^{3} + 3 \, A a e^{3} +{\left (B b + A c\right )} d^{2} e +{\left (B a + A b\right )} d e^{2} + 6 \,{\left (3 \, B c d e^{2} +{\left (B b + A c\right )} e^{3}\right )} x^{2} + 4 \,{\left (3 \, B c d^{2} e +{\left (B b + A c\right )} d e^{2} +{\left (B a + A b\right )} e^{3}\right )} x}{12 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.18555, size = 338, normalized size = 2.56 \begin{align*} -\frac{12 \, B c e^{3} x^{3} + 3 \, B c d^{3} + 3 \, A a e^{3} +{\left (B b + A c\right )} d^{2} e +{\left (B a + A b\right )} d e^{2} + 6 \,{\left (3 \, B c d e^{2} +{\left (B b + A c\right )} e^{3}\right )} x^{2} + 4 \,{\left (3 \, B c d^{2} e +{\left (B b + A c\right )} d e^{2} +{\left (B a + A b\right )} e^{3}\right )} x}{12 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 52.2882, size = 192, normalized size = 1.45 \begin{align*} - \frac{3 A a e^{3} + A b d e^{2} + A c d^{2} e + B a d e^{2} + B b d^{2} e + 3 B c d^{3} + 12 B c e^{3} x^{3} + x^{2} \left (6 A c e^{3} + 6 B b e^{3} + 18 B c d e^{2}\right ) + x \left (4 A b e^{3} + 4 A c d e^{2} + 4 B a e^{3} + 4 B b d e^{2} + 12 B c d^{2} e\right )}{12 d^{4} e^{4} + 48 d^{3} e^{5} x + 72 d^{2} e^{6} x^{2} + 48 d e^{7} x^{3} + 12 e^{8} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*A*a*e**3 + A*b*d*e**2 + A*c*d**2*e + B*a*d*e**2 + B*b*d**2*e + 3*B*c*d**3 + 12*B*c*e**3*x**3 + x**2*(6*A*c
*e**3 + 6*B*b*e**3 + 18*B*c*d*e**2) + x*(4*A*b*e**3 + 4*A*c*d*e**2 + 4*B*a*e**3 + 4*B*b*d*e**2 + 12*B*c*d**2*e
))/(12*d**4*e**4 + 48*d**3*e**5*x + 72*d**2*e**6*x**2 + 48*d*e**7*x**3 + 12*e**8*x**4)

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Giac [A]  time = 1.11537, size = 297, normalized size = 2.25 \begin{align*} -\frac{1}{12} \,{\left (\frac{12 \, B c e^{\left (-1\right )}}{x e + d} - \frac{18 \, B c d e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}} + \frac{12 \, B c d^{2} e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}} - \frac{3 \, B c d^{3} e^{\left (-1\right )}}{{\left (x e + d\right )}^{4}} + \frac{6 \, B b}{{\left (x e + d\right )}^{2}} + \frac{6 \, A c}{{\left (x e + d\right )}^{2}} - \frac{8 \, B b d}{{\left (x e + d\right )}^{3}} - \frac{8 \, A c d}{{\left (x e + d\right )}^{3}} + \frac{3 \, B b d^{2}}{{\left (x e + d\right )}^{4}} + \frac{3 \, A c d^{2}}{{\left (x e + d\right )}^{4}} + \frac{4 \, B a e}{{\left (x e + d\right )}^{3}} + \frac{4 \, A b e}{{\left (x e + d\right )}^{3}} - \frac{3 \, B a d e}{{\left (x e + d\right )}^{4}} - \frac{3 \, A b d e}{{\left (x e + d\right )}^{4}} + \frac{3 \, A a e^{2}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-3\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(12*B*c*e^(-1)/(x*e + d) - 18*B*c*d*e^(-1)/(x*e + d)^2 + 12*B*c*d^2*e^(-1)/(x*e + d)^3 - 3*B*c*d^3*e^(-1
)/(x*e + d)^4 + 6*B*b/(x*e + d)^2 + 6*A*c/(x*e + d)^2 - 8*B*b*d/(x*e + d)^3 - 8*A*c*d/(x*e + d)^3 + 3*B*b*d^2/
(x*e + d)^4 + 3*A*c*d^2/(x*e + d)^4 + 4*B*a*e/(x*e + d)^3 + 4*A*b*e/(x*e + d)^3 - 3*B*a*d*e/(x*e + d)^4 - 3*A*
b*d*e/(x*e + d)^4 + 3*A*a*e^2/(x*e + d)^4)*e^(-3)