3.1113 \(\int \frac{(A+B x) (b x+c x^2)}{(d+e x)^6} \, dx\)

Optimal. Leaf size=118 \[ \frac{-A c e-b B e+3 B c d}{3 e^4 (d+e x)^3}-\frac{B d (3 c d-2 b e)-A e (2 c d-b e)}{4 e^4 (d+e x)^4}+\frac{d (B d-A e) (c d-b e)}{5 e^4 (d+e x)^5}-\frac{B c}{2 e^4 (d+e x)^2} \]

[Out]

(d*(B*d - A*e)*(c*d - b*e))/(5*e^4*(d + e*x)^5) - (B*d*(3*c*d - 2*b*e) - A*e*(2*c*d - b*e))/(4*e^4*(d + e*x)^4
) + (3*B*c*d - b*B*e - A*c*e)/(3*e^4*(d + e*x)^3) - (B*c)/(2*e^4*(d + e*x)^2)

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Rubi [A]  time = 0.100303, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {771} \[ \frac{-A c e-b B e+3 B c d}{3 e^4 (d+e x)^3}-\frac{B d (3 c d-2 b e)-A e (2 c d-b e)}{4 e^4 (d+e x)^4}+\frac{d (B d-A e) (c d-b e)}{5 e^4 (d+e x)^5}-\frac{B c}{2 e^4 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2))/(d + e*x)^6,x]

[Out]

(d*(B*d - A*e)*(c*d - b*e))/(5*e^4*(d + e*x)^5) - (B*d*(3*c*d - 2*b*e) - A*e*(2*c*d - b*e))/(4*e^4*(d + e*x)^4
) + (3*B*c*d - b*B*e - A*c*e)/(3*e^4*(d + e*x)^3) - (B*c)/(2*e^4*(d + e*x)^2)

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

Mathematica [A]  time = 0.0493156, size = 104, normalized size = 0.88 \[ -\frac{A e \left (3 b e (d+5 e x)+2 c \left (d^2+5 d e x+10 e^2 x^2\right )\right )+B \left (2 b e \left (d^2+5 d e x+10 e^2 x^2\right )+3 c \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )\right )}{60 e^4 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2))/(d + e*x)^6,x]

[Out]

-(A*e*(3*b*e*(d + 5*e*x) + 2*c*(d^2 + 5*d*e*x + 10*e^2*x^2)) + B*(2*b*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*c*(d^
3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3)))/(60*e^4*(d + e*x)^5)

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Maple [A]  time = 0.004, size = 118, normalized size = 1. \begin{align*} -{\frac{Ab{e}^{2}-2\,Acde-2\,bBde+3\,Bc{d}^{2}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{Ace+bBe-3\,Bcd}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{Bc}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}+{\frac{d \left ( Ab{e}^{2}-Acde-bBde+Bc{d}^{2} \right ) }{5\,{e}^{4} \left ( ex+d \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)/(e*x+d)^6,x)

[Out]

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

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Maxima [A]  time = 1.19843, size = 211, normalized size = 1.79 \begin{align*} -\frac{30 \, B c e^{3} x^{3} + 3 \, B c d^{3} + 3 \, A b d e^{2} + 2 \,{\left (B b + A c\right )} d^{2} e + 10 \,{\left (3 \, B c d e^{2} + 2 \,{\left (B b + A c\right )} e^{3}\right )} x^{2} + 5 \,{\left (3 \, B c d^{2} e + 3 \, A b e^{3} + 2 \,{\left (B b + A c\right )} d e^{2}\right )} x}{60 \,{\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} 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)/(e*x+d)^6,x, algorithm="maxima")

[Out]

-1/60*(30*B*c*e^3*x^3 + 3*B*c*d^3 + 3*A*b*d*e^2 + 2*(B*b + A*c)*d^2*e + 10*(3*B*c*d*e^2 + 2*(B*b + A*c)*e^3)*x
^2 + 5*(3*B*c*d^2*e + 3*A*b*e^3 + 2*(B*b + A*c)*d*e^2)*x)/(e^9*x^5 + 5*d*e^8*x^4 + 10*d^2*e^7*x^3 + 10*d^3*e^6
*x^2 + 5*d^4*e^5*x + d^5*e^4)

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Fricas [A]  time = 1.51897, size = 339, normalized size = 2.87 \begin{align*} -\frac{30 \, B c e^{3} x^{3} + 3 \, B c d^{3} + 3 \, A b d e^{2} + 2 \,{\left (B b + A c\right )} d^{2} e + 10 \,{\left (3 \, B c d e^{2} + 2 \,{\left (B b + A c\right )} e^{3}\right )} x^{2} + 5 \,{\left (3 \, B c d^{2} e + 3 \, A b e^{3} + 2 \,{\left (B b + A c\right )} d e^{2}\right )} x}{60 \,{\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} 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)/(e*x+d)^6,x, algorithm="fricas")

[Out]

-1/60*(30*B*c*e^3*x^3 + 3*B*c*d^3 + 3*A*b*d*e^2 + 2*(B*b + A*c)*d^2*e + 10*(3*B*c*d*e^2 + 2*(B*b + A*c)*e^3)*x
^2 + 5*(3*B*c*d^2*e + 3*A*b*e^3 + 2*(B*b + A*c)*d*e^2)*x)/(e^9*x^5 + 5*d*e^8*x^4 + 10*d^2*e^7*x^3 + 10*d^3*e^6
*x^2 + 5*d^4*e^5*x + d^5*e^4)

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Sympy [A]  time = 17.757, size = 184, normalized size = 1.56 \begin{align*} - \frac{3 A b d e^{2} + 2 A c d^{2} e + 2 B b d^{2} e + 3 B c d^{3} + 30 B c e^{3} x^{3} + x^{2} \left (20 A c e^{3} + 20 B b e^{3} + 30 B c d e^{2}\right ) + x \left (15 A b e^{3} + 10 A c d e^{2} + 10 B b d e^{2} + 15 B c d^{2} e\right )}{60 d^{5} e^{4} + 300 d^{4} e^{5} x + 600 d^{3} e^{6} x^{2} + 600 d^{2} e^{7} x^{3} + 300 d e^{8} x^{4} + 60 e^{9} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)/(e*x+d)**6,x)

[Out]

-(3*A*b*d*e**2 + 2*A*c*d**2*e + 2*B*b*d**2*e + 3*B*c*d**3 + 30*B*c*e**3*x**3 + x**2*(20*A*c*e**3 + 20*B*b*e**3
+ 30*B*c*d*e**2) + x*(15*A*b*e**3 + 10*A*c*d*e**2 + 10*B*b*d*e**2 + 15*B*c*d**2*e))/(60*d**5*e**4 + 300*d**4*
e**5*x + 600*d**3*e**6*x**2 + 600*d**2*e**7*x**3 + 300*d*e**8*x**4 + 60*e**9*x**5)

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Giac [A]  time = 1.19408, size = 155, normalized size = 1.31 \begin{align*} -\frac{{\left (30 \, B c x^{3} e^{3} + 30 \, B c d x^{2} e^{2} + 15 \, B c d^{2} x e + 3 \, B c d^{3} + 20 \, B b x^{2} e^{3} + 20 \, A c x^{2} e^{3} + 10 \, B b d x e^{2} + 10 \, A c d x e^{2} + 2 \, B b d^{2} e + 2 \, A c d^{2} e + 15 \, A b x e^{3} + 3 \, A b d e^{2}\right )} e^{\left (-4\right )}}{60 \,{\left (x e + d\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/(e*x+d)^6,x, algorithm="giac")

[Out]

-1/60*(30*B*c*x^3*e^3 + 30*B*c*d*x^2*e^2 + 15*B*c*d^2*x*e + 3*B*c*d^3 + 20*B*b*x^2*e^3 + 20*A*c*x^2*e^3 + 10*B
*b*d*x*e^2 + 10*A*c*d*x*e^2 + 2*B*b*d^2*e + 2*A*c*d^2*e + 15*A*b*x*e^3 + 3*A*b*d*e^2)*e^(-4)/(x*e + d)^5