3.2317 \(\int \frac{(A+B x) (a+b x+c x^2)}{(d+e x)^7} \, dx\)

Optimal. Leaf size=134 \[ \frac{A e (2 c d-b e)-B \left (3 c d^2-e (2 b d-a e)\right )}{5 e^4 (d+e x)^5}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )}{6 e^4 (d+e x)^6}+\frac{-A c e-b B e+3 B c d}{4 e^4 (d+e x)^4}-\frac{B c}{3 e^4 (d+e x)^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.102488, antiderivative size = 133, 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} \[ -\frac{-B e (2 b d-a e)-A e (2 c d-b e)+3 B c d^2}{5 e^4 (d+e x)^5}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )}{6 e^4 (d+e x)^6}+\frac{-A c e-b B e+3 B c d}{4 e^4 (d+e x)^4}-\frac{B c}{3 e^4 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2))/(d + e*x)^7,x]

[Out]

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

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)^7} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^7}+\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)^6}+\frac{-3 B c d+b B e+A c e}{e^3 (d+e x)^5}+\frac{B c}{e^3 (d+e x)^4}\right ) \, dx\\ &=\frac{(B d-A e) \left (c d^2-b d e+a e^2\right )}{6 e^4 (d+e x)^6}-\frac{3 B c d^2-B e (2 b d-a e)-A e (2 c d-b e)}{5 e^4 (d+e x)^5}+\frac{3 B c d-b B e-A c e}{4 e^4 (d+e x)^4}-\frac{B c}{3 e^4 (d+e x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0673116, size = 119, normalized size = 0.89 \[ -\frac{A e \left (2 e (5 a e+b d+6 b e x)+c \left (d^2+6 d e x+15 e^2 x^2\right )\right )+B \left (e \left (2 a e (d+6 e x)+b \left (d^2+6 d e x+15 e^2 x^2\right )\right )+c \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )\right )}{60 e^4 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2))/(d + e*x)^7,x]

[Out]

-(A*e*(2*e*(b*d + 5*a*e + 6*b*e*x) + c*(d^2 + 6*d*e*x + 15*e^2*x^2)) + B*(c*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 +
20*e^3*x^3) + e*(2*a*e*(d + 6*e*x) + b*(d^2 + 6*d*e*x + 15*e^2*x^2))))/(60*e^4*(d + e*x)^6)

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 142, normalized size = 1.1 \begin{align*} -{\frac{aA{e}^{3}-Abd{e}^{2}+Ac{d}^{2}e-aBd{e}^{2}+B{d}^{2}be-Bc{d}^{3}}{6\,{e}^{4} \left ( ex+d \right ) ^{6}}}-{\frac{Ab{e}^{2}-2\,Acde+aB{e}^{2}-2\,Bbde+3\,Bc{d}^{2}}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}}-{\frac{Bc}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{Ace+bBe-3\,Bcd}{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)^7,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.05701, size = 240, normalized size = 1.79 \begin{align*} -\frac{20 \, B c e^{3} x^{3} + B c d^{3} + 10 \, A a e^{3} +{\left (B b + A c\right )} d^{2} e + 2 \,{\left (B a + A b\right )} d e^{2} + 15 \,{\left (B c d e^{2} +{\left (B b + A c\right )} e^{3}\right )} x^{2} + 6 \,{\left (B c d^{2} e +{\left (B b + A c\right )} d e^{2} + 2 \,{\left (B a + A b\right )} e^{3}\right )} x}{60 \,{\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} 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)^7,x, algorithm="maxima")

[Out]

-1/60*(20*B*c*e^3*x^3 + B*c*d^3 + 10*A*a*e^3 + (B*b + A*c)*d^2*e + 2*(B*a + A*b)*d*e^2 + 15*(B*c*d*e^2 + (B*b
+ A*c)*e^3)*x^2 + 6*(B*c*d^2*e + (B*b + A*c)*d*e^2 + 2*(B*a + A*b)*e^3)*x)/(e^10*x^6 + 6*d*e^9*x^5 + 15*d^2*e^
8*x^4 + 20*d^3*e^7*x^3 + 15*d^4*e^6*x^2 + 6*d^5*e^5*x + d^6*e^4)

________________________________________________________________________________________

Fricas [A]  time = 1.2487, size = 386, normalized size = 2.88 \begin{align*} -\frac{20 \, B c e^{3} x^{3} + B c d^{3} + 10 \, A a e^{3} +{\left (B b + A c\right )} d^{2} e + 2 \,{\left (B a + A b\right )} d e^{2} + 15 \,{\left (B c d e^{2} +{\left (B b + A c\right )} e^{3}\right )} x^{2} + 6 \,{\left (B c d^{2} e +{\left (B b + A c\right )} d e^{2} + 2 \,{\left (B a + A b\right )} e^{3}\right )} x}{60 \,{\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} 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)^7,x, algorithm="fricas")

[Out]

-1/60*(20*B*c*e^3*x^3 + B*c*d^3 + 10*A*a*e^3 + (B*b + A*c)*d^2*e + 2*(B*a + A*b)*d*e^2 + 15*(B*c*d*e^2 + (B*b
+ A*c)*e^3)*x^2 + 6*(B*c*d^2*e + (B*b + A*c)*d*e^2 + 2*(B*a + A*b)*e^3)*x)/(e^10*x^6 + 6*d*e^9*x^5 + 15*d^2*e^
8*x^4 + 20*d^3*e^7*x^3 + 15*d^4*e^6*x^2 + 6*d^5*e^5*x + d^6*e^4)

________________________________________________________________________________________

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)*(c*x**2+b*x+a)/(e*x+d)**7,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.11216, size = 178, normalized size = 1.33 \begin{align*} -\frac{{\left (20 \, B c x^{3} e^{3} + 15 \, B c d x^{2} e^{2} + 6 \, B c d^{2} x e + B c d^{3} + 15 \, B b x^{2} e^{3} + 15 \, A c x^{2} e^{3} + 6 \, B b d x e^{2} + 6 \, A c d x e^{2} + B b d^{2} e + A c d^{2} e + 12 \, B a x e^{3} + 12 \, A b x e^{3} + 2 \, B a d e^{2} + 2 \, A b d e^{2} + 10 \, A a e^{3}\right )} e^{\left (-4\right )}}{60 \,{\left (x e + d\right )}^{6}} \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)^7,x, algorithm="giac")

[Out]

-1/60*(20*B*c*x^3*e^3 + 15*B*c*d*x^2*e^2 + 6*B*c*d^2*x*e + B*c*d^3 + 15*B*b*x^2*e^3 + 15*A*c*x^2*e^3 + 6*B*b*d
*x*e^2 + 6*A*c*d*x*e^2 + B*b*d^2*e + A*c*d^2*e + 12*B*a*x*e^3 + 12*A*b*x*e^3 + 2*B*a*d*e^2 + 2*A*b*d*e^2 + 10*
A*a*e^3)*e^(-4)/(x*e + d)^6