3.1294 \(\int \frac{(A+B x) (a+c x^2)}{(d+e x)^5} \, dx\)

Optimal. Leaf size=106 \[ -\frac{a B e^2-2 A c d e+3 B c d^2}{3 e^4 (d+e x)^3}+\frac{\left (a e^2+c d^2\right ) (B d-A e)}{4 e^4 (d+e x)^4}+\frac{c (3 B d-A e)}{2 e^4 (d+e x)^2}-\frac{B c}{e^4 (d+e x)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0691662, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {772} \[ -\frac{a B e^2-2 A c d e+3 B c d^2}{3 e^4 (d+e x)^3}+\frac{\left (a e^2+c d^2\right ) (B d-A e)}{4 e^4 (d+e x)^4}+\frac{c (3 B d-A e)}{2 e^4 (d+e x)^2}-\frac{B c}{e^4 (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0468198, size = 87, normalized size = 0.82 \[ -\frac{3 a A e^3+a B e^2 (d+4 e x)+A c e \left (d^2+4 d e x+6 e^2 x^2\right )+3 B c \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )}{12 e^4 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 110, normalized size = 1. \begin{align*} -{\frac{Bc}{{e}^{4} \left ( ex+d \right ) }}-{\frac{c \left ( Ae-3\,Bd \right ) }{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{-2\,Acde+aB{e}^{2}+3\,Bc{d}^{2}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{aA{e}^{3}+Ac{d}^{2}e-aBd{e}^{2}-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+a)/(e*x+d)^5,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.07852, size = 178, normalized size = 1.68 \begin{align*} -\frac{12 \, B c e^{3} x^{3} + 3 \, B c d^{3} + A c d^{2} e + B a d e^{2} + 3 \, A a e^{3} + 6 \,{\left (3 \, B c d e^{2} + A c e^{3}\right )} x^{2} + 4 \,{\left (3 \, B c d^{2} e + A c d e^{2} + B a 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+a)/(e*x+d)^5,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.83384, size = 284, normalized size = 2.68 \begin{align*} -\frac{12 \, B c e^{3} x^{3} + 3 \, B c d^{3} + A c d^{2} e + B a d e^{2} + 3 \, A a e^{3} + 6 \,{\left (3 \, B c d e^{2} + A c e^{3}\right )} x^{2} + 4 \,{\left (3 \, B c d^{2} e + A c d e^{2} + B a 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+a)/(e*x+d)^5,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 11.7133, size = 148, normalized size = 1.4 \begin{align*} - \frac{3 A a e^{3} + A c d^{2} e + B a d e^{2} + 3 B c d^{3} + 12 B c e^{3} x^{3} + x^{2} \left (6 A c e^{3} + 18 B c d e^{2}\right ) + x \left (4 A c d e^{2} + 4 B a e^{3} + 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+a)/(e*x+d)**5,x)

[Out]

-(3*A*a*e**3 + A*c*d**2*e + B*a*d*e**2 + 3*B*c*d**3 + 12*B*c*e**3*x**3 + x**2*(6*A*c*e**3 + 18*B*c*d*e**2) + x
*(4*A*c*d*e**2 + 4*B*a*e**3 + 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)

________________________________________________________________________________________

Giac [A]  time = 1.28296, size = 204, normalized size = 1.92 \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 \, A c}{{\left (x e + d\right )}^{2}} - \frac{8 \, A c d}{{\left (x e + d\right )}^{3}} + \frac{3 \, A c d^{2}}{{\left (x e + d\right )}^{4}} + \frac{4 \, B a e}{{\left (x e + d\right )}^{3}} - \frac{3 \, B a 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+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*A*c/(x*e + d)^2 - 8*A*c*d/(x*e + d)^3 + 3*A*c*d^2/(x*e + d)^4 + 4*B*a*e/(x*e + d)^3 - 3*B*a*
d*e/(x*e + d)^4 + 3*A*a*e^2/(x*e + d)^4)*e^(-3)