3.1110 \(\int \frac{(A+B x) (b x+c x^2)}{(d+e x)^3} \, dx\)

Optimal. Leaf size=104 \[ \frac{d (B d-A e) (c d-b e)}{2 e^4 (d+e x)^2}-\frac{B d (3 c d-2 b e)-A e (2 c d-b e)}{e^4 (d+e x)}-\frac{\log (d+e x) (-A c e-b B e+3 B c d)}{e^4}+\frac{B c x}{e^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.107333, antiderivative size = 104, 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} \[ \frac{d (B d-A e) (c d-b e)}{2 e^4 (d+e x)^2}-\frac{B d (3 c d-2 b e)-A e (2 c d-b e)}{e^4 (d+e x)}-\frac{\log (d+e x) (-A c e-b B e+3 B c d)}{e^4}+\frac{B c x}{e^3} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2))/(d + e*x)^3,x]

[Out]

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

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

Mathematica [A]  time = 0.0915148, size = 96, normalized size = 0.92 \[ \frac{\frac{-2 A b e^2+4 A c d e+4 b B d e-6 B c d^2}{d+e x}+\frac{d (B d-A e) (c d-b e)}{(d+e x)^2}+2 \log (d+e x) (A c e+b B e-3 B c d)+2 B c e x}{2 e^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2))/(d + e*x)^3,x]

[Out]

(2*B*c*e*x + (d*(B*d - A*e)*(c*d - b*e))/(d + e*x)^2 + (-6*B*c*d^2 + 4*b*B*d*e + 4*A*c*d*e - 2*A*b*e^2)/(d + e
*x) + 2*(-3*B*c*d + b*B*e + A*c*e)*Log[d + e*x])/(2*e^4)

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 174, normalized size = 1.7 \begin{align*}{\frac{Bcx}{{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) Ac}{{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) bB}{{e}^{3}}}-3\,{\frac{\ln \left ( ex+d \right ) Bcd}{{e}^{4}}}+{\frac{dAb}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{Ac{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{B{d}^{2}b}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{Bc{d}^{3}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{Ab}{{e}^{2} \left ( ex+d \right ) }}+2\,{\frac{Acd}{{e}^{3} \left ( ex+d \right ) }}+2\,{\frac{bBd}{{e}^{3} \left ( ex+d \right ) }}-3\,{\frac{Bc{d}^{2}}{{e}^{4} \left ( ex+d \right ) }} \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)^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.06099, size = 162, normalized size = 1.56 \begin{align*} -\frac{5 \, B c d^{3} + A b d e^{2} - 3 \,{\left (B b + A c\right )} d^{2} e + 2 \,{\left (3 \, B c d^{2} e + A b e^{3} - 2 \,{\left (B b + A c\right )} d e^{2}\right )} x}{2 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} + \frac{B c x}{e^{3}} - \frac{{\left (3 \, B c d -{\left (B b + A c\right )} e\right )} \log \left (e x + d\right )}{e^{4}} \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)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.47437, size = 400, normalized size = 3.85 \begin{align*} \frac{2 \, B c e^{3} x^{3} + 4 \, B c d e^{2} x^{2} - 5 \, B c d^{3} - A b d e^{2} + 3 \,{\left (B b + A c\right )} d^{2} e - 2 \,{\left (2 \, B c d^{2} e + A b e^{3} - 2 \,{\left (B b + A c\right )} d e^{2}\right )} x - 2 \,{\left (3 \, B c d^{3} -{\left (B b + A c\right )} d^{2} e +{\left (3 \, B c d e^{2} -{\left (B b + A c\right )} e^{3}\right )} x^{2} + 2 \,{\left (3 \, B c d^{2} e -{\left (B b + A c\right )} d e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} 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)^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 2.06359, size = 138, normalized size = 1.33 \begin{align*} \frac{B c x}{e^{3}} + \frac{- A b d e^{2} + 3 A c d^{2} e + 3 B b d^{2} e - 5 B c d^{3} + x \left (- 2 A b e^{3} + 4 A c d e^{2} + 4 B b d e^{2} - 6 B c d^{2} e\right )}{2 d^{2} e^{4} + 4 d e^{5} x + 2 e^{6} x^{2}} + \frac{\left (A c e + B b e - 3 B c d\right ) \log{\left (d + e x \right )}}{e^{4}} \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)**3,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.19104, size = 153, normalized size = 1.47 \begin{align*} B c x e^{\left (-3\right )} -{\left (3 \, B c d - B b e - A c e\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) - \frac{{\left (5 \, B c d^{3} - 3 \, B b d^{2} e - 3 \, A c d^{2} e + A b d e^{2} + 2 \,{\left (3 \, B c d^{2} e - 2 \, B b d e^{2} - 2 \, A c d e^{2} + A b e^{3}\right )} x\right )} e^{\left (-4\right )}}{2 \,{\left (x e + d\right )}^{2}} \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)^3,x, algorithm="giac")

[Out]

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