3.1501 \(\int \frac{(b+2 c x) (a+b x+c x^2)}{(d+e x)^3} \, dx\)

Optimal. Leaf size=111 \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{e^4 (d+e x)}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{2 e^4 (d+e x)^2}-\frac{3 c (2 c d-b e) \log (d+e x)}{e^4}+\frac{2 c^2 x}{e^3} \]

[Out]

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

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Rubi [A]  time = 0.0937636, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{e^4 (d+e x)}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{2 e^4 (d+e x)^2}-\frac{3 c (2 c d-b e) \log (d+e x)}{e^4}+\frac{2 c^2 x}{e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0498864, size = 118, normalized size = 1.06 \[ \frac{c e (3 b d (3 d+4 e x)-2 a e (d+2 e x))-b e^2 (a e+b (d+2 e x))-6 c (d+e x)^2 (2 c d-b e) \log (d+e x)+c^2 \left (-8 d^2 e x-10 d^3+8 d e^2 x^2+4 e^3 x^3\right )}{2 e^4 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(-10*d^3 - 8*d^2*e*x + 8*d*e^2*x^2 + 4*e^3*x^3) - b*e^2*(a*e + b*(d + 2*e*x)) + c*e*(-2*a*e*(d + 2*e*x) +
3*b*d*(3*d + 4*e*x)) - 6*c*(2*c*d - b*e)*(d + e*x)^2*Log[d + e*x])/(2*e^4*(d + e*x)^2)

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Maple [A]  time = 0.007, size = 179, normalized size = 1.6 \begin{align*} 2\,{\frac{{c}^{2}x}{{e}^{3}}}-{\frac{ab}{2\,e \left ( ex+d \right ) ^{2}}}+{\frac{acd}{{e}^{2} \left ( ex+d \right ) ^{2}}}+{\frac{{b}^{2}d}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{3\,b{d}^{2}c}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{{c}^{2}{d}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}+3\,{\frac{c\ln \left ( ex+d \right ) b}{{e}^{3}}}-6\,{\frac{{c}^{2}\ln \left ( ex+d \right ) d}{{e}^{4}}}-2\,{\frac{ac}{{e}^{2} \left ( ex+d \right ) }}-{\frac{{b}^{2}}{{e}^{2} \left ( ex+d \right ) }}+6\,{\frac{bcd}{{e}^{3} \left ( ex+d \right ) }}-6\,{\frac{{c}^{2}{d}^{2}}{{e}^{4} \left ( ex+d \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(c*x^2+b*x+a)/(e*x+d)^3,x)

[Out]

2*c^2*x/e^3-1/2/e/(e*x+d)^2*a*b+1/e^2/(e*x+d)^2*d*a*c+1/2/e^2/(e*x+d)^2*d*b^2-3/2/e^3/(e*x+d)^2*d^2*b*c+1/e^4/
(e*x+d)^2*c^2*d^3+3*c/e^3*ln(e*x+d)*b-6*c^2/e^4*ln(e*x+d)*d-2/e^2/(e*x+d)*a*c-1/e^2/(e*x+d)*b^2+6/e^3/(e*x+d)*
b*c*d-6/e^4/(e*x+d)*c^2*d^2

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Maxima [A]  time = 1.02747, size = 173, normalized size = 1.56 \begin{align*} -\frac{10 \, c^{2} d^{3} - 9 \, b c d^{2} e + a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2} + 2 \,{\left (6 \, c^{2} d^{2} e - 6 \, b c d e^{2} +{\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x}{2 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} + \frac{2 \, c^{2} x}{e^{3}} - \frac{3 \,{\left (2 \, c^{2} d - b c e\right )} \log \left (e x + d\right )}{e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)/(e*x+d)^3,x, algorithm="maxima")

[Out]

-1/2*(10*c^2*d^3 - 9*b*c*d^2*e + a*b*e^3 + (b^2 + 2*a*c)*d*e^2 + 2*(6*c^2*d^2*e - 6*b*c*d*e^2 + (b^2 + 2*a*c)*
e^3)*x)/(e^6*x^2 + 2*d*e^5*x + d^2*e^4) + 2*c^2*x/e^3 - 3*(2*c^2*d - b*c*e)*log(e*x + d)/e^4

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Fricas [A]  time = 1.72678, size = 387, normalized size = 3.49 \begin{align*} \frac{4 \, c^{2} e^{3} x^{3} + 8 \, c^{2} d e^{2} x^{2} - 10 \, c^{2} d^{3} + 9 \, b c d^{2} e - a b e^{3} -{\left (b^{2} + 2 \, a c\right )} d e^{2} - 2 \,{\left (4 \, c^{2} d^{2} e - 6 \, b c d e^{2} +{\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x - 6 \,{\left (2 \, c^{2} d^{3} - b c d^{2} e +{\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} x^{2} + 2 \,{\left (2 \, c^{2} d^{2} e - b c 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((2*c*x+b)*(c*x^2+b*x+a)/(e*x+d)^3,x, algorithm="fricas")

[Out]

1/2*(4*c^2*e^3*x^3 + 8*c^2*d*e^2*x^2 - 10*c^2*d^3 + 9*b*c*d^2*e - a*b*e^3 - (b^2 + 2*a*c)*d*e^2 - 2*(4*c^2*d^2
*e - 6*b*c*d*e^2 + (b^2 + 2*a*c)*e^3)*x - 6*(2*c^2*d^3 - b*c*d^2*e + (2*c^2*d*e^2 - b*c*e^3)*x^2 + 2*(2*c^2*d^
2*e - b*c*d*e^2)*x)*log(e*x + d))/(e^6*x^2 + 2*d*e^5*x + d^2*e^4)

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Sympy [A]  time = 1.94575, size = 139, normalized size = 1.25 \begin{align*} \frac{2 c^{2} x}{e^{3}} + \frac{3 c \left (b e - 2 c d\right ) \log{\left (d + e x \right )}}{e^{4}} - \frac{a b e^{3} + 2 a c d e^{2} + b^{2} d e^{2} - 9 b c d^{2} e + 10 c^{2} d^{3} + x \left (4 a c e^{3} + 2 b^{2} e^{3} - 12 b c d e^{2} + 12 c^{2} d^{2} e\right )}{2 d^{2} e^{4} + 4 d e^{5} x + 2 e^{6} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x**2+b*x+a)/(e*x+d)**3,x)

[Out]

2*c**2*x/e**3 + 3*c*(b*e - 2*c*d)*log(d + e*x)/e**4 - (a*b*e**3 + 2*a*c*d*e**2 + b**2*d*e**2 - 9*b*c*d**2*e +
10*c**2*d**3 + x*(4*a*c*e**3 + 2*b**2*e**3 - 12*b*c*d*e**2 + 12*c**2*d**2*e))/(2*d**2*e**4 + 4*d*e**5*x + 2*e*
*6*x**2)

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Giac [A]  time = 1.19507, size = 157, normalized size = 1.41 \begin{align*} 2 \, c^{2} x e^{\left (-3\right )} - 3 \,{\left (2 \, c^{2} d - b c e\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) - \frac{{\left (10 \, c^{2} d^{3} - 9 \, b c d^{2} e + b^{2} d e^{2} + 2 \, a c d e^{2} + a b e^{3} + 2 \,{\left (6 \, c^{2} d^{2} e - 6 \, b c d e^{2} + b^{2} e^{3} + 2 \, a c 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((2*c*x+b)*(c*x^2+b*x+a)/(e*x+d)^3,x, algorithm="giac")

[Out]

2*c^2*x*e^(-3) - 3*(2*c^2*d - b*c*e)*e^(-4)*log(abs(x*e + d)) - 1/2*(10*c^2*d^3 - 9*b*c*d^2*e + b^2*d*e^2 + 2*
a*c*d*e^2 + a*b*e^3 + 2*(6*c^2*d^2*e - 6*b*c*d*e^2 + b^2*e^3 + 2*a*c*e^3)*x)*e^(-4)/(x*e + d)^2