3.239 \(\int \frac{(b x+c x^2)^2}{(d+e x)^4} \, dx\)

Optimal. Leaf size=120 \[ -\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{e^5 (d+e x)}-\frac{d^2 (c d-b e)^2}{3 e^5 (d+e x)^3}+\frac{d (c d-b e) (2 c d-b e)}{e^5 (d+e x)^2}-\frac{2 c (2 c d-b e) \log (d+e x)}{e^5}+\frac{c^2 x}{e^4} \]

[Out]

(c^2*x)/e^4 - (d^2*(c*d - b*e)^2)/(3*e^5*(d + e*x)^3) + (d*(c*d - b*e)*(2*c*d - b*e))/(e^5*(d + e*x)^2) - (6*c
^2*d^2 - 6*b*c*d*e + b^2*e^2)/(e^5*(d + e*x)) - (2*c*(2*c*d - b*e)*Log[d + e*x])/e^5

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Rubi [A]  time = 0.0962984, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {698} \[ -\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{e^5 (d+e x)}-\frac{d^2 (c d-b e)^2}{3 e^5 (d+e x)^3}+\frac{d (c d-b e) (2 c d-b e)}{e^5 (d+e x)^2}-\frac{2 c (2 c d-b e) \log (d+e x)}{e^5}+\frac{c^2 x}{e^4} \]

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^2/(d + e*x)^4,x]

[Out]

(c^2*x)/e^4 - (d^2*(c*d - b*e)^2)/(3*e^5*(d + e*x)^3) + (d*(c*d - b*e)*(2*c*d - b*e))/(e^5*(d + e*x)^2) - (6*c
^2*d^2 - 6*b*c*d*e + b^2*e^2)/(e^5*(d + e*x)) - (2*c*(2*c*d - b*e)*Log[d + e*x])/e^5

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.0558831, size = 134, normalized size = 1.12 \[ \frac{-b^2 e^2 \left (d^2+3 d e x+3 e^2 x^2\right )+b c d e \left (11 d^2+27 d e x+18 e^2 x^2\right )-6 c (d+e x)^3 (2 c d-b e) \log (d+e x)+c^2 \left (-9 d^2 e^2 x^2-27 d^3 e x-13 d^4+9 d e^3 x^3+3 e^4 x^4\right )}{3 e^5 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^2/(d + e*x)^4,x]

[Out]

(-(b^2*e^2*(d^2 + 3*d*e*x + 3*e^2*x^2)) + b*c*d*e*(11*d^2 + 27*d*e*x + 18*e^2*x^2) + c^2*(-13*d^4 - 27*d^3*e*x
- 9*d^2*e^2*x^2 + 9*d*e^3*x^3 + 3*e^4*x^4) - 6*c*(2*c*d - b*e)*(d + e*x)^3*Log[d + e*x])/(3*e^5*(d + e*x)^3)

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Maple [A]  time = 0.05, size = 189, normalized size = 1.6 \begin{align*}{\frac{{c}^{2}x}{{e}^{4}}}+{\frac{{b}^{2}d}{{e}^{3} \left ( ex+d \right ) ^{2}}}-3\,{\frac{{d}^{2}bc}{{e}^{4} \left ( ex+d \right ) ^{2}}}+2\,{\frac{{c}^{2}{d}^{3}}{{e}^{5} \left ( ex+d \right ) ^{2}}}+2\,{\frac{c\ln \left ( ex+d \right ) b}{{e}^{4}}}-4\,{\frac{{c}^{2}d\ln \left ( ex+d \right ) }{{e}^{5}}}-{\frac{{b}^{2}{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{2\,{d}^{3}bc}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{{c}^{2}{d}^{4}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{{b}^{2}}{{e}^{3} \left ( ex+d \right ) }}+6\,{\frac{bcd}{{e}^{4} \left ( ex+d \right ) }}-6\,{\frac{{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^2/(e*x+d)^4,x)

[Out]

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

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Maxima [A]  time = 1.05529, size = 215, normalized size = 1.79 \begin{align*} -\frac{13 \, c^{2} d^{4} - 11 \, b c d^{3} e + b^{2} d^{2} e^{2} + 3 \,{\left (6 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 3 \,{\left (10 \, c^{2} d^{3} e - 9 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x}{3 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} + \frac{c^{2} x}{e^{4}} - \frac{2 \,{\left (2 \, c^{2} d - b c e\right )} \log \left (e x + d\right )}{e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^2/(e*x+d)^4,x, algorithm="maxima")

[Out]

-1/3*(13*c^2*d^4 - 11*b*c*d^3*e + b^2*d^2*e^2 + 3*(6*c^2*d^2*e^2 - 6*b*c*d*e^3 + b^2*e^4)*x^2 + 3*(10*c^2*d^3*
e - 9*b*c*d^2*e^2 + b^2*d*e^3)*x)/(e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5) + c^2*x/e^4 - 2*(2*c^2*d - b
*c*e)*log(e*x + d)/e^5

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Fricas [B]  time = 1.61198, size = 494, normalized size = 4.12 \begin{align*} \frac{3 \, c^{2} e^{4} x^{4} + 9 \, c^{2} d e^{3} x^{3} - 13 \, c^{2} d^{4} + 11 \, b c d^{3} e - b^{2} d^{2} e^{2} - 3 \,{\left (3 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} - 3 \,{\left (9 \, c^{2} d^{3} e - 9 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x - 6 \,{\left (2 \, c^{2} d^{4} - b c d^{3} e +{\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x^{3} + 3 \,{\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} x^{2} + 3 \,{\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^2/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/3*(3*c^2*e^4*x^4 + 9*c^2*d*e^3*x^3 - 13*c^2*d^4 + 11*b*c*d^3*e - b^2*d^2*e^2 - 3*(3*c^2*d^2*e^2 - 6*b*c*d*e^
3 + b^2*e^4)*x^2 - 3*(9*c^2*d^3*e - 9*b*c*d^2*e^2 + b^2*d*e^3)*x - 6*(2*c^2*d^4 - b*c*d^3*e + (2*c^2*d*e^3 - b
*c*e^4)*x^3 + 3*(2*c^2*d^2*e^2 - b*c*d*e^3)*x^2 + 3*(2*c^2*d^3*e - b*c*d^2*e^2)*x)*log(e*x + d))/(e^8*x^3 + 3*
d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5)

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Sympy [A]  time = 4.22294, size = 163, normalized size = 1.36 \begin{align*} \frac{c^{2} x}{e^{4}} + \frac{2 c \left (b e - 2 c d\right ) \log{\left (d + e x \right )}}{e^{5}} - \frac{b^{2} d^{2} e^{2} - 11 b c d^{3} e + 13 c^{2} d^{4} + x^{2} \left (3 b^{2} e^{4} - 18 b c d e^{3} + 18 c^{2} d^{2} e^{2}\right ) + x \left (3 b^{2} d e^{3} - 27 b c d^{2} e^{2} + 30 c^{2} d^{3} e\right )}{3 d^{3} e^{5} + 9 d^{2} e^{6} x + 9 d e^{7} x^{2} + 3 e^{8} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**2/(e*x+d)**4,x)

[Out]

c**2*x/e**4 + 2*c*(b*e - 2*c*d)*log(d + e*x)/e**5 - (b**2*d**2*e**2 - 11*b*c*d**3*e + 13*c**2*d**4 + x**2*(3*b
**2*e**4 - 18*b*c*d*e**3 + 18*c**2*d**2*e**2) + x*(3*b**2*d*e**3 - 27*b*c*d**2*e**2 + 30*c**2*d**3*e))/(3*d**3
*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3)

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Giac [A]  time = 1.3237, size = 177, normalized size = 1.48 \begin{align*} c^{2} x e^{\left (-4\right )} - 2 \,{\left (2 \, c^{2} d - b c e\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) - \frac{{\left (13 \, c^{2} d^{4} - 11 \, b c d^{3} e + b^{2} d^{2} e^{2} + 3 \,{\left (6 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 3 \,{\left (10 \, c^{2} d^{3} e - 9 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )} e^{\left (-5\right )}}{3 \,{\left (x e + d\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^2/(e*x+d)^4,x, algorithm="giac")

[Out]

c^2*x*e^(-4) - 2*(2*c^2*d - b*c*e)*e^(-5)*log(abs(x*e + d)) - 1/3*(13*c^2*d^4 - 11*b*c*d^3*e + b^2*d^2*e^2 + 3
*(6*c^2*d^2*e^2 - 6*b*c*d*e^3 + b^2*e^4)*x^2 + 3*(10*c^2*d^3*e - 9*b*c*d^2*e^2 + b^2*d*e^3)*x)*e^(-5)/(x*e + d
)^3