### 3.2127 $$\int \frac{(a+b x+c x^2)^2}{(d+e x)^5} \, dx$$

Optimal. Leaf size=150 $-\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{2 e^5 (d+e x)^2}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5 (d+e x)^3}-\frac{\left (a e^2-b d e+c d^2\right )^2}{4 e^5 (d+e x)^4}+\frac{2 c (2 c d-b e)}{e^5 (d+e x)}+\frac{c^2 \log (d+e x)}{e^5}$

[Out]

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

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Rubi [A]  time = 0.12168, antiderivative size = 150, 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 = {698} $-\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{2 e^5 (d+e x)^2}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5 (d+e x)^3}-\frac{\left (a e^2-b d e+c d^2\right )^2}{4 e^5 (d+e x)^4}+\frac{2 c (2 c d-b e)}{e^5 (d+e x)}+\frac{c^2 \log (d+e x)}{e^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0728061, size = 170, normalized size = 1.13 $\frac{-e^2 \left (3 a^2 e^2+2 a b e (d+4 e x)+b^2 \left (d^2+4 d e x+6 e^2 x^2\right )\right )-2 c e \left (a e \left (d^2+4 d e x+6 e^2 x^2\right )+3 b \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )\right )+c^2 d \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )+12 c^2 (d+e x)^4 \log (d+e x)}{12 e^5 (d+e x)^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(c^2*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3) - e^2*(3*a^2*e^2 + 2*a*b*e*(d + 4*e*x) + b^2*(d^2 +
4*d*e*x + 6*e^2*x^2)) - 2*c*e*(a*e*(d^2 + 4*d*e*x + 6*e^2*x^2) + 3*b*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^
3)) + 12*c^2*(d + e*x)^4*Log[d + e*x])/(12*e^5*(d + e*x)^4)

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Maple [A]  time = 0.048, size = 287, normalized size = 1.9 \begin{align*} -{\frac{{a}^{2}}{4\,e \left ( ex+d \right ) ^{4}}}+{\frac{abd}{2\,{e}^{2} \left ( ex+d \right ) ^{4}}}-{\frac{ac{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{{b}^{2}{d}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}+{\frac{{d}^{3}bc}{2\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{{c}^{2}{d}^{4}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}}-{\frac{2\,ab}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}+{\frac{4\,acd}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{2\,{b}^{2}d}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-2\,{\frac{bc{d}^{2}}{{e}^{4} \left ( ex+d \right ) ^{3}}}+{\frac{4\,{c}^{2}{d}^{3}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{ac}{{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+3\,{\frac{bcd}{{e}^{4} \left ( ex+d \right ) ^{2}}}-3\,{\frac{{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{{c}^{2}\ln \left ( ex+d \right ) }{{e}^{5}}}-2\,{\frac{bc}{{e}^{4} \left ( ex+d \right ) }}+4\,{\frac{{c}^{2}d}{{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+a)^2/(e*x+d)^5,x)

[Out]

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

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Maxima [A]  time = 1.00033, size = 290, normalized size = 1.93 \begin{align*} \frac{25 \, c^{2} d^{4} - 6 \, b c d^{3} e - 2 \, a b d e^{3} - 3 \, a^{2} e^{4} -{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 24 \,{\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x^{3} + 6 \,{\left (18 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} -{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 4 \,{\left (22 \, c^{2} d^{3} e - 6 \, b c d^{2} e^{2} - 2 \, a b e^{4} -{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{12 \,{\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} + \frac{c^{2} \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+a)^2/(e*x+d)^5,x, algorithm="maxima")

[Out]

1/12*(25*c^2*d^4 - 6*b*c*d^3*e - 2*a*b*d*e^3 - 3*a^2*e^4 - (b^2 + 2*a*c)*d^2*e^2 + 24*(2*c^2*d*e^3 - b*c*e^4)*
x^3 + 6*(18*c^2*d^2*e^2 - 6*b*c*d*e^3 - (b^2 + 2*a*c)*e^4)*x^2 + 4*(22*c^2*d^3*e - 6*b*c*d^2*e^2 - 2*a*b*e^4 -
(b^2 + 2*a*c)*d*e^3)*x)/(e^9*x^4 + 4*d*e^8*x^3 + 6*d^2*e^7*x^2 + 4*d^3*e^6*x + d^4*e^5) + c^2*log(e*x + d)/e^
5

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Fricas [A]  time = 2.02727, size = 548, normalized size = 3.65 \begin{align*} \frac{25 \, c^{2} d^{4} - 6 \, b c d^{3} e - 2 \, a b d e^{3} - 3 \, a^{2} e^{4} -{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 24 \,{\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x^{3} + 6 \,{\left (18 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} -{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 4 \,{\left (22 \, c^{2} d^{3} e - 6 \, b c d^{2} e^{2} - 2 \, a b e^{4} -{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x + 12 \,{\left (c^{2} e^{4} x^{4} + 4 \, c^{2} d e^{3} x^{3} + 6 \, c^{2} d^{2} e^{2} x^{2} + 4 \, c^{2} d^{3} e x + c^{2} d^{4}\right )} \log \left (e x + d\right )}{12 \,{\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)^5,x, algorithm="fricas")

[Out]

1/12*(25*c^2*d^4 - 6*b*c*d^3*e - 2*a*b*d*e^3 - 3*a^2*e^4 - (b^2 + 2*a*c)*d^2*e^2 + 24*(2*c^2*d*e^3 - b*c*e^4)*
x^3 + 6*(18*c^2*d^2*e^2 - 6*b*c*d*e^3 - (b^2 + 2*a*c)*e^4)*x^2 + 4*(22*c^2*d^3*e - 6*b*c*d^2*e^2 - 2*a*b*e^4 -
(b^2 + 2*a*c)*d*e^3)*x + 12*(c^2*e^4*x^4 + 4*c^2*d*e^3*x^3 + 6*c^2*d^2*e^2*x^2 + 4*c^2*d^3*e*x + c^2*d^4)*log
(e*x + d))/(e^9*x^4 + 4*d*e^8*x^3 + 6*d^2*e^7*x^2 + 4*d^3*e^6*x + d^4*e^5)

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Sympy [A]  time = 17.1165, size = 238, normalized size = 1.59 \begin{align*} \frac{c^{2} \log{\left (d + e x \right )}}{e^{5}} - \frac{3 a^{2} e^{4} + 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} + 6 b c d^{3} e - 25 c^{2} d^{4} + x^{3} \left (24 b c e^{4} - 48 c^{2} d e^{3}\right ) + x^{2} \left (12 a c e^{4} + 6 b^{2} e^{4} + 36 b c d e^{3} - 108 c^{2} d^{2} e^{2}\right ) + x \left (8 a b e^{4} + 8 a c d e^{3} + 4 b^{2} d e^{3} + 24 b c d^{2} e^{2} - 88 c^{2} d^{3} e\right )}{12 d^{4} e^{5} + 48 d^{3} e^{6} x + 72 d^{2} e^{7} x^{2} + 48 d e^{8} x^{3} + 12 e^{9} x^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**2/(e*x+d)**5,x)

[Out]

c**2*log(d + e*x)/e**5 - (3*a**2*e**4 + 2*a*b*d*e**3 + 2*a*c*d**2*e**2 + b**2*d**2*e**2 + 6*b*c*d**3*e - 25*c*
*2*d**4 + x**3*(24*b*c*e**4 - 48*c**2*d*e**3) + x**2*(12*a*c*e**4 + 6*b**2*e**4 + 36*b*c*d*e**3 - 108*c**2*d**
2*e**2) + x*(8*a*b*e**4 + 8*a*c*d*e**3 + 4*b**2*d*e**3 + 24*b*c*d**2*e**2 - 88*c**2*d**3*e))/(12*d**4*e**5 + 4
8*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4)

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Giac [B]  time = 1.12609, size = 410, normalized size = 2.73 \begin{align*} -c^{2} e^{\left (-5\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac{1}{12} \,{\left (\frac{48 \, c^{2} d e^{15}}{x e + d} - \frac{36 \, c^{2} d^{2} e^{15}}{{\left (x e + d\right )}^{2}} + \frac{16 \, c^{2} d^{3} e^{15}}{{\left (x e + d\right )}^{3}} - \frac{3 \, c^{2} d^{4} e^{15}}{{\left (x e + d\right )}^{4}} - \frac{24 \, b c e^{16}}{x e + d} + \frac{36 \, b c d e^{16}}{{\left (x e + d\right )}^{2}} - \frac{24 \, b c d^{2} e^{16}}{{\left (x e + d\right )}^{3}} + \frac{6 \, b c d^{3} e^{16}}{{\left (x e + d\right )}^{4}} - \frac{6 \, b^{2} e^{17}}{{\left (x e + d\right )}^{2}} - \frac{12 \, a c e^{17}}{{\left (x e + d\right )}^{2}} + \frac{8 \, b^{2} d e^{17}}{{\left (x e + d\right )}^{3}} + \frac{16 \, a c d e^{17}}{{\left (x e + d\right )}^{3}} - \frac{3 \, b^{2} d^{2} e^{17}}{{\left (x e + d\right )}^{4}} - \frac{6 \, a c d^{2} e^{17}}{{\left (x e + d\right )}^{4}} - \frac{8 \, a b e^{18}}{{\left (x e + d\right )}^{3}} + \frac{6 \, a b d e^{18}}{{\left (x e + d\right )}^{4}} - \frac{3 \, a^{2} e^{19}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-20\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)^5,x, algorithm="giac")

[Out]

-c^2*e^(-5)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) + 1/12*(48*c^2*d*e^15/(x*e + d) - 36*c^2*d^2*e^15/(x*e + d)^2
+ 16*c^2*d^3*e^15/(x*e + d)^3 - 3*c^2*d^4*e^15/(x*e + d)^4 - 24*b*c*e^16/(x*e + d) + 36*b*c*d*e^16/(x*e + d)^
2 - 24*b*c*d^2*e^16/(x*e + d)^3 + 6*b*c*d^3*e^16/(x*e + d)^4 - 6*b^2*e^17/(x*e + d)^2 - 12*a*c*e^17/(x*e + d)^
2 + 8*b^2*d*e^17/(x*e + d)^3 + 16*a*c*d*e^17/(x*e + d)^3 - 3*b^2*d^2*e^17/(x*e + d)^4 - 6*a*c*d^2*e^17/(x*e +
d)^4 - 8*a*b*e^18/(x*e + d)^3 + 6*a*b*d*e^18/(x*e + d)^4 - 3*a^2*e^19/(x*e + d)^4)*e^(-20)