### 3.240 $$\int \frac{(b x+c x^2)^2}{(d+e x)^5} \, dx$$

Optimal. Leaf size=131 $-\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{2 e^5 (d+e x)^2}-\frac{d^2 (c d-b e)^2}{4 e^5 (d+e x)^4}+\frac{2 c (2 c d-b e)}{e^5 (d+e x)}+\frac{2 d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^3}+\frac{c^2 \log (d+e x)}{e^5}$

[Out]

-(d^2*(c*d - b*e)^2)/(4*e^5*(d + e*x)^4) + (2*d*(c*d - b*e)*(2*c*d - b*e))/(3*e^5*(d + e*x)^3) - (6*c^2*d^2 -
6*b*c*d*e + b^2*e^2)/(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 = 0.091824, antiderivative size = 131, 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} $-\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{2 e^5 (d+e x)^2}-\frac{d^2 (c d-b e)^2}{4 e^5 (d+e x)^4}+\frac{2 c (2 c d-b e)}{e^5 (d+e x)}+\frac{2 d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^3}+\frac{c^2 \log (d+e x)}{e^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*x + c*x^2)^2/(d + e*x)^5,x]

[Out]

-(d^2*(c*d - b*e)^2)/(4*e^5*(d + e*x)^4) + (2*d*(c*d - b*e)*(2*c*d - b*e))/(3*e^5*(d + e*x)^3) - (6*c^2*d^2 -
6*b*c*d*e + b^2*e^2)/(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 (b x+c x^2\right )^2}{(d+e x)^5} \, dx &=\int \left (\frac{d^2 (c d-b e)^2}{e^4 (d+e x)^5}+\frac{2 d (c d-b e) (-2 c d+b e)}{e^4 (d+e x)^4}+\frac{6 c^2 d^2-6 b c d e+b^2 e^2}{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{d^2 (c d-b e)^2}{4 e^5 (d+e x)^4}+\frac{2 d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^3}-\frac{6 c^2 d^2-6 b c d e+b^2 e^2}{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 = 0.0447406, size = 126, normalized size = 0.96 $\frac{-b^2 e^2 \left (d^2+4 d e x+6 e^2 x^2\right )-6 b c e \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )+c^2 d \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )+12 c^2 (d+e x)^4 \log (d+e x)}{12 e^5 (d+e x)^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-(b^2*e^2*(d^2 + 4*d*e*x + 6*e^2*x^2)) - 6*b*c*e*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) + c^2*d*(25*d^3
+ 88*d^2*e*x + 108*d*e^2*x^2 + 48*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 = 0.05, size = 197, normalized size = 1.5 \begin{align*} -{\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}}}+{\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}}}-2\,{\frac{bc}{{e}^{4} \left ( ex+d \right ) }}+4\,{\frac{{c}^{2}d}{{e}^{5} \left ( ex+d \right ) }}-{\frac{{b}^{2}{d}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}+{\frac{bc{d}^{3}}{2\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{{c}^{2}{d}^{4}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^2/(e*x+d)^5,x)

[Out]

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

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Maxima [A]  time = 1.17247, size = 239, normalized size = 1.82 \begin{align*} \frac{25 \, c^{2} d^{4} - 6 \, b c d^{3} e - b^{2} 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} - b^{2} e^{4}\right )} x^{2} + 4 \,{\left (22 \, c^{2} d^{3} e - 6 \, b c d^{2} e^{2} - b^{2} 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)^2/(e*x+d)^5,x, algorithm="maxima")

[Out]

1/12*(25*c^2*d^4 - 6*b*c*d^3*e - b^2*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*e^4)*x^2 + 4*(22*c^2*d^3*e - 6*b*c*d^2*e^2 - b^2*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 = 1.57932, size = 456, normalized size = 3.48 \begin{align*} \frac{25 \, c^{2} d^{4} - 6 \, b c d^{3} e - b^{2} 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} - b^{2} e^{4}\right )} x^{2} + 4 \,{\left (22 \, c^{2} d^{3} e - 6 \, b c d^{2} e^{2} - b^{2} 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)^2/(e*x+d)^5,x, algorithm="fricas")

[Out]

1/12*(25*c^2*d^4 - 6*b*c*d^3*e - b^2*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*e^4)*x^2 + 4*(22*c^2*d^3*e - 6*b*c*d^2*e^2 - b^2*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 = 5.06712, size = 180, normalized size = 1.37 \begin{align*} \frac{c^{2} \log{\left (d + e x \right )}}{e^{5}} - \frac{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 (6 b^{2} e^{4} + 36 b c d e^{3} - 108 c^{2} d^{2} e^{2}\right ) + x \left (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)**2/(e*x+d)**5,x)

[Out]

c**2*log(d + e*x)/e**5 - (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*(6*b**2*e**4 + 36*b*c*d*e**3 - 108*c**2*d**2*e**2) + x*(4*b**2*d*e**3 + 24*b*c*d**2*e**2 - 88*c**2*d**3*
e))/(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)

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Giac [A]  time = 1.34748, size = 289, normalized size = 2.21 \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{8 \, b^{2} d e^{17}}{{\left (x e + d\right )}^{3}} - \frac{3 \, b^{2} d^{2} e^{17}}{{\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)^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 + 8*b^2*d*e^17/(x*e + d)
^3 - 3*b^2*d^2*e^17/(x*e + d)^4)*e^(-20)