### 3.238 $$\int \frac{(b x+c x^2)^2}{(d+e x)^3} \, dx$$

Optimal. Leaf size=119 $\frac{\left (b^2 e^2-6 b c d e+6 c^2 d^2\right ) \log (d+e x)}{e^5}-\frac{d^2 (c d-b e)^2}{2 e^5 (d+e x)^2}+\frac{2 d (2 c d-b e) (c d-b e)}{e^5 (d+e x)}-\frac{c x (3 c d-2 b e)}{e^4}+\frac{c^2 x^2}{2 e^3}$

[Out]

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

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Rubi [A]  time = 0.106206, antiderivative size = 119, 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} $\frac{\left (b^2 e^2-6 b c d e+6 c^2 d^2\right ) \log (d+e x)}{e^5}-\frac{d^2 (c d-b e)^2}{2 e^5 (d+e x)^2}+\frac{2 d (2 c d-b e) (c d-b e)}{e^5 (d+e x)}-\frac{c x (3 c d-2 b e)}{e^4}+\frac{c^2 x^2}{2 e^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*x + c*x^2)^2/(d + e*x)^3,x]

[Out]

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

Mathematica [A]  time = 0.0690577, size = 116, normalized size = 0.97 $\frac{\frac{4 d \left (b^2 e^2-3 b c d e+2 c^2 d^2\right )}{d+e x}+2 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right ) \log (d+e x)-\frac{d^2 (c d-b e)^2}{(d+e x)^2}-2 c e x (3 c d-2 b e)+c^2 e^2 x^2}{2 e^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.05, size = 178, normalized size = 1.5 \begin{align*}{\frac{{c}^{2}{x}^{2}}{2\,{e}^{3}}}+2\,{\frac{bcx}{{e}^{3}}}-3\,{\frac{{c}^{2}dx}{{e}^{4}}}-{\frac{{b}^{2}{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{{d}^{3}bc}{{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{{c}^{2}{d}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{{b}^{2}\ln \left ( ex+d \right ) }{{e}^{3}}}-6\,{\frac{\ln \left ( ex+d \right ) bcd}{{e}^{4}}}+6\,{\frac{\ln \left ( ex+d \right ){c}^{2}{d}^{2}}{{e}^{5}}}+2\,{\frac{{b}^{2}d}{{e}^{3} \left ( ex+d \right ) }}-6\,{\frac{bc{d}^{2}}{{e}^{4} \left ( ex+d \right ) }}+4\,{\frac{{c}^{2}{d}^{3}}{{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)^3,x)

[Out]

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

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Maxima [A]  time = 1.13679, size = 198, normalized size = 1.66 \begin{align*} \frac{7 \, c^{2} d^{4} - 10 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} + 4 \,{\left (2 \, c^{2} d^{3} e - 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x}{2 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} + \frac{c^{2} e x^{2} - 2 \,{\left (3 \, c^{2} d - 2 \, b c e\right )} x}{2 \, e^{4}} + \frac{{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.59996, size = 486, normalized size = 4.08 \begin{align*} \frac{c^{2} e^{4} x^{4} + 7 \, c^{2} d^{4} - 10 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} - 4 \,{\left (c^{2} d e^{3} - b c e^{4}\right )} x^{3} -{\left (11 \, c^{2} d^{2} e^{2} - 8 \, b c d e^{3}\right )} x^{2} + 2 \,{\left (c^{2} d^{3} e - 4 \, b c d^{2} e^{2} + 2 \, b^{2} d e^{3}\right )} x + 2 \,{\left (6 \, c^{2} d^{4} - 6 \, b c d^{3} e + b^{2} d^{2} e^{2} +{\left (6 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 2 \,{\left (6 \, c^{2} d^{3} e - 6 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} 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)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.5355, size = 153, normalized size = 1.29 \begin{align*} \frac{c^{2} x^{2}}{2 e^{3}} + \frac{3 b^{2} d^{2} e^{2} - 10 b c d^{3} e + 7 c^{2} d^{4} + x \left (4 b^{2} d e^{3} - 12 b c d^{2} e^{2} + 8 c^{2} d^{3} e\right )}{2 d^{2} e^{5} + 4 d e^{6} x + 2 e^{7} x^{2}} + \frac{x \left (2 b c e - 3 c^{2} d\right )}{e^{4}} + \frac{\left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) \log{\left (d + e x \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)**3,x)

[Out]

c**2*x**2/(2*e**3) + (3*b**2*d**2*e**2 - 10*b*c*d**3*e + 7*c**2*d**4 + x*(4*b**2*d*e**3 - 12*b*c*d**2*e**2 + 8
*c**2*d**3*e))/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + x*(2*b*c*e - 3*c**2*d)/e**4 + (b**2*e**2 - 6*b*c*d*e
+ 6*c**2*d**2)*log(d + e*x)/e**5

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Giac [A]  time = 1.29311, size = 181, normalized size = 1.52 \begin{align*}{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (c^{2} x^{2} e^{3} - 6 \, c^{2} d x e^{2} + 4 \, b c x e^{3}\right )} e^{\left (-6\right )} + \frac{{\left (7 \, c^{2} d^{4} - 10 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} + 4 \,{\left (2 \, c^{2} d^{3} e - 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )} e^{\left (-5\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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