### 3.1171 $$\int \frac{(b d+2 c d x)^3}{(a+b x+c x^2)^2} \, dx$$

Optimal. Leaf size=43 $4 c d^3 \log \left (a+b x+c x^2\right )-\frac{d^3 (b+2 c x)^2}{a+b x+c x^2}$

[Out]

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

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Rubi [A]  time = 0.0202037, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.083, Rules used = {686, 628} $4 c d^3 \log \left (a+b x+c x^2\right )-\frac{d^3 (b+2 c x)^2}{a+b x+c x^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 686

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*(d + e*x)^(m - 1)*
(a + b*x + c*x^2)^(p + 1))/(b*(p + 1)), x] - Dist[(d*e*(m - 1))/(b*(p + 1)), Int[(d + e*x)^(m - 2)*(a + b*x +
c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2
*p + 3, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

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

Mathematica [A]  time = 0.020575, size = 42, normalized size = 0.98 $d^3 \left (\frac{4 a c-b^2}{a+b x+c x^2}+4 c \log \left (a+b x+c x^2\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.045, size = 58, normalized size = 1.4 \begin{align*} 4\,{\frac{{d}^{3}ac}{c{x}^{2}+bx+a}}-{\frac{{d}^{3}{b}^{2}}{c{x}^{2}+bx+a}}+4\,c{d}^{3}\ln \left ( c{x}^{2}+bx+a \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01148, size = 58, normalized size = 1.35 \begin{align*} 4 \, c d^{3} \log \left (c x^{2} + b x + a\right ) - \frac{{\left (b^{2} - 4 \, a c\right )} d^{3}}{c x^{2} + b x + a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*c*d^3*log(c*x^2 + b*x + a) - (b^2 - 4*a*c)*d^3/(c*x^2 + b*x + a)

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Fricas [A]  time = 2.01536, size = 136, normalized size = 3.16 \begin{align*} -\frac{{\left (b^{2} - 4 \, a c\right )} d^{3} - 4 \,{\left (c^{2} d^{3} x^{2} + b c d^{3} x + a c d^{3}\right )} \log \left (c x^{2} + b x + a\right )}{c x^{2} + b x + a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((b^2 - 4*a*c)*d^3 - 4*(c^2*d^3*x^2 + b*c*d^3*x + a*c*d^3)*log(c*x^2 + b*x + a))/(c*x^2 + b*x + a)

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Sympy [A]  time = 1.21328, size = 42, normalized size = 0.98 \begin{align*} 4 c d^{3} \log{\left (a + b x + c x^{2} \right )} + \frac{4 a c d^{3} - b^{2} d^{3}}{a + b x + c x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*c*d**3*log(a + b*x + c*x**2) + (4*a*c*d**3 - b**2*d**3)/(a + b*x + c*x**2)

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Giac [A]  time = 1.14961, size = 63, normalized size = 1.47 \begin{align*} 4 \, c d^{3} \log \left (c x^{2} + b x + a\right ) - \frac{b^{2} d^{3} - 4 \, a c d^{3}}{c x^{2} + b x + a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*c*d^3*log(c*x^2 + b*x + a) - (b^2*d^3 - 4*a*c*d^3)/(c*x^2 + b*x + a)