### 3.1159 $$\int \frac{(b d+2 c d x)^3}{a+b x+c x^2} \, dx$$

Optimal. Leaf size=36 $d^3 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )+d^3 (b+2 c x)^2$

[Out]

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

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Rubi [A]  time = 0.0228052, antiderivative size = 36, 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 = {692, 628} $d^3 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )+d^3 (b+2 c x)^2$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 692

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

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

Mathematica [A]  time = 0.0123265, size = 33, normalized size = 0.92 $d^3 \left (\left (b^2-4 a c\right ) \log (a+x (b+c x))+4 c x (b+c x)\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.042, size = 57, normalized size = 1.6 \begin{align*} 4\,{x}^{2}{c}^{2}{d}^{3}-4\,\ln \left ( c{x}^{2}+bx+a \right ) ac{d}^{3}+\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}{d}^{3}+4\,xbc{d}^{3} \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),x)

[Out]

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

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Maxima [A]  time = 1.14233, size = 58, normalized size = 1.61 \begin{align*} 4 \, c^{2} d^{3} x^{2} + 4 \, b c d^{3} x +{\left (b^{2} - 4 \, a c\right )} d^{3} \log \left (c x^{2} + b x + a\right ) \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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.67158, size = 95, normalized size = 2.64 \begin{align*} 4 \, c^{2} d^{3} x^{2} + 4 \, b c d^{3} x +{\left (b^{2} - 4 \, a c\right )} d^{3} \log \left (c x^{2} + b x + a\right ) \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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.624678, size = 44, normalized size = 1.22 \begin{align*} 4 b c d^{3} x + 4 c^{2} d^{3} x^{2} - d^{3} \left (4 a c - b^{2}\right ) \log{\left (a + b x + c x^{2} \right )} \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),x)

[Out]

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

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Giac [A]  time = 1.12149, size = 72, normalized size = 2. \begin{align*}{\left (b^{2} d^{3} - 4 \, a c d^{3}\right )} \log \left (c x^{2} + b x + a\right ) + \frac{4 \,{\left (c^{4} d^{3} x^{2} + b c^{3} d^{3} x\right )}}{c^{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),x, algorithm="giac")

[Out]

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