### 3.1169 $$\int \frac{(b d+2 c d x)^5}{(a+b x+c x^2)^2} \, dx$$

Optimal. Leaf size=65 $8 c d^5 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )-\frac{d^5 (b+2 c x)^4}{a+b x+c x^2}+8 c d^5 (b+2 c x)^2$

[Out]

8*c*d^5*(b + 2*c*x)^2 - (d^5*(b + 2*c*x)^4)/(a + b*x + c*x^2) + 8*c*(b^2 - 4*a*c)*d^5*Log[a + b*x + c*x^2]

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Rubi [A]  time = 0.039496, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.125, Rules used = {686, 692, 628} $8 c d^5 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )-\frac{d^5 (b+2 c x)^4}{a+b x+c x^2}+8 c d^5 (b+2 c x)^2$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

8*c*d^5*(b + 2*c*x)^2 - (d^5*(b + 2*c*x)^4)/(a + b*x + c*x^2) + 8*c*(b^2 - 4*a*c)*d^5*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 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)^5}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac{d^5 (b+2 c x)^4}{a+b x+c x^2}+\left (8 c d^2\right ) \int \frac{(b d+2 c d x)^3}{a+b x+c x^2} \, dx\\ &=8 c d^5 (b+2 c x)^2-\frac{d^5 (b+2 c x)^4}{a+b x+c x^2}+\left (8 c \left (b^2-4 a c\right ) d^4\right ) \int \frac{b d+2 c d x}{a+b x+c x^2} \, dx\\ &=8 c d^5 (b+2 c x)^2-\frac{d^5 (b+2 c x)^4}{a+b x+c x^2}+8 c \left (b^2-4 a c\right ) d^5 \log \left (a+b x+c x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0342878, size = 64, normalized size = 0.98 $d^5 \left (-\frac{\left (b^2-4 a c\right )^2}{a+x (b+c x)}+8 c \left (b^2-4 a c\right ) \log (a+x (b+c x))+16 b c^2 x+16 c^3 x^2\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

d^5*(16*b*c^2*x + 16*c^3*x^2 - (b^2 - 4*a*c)^2/(a + x*(b + c*x)) + 8*c*(b^2 - 4*a*c)*Log[a + x*(b + c*x)])

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Maple [A]  time = 0.047, size = 128, normalized size = 2. \begin{align*} 16\,{d}^{5}{c}^{3}{x}^{2}+16\,{d}^{5}b{c}^{2}x-16\,{\frac{{d}^{5}{a}^{2}{c}^{2}}{c{x}^{2}+bx+a}}+8\,{\frac{{d}^{5}ac{b}^{2}}{c{x}^{2}+bx+a}}-{\frac{{d}^{5}{b}^{4}}{c{x}^{2}+bx+a}}-32\,{d}^{5}\ln \left ( c{x}^{2}+bx+a \right ) a{c}^{2}+8\,{d}^{5}\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}c \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16*d^5*c^3*x^2+16*d^5*b*c^2*x-16*d^5/(c*x^2+b*x+a)*a^2*c^2+8*d^5/(c*x^2+b*x+a)*a*c*b^2-d^5/(c*x^2+b*x+a)*b^4-3
2*d^5*ln(c*x^2+b*x+a)*a*c^2+8*d^5*ln(c*x^2+b*x+a)*b^2*c

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Maxima [A]  time = 1.1619, size = 116, normalized size = 1.78 \begin{align*} 16 \, c^{3} d^{5} x^{2} + 16 \, b c^{2} d^{5} x + 8 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{5} \log \left (c x^{2} + b x + a\right ) - \frac{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{5}}{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)^5/(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

16*c^3*d^5*x^2 + 16*b*c^2*d^5*x + 8*(b^2*c - 4*a*c^2)*d^5*log(c*x^2 + b*x + a) - (b^4 - 8*a*b^2*c + 16*a^2*c^2
)*d^5/(c*x^2 + b*x + a)

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Fricas [B]  time = 1.96973, size = 342, normalized size = 5.26 \begin{align*} \frac{16 \, c^{4} d^{5} x^{4} + 32 \, b c^{3} d^{5} x^{3} + 16 \, a b c^{2} d^{5} x + 16 \,{\left (b^{2} c^{2} + a c^{3}\right )} d^{5} x^{2} -{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{5} + 8 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{5} x^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} d^{5} x +{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{5}\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)^5/(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

(16*c^4*d^5*x^4 + 32*b*c^3*d^5*x^3 + 16*a*b*c^2*d^5*x + 16*(b^2*c^2 + a*c^3)*d^5*x^2 - (b^4 - 8*a*b^2*c + 16*a
^2*c^2)*d^5 + 8*((b^2*c^2 - 4*a*c^3)*d^5*x^2 + (b^3*c - 4*a*b*c^2)*d^5*x + (a*b^2*c - 4*a^2*c^2)*d^5)*log(c*x^
2 + b*x + a))/(c*x^2 + b*x + a)

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Sympy [A]  time = 2.30193, size = 90, normalized size = 1.38 \begin{align*} 16 b c^{2} d^{5} x + 16 c^{3} d^{5} x^{2} - 8 c d^{5} \left (4 a c - b^{2}\right ) \log{\left (a + b x + c x^{2} \right )} - \frac{16 a^{2} c^{2} d^{5} - 8 a b^{2} c d^{5} + b^{4} d^{5}}{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)**5/(c*x**2+b*x+a)**2,x)

[Out]

16*b*c**2*d**5*x + 16*c**3*d**5*x**2 - 8*c*d**5*(4*a*c - b**2)*log(a + b*x + c*x**2) - (16*a**2*c**2*d**5 - 8*
a*b**2*c*d**5 + b**4*d**5)/(a + b*x + c*x**2)

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Giac [A]  time = 1.16529, size = 135, normalized size = 2.08 \begin{align*} 8 \,{\left (b^{2} c d^{5} - 4 \, a c^{2} d^{5}\right )} \log \left (c x^{2} + b x + a\right ) - \frac{b^{4} d^{5} - 8 \, a b^{2} c d^{5} + 16 \, a^{2} c^{2} d^{5}}{c x^{2} + b x + a} + \frac{16 \,{\left (c^{7} d^{5} x^{2} + b c^{6} d^{5} x\right )}}{c^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8*(b^2*c*d^5 - 4*a*c^2*d^5)*log(c*x^2 + b*x + a) - (b^4*d^5 - 8*a*b^2*c*d^5 + 16*a^2*c^2*d^5)/(c*x^2 + b*x + a
) + 16*(c^7*d^5*x^2 + b*c^6*d^5*x)/c^4