### 3.1167 $$\int \frac{(b d+2 c d x)^7}{(a+b x+c x^2)^2} \, dx$$

Optimal. Leaf size=89 $12 c d^7 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )+12 c d^7 \left (b^2-4 a c\right ) (b+2 c x)^2-\frac{d^7 (b+2 c x)^6}{a+b x+c x^2}+6 c d^7 (b+2 c x)^4$

[Out]

12*c*(b^2 - 4*a*c)*d^7*(b + 2*c*x)^2 + 6*c*d^7*(b + 2*c*x)^4 - (d^7*(b + 2*c*x)^6)/(a + b*x + c*x^2) + 12*c*(b
^2 - 4*a*c)^2*d^7*Log[a + b*x + c*x^2]

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Rubi [A]  time = 0.06212, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.125, Rules used = {686, 692, 628} $12 c d^7 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )+12 c d^7 \left (b^2-4 a c\right ) (b+2 c x)^2-\frac{d^7 (b+2 c x)^6}{a+b x+c x^2}+6 c d^7 (b+2 c x)^4$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*d + 2*c*d*x)^7/(a + b*x + c*x^2)^2,x]

[Out]

12*c*(b^2 - 4*a*c)*d^7*(b + 2*c*x)^2 + 6*c*d^7*(b + 2*c*x)^4 - (d^7*(b + 2*c*x)^6)/(a + b*x + c*x^2) + 12*c*(b
^2 - 4*a*c)^2*d^7*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)^7}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac{d^7 (b+2 c x)^6}{a+b x+c x^2}+\left (12 c d^2\right ) \int \frac{(b d+2 c d x)^5}{a+b x+c x^2} \, dx\\ &=6 c d^7 (b+2 c x)^4-\frac{d^7 (b+2 c x)^6}{a+b x+c x^2}+\left (12 c \left (b^2-4 a c\right ) d^4\right ) \int \frac{(b d+2 c d x)^3}{a+b x+c x^2} \, dx\\ &=12 c \left (b^2-4 a c\right ) d^7 (b+2 c x)^2+6 c d^7 (b+2 c x)^4-\frac{d^7 (b+2 c x)^6}{a+b x+c x^2}+\left (12 c \left (b^2-4 a c\right )^2 d^6\right ) \int \frac{b d+2 c d x}{a+b x+c x^2} \, dx\\ &=12 c \left (b^2-4 a c\right ) d^7 (b+2 c x)^2+6 c d^7 (b+2 c x)^4-\frac{d^7 (b+2 c x)^6}{a+b x+c x^2}+12 c \left (b^2-4 a c\right )^2 d^7 \log \left (a+b x+c x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0880245, size = 103, normalized size = 1.16 $d^7 \left (-16 c^3 x^2 \left (8 a c-5 b^2\right )+16 b c^2 x \left (3 b^2-8 a c\right )-\frac{\left (b^2-4 a c\right )^3}{a+x (b+c x)}+12 c \left (b^2-4 a c\right )^2 \log (a+x (b+c x))+64 b c^4 x^3+32 c^5 x^4\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*d + 2*c*d*x)^7/(a + b*x + c*x^2)^2,x]

[Out]

d^7*(16*b*c^2*(3*b^2 - 8*a*c)*x - 16*c^3*(-5*b^2 + 8*a*c)*x^2 + 64*b*c^4*x^3 + 32*c^5*x^4 - (b^2 - 4*a*c)^3/(a
+ x*(b + c*x)) + 12*c*(b^2 - 4*a*c)^2*Log[a + x*(b + c*x)])

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Maple [B]  time = 0.048, size = 230, normalized size = 2.6 \begin{align*} 32\,{d}^{7}{c}^{5}{x}^{4}+64\,{d}^{7}b{c}^{4}{x}^{3}-128\,{d}^{7}{x}^{2}a{c}^{4}+80\,{d}^{7}{x}^{2}{b}^{2}{c}^{3}-128\,{d}^{7}ab{c}^{3}x+48\,{d}^{7}{b}^{3}{c}^{2}x+64\,{\frac{{d}^{7}{a}^{3}{c}^{3}}{c{x}^{2}+bx+a}}-48\,{\frac{{d}^{7}{a}^{2}{b}^{2}{c}^{2}}{c{x}^{2}+bx+a}}+12\,{\frac{{d}^{7}a{b}^{4}c}{c{x}^{2}+bx+a}}-{\frac{{d}^{7}{b}^{6}}{c{x}^{2}+bx+a}}+192\,{d}^{7}\ln \left ( c{x}^{2}+bx+a \right ){a}^{2}{c}^{3}-96\,{d}^{7}\ln \left ( c{x}^{2}+bx+a \right ) a{b}^{2}{c}^{2}+12\,{d}^{7}\ln \left ( c{x}^{2}+bx+a \right ){b}^{4}c \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^7/(c*x^2+b*x+a)^2,x)

[Out]

32*d^7*c^5*x^4+64*d^7*b*c^4*x^3-128*d^7*x^2*a*c^4+80*d^7*x^2*b^2*c^3-128*d^7*a*b*c^3*x+48*d^7*b^3*c^2*x+64*d^7
/(c*x^2+b*x+a)*a^3*c^3-48*d^7/(c*x^2+b*x+a)*a^2*b^2*c^2+12*d^7/(c*x^2+b*x+a)*a*b^4*c-d^7/(c*x^2+b*x+a)*b^6+192
*d^7*ln(c*x^2+b*x+a)*a^2*c^3-96*d^7*ln(c*x^2+b*x+a)*a*b^2*c^2+12*d^7*ln(c*x^2+b*x+a)*b^4*c

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Maxima [A]  time = 1.11781, size = 209, normalized size = 2.35 \begin{align*} 32 \, c^{5} d^{7} x^{4} + 64 \, b c^{4} d^{7} x^{3} + 16 \,{\left (5 \, b^{2} c^{3} - 8 \, a c^{4}\right )} d^{7} x^{2} + 16 \,{\left (3 \, b^{3} c^{2} - 8 \, a b c^{3}\right )} d^{7} x + 12 \,{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{7} \log \left (c x^{2} + b x + a\right ) - \frac{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{7}}{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)^7/(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

32*c^5*d^7*x^4 + 64*b*c^4*d^7*x^3 + 16*(5*b^2*c^3 - 8*a*c^4)*d^7*x^2 + 16*(3*b^3*c^2 - 8*a*b*c^3)*d^7*x + 12*(
b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^7*log(c*x^2 + b*x + a) - (b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*
d^7/(c*x^2 + b*x + a)

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Fricas [B]  time = 1.9632, size = 587, normalized size = 6.6 \begin{align*} \frac{32 \, c^{6} d^{7} x^{6} + 96 \, b c^{5} d^{7} x^{5} + 48 \,{\left (3 \, b^{2} c^{4} - 2 \, a c^{5}\right )} d^{7} x^{4} + 64 \,{\left (2 \, b^{3} c^{3} - 3 \, a b c^{4}\right )} d^{7} x^{3} + 16 \,{\left (3 \, b^{4} c^{2} - 3 \, a b^{2} c^{3} - 8 \, a^{2} c^{4}\right )} d^{7} x^{2} + 16 \,{\left (3 \, a b^{3} c^{2} - 8 \, a^{2} b c^{3}\right )} d^{7} x -{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{7} + 12 \,{\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{7} x^{2} +{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{7} x +{\left (a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3}\right )} d^{7}\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)^7/(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

(32*c^6*d^7*x^6 + 96*b*c^5*d^7*x^5 + 48*(3*b^2*c^4 - 2*a*c^5)*d^7*x^4 + 64*(2*b^3*c^3 - 3*a*b*c^4)*d^7*x^3 + 1
6*(3*b^4*c^2 - 3*a*b^2*c^3 - 8*a^2*c^4)*d^7*x^2 + 16*(3*a*b^3*c^2 - 8*a^2*b*c^3)*d^7*x - (b^6 - 12*a*b^4*c + 4
8*a^2*b^2*c^2 - 64*a^3*c^3)*d^7 + 12*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^7*x^2 + (b^5*c - 8*a*b^3*c^2 + 16
*a^2*b*c^3)*d^7*x + (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*d^7)*log(c*x^2 + b*x + a))/(c*x^2 + b*x + a)

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Sympy [A]  time = 4.28406, size = 160, normalized size = 1.8 \begin{align*} 64 b c^{4} d^{7} x^{3} + 32 c^{5} d^{7} x^{4} + 12 c d^{7} \left (4 a c - b^{2}\right )^{2} \log{\left (a + b x + c x^{2} \right )} + x^{2} \left (- 128 a c^{4} d^{7} + 80 b^{2} c^{3} d^{7}\right ) + x \left (- 128 a b c^{3} d^{7} + 48 b^{3} c^{2} d^{7}\right ) + \frac{64 a^{3} c^{3} d^{7} - 48 a^{2} b^{2} c^{2} d^{7} + 12 a b^{4} c d^{7} - b^{6} d^{7}}{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)**7/(c*x**2+b*x+a)**2,x)

[Out]

64*b*c**4*d**7*x**3 + 32*c**5*d**7*x**4 + 12*c*d**7*(4*a*c - b**2)**2*log(a + b*x + c*x**2) + x**2*(-128*a*c**
4*d**7 + 80*b**2*c**3*d**7) + x*(-128*a*b*c**3*d**7 + 48*b**3*c**2*d**7) + (64*a**3*c**3*d**7 - 48*a**2*b**2*c
**2*d**7 + 12*a*b**4*c*d**7 - b**6*d**7)/(a + b*x + c*x**2)

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Giac [B]  time = 1.15258, size = 244, normalized size = 2.74 \begin{align*} 12 \,{\left (b^{4} c d^{7} - 8 \, a b^{2} c^{2} d^{7} + 16 \, a^{2} c^{3} d^{7}\right )} \log \left (c x^{2} + b x + a\right ) - \frac{b^{6} d^{7} - 12 \, a b^{4} c d^{7} + 48 \, a^{2} b^{2} c^{2} d^{7} - 64 \, a^{3} c^{3} d^{7}}{c x^{2} + b x + a} + \frac{16 \,{\left (2 \, c^{13} d^{7} x^{4} + 4 \, b c^{12} d^{7} x^{3} + 5 \, b^{2} c^{11} d^{7} x^{2} - 8 \, a c^{12} d^{7} x^{2} + 3 \, b^{3} c^{10} d^{7} x - 8 \, a b c^{11} d^{7} x\right )}}{c^{8}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^7/(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

12*(b^4*c*d^7 - 8*a*b^2*c^2*d^7 + 16*a^2*c^3*d^7)*log(c*x^2 + b*x + a) - (b^6*d^7 - 12*a*b^4*c*d^7 + 48*a^2*b^
2*c^2*d^7 - 64*a^3*c^3*d^7)/(c*x^2 + b*x + a) + 16*(2*c^13*d^7*x^4 + 4*b*c^12*d^7*x^3 + 5*b^2*c^11*d^7*x^2 - 8
*a*c^12*d^7*x^2 + 3*b^3*c^10*d^7*x - 8*a*b*c^11*d^7*x)/c^8