### 3.1155 $$\int \frac{(b d+2 c d x)^7}{a+b x+c x^2} \, dx$$

Optimal. Leaf size=86 $d^7 \left (b^2-4 a c\right )^3 \log \left (a+b x+c x^2\right )+\frac{1}{2} d^7 \left (b^2-4 a c\right ) (b+2 c x)^4+d^7 \left (b^2-4 a c\right )^2 (b+2 c x)^2+\frac{1}{3} d^7 (b+2 c x)^6$

[Out]

(b^2 - 4*a*c)^2*d^7*(b + 2*c*x)^2 + ((b^2 - 4*a*c)*d^7*(b + 2*c*x)^4)/2 + (d^7*(b + 2*c*x)^6)/3 + (b^2 - 4*a*c
)^3*d^7*Log[a + b*x + c*x^2]

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Rubi [A]  time = 0.0701957, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.083, Rules used = {692, 628} $d^7 \left (b^2-4 a c\right )^3 \log \left (a+b x+c x^2\right )+\frac{1}{2} d^7 \left (b^2-4 a c\right ) (b+2 c x)^4+d^7 \left (b^2-4 a c\right )^2 (b+2 c x)^2+\frac{1}{3} d^7 (b+2 c x)^6$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(b^2 - 4*a*c)^2*d^7*(b + 2*c*x)^2 + ((b^2 - 4*a*c)*d^7*(b + 2*c*x)^4)/2 + (d^7*(b + 2*c*x)^6)/3 + (b^2 - 4*a*c
)^3*d^7*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)^7}{a+b x+c x^2} \, dx &=\frac{1}{3} d^7 (b+2 c x)^6+\left (\left (b^2-4 a c\right ) d^2\right ) \int \frac{(b d+2 c d x)^5}{a+b x+c x^2} \, dx\\ &=\frac{1}{2} \left (b^2-4 a c\right ) d^7 (b+2 c x)^4+\frac{1}{3} d^7 (b+2 c x)^6+\left (\left (b^2-4 a c\right )^2 d^4\right ) \int \frac{(b d+2 c d x)^3}{a+b x+c x^2} \, dx\\ &=\left (b^2-4 a c\right )^2 d^7 (b+2 c x)^2+\frac{1}{2} \left (b^2-4 a c\right ) d^7 (b+2 c x)^4+\frac{1}{3} d^7 (b+2 c x)^6+\left (\left (b^2-4 a c\right )^3 d^6\right ) \int \frac{b d+2 c d x}{a+b x+c x^2} \, dx\\ &=\left (b^2-4 a c\right )^2 d^7 (b+2 c x)^2+\frac{1}{2} \left (b^2-4 a c\right ) d^7 (b+2 c x)^4+\frac{1}{3} d^7 (b+2 c x)^6+\left (b^2-4 a c\right )^3 d^7 \log \left (a+b x+c x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0552827, size = 110, normalized size = 1.28 $d^7 \left (\frac{4}{3} c x (b+c x) \left (8 c^2 \left (6 a^2-3 a c x^2+2 c^2 x^4\right )+b^2 \left (34 c^2 x^2-36 a c\right )+8 b c^2 x \left (4 c x^2-3 a\right )+18 b^3 c x+9 b^4\right )+\left (b^2-4 a c\right )^3 \log (a+x (b+c x))\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

d^7*((4*c*x*(b + c*x)*(9*b^4 + 18*b^3*c*x + 8*b*c^2*x*(-3*a + 4*c*x^2) + b^2*(-36*a*c + 34*c^2*x^2) + 8*c^2*(6
*a^2 - 3*a*c*x^2 + 2*c^2*x^4)))/3 + (b^2 - 4*a*c)^3*Log[a + x*(b + c*x)])

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Maple [B]  time = 0.042, size = 243, normalized size = 2.8 \begin{align*}{\frac{64\,{d}^{7}{c}^{6}{x}^{6}}{3}}+64\,{d}^{7}b{c}^{5}{x}^{5}-32\,{d}^{7}{x}^{4}a{c}^{5}+88\,{d}^{7}{x}^{4}{b}^{2}{c}^{4}-64\,{d}^{7}{x}^{3}ab{c}^{4}+{\frac{208\,{d}^{7}{x}^{3}{b}^{3}{c}^{3}}{3}}+64\,{d}^{7}{x}^{2}{a}^{2}{c}^{4}-80\,{d}^{7}{x}^{2}a{b}^{2}{c}^{3}+36\,{d}^{7}{x}^{2}{b}^{4}{c}^{2}+64\,{d}^{7}b{a}^{2}{c}^{3}x-48\,{d}^{7}a{b}^{3}{c}^{2}x+12\,{d}^{7}{b}^{5}cx-64\,{d}^{7}\ln \left ( c{x}^{2}+bx+a \right ){a}^{3}{c}^{3}+48\,{d}^{7}\ln \left ( c{x}^{2}+bx+a \right ){a}^{2}{b}^{2}{c}^{2}-12\,{d}^{7}\ln \left ( c{x}^{2}+bx+a \right ) a{b}^{4}c+{d}^{7}\ln \left ( c{x}^{2}+bx+a \right ){b}^{6} \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),x)

[Out]

64/3*d^7*c^6*x^6+64*d^7*b*c^5*x^5-32*d^7*x^4*a*c^5+88*d^7*x^4*b^2*c^4-64*d^7*x^3*a*b*c^4+208/3*d^7*x^3*b^3*c^3
+64*d^7*x^2*a^2*c^4-80*d^7*x^2*a*b^2*c^3+36*d^7*x^2*b^4*c^2+64*d^7*b*a^2*c^3*x-48*d^7*a*b^3*c^2*x+12*d^7*b^5*c
*x-64*d^7*ln(c*x^2+b*x+a)*a^3*c^3+48*d^7*ln(c*x^2+b*x+a)*a^2*b^2*c^2-12*d^7*ln(c*x^2+b*x+a)*a*b^4*c+d^7*ln(c*x
^2+b*x+a)*b^6

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Maxima [B]  time = 2.38226, size = 244, normalized size = 2.84 \begin{align*} \frac{64}{3} \, c^{6} d^{7} x^{6} + 64 \, b c^{5} d^{7} x^{5} + 8 \,{\left (11 \, b^{2} c^{4} - 4 \, a c^{5}\right )} d^{7} x^{4} + \frac{16}{3} \,{\left (13 \, b^{3} c^{3} - 12 \, a b c^{4}\right )} d^{7} x^{3} + 4 \,{\left (9 \, b^{4} c^{2} - 20 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{7} x^{2} + 4 \,{\left (3 \, b^{5} c - 12 \, a b^{3} c^{2} + 16 \, 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} \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)^7/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

64/3*c^6*d^7*x^6 + 64*b*c^5*d^7*x^5 + 8*(11*b^2*c^4 - 4*a*c^5)*d^7*x^4 + 16/3*(13*b^3*c^3 - 12*a*b*c^4)*d^7*x^
3 + 4*(9*b^4*c^2 - 20*a*b^2*c^3 + 16*a^2*c^4)*d^7*x^2 + 4*(3*b^5*c - 12*a*b^3*c^2 + 16*a^2*b*c^3)*d^7*x + (b^6
- 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*d^7*log(c*x^2 + b*x + a)

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Fricas [B]  time = 1.63438, size = 393, normalized size = 4.57 \begin{align*} \frac{64}{3} \, c^{6} d^{7} x^{6} + 64 \, b c^{5} d^{7} x^{5} + 8 \,{\left (11 \, b^{2} c^{4} - 4 \, a c^{5}\right )} d^{7} x^{4} + \frac{16}{3} \,{\left (13 \, b^{3} c^{3} - 12 \, a b c^{4}\right )} d^{7} x^{3} + 4 \,{\left (9 \, b^{4} c^{2} - 20 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{7} x^{2} + 4 \,{\left (3 \, b^{5} c - 12 \, a b^{3} c^{2} + 16 \, 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} \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)^7/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

64/3*c^6*d^7*x^6 + 64*b*c^5*d^7*x^5 + 8*(11*b^2*c^4 - 4*a*c^5)*d^7*x^4 + 16/3*(13*b^3*c^3 - 12*a*b*c^4)*d^7*x^
3 + 4*(9*b^4*c^2 - 20*a*b^2*c^3 + 16*a^2*c^4)*d^7*x^2 + 4*(3*b^5*c - 12*a*b^3*c^2 + 16*a^2*b*c^3)*d^7*x + (b^6
- 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*d^7*log(c*x^2 + b*x + a)

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Sympy [B]  time = 1.27699, size = 185, normalized size = 2.15 \begin{align*} 64 b c^{5} d^{7} x^{5} + \frac{64 c^{6} d^{7} x^{6}}{3} - d^{7} \left (4 a c - b^{2}\right )^{3} \log{\left (a + b x + c x^{2} \right )} + x^{4} \left (- 32 a c^{5} d^{7} + 88 b^{2} c^{4} d^{7}\right ) + x^{3} \left (- 64 a b c^{4} d^{7} + \frac{208 b^{3} c^{3} d^{7}}{3}\right ) + x^{2} \left (64 a^{2} c^{4} d^{7} - 80 a b^{2} c^{3} d^{7} + 36 b^{4} c^{2} d^{7}\right ) + x \left (64 a^{2} b c^{3} d^{7} - 48 a b^{3} c^{2} d^{7} + 12 b^{5} c d^{7}\right ) \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),x)

[Out]

64*b*c**5*d**7*x**5 + 64*c**6*d**7*x**6/3 - d**7*(4*a*c - b**2)**3*log(a + b*x + c*x**2) + x**4*(-32*a*c**5*d*
*7 + 88*b**2*c**4*d**7) + x**3*(-64*a*b*c**4*d**7 + 208*b**3*c**3*d**7/3) + x**2*(64*a**2*c**4*d**7 - 80*a*b**
2*c**3*d**7 + 36*b**4*c**2*d**7) + x*(64*a**2*b*c**3*d**7 - 48*a*b**3*c**2*d**7 + 12*b**5*c*d**7)

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Giac [B]  time = 1.18118, size = 296, normalized size = 3.44 \begin{align*}{\left (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}\right )} \log \left (c x^{2} + b x + a\right ) + \frac{4 \,{\left (16 \, c^{12} d^{7} x^{6} + 48 \, b c^{11} d^{7} x^{5} + 66 \, b^{2} c^{10} d^{7} x^{4} - 24 \, a c^{11} d^{7} x^{4} + 52 \, b^{3} c^{9} d^{7} x^{3} - 48 \, a b c^{10} d^{7} x^{3} + 27 \, b^{4} c^{8} d^{7} x^{2} - 60 \, a b^{2} c^{9} d^{7} x^{2} + 48 \, a^{2} c^{10} d^{7} x^{2} + 9 \, b^{5} c^{7} d^{7} x - 36 \, a b^{3} c^{8} d^{7} x + 48 \, a^{2} b c^{9} d^{7} x\right )}}{3 \, c^{6}} \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),x, algorithm="giac")

[Out]

(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)*log(c*x^2 + b*x + a) + 4/3*(16*c^12*d^7*x^6 +
48*b*c^11*d^7*x^5 + 66*b^2*c^10*d^7*x^4 - 24*a*c^11*d^7*x^4 + 52*b^3*c^9*d^7*x^3 - 48*a*b*c^10*d^7*x^3 + 27*b
^4*c^8*d^7*x^2 - 60*a*b^2*c^9*d^7*x^2 + 48*a^2*c^10*d^7*x^2 + 9*b^5*c^7*d^7*x - 36*a*b^3*c^8*d^7*x + 48*a^2*b*
c^9*d^7*x)/c^6