### 3.1180 $$\int \frac{(b d+2 c d x)^7}{(a+b x+c x^2)^3} \, dx$$

Optimal. Leaf size=97 $48 c^2 d^7 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )-\frac{6 c d^7 (b+2 c x)^4}{a+b x+c x^2}-\frac{d^7 (b+2 c x)^6}{2 \left (a+b x+c x^2\right )^2}+48 c^2 d^7 (b+2 c x)^2$

[Out]

48*c^2*d^7*(b + 2*c*x)^2 - (d^7*(b + 2*c*x)^6)/(2*(a + b*x + c*x^2)^2) - (6*c*d^7*(b + 2*c*x)^4)/(a + b*x + c*
x^2) + 48*c^2*(b^2 - 4*a*c)*d^7*Log[a + b*x + c*x^2]

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Rubi [A]  time = 0.0604783, antiderivative size = 97, 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} $48 c^2 d^7 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )-\frac{6 c d^7 (b+2 c x)^4}{a+b x+c x^2}-\frac{d^7 (b+2 c x)^6}{2 \left (a+b x+c x^2\right )^2}+48 c^2 d^7 (b+2 c x)^2$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*d + 2*c*d*x)^7/(a + b*x + c*x^2)^3,x]

[Out]

48*c^2*d^7*(b + 2*c*x)^2 - (d^7*(b + 2*c*x)^6)/(2*(a + b*x + c*x^2)^2) - (6*c*d^7*(b + 2*c*x)^4)/(a + b*x + c*
x^2) + 48*c^2*(b^2 - 4*a*c)*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 )^3} \, dx &=-\frac{d^7 (b+2 c x)^6}{2 \left (a+b x+c x^2\right )^2}+\left (6 c d^2\right ) \int \frac{(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac{d^7 (b+2 c x)^6}{2 \left (a+b x+c x^2\right )^2}-\frac{6 c d^7 (b+2 c x)^4}{a+b x+c x^2}+\left (48 c^2 d^4\right ) \int \frac{(b d+2 c d x)^3}{a+b x+c x^2} \, dx\\ &=48 c^2 d^7 (b+2 c x)^2-\frac{d^7 (b+2 c x)^6}{2 \left (a+b x+c x^2\right )^2}-\frac{6 c d^7 (b+2 c x)^4}{a+b x+c x^2}+\left (48 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac{b d+2 c d x}{a+b x+c x^2} \, dx\\ &=48 c^2 d^7 (b+2 c x)^2-\frac{d^7 (b+2 c x)^6}{2 \left (a+b x+c x^2\right )^2}-\frac{6 c d^7 (b+2 c x)^4}{a+b x+c x^2}+48 c^2 \left (b^2-4 a c\right ) d^7 \log \left (a+b x+c x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0376434, size = 92, normalized size = 0.95 $d^7 \left (48 c^2 \left (b^2-4 a c\right ) \log (a+x (b+c x))-\frac{12 c \left (b^2-4 a c\right )^2}{a+x (b+c x)}-\frac{\left (b^2-4 a c\right )^3}{2 (a+x (b+c x))^2}+64 b c^3 x+64 c^4 x^2\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*d + 2*c*d*x)^7/(a + b*x + c*x^2)^3,x]

[Out]

d^7*(64*b*c^3*x + 64*c^4*x^2 - (b^2 - 4*a*c)^3/(2*(a + x*(b + c*x))^2) - (12*c*(b^2 - 4*a*c)^2)/(a + x*(b + c*
x)) + 48*c^2*(b^2 - 4*a*c)*Log[a + x*(b + c*x)])

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Maple [B]  time = 0.051, size = 307, normalized size = 3.2 \begin{align*} 64\,{d}^{7}{c}^{4}{x}^{2}+64\,{d}^{7}b{c}^{3}x-192\,{\frac{{d}^{7}{x}^{2}{a}^{2}{c}^{4}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+96\,{\frac{{d}^{7}{x}^{2}a{b}^{2}{c}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-12\,{\frac{{d}^{7}{x}^{2}{b}^{4}{c}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-192\,{\frac{{d}^{7}b{a}^{2}{c}^{3}x}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+96\,{\frac{{d}^{7}a{b}^{3}{c}^{2}x}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-12\,{\frac{{d}^{7}{b}^{5}cx}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-160\,{\frac{{d}^{7}{a}^{3}{c}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+72\,{\frac{{d}^{7}{a}^{2}{b}^{2}{c}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-6\,{\frac{{d}^{7}a{b}^{4}c}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-{\frac{{d}^{7}{b}^{6}}{2\, \left ( c{x}^{2}+bx+a \right ) ^{2}}}-192\,{d}^{7}\ln \left ( c{x}^{2}+bx+a \right ) a{c}^{3}+48\,{d}^{7}\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}{c}^{2} \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)^3,x)

[Out]

64*d^7*c^4*x^2+64*d^7*b*c^3*x-192*d^7/(c*x^2+b*x+a)^2*x^2*a^2*c^4+96*d^7/(c*x^2+b*x+a)^2*x^2*a*b^2*c^3-12*d^7/
(c*x^2+b*x+a)^2*x^2*b^4*c^2-192*d^7/(c*x^2+b*x+a)^2*x*a^2*b*c^3+96*d^7/(c*x^2+b*x+a)^2*x*a*b^3*c^2-12*d^7/(c*x
^2+b*x+a)^2*x*b^5*c-160*d^7/(c*x^2+b*x+a)^2*a^3*c^3+72*d^7/(c*x^2+b*x+a)^2*a^2*b^2*c^2-6*d^7/(c*x^2+b*x+a)^2*a
*b^4*c-1/2*d^7/(c*x^2+b*x+a)^2*b^6-192*d^7*ln(c*x^2+b*x+a)*a*c^3+48*d^7*ln(c*x^2+b*x+a)*b^2*c^2

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Maxima [A]  time = 1.12292, size = 255, normalized size = 2.63 \begin{align*} 64 \, c^{4} d^{7} x^{2} + 64 \, b c^{3} d^{7} x + 48 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{7} \log \left (c x^{2} + b x + a\right ) - \frac{24 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{7} x^{2} + 24 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{7} x +{\left (b^{6} + 12 \, a b^{4} c - 144 \, a^{2} b^{2} c^{2} + 320 \, a^{3} c^{3}\right )} d^{7}}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\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)^3,x, algorithm="maxima")

[Out]

64*c^4*d^7*x^2 + 64*b*c^3*d^7*x + 48*(b^2*c^2 - 4*a*c^3)*d^7*log(c*x^2 + b*x + a) - 1/2*(24*(b^4*c^2 - 8*a*b^2
*c^3 + 16*a^2*c^4)*d^7*x^2 + 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^7*x + (b^6 + 12*a*b^4*c - 144*a^2*b^2*c
^2 + 320*a^3*c^3)*d^7)/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)

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Fricas [B]  time = 2.11628, size = 724, normalized size = 7.46 \begin{align*} \frac{128 \, c^{6} d^{7} x^{6} + 384 \, b c^{5} d^{7} x^{5} + 128 \,{\left (3 \, b^{2} c^{4} + 2 \, a c^{5}\right )} d^{7} x^{4} + 128 \,{\left (b^{3} c^{3} + 4 \, a b c^{4}\right )} d^{7} x^{3} - 8 \,{\left (3 \, b^{4} c^{2} - 56 \, a b^{2} c^{3} + 32 \, a^{2} c^{4}\right )} d^{7} x^{2} - 8 \,{\left (3 \, b^{5} c - 24 \, a b^{3} c^{2} + 32 \, a^{2} b c^{3}\right )} d^{7} x -{\left (b^{6} + 12 \, a b^{4} c - 144 \, a^{2} b^{2} c^{2} + 320 \, a^{3} c^{3}\right )} d^{7} + 96 \,{\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{7} x^{4} + 2 \,{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d^{7} x^{3} +{\left (b^{4} c^{2} - 2 \, a b^{2} c^{3} - 8 \, a^{2} c^{4}\right )} d^{7} x^{2} + 2 \,{\left (a b^{3} c^{2} - 4 \, a^{2} b c^{3}\right )} d^{7} x +{\left (a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}\right )} d^{7}\right )} \log \left (c x^{2} + b x + a\right )}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\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)^3,x, algorithm="fricas")

[Out]

1/2*(128*c^6*d^7*x^6 + 384*b*c^5*d^7*x^5 + 128*(3*b^2*c^4 + 2*a*c^5)*d^7*x^4 + 128*(b^3*c^3 + 4*a*b*c^4)*d^7*x
^3 - 8*(3*b^4*c^2 - 56*a*b^2*c^3 + 32*a^2*c^4)*d^7*x^2 - 8*(3*b^5*c - 24*a*b^3*c^2 + 32*a^2*b*c^3)*d^7*x - (b^
6 + 12*a*b^4*c - 144*a^2*b^2*c^2 + 320*a^3*c^3)*d^7 + 96*((b^2*c^4 - 4*a*c^5)*d^7*x^4 + 2*(b^3*c^3 - 4*a*b*c^4
)*d^7*x^3 + (b^4*c^2 - 2*a*b^2*c^3 - 8*a^2*c^4)*d^7*x^2 + 2*(a*b^3*c^2 - 4*a^2*b*c^3)*d^7*x + (a^2*b^2*c^2 - 4
*a^3*c^3)*d^7)*log(c*x^2 + b*x + a))/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)

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Sympy [B]  time = 13.4142, size = 219, normalized size = 2.26 \begin{align*} 64 b c^{3} d^{7} x + 64 c^{4} d^{7} x^{2} - 48 c^{2} d^{7} \left (4 a c - b^{2}\right ) \log{\left (a + b x + c x^{2} \right )} - \frac{320 a^{3} c^{3} d^{7} - 144 a^{2} b^{2} c^{2} d^{7} + 12 a b^{4} c d^{7} + b^{6} d^{7} + x^{2} \left (384 a^{2} c^{4} d^{7} - 192 a b^{2} c^{3} d^{7} + 24 b^{4} c^{2} d^{7}\right ) + x \left (384 a^{2} b c^{3} d^{7} - 192 a b^{3} c^{2} d^{7} + 24 b^{5} c d^{7}\right )}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \left (4 a c + 2 b^{2}\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)**3,x)

[Out]

64*b*c**3*d**7*x + 64*c**4*d**7*x**2 - 48*c**2*d**7*(4*a*c - b**2)*log(a + b*x + c*x**2) - (320*a**3*c**3*d**7
- 144*a**2*b**2*c**2*d**7 + 12*a*b**4*c*d**7 + b**6*d**7 + x**2*(384*a**2*c**4*d**7 - 192*a*b**2*c**3*d**7 +
24*b**4*c**2*d**7) + x*(384*a**2*b*c**3*d**7 - 192*a*b**3*c**2*d**7 + 24*b**5*c*d**7))/(2*a**2 + 4*a*b*x + 4*b
*c*x**3 + 2*c**2*x**4 + x**2*(4*a*c + 2*b**2))

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Giac [B]  time = 1.1791, size = 258, normalized size = 2.66 \begin{align*} 48 \,{\left (b^{2} c^{2} d^{7} - 4 \, a 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} - 144 \, a^{2} b^{2} c^{2} d^{7} + 320 \, a^{3} c^{3} d^{7} + 24 \,{\left (b^{4} c^{2} d^{7} - 8 \, a b^{2} c^{3} d^{7} + 16 \, a^{2} c^{4} d^{7}\right )} x^{2} + 24 \,{\left (b^{5} c d^{7} - 8 \, a b^{3} c^{2} d^{7} + 16 \, a^{2} b c^{3} d^{7}\right )} x}{2 \,{\left (c x^{2} + b x + a\right )}^{2}} + \frac{64 \,{\left (c^{10} d^{7} x^{2} + b c^{9} d^{7} x\right )}}{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)^3,x, algorithm="giac")

[Out]

48*(b^2*c^2*d^7 - 4*a*c^3*d^7)*log(c*x^2 + b*x + a) - 1/2*(b^6*d^7 + 12*a*b^4*c*d^7 - 144*a^2*b^2*c^2*d^7 + 32
0*a^3*c^3*d^7 + 24*(b^4*c^2*d^7 - 8*a*b^2*c^3*d^7 + 16*a^2*c^4*d^7)*x^2 + 24*(b^5*c*d^7 - 8*a*b^3*c^2*d^7 + 16
*a^2*b*c^3*d^7)*x)/(c*x^2 + b*x + a)^2 + 64*(c^10*d^7*x^2 + b*c^9*d^7*x)/c^6