3.1179 \(\int \frac{(b d+2 c d x)^8}{(a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=134 \[ 140 c^2 d^8 \left (b^2-4 a c\right ) (b+2 c x)-140 c^2 d^8 \left (b^2-4 a c\right )^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{7 c d^8 (b+2 c x)^5}{a+b x+c x^2}-\frac{d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}+\frac{140}{3} c^2 d^8 (b+2 c x)^3 \]

[Out]

140*c^2*(b^2 - 4*a*c)*d^8*(b + 2*c*x) + (140*c^2*d^8*(b + 2*c*x)^3)/3 - (d^8*(b + 2*c*x)^7)/(2*(a + b*x + c*x^
2)^2) - (7*c*d^8*(b + 2*c*x)^5)/(a + b*x + c*x^2) - 140*c^2*(b^2 - 4*a*c)^(3/2)*d^8*ArcTanh[(b + 2*c*x)/Sqrt[b
^2 - 4*a*c]]

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Rubi [A]  time = 0.101095, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {686, 692, 618, 206} \[ 140 c^2 d^8 \left (b^2-4 a c\right ) (b+2 c x)-140 c^2 d^8 \left (b^2-4 a c\right )^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{7 c d^8 (b+2 c x)^5}{a+b x+c x^2}-\frac{d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}+\frac{140}{3} c^2 d^8 (b+2 c x)^3 \]

Antiderivative was successfully verified.

[In]

Int[(b*d + 2*c*d*x)^8/(a + b*x + c*x^2)^3,x]

[Out]

140*c^2*(b^2 - 4*a*c)*d^8*(b + 2*c*x) + (140*c^2*d^8*(b + 2*c*x)^3)/3 - (d^8*(b + 2*c*x)^7)/(2*(a + b*x + c*x^
2)^2) - (7*c*d^8*(b + 2*c*x)^5)/(a + b*x + c*x^2) - 140*c^2*(b^2 - 4*a*c)^(3/2)*d^8*ArcTanh[(b + 2*c*x)/Sqrt[b
^2 - 4*a*c]]

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 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac{d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}+\left (7 c d^2\right ) \int \frac{(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac{d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}-\frac{7 c d^8 (b+2 c x)^5}{a+b x+c x^2}+\left (70 c^2 d^4\right ) \int \frac{(b d+2 c d x)^4}{a+b x+c x^2} \, dx\\ &=\frac{140}{3} c^2 d^8 (b+2 c x)^3-\frac{d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}-\frac{7 c d^8 (b+2 c x)^5}{a+b x+c x^2}+\left (70 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac{(b d+2 c d x)^2}{a+b x+c x^2} \, dx\\ &=140 c^2 \left (b^2-4 a c\right ) d^8 (b+2 c x)+\frac{140}{3} c^2 d^8 (b+2 c x)^3-\frac{d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}-\frac{7 c d^8 (b+2 c x)^5}{a+b x+c x^2}+\left (70 c^2 \left (b^2-4 a c\right )^2 d^8\right ) \int \frac{1}{a+b x+c x^2} \, dx\\ &=140 c^2 \left (b^2-4 a c\right ) d^8 (b+2 c x)+\frac{140}{3} c^2 d^8 (b+2 c x)^3-\frac{d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}-\frac{7 c d^8 (b+2 c x)^5}{a+b x+c x^2}-\left (140 c^2 \left (b^2-4 a c\right )^2 d^8\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )\\ &=140 c^2 \left (b^2-4 a c\right ) d^8 (b+2 c x)+\frac{140}{3} c^2 d^8 (b+2 c x)^3-\frac{d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}-\frac{7 c d^8 (b+2 c x)^5}{a+b x+c x^2}-140 c^2 \left (b^2-4 a c\right )^{3/2} d^8 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0861327, size = 142, normalized size = 1.06 \[ d^8 \left (-256 c^3 x \left (3 a c-b^2\right )+140 c^2 \left (4 a c-b^2\right )^{3/2} \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )-\frac{13 c \left (b^2-4 a c\right )^2 (b+2 c x)}{a+x (b+c x)}-\frac{\left (b^2-4 a c\right )^3 (b+2 c x)}{2 (a+x (b+c x))^2}+128 b c^4 x^2+\frac{256 c^5 x^3}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(b*d + 2*c*d*x)^8/(a + b*x + c*x^2)^3,x]

[Out]

d^8*(-256*c^3*(-b^2 + 3*a*c)*x + 128*b*c^4*x^2 + (256*c^5*x^3)/3 - ((b^2 - 4*a*c)^3*(b + 2*c*x))/(2*(a + x*(b
+ c*x))^2) - (13*c*(b^2 - 4*a*c)^2*(b + 2*c*x))/(a + x*(b + c*x)) + 140*c^2*(-b^2 + 4*a*c)^(3/2)*ArcTan[(b + 2
*c*x)/Sqrt[-b^2 + 4*a*c]])

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Maple [B]  time = 0.162, size = 526, normalized size = 3.9 \begin{align*}{\frac{256\,{d}^{8}{c}^{5}{x}^{3}}{3}}+128\,{d}^{8}b{c}^{4}{x}^{2}-768\,{d}^{8}a{c}^{4}x+256\,{d}^{8}{b}^{2}{c}^{3}x-416\,{\frac{{d}^{8}{x}^{3}{a}^{2}{c}^{5}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+208\,{\frac{{d}^{8}{x}^{3}a{b}^{2}{c}^{4}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-26\,{\frac{{d}^{8}{x}^{3}{b}^{4}{c}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-624\,{\frac{{d}^{8}{x}^{2}{a}^{2}b{c}^{4}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+312\,{\frac{{d}^{8}{x}^{2}a{b}^{3}{c}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-39\,{\frac{{d}^{8}{x}^{2}{b}^{5}{c}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-352\,{\frac{{d}^{8}{a}^{3}{c}^{4}x}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-48\,{\frac{{d}^{8}{b}^{2}{a}^{2}{c}^{3}x}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+90\,{\frac{{d}^{8}a{b}^{4}{c}^{2}x}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-14\,{\frac{{d}^{8}{b}^{6}cx}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-176\,{\frac{{d}^{8}{a}^{3}b{c}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+80\,{\frac{{d}^{8}{a}^{2}{b}^{3}{c}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-7\,{\frac{{d}^{8}a{b}^{5}c}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-{\frac{{d}^{8}{b}^{7}}{2\, \left ( c{x}^{2}+bx+a \right ) ^{2}}}+2240\,{\frac{{c}^{4}{d}^{8}{a}^{2}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-1120\,{\frac{{d}^{8}{c}^{3}a{b}^{2}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+140\,{\frac{{d}^{8}{c}^{2}{b}^{4}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^8/(c*x^2+b*x+a)^3,x)

[Out]

256/3*d^8*c^5*x^3+128*d^8*b*c^4*x^2-768*d^8*a*c^4*x+256*d^8*b^2*c^3*x-416*d^8/(c*x^2+b*x+a)^2*x^3*a^2*c^5+208*
d^8/(c*x^2+b*x+a)^2*x^3*a*b^2*c^4-26*d^8/(c*x^2+b*x+a)^2*x^3*b^4*c^3-624*d^8/(c*x^2+b*x+a)^2*x^2*a^2*b*c^4+312
*d^8/(c*x^2+b*x+a)^2*x^2*a*b^3*c^3-39*d^8/(c*x^2+b*x+a)^2*x^2*b^5*c^2-352*d^8/(c*x^2+b*x+a)^2*a^3*c^4*x-48*d^8
/(c*x^2+b*x+a)^2*b^2*a^2*c^3*x+90*d^8/(c*x^2+b*x+a)^2*a*b^4*c^2*x-14*d^8/(c*x^2+b*x+a)^2*b^6*c*x-176*d^8/(c*x^
2+b*x+a)^2*a^3*b*c^3+80*d^8/(c*x^2+b*x+a)^2*a^2*b^3*c^2-7*d^8/(c*x^2+b*x+a)^2*a*b^5*c-1/2*d^8/(c*x^2+b*x+a)^2*
b^7+2240*d^8*c^4/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2-1120*d^8*c^3/(4*a*c-b^2)^(1/2)*arct
an((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^2+140*d^8*c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^4

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^8/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.15014, size = 1879, normalized size = 14.02 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^8/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

[1/6*(512*c^7*d^8*x^7 + 1792*b*c^6*d^8*x^6 + 3584*(b^2*c^5 - a*c^6)*d^8*x^5 + 256*(15*b^3*c^4 - 26*a*b*c^5)*d^
8*x^4 + 4*(345*b^4*c^3 + 312*a*b^2*c^4 - 2800*a^2*c^5)*d^8*x^3 - 6*(39*b^5*c^2 - 824*a*b^3*c^3 + 2032*a^2*b*c^
4)*d^8*x^2 - 12*(7*b^6*c - 45*a*b^4*c^2 - 104*a^2*b^2*c^3 + 560*a^3*c^4)*d^8*x - 3*(b^7 + 14*a*b^5*c - 160*a^2
*b^3*c^2 + 352*a^3*b*c^3)*d^8 - 420*((b^2*c^4 - 4*a*c^5)*d^8*x^4 + 2*(b^3*c^3 - 4*a*b*c^4)*d^8*x^3 + (b^4*c^2
- 2*a*b^2*c^3 - 8*a^2*c^4)*d^8*x^2 + 2*(a*b^3*c^2 - 4*a^2*b*c^3)*d^8*x + (a^2*b^2*c^2 - 4*a^3*c^3)*d^8)*sqrt(b
^2 - 4*a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c + sqrt(b^2 - 4*a*c)*(2*c*x + b))/(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), 1/6*(512*c^7*d^8*x^7 + 1792*b*c^6*d^8*x^6 + 3584*(b^2*c^5
- a*c^6)*d^8*x^5 + 256*(15*b^3*c^4 - 26*a*b*c^5)*d^8*x^4 + 4*(345*b^4*c^3 + 312*a*b^2*c^4 - 2800*a^2*c^5)*d^8
*x^3 - 6*(39*b^5*c^2 - 824*a*b^3*c^3 + 2032*a^2*b*c^4)*d^8*x^2 - 12*(7*b^6*c - 45*a*b^4*c^2 - 104*a^2*b^2*c^3
+ 560*a^3*c^4)*d^8*x - 3*(b^7 + 14*a*b^5*c - 160*a^2*b^3*c^2 + 352*a^3*b*c^3)*d^8 - 840*((b^2*c^4 - 4*a*c^5)*d
^8*x^4 + 2*(b^3*c^3 - 4*a*b*c^4)*d^8*x^3 + (b^4*c^2 - 2*a*b^2*c^3 - 8*a^2*c^4)*d^8*x^2 + 2*(a*b^3*c^2 - 4*a^2*
b*c^3)*d^8*x + (a^2*b^2*c^2 - 4*a^3*c^3)*d^8)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 -
4*a*c)))/(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 = 12.3837, size = 469, normalized size = 3.5 \begin{align*} 128 b c^{4} d^{8} x^{2} + \frac{256 c^{5} d^{8} x^{3}}{3} - 70 c^{2} d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{3}} \log{\left (x + \frac{280 a b c^{3} d^{8} - 70 b^{3} c^{2} d^{8} - 70 c^{2} d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{3}}}{560 a c^{4} d^{8} - 140 b^{2} c^{3} d^{8}} \right )} + 70 c^{2} d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{3}} \log{\left (x + \frac{280 a b c^{3} d^{8} - 70 b^{3} c^{2} d^{8} + 70 c^{2} d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{3}}}{560 a c^{4} d^{8} - 140 b^{2} c^{3} d^{8}} \right )} + x \left (- 768 a c^{4} d^{8} + 256 b^{2} c^{3} d^{8}\right ) - \frac{352 a^{3} b c^{3} d^{8} - 160 a^{2} b^{3} c^{2} d^{8} + 14 a b^{5} c d^{8} + b^{7} d^{8} + x^{3} \left (832 a^{2} c^{5} d^{8} - 416 a b^{2} c^{4} d^{8} + 52 b^{4} c^{3} d^{8}\right ) + x^{2} \left (1248 a^{2} b c^{4} d^{8} - 624 a b^{3} c^{3} d^{8} + 78 b^{5} c^{2} d^{8}\right ) + x \left (704 a^{3} c^{4} d^{8} + 96 a^{2} b^{2} c^{3} d^{8} - 180 a b^{4} c^{2} d^{8} + 28 b^{6} c d^{8}\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)**8/(c*x**2+b*x+a)**3,x)

[Out]

128*b*c**4*d**8*x**2 + 256*c**5*d**8*x**3/3 - 70*c**2*d**8*sqrt(-(4*a*c - b**2)**3)*log(x + (280*a*b*c**3*d**8
- 70*b**3*c**2*d**8 - 70*c**2*d**8*sqrt(-(4*a*c - b**2)**3))/(560*a*c**4*d**8 - 140*b**2*c**3*d**8)) + 70*c**
2*d**8*sqrt(-(4*a*c - b**2)**3)*log(x + (280*a*b*c**3*d**8 - 70*b**3*c**2*d**8 + 70*c**2*d**8*sqrt(-(4*a*c - b
**2)**3))/(560*a*c**4*d**8 - 140*b**2*c**3*d**8)) + x*(-768*a*c**4*d**8 + 256*b**2*c**3*d**8) - (352*a**3*b*c*
*3*d**8 - 160*a**2*b**3*c**2*d**8 + 14*a*b**5*c*d**8 + b**7*d**8 + x**3*(832*a**2*c**5*d**8 - 416*a*b**2*c**4*
d**8 + 52*b**4*c**3*d**8) + x**2*(1248*a**2*b*c**4*d**8 - 624*a*b**3*c**3*d**8 + 78*b**5*c**2*d**8) + x*(704*a
**3*c**4*d**8 + 96*a**2*b**2*c**3*d**8 - 180*a*b**4*c**2*d**8 + 28*b**6*c*d**8))/(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 = 1.19335, size = 425, normalized size = 3.17 \begin{align*} \frac{140 \,{\left (b^{4} c^{2} d^{8} - 8 \, a b^{2} c^{3} d^{8} + 16 \, a^{2} c^{4} d^{8}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} - \frac{52 \, b^{4} c^{3} d^{8} x^{3} - 416 \, a b^{2} c^{4} d^{8} x^{3} + 832 \, a^{2} c^{5} d^{8} x^{3} + 78 \, b^{5} c^{2} d^{8} x^{2} - 624 \, a b^{3} c^{3} d^{8} x^{2} + 1248 \, a^{2} b c^{4} d^{8} x^{2} + 28 \, b^{6} c d^{8} x - 180 \, a b^{4} c^{2} d^{8} x + 96 \, a^{2} b^{2} c^{3} d^{8} x + 704 \, a^{3} c^{4} d^{8} x + b^{7} d^{8} + 14 \, a b^{5} c d^{8} - 160 \, a^{2} b^{3} c^{2} d^{8} + 352 \, a^{3} b c^{3} d^{8}}{2 \,{\left (c x^{2} + b x + a\right )}^{2}} + \frac{128 \,{\left (2 \, c^{14} d^{8} x^{3} + 3 \, b c^{13} d^{8} x^{2} + 6 \, b^{2} c^{12} d^{8} x - 18 \, a c^{13} d^{8} x\right )}}{3 \, c^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^8/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

140*(b^4*c^2*d^8 - 8*a*b^2*c^3*d^8 + 16*a^2*c^4*d^8)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/sqrt(-b^2 + 4*a*c)
- 1/2*(52*b^4*c^3*d^8*x^3 - 416*a*b^2*c^4*d^8*x^3 + 832*a^2*c^5*d^8*x^3 + 78*b^5*c^2*d^8*x^2 - 624*a*b^3*c^3*
d^8*x^2 + 1248*a^2*b*c^4*d^8*x^2 + 28*b^6*c*d^8*x - 180*a*b^4*c^2*d^8*x + 96*a^2*b^2*c^3*d^8*x + 704*a^3*c^4*d
^8*x + b^7*d^8 + 14*a*b^5*c*d^8 - 160*a^2*b^3*c^2*d^8 + 352*a^3*b*c^3*d^8)/(c*x^2 + b*x + a)^2 + 128/3*(2*c^14
*d^8*x^3 + 3*b*c^13*d^8*x^2 + 6*b^2*c^12*d^8*x - 18*a*c^13*d^8*x)/c^9