3.1155 \(\int \frac{(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=126 \[ \frac{28}{3} c d^8 \left (b^2-4 a c\right ) (b+2 c x)^3+28 c d^8 \left (b^2-4 a c\right )^2 (b+2 c x)-28 c d^8 \left (b^2-4 a c\right )^{5/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{d^8 (b+2 c x)^7}{a+b x+c x^2}+\frac{28}{5} c d^8 (b+2 c x)^5 \]

[Out]

28*c*(b^2 - 4*a*c)^2*d^8*(b + 2*c*x) + (28*c*(b^2 - 4*a*c)*d^8*(b + 2*c*x)^3)/3
+ (28*c*d^8*(b + 2*c*x)^5)/5 - (d^8*(b + 2*c*x)^7)/(a + b*x + c*x^2) - 28*c*(b^2
- 4*a*c)^(5/2)*d^8*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]

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Rubi [A]  time = 0.268455, antiderivative size = 126, 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 \[ \frac{28}{3} c d^8 \left (b^2-4 a c\right ) (b+2 c x)^3+28 c d^8 \left (b^2-4 a c\right )^2 (b+2 c x)-28 c d^8 \left (b^2-4 a c\right )^{5/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{d^8 (b+2 c x)^7}{a+b x+c x^2}+\frac{28}{5} c d^8 (b+2 c x)^5 \]

Antiderivative was successfully verified.

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

[Out]

28*c*(b^2 - 4*a*c)^2*d^8*(b + 2*c*x) + (28*c*(b^2 - 4*a*c)*d^8*(b + 2*c*x)^3)/3
+ (28*c*d^8*(b + 2*c*x)^5)/5 - (d^8*(b + 2*c*x)^7)/(a + b*x + c*x^2) - 28*c*(b^2
- 4*a*c)^(5/2)*d^8*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]

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Rubi in Sympy [A]  time = 70.8429, size = 141, normalized size = 1.12 \[ 28 b c d^{8} \left (- 4 a c + b^{2}\right )^{2} + 56 c^{2} d^{8} x \left (- 4 a c + b^{2}\right )^{2} + \frac{28 c d^{8} \left (b + 2 c x\right )^{5}}{5} + \frac{28 c d^{8} \left (b + 2 c x\right )^{3} \left (- 4 a c + b^{2}\right )}{3} - 28 c d^{8} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )} - \frac{d^{8} \left (b + 2 c x\right )^{7}}{a + b x + c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**8/(c*x**2+b*x+a)**2,x)

[Out]

28*b*c*d**8*(-4*a*c + b**2)**2 + 56*c**2*d**8*x*(-4*a*c + b**2)**2 + 28*c*d**8*(
b + 2*c*x)**5/5 + 28*c*d**8*(b + 2*c*x)**3*(-4*a*c + b**2)/3 - 28*c*d**8*(-4*a*c
+ b**2)**(5/2)*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2)) - d**8*(b + 2*c*x)**7/(a
+ b*x + c*x**2)

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Mathematica [A]  time = 0.153616, size = 155, normalized size = 1.23 \[ d^8 \left (32 c^2 x \left (24 a^2 c^2-16 a b^2 c+3 b^4\right )-\frac{512}{3} c^4 x^3 \left (a c-b^2\right )+128 b c^3 x^2 \left (b^2-2 a c\right )-\frac{\left (b^2-4 a c\right )^3 (b+2 c x)}{a+x (b+c x)}-28 c \left (4 a c-b^2\right )^{5/2} \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )+128 b c^5 x^4+\frac{256 c^6 x^5}{5}\right ) \]

Antiderivative was successfully verified.

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

[Out]

d^8*(32*c^2*(3*b^4 - 16*a*b^2*c + 24*a^2*c^2)*x + 128*b*c^3*(b^2 - 2*a*c)*x^2 -
(512*c^4*(-b^2 + a*c)*x^3)/3 + 128*b*c^5*x^4 + (256*c^6*x^5)/5 - ((b^2 - 4*a*c)^
3*(b + 2*c*x))/(a + x*(b + c*x)) - 28*c*(-b^2 + 4*a*c)^(5/2)*ArcTan[(b + 2*c*x)/
Sqrt[-b^2 + 4*a*c]])

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Maple [B]  time = 0.015, size = 479, normalized size = 3.8 \[{\frac{256\,{d}^{8}{c}^{6}{x}^{5}}{5}}+128\,{d}^{8}b{c}^{5}{x}^{4}-{\frac{512\,{d}^{8}{x}^{3}a{c}^{5}}{3}}+{\frac{512\,{d}^{8}{x}^{3}{b}^{2}{c}^{4}}{3}}-256\,{d}^{8}{x}^{2}ab{c}^{4}+128\,{d}^{8}{b}^{3}{c}^{3}{x}^{2}+768\,{d}^{8}x{a}^{2}{c}^{4}-512\,{d}^{8}xa{b}^{2}{c}^{3}+96\,{d}^{8}x{b}^{4}{c}^{2}+128\,{\frac{{d}^{8}{a}^{3}{c}^{4}x}{c{x}^{2}+bx+a}}-96\,{\frac{{d}^{8}{a}^{2}{b}^{2}{c}^{3}x}{c{x}^{2}+bx+a}}+24\,{\frac{{d}^{8}{c}^{2}a{b}^{4}x}{c{x}^{2}+bx+a}}-2\,{\frac{{d}^{8}{b}^{6}cx}{c{x}^{2}+bx+a}}+64\,{\frac{{d}^{8}{a}^{3}b{c}^{3}}{c{x}^{2}+bx+a}}-48\,{\frac{{d}^{8}{a}^{2}{b}^{3}{c}^{2}}{c{x}^{2}+bx+a}}+12\,{\frac{{d}^{8}a{b}^{5}c}{c{x}^{2}+bx+a}}-{\frac{{d}^{8}{b}^{7}}{c{x}^{2}+bx+a}}-1792\,{\frac{{d}^{8}{a}^{3}{c}^{4}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+1344\,{\frac{{d}^{8}{a}^{2}{b}^{2}{c}^{3}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-336\,{\frac{{d}^{8}{c}^{2}a{b}^{4}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+28\,{\frac{{d}^{8}{b}^{6}c}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

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)^2,x)

[Out]

256/5*d^8*c^6*x^5+128*d^8*b*c^5*x^4-512/3*d^8*x^3*a*c^5+512/3*d^8*x^3*b^2*c^4-25
6*d^8*x^2*a*b*c^4+128*d^8*b^3*c^3*x^2+768*d^8*x*a^2*c^4-512*d^8*x*a*b^2*c^3+96*d
^8*x*b^4*c^2+128*d^8/(c*x^2+b*x+a)*a^3*c^4*x-96*d^8/(c*x^2+b*x+a)*a^2*b^2*c^3*x+
24*d^8/(c*x^2+b*x+a)*c^2*a*b^4*x-2*d^8/(c*x^2+b*x+a)*b^6*c*x+64*d^8/(c*x^2+b*x+a
)*a^3*b*c^3-48*d^8/(c*x^2+b*x+a)*a^2*b^3*c^2+12*d^8/(c*x^2+b*x+a)*a*b^5*c-d^8/(c
*x^2+b*x+a)*b^7-1792*d^8*c^4/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2
))*a^3+1344*d^8*c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*b^
2-336*d^8*c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^4+28*d^8
*c/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^6

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220508, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

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)^2,x, algorithm="fricas")

[Out]

[1/15*(768*c^7*d^8*x^7 + 2688*b*c^6*d^8*x^6 + 896*(5*b^2*c^5 - 2*a*c^6)*d^8*x^5
+ 4480*(b^3*c^4 - a*b*c^5)*d^8*x^4 + 1120*(3*b^4*c^3 - 8*a*b^2*c^4 + 8*a^2*c^5)*
d^8*x^3 + 480*(3*b^5*c^2 - 12*a*b^3*c^3 + 16*a^2*b*c^4)*d^8*x^2 - 30*(b^6*c - 60
*a*b^4*c^2 + 304*a^2*b^2*c^3 - 448*a^3*c^4)*d^8*x - 15*(b^7 - 12*a*b^5*c + 48*a^
2*b^3*c^2 - 64*a^3*b*c^3)*d^8 + 210*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^8*x^
2 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^8*x + (a*b^4*c - 8*a^2*b^2*c^2 + 16*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*x^2 + b*x + a), 1/15*(768*c^7*d^8*
x^7 + 2688*b*c^6*d^8*x^6 + 896*(5*b^2*c^5 - 2*a*c^6)*d^8*x^5 + 4480*(b^3*c^4 - a
*b*c^5)*d^8*x^4 + 1120*(3*b^4*c^3 - 8*a*b^2*c^4 + 8*a^2*c^5)*d^8*x^3 + 480*(3*b^
5*c^2 - 12*a*b^3*c^3 + 16*a^2*b*c^4)*d^8*x^2 - 30*(b^6*c - 60*a*b^4*c^2 + 304*a^
2*b^2*c^3 - 448*a^3*c^4)*d^8*x - 15*(b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*
b*c^3)*d^8 - 420*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^8*x^2 + (b^5*c - 8*a*b^
3*c^2 + 16*a^2*b*c^3)*d^8*x + (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*d^8)*sqrt(-
b^2 + 4*a*c)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c)))/(c*x^2 + b*x + a)]

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Sympy [A]  time = 10.3216, size = 476, normalized size = 3.78 \[ 128 b c^{5} d^{8} x^{4} + \frac{256 c^{6} d^{8} x^{5}}{5} + 14 c d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{5}} \log{\left (x + \frac{224 a^{2} b c^{3} d^{8} - 112 a b^{3} c^{2} d^{8} + 14 b^{5} c d^{8} - 14 c d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{5}}}{448 a^{2} c^{4} d^{8} - 224 a b^{2} c^{3} d^{8} + 28 b^{4} c^{2} d^{8}} \right )} - 14 c d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{5}} \log{\left (x + \frac{224 a^{2} b c^{3} d^{8} - 112 a b^{3} c^{2} d^{8} + 14 b^{5} c d^{8} + 14 c d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{5}}}{448 a^{2} c^{4} d^{8} - 224 a b^{2} c^{3} d^{8} + 28 b^{4} c^{2} d^{8}} \right )} + x^{3} \left (- \frac{512 a c^{5} d^{8}}{3} + \frac{512 b^{2} c^{4} d^{8}}{3}\right ) + x^{2} \left (- 256 a b c^{4} d^{8} + 128 b^{3} c^{3} d^{8}\right ) + x \left (768 a^{2} c^{4} d^{8} - 512 a b^{2} c^{3} d^{8} + 96 b^{4} c^{2} d^{8}\right ) + \frac{64 a^{3} b c^{3} d^{8} - 48 a^{2} b^{3} c^{2} d^{8} + 12 a b^{5} c d^{8} - b^{7} d^{8} + x \left (128 a^{3} c^{4} d^{8} - 96 a^{2} b^{2} c^{3} d^{8} + 24 a b^{4} c^{2} d^{8} - 2 b^{6} c d^{8}\right )}{a + b x + c x^{2}} \]

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)**2,x)

[Out]

128*b*c**5*d**8*x**4 + 256*c**6*d**8*x**5/5 + 14*c*d**8*sqrt(-(4*a*c - b**2)**5)
*log(x + (224*a**2*b*c**3*d**8 - 112*a*b**3*c**2*d**8 + 14*b**5*c*d**8 - 14*c*d*
*8*sqrt(-(4*a*c - b**2)**5))/(448*a**2*c**4*d**8 - 224*a*b**2*c**3*d**8 + 28*b**
4*c**2*d**8)) - 14*c*d**8*sqrt(-(4*a*c - b**2)**5)*log(x + (224*a**2*b*c**3*d**8
- 112*a*b**3*c**2*d**8 + 14*b**5*c*d**8 + 14*c*d**8*sqrt(-(4*a*c - b**2)**5))/(
448*a**2*c**4*d**8 - 224*a*b**2*c**3*d**8 + 28*b**4*c**2*d**8)) + x**3*(-512*a*c
**5*d**8/3 + 512*b**2*c**4*d**8/3) + x**2*(-256*a*b*c**4*d**8 + 128*b**3*c**3*d*
*8) + x*(768*a**2*c**4*d**8 - 512*a*b**2*c**3*d**8 + 96*b**4*c**2*d**8) + (64*a*
*3*b*c**3*d**8 - 48*a**2*b**3*c**2*d**8 + 12*a*b**5*c*d**8 - b**7*d**8 + x*(128*
a**3*c**4*d**8 - 96*a**2*b**2*c**3*d**8 + 24*a*b**4*c**2*d**8 - 2*b**6*c*d**8))/
(a + b*x + c*x**2)

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GIAC/XCAS [A]  time = 0.217138, size = 416, normalized size = 3.3 \[ \frac{28 \,{\left (b^{6} c d^{8} - 12 \, a b^{4} c^{2} d^{8} + 48 \, a^{2} b^{2} c^{3} d^{8} - 64 \, a^{3} c^{4} d^{8}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} - \frac{2 \, b^{6} c d^{8} x - 24 \, a b^{4} c^{2} d^{8} x + 96 \, a^{2} b^{2} c^{3} d^{8} x - 128 \, a^{3} c^{4} d^{8} x + b^{7} d^{8} - 12 \, a b^{5} c d^{8} + 48 \, a^{2} b^{3} c^{2} d^{8} - 64 \, a^{3} b c^{3} d^{8}}{c x^{2} + b x + a} + \frac{32 \,{\left (24 \, c^{16} d^{8} x^{5} + 60 \, b c^{15} d^{8} x^{4} + 80 \, b^{2} c^{14} d^{8} x^{3} - 80 \, a c^{15} d^{8} x^{3} + 60 \, b^{3} c^{13} d^{8} x^{2} - 120 \, a b c^{14} d^{8} x^{2} + 45 \, b^{4} c^{12} d^{8} x - 240 \, a b^{2} c^{13} d^{8} x + 360 \, a^{2} c^{14} d^{8} x\right )}}{15 \, c^{10}} \]

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)^2,x, algorithm="giac")

[Out]

28*(b^6*c*d^8 - 12*a*b^4*c^2*d^8 + 48*a^2*b^2*c^3*d^8 - 64*a^3*c^4*d^8)*arctan((
2*c*x + b)/sqrt(-b^2 + 4*a*c))/sqrt(-b^2 + 4*a*c) - (2*b^6*c*d^8*x - 24*a*b^4*c^
2*d^8*x + 96*a^2*b^2*c^3*d^8*x - 128*a^3*c^4*d^8*x + b^7*d^8 - 12*a*b^5*c*d^8 +
48*a^2*b^3*c^2*d^8 - 64*a^3*b*c^3*d^8)/(c*x^2 + b*x + a) + 32/15*(24*c^16*d^8*x^
5 + 60*b*c^15*d^8*x^4 + 80*b^2*c^14*d^8*x^3 - 80*a*c^15*d^8*x^3 + 60*b^3*c^13*d^
8*x^2 - 120*a*b*c^14*d^8*x^2 + 45*b^4*c^12*d^8*x - 240*a*b^2*c^13*d^8*x + 360*a^
2*c^14*d^8*x)/c^10