3.1143 \(\int \frac{(b d+2 c d x)^8}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=122 \[ \frac{2}{5} d^8 \left (b^2-4 a c\right ) (b+2 c x)^5+\frac{2}{3} d^8 \left (b^2-4 a c\right )^2 (b+2 c x)^3+2 d^8 \left (b^2-4 a c\right )^3 (b+2 c x)-2 d^8 \left (b^2-4 a c\right )^{7/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )+\frac{2}{7} d^8 (b+2 c x)^7 \]

[Out]

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

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Rubi [A]  time = 0.339336, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{2}{5} d^8 \left (b^2-4 a c\right ) (b+2 c x)^5+\frac{2}{3} d^8 \left (b^2-4 a c\right )^2 (b+2 c x)^3+2 d^8 \left (b^2-4 a c\right )^3 (b+2 c x)-2 d^8 \left (b^2-4 a c\right )^{7/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )+\frac{2}{7} d^8 (b+2 c x)^7 \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 71.3193, size = 136, normalized size = 1.11 \[ 2 b d^{8} \left (- 4 a c + b^{2}\right )^{3} + 4 c d^{8} x \left (- 4 a c + b^{2}\right )^{3} + \frac{2 d^{8} \left (b + 2 c x\right )^{7}}{7} + \frac{2 d^{8} \left (b + 2 c x\right )^{5} \left (- 4 a c + b^{2}\right )}{5} + \frac{2 d^{8} \left (b + 2 c x\right )^{3} \left (- 4 a c + b^{2}\right )^{2}}{3} - 2 d^{8} \left (- 4 a c + b^{2}\right )^{\frac{7}{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.268146, size = 188, normalized size = 1.54 \[ d^8 \left (\frac{16}{105} c x \left (112 b^2 c^2 \left (15 a^2-10 a c x^2+12 c^2 x^4\right )+840 b c^3 x \left (a^2-a c x^2+c^2 x^4\right )+16 c^3 \left (-105 a^3+35 a^2 c x^2-21 a c^2 x^4+15 c^3 x^6\right )+70 b^4 c \left (11 c x^2-9 a\right )+420 b^3 c^2 x \left (3 c x^2-2 a\right )+105 b^6+315 b^5 c x\right )+2 \left (4 a c-b^2\right )^{7/2} \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

d^8*((16*c*x*(105*b^6 + 315*b^5*c*x + 420*b^3*c^2*x*(-2*a + 3*c*x^2) + 70*b^4*c*
(-9*a + 11*c*x^2) + 840*b*c^3*x*(a^2 - a*c*x^2 + c^2*x^4) + 112*b^2*c^2*(15*a^2
- 10*a*c*x^2 + 12*c^2*x^4) + 16*c^3*(-105*a^3 + 35*a^2*c*x^2 - 21*a*c^2*x^4 + 15
*c^3*x^6)))/105 + 2*(-b^2 + 4*a*c)^(7/2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])

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Maple [B]  time = 0.012, size = 432, normalized size = 3.5 \[{\frac{256\,{d}^{8}{c}^{7}{x}^{7}}{7}}+128\,{d}^{8}b{c}^{6}{x}^{6}-{\frac{256\,{d}^{8}{x}^{5}a{c}^{6}}{5}}+{\frac{1024\,{d}^{8}{x}^{5}{b}^{2}{c}^{5}}{5}}-128\,{d}^{8}{x}^{4}ab{c}^{5}+192\,{d}^{8}{x}^{4}{b}^{3}{c}^{4}+{\frac{256\,{d}^{8}{x}^{3}{a}^{2}{c}^{5}}{3}}-{\frac{512\,{d}^{8}{x}^{3}a{b}^{2}{c}^{4}}{3}}+{\frac{352\,{d}^{8}{x}^{3}{b}^{4}{c}^{3}}{3}}+128\,{d}^{8}{x}^{2}{a}^{2}b{c}^{4}-128\,{d}^{8}a{b}^{3}{c}^{3}{x}^{2}+48\,{d}^{8}{x}^{2}{b}^{5}{c}^{2}-256\,{d}^{8}{a}^{3}{c}^{4}x+256\,{d}^{8}{a}^{2}{b}^{2}{c}^{3}x-96\,{d}^{8}{c}^{2}a{b}^{4}x+16\,{d}^{8}{b}^{6}cx+512\,{\frac{{d}^{8}{a}^{4}{c}^{4}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-512\,{\frac{{d}^{8}{a}^{3}{b}^{2}{c}^{3}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+192\,{\frac{{d}^{8}{a}^{2}{b}^{4}{c}^{2}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-32\,{\frac{{d}^{8}a{b}^{6}c}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{{d}^{8}{b}^{8}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

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),x)

[Out]

256/7*d^8*c^7*x^7+128*d^8*b*c^6*x^6-256/5*d^8*x^5*a*c^6+1024/5*d^8*x^5*b^2*c^5-1
28*d^8*x^4*a*b*c^5+192*d^8*x^4*b^3*c^4+256/3*d^8*x^3*a^2*c^5-512/3*d^8*x^3*a*b^2
*c^4+352/3*d^8*x^3*b^4*c^3+128*d^8*x^2*a^2*b*c^4-128*d^8*a*b^3*c^3*x^2+48*d^8*x^
2*b^5*c^2-256*d^8*a^3*c^4*x+256*d^8*a^2*b^2*c^3*x-96*d^8*c^2*a*b^4*x+16*d^8*b^6*
c*x+512*d^8/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^4*c^4-512*d^
8/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^3*b^2*c^3+192*d^8/(4*a
*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*b^4*c^2-32*d^8/(4*a*c-b^2)
^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^6*c+2*d^8/(4*a*c-b^2)^(1/2)*arcta
n((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^8

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218746, size = 1, normalized size = 0.01 \[ \left [\frac{256}{7} \, c^{7} d^{8} x^{7} + 128 \, b c^{6} d^{8} x^{6} + \frac{256}{5} \,{\left (4 \, b^{2} c^{5} - a c^{6}\right )} d^{8} x^{5} + 64 \,{\left (3 \, b^{3} c^{4} - 2 \, a b c^{5}\right )} d^{8} x^{4} + \frac{32}{3} \,{\left (11 \, b^{4} c^{3} - 16 \, a b^{2} c^{4} + 8 \, a^{2} c^{5}\right )} d^{8} x^{3} + 16 \,{\left (3 \, b^{5} c^{2} - 8 \, a b^{3} c^{3} + 8 \, a^{2} b c^{4}\right )} d^{8} x^{2} -{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt{b^{2} - 4 \, a c} d^{8} \log \left (\frac{2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt{b^{2} - 4 \, a c}{\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 16 \,{\left (b^{6} c - 6 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3} - 16 \, a^{3} c^{4}\right )} d^{8} x, \frac{256}{7} \, c^{7} d^{8} x^{7} + 128 \, b c^{6} d^{8} x^{6} + \frac{256}{5} \,{\left (4 \, b^{2} c^{5} - a c^{6}\right )} d^{8} x^{5} + 64 \,{\left (3 \, b^{3} c^{4} - 2 \, a b c^{5}\right )} d^{8} x^{4} + \frac{32}{3} \,{\left (11 \, b^{4} c^{3} - 16 \, a b^{2} c^{4} + 8 \, a^{2} c^{5}\right )} d^{8} x^{3} + 16 \,{\left (3 \, b^{5} c^{2} - 8 \, a b^{3} c^{3} + 8 \, a^{2} b c^{4}\right )} d^{8} x^{2} - 2 \,{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt{-b^{2} + 4 \, a c} d^{8} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right ) + 16 \,{\left (b^{6} c - 6 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3} - 16 \, a^{3} c^{4}\right )} d^{8} x\right ] \]

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),x, algorithm="fricas")

[Out]

[256/7*c^7*d^8*x^7 + 128*b*c^6*d^8*x^6 + 256/5*(4*b^2*c^5 - a*c^6)*d^8*x^5 + 64*
(3*b^3*c^4 - 2*a*b*c^5)*d^8*x^4 + 32/3*(11*b^4*c^3 - 16*a*b^2*c^4 + 8*a^2*c^5)*d
^8*x^3 + 16*(3*b^5*c^2 - 8*a*b^3*c^3 + 8*a^2*b*c^4)*d^8*x^2 - (b^6 - 12*a*b^4*c
+ 48*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(b^2 - 4*a*c)*d^8*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)) + 16*(b^6*c - 6*
a*b^4*c^2 + 16*a^2*b^2*c^3 - 16*a^3*c^4)*d^8*x, 256/7*c^7*d^8*x^7 + 128*b*c^6*d^
8*x^6 + 256/5*(4*b^2*c^5 - a*c^6)*d^8*x^5 + 64*(3*b^3*c^4 - 2*a*b*c^5)*d^8*x^4 +
32/3*(11*b^4*c^3 - 16*a*b^2*c^4 + 8*a^2*c^5)*d^8*x^3 + 16*(3*b^5*c^2 - 8*a*b^3*
c^3 + 8*a^2*b*c^4)*d^8*x^2 - 2*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*
sqrt(-b^2 + 4*a*c)*d^8*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c)) + 16*(b^6*c - 6*a*
b^4*c^2 + 16*a^2*b^2*c^3 - 16*a^3*c^4)*d^8*x]

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Sympy [A]  time = 4.533, size = 502, normalized size = 4.11 \[ 128 b c^{6} d^{8} x^{6} + \frac{256 c^{7} d^{8} x^{7}}{7} - d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{7}} \log{\left (x + \frac{64 a^{3} b c^{3} d^{8} - 48 a^{2} b^{3} c^{2} d^{8} + 12 a b^{5} c d^{8} - b^{7} d^{8} - d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{7}}}{128 a^{3} c^{4} d^{8} - 96 a^{2} b^{2} c^{3} d^{8} + 24 a b^{4} c^{2} d^{8} - 2 b^{6} c d^{8}} \right )} + d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{7}} \log{\left (x + \frac{64 a^{3} b c^{3} d^{8} - 48 a^{2} b^{3} c^{2} d^{8} + 12 a b^{5} c d^{8} - b^{7} d^{8} + d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{7}}}{128 a^{3} c^{4} d^{8} - 96 a^{2} b^{2} c^{3} d^{8} + 24 a b^{4} c^{2} d^{8} - 2 b^{6} c d^{8}} \right )} + x^{5} \left (- \frac{256 a c^{6} d^{8}}{5} + \frac{1024 b^{2} c^{5} d^{8}}{5}\right ) + x^{4} \left (- 128 a b c^{5} d^{8} + 192 b^{3} c^{4} d^{8}\right ) + x^{3} \left (\frac{256 a^{2} c^{5} d^{8}}{3} - \frac{512 a b^{2} c^{4} d^{8}}{3} + \frac{352 b^{4} c^{3} d^{8}}{3}\right ) + x^{2} \left (128 a^{2} b c^{4} d^{8} - 128 a b^{3} c^{3} d^{8} + 48 b^{5} c^{2} d^{8}\right ) + x \left (- 256 a^{3} c^{4} d^{8} + 256 a^{2} b^{2} c^{3} d^{8} - 96 a b^{4} c^{2} d^{8} + 16 b^{6} c d^{8}\right ) \]

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),x)

[Out]

128*b*c**6*d**8*x**6 + 256*c**7*d**8*x**7/7 - d**8*sqrt(-(4*a*c - b**2)**7)*log(
x + (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
- d**8*sqrt(-(4*a*c - b**2)**7))/(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)) + d**8*sqrt(-(4*a*c - b**2)**7)*log(x + (
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 + d*
*8*sqrt(-(4*a*c - b**2)**7))/(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)) + x**5*(-256*a*c**6*d**8/5 + 1024*b**2*c**5*d*
*8/5) + x**4*(-128*a*b*c**5*d**8 + 192*b**3*c**4*d**8) + x**3*(256*a**2*c**5*d**
8/3 - 512*a*b**2*c**4*d**8/3 + 352*b**4*c**3*d**8/3) + x**2*(128*a**2*b*c**4*d**
8 - 128*a*b**3*c**3*d**8 + 48*b**5*c**2*d**8) + x*(-256*a**3*c**4*d**8 + 256*a**
2*b**2*c**3*d**8 - 96*a*b**4*c**2*d**8 + 16*b**6*c*d**8)

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GIAC/XCAS [A]  time = 0.216901, size = 423, normalized size = 3.47 \[ \frac{2 \,{\left (b^{8} d^{8} - 16 \, a b^{6} c d^{8} + 96 \, a^{2} b^{4} c^{2} d^{8} - 256 \, a^{3} b^{2} c^{3} d^{8} + 256 \, a^{4} c^{4} d^{8}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} + \frac{16 \,{\left (240 \, c^{14} d^{8} x^{7} + 840 \, b c^{13} d^{8} x^{6} + 1344 \, b^{2} c^{12} d^{8} x^{5} - 336 \, a c^{13} d^{8} x^{5} + 1260 \, b^{3} c^{11} d^{8} x^{4} - 840 \, a b c^{12} d^{8} x^{4} + 770 \, b^{4} c^{10} d^{8} x^{3} - 1120 \, a b^{2} c^{11} d^{8} x^{3} + 560 \, a^{2} c^{12} d^{8} x^{3} + 315 \, b^{5} c^{9} d^{8} x^{2} - 840 \, a b^{3} c^{10} d^{8} x^{2} + 840 \, a^{2} b c^{11} d^{8} x^{2} + 105 \, b^{6} c^{8} d^{8} x - 630 \, a b^{4} c^{9} d^{8} x + 1680 \, a^{2} b^{2} c^{10} d^{8} x - 1680 \, a^{3} c^{11} d^{8} x\right )}}{105 \, c^{7}} \]

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),x, algorithm="giac")

[Out]

2*(b^8*d^8 - 16*a*b^6*c*d^8 + 96*a^2*b^4*c^2*d^8 - 256*a^3*b^2*c^3*d^8 + 256*a^4
*c^4*d^8)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/sqrt(-b^2 + 4*a*c) + 16/105*(24
0*c^14*d^8*x^7 + 840*b*c^13*d^8*x^6 + 1344*b^2*c^12*d^8*x^5 - 336*a*c^13*d^8*x^5
+ 1260*b^3*c^11*d^8*x^4 - 840*a*b*c^12*d^8*x^4 + 770*b^4*c^10*d^8*x^3 - 1120*a*
b^2*c^11*d^8*x^3 + 560*a^2*c^12*d^8*x^3 + 315*b^5*c^9*d^8*x^2 - 840*a*b^3*c^10*d
^8*x^2 + 840*a^2*b*c^11*d^8*x^2 + 105*b^6*c^8*d^8*x - 630*a*b^4*c^9*d^8*x + 1680
*a^2*b^2*c^10*d^8*x - 1680*a^3*c^11*d^8*x)/c^7