3.1166 \(\int \frac{(b d+2 c d x)^{10}}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=160 \[ 84 c^2 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^3+252 c^2 d^{10} \left (b^2-4 a c\right )^2 (b+2 c x)-252 c^2 d^{10} \left (b^2-4 a c\right )^{5/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}-\frac{d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}+\frac{252}{5} c^2 d^{10} (b+2 c x)^5 \]

[Out]

252*c^2*(b^2 - 4*a*c)^2*d^10*(b + 2*c*x) + 84*c^2*(b^2 - 4*a*c)*d^10*(b + 2*c*x)
^3 + (252*c^2*d^10*(b + 2*c*x)^5)/5 - (d^10*(b + 2*c*x)^9)/(2*(a + b*x + c*x^2)^
2) - (9*c*d^10*(b + 2*c*x)^7)/(a + b*x + c*x^2) - 252*c^2*(b^2 - 4*a*c)^(5/2)*d^
10*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]

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Rubi [A]  time = 0.344062, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ 84 c^2 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^3+252 c^2 d^{10} \left (b^2-4 a c\right )^2 (b+2 c x)-252 c^2 d^{10} \left (b^2-4 a c\right )^{5/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}-\frac{d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}+\frac{252}{5} c^2 d^{10} (b+2 c x)^5 \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^10/(a + b*x + c*x^2)^3,x]

[Out]

252*c^2*(b^2 - 4*a*c)^2*d^10*(b + 2*c*x) + 84*c^2*(b^2 - 4*a*c)*d^10*(b + 2*c*x)
^3 + (252*c^2*d^10*(b + 2*c*x)^5)/5 - (d^10*(b + 2*c*x)^9)/(2*(a + b*x + c*x^2)^
2) - (9*c*d^10*(b + 2*c*x)^7)/(a + b*x + c*x^2) - 252*c^2*(b^2 - 4*a*c)^(5/2)*d^
10*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]

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Rubi in Sympy [A]  time = 92.1499, size = 175, normalized size = 1.09 \[ 252 b c^{2} d^{10} \left (- 4 a c + b^{2}\right )^{2} + 504 c^{3} d^{10} x \left (- 4 a c + b^{2}\right )^{2} + \frac{252 c^{2} d^{10} \left (b + 2 c x\right )^{5}}{5} + 84 c^{2} d^{10} \left (b + 2 c x\right )^{3} \left (- 4 a c + b^{2}\right ) - 252 c^{2} d^{10} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )} - \frac{9 c d^{10} \left (b + 2 c x\right )^{7}}{a + b x + c x^{2}} - \frac{d^{10} \left (b + 2 c x\right )^{9}}{2 \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**10/(c*x**2+b*x+a)**3,x)

[Out]

252*b*c**2*d**10*(-4*a*c + b**2)**2 + 504*c**3*d**10*x*(-4*a*c + b**2)**2 + 252*
c**2*d**10*(b + 2*c*x)**5/5 + 84*c**2*d**10*(b + 2*c*x)**3*(-4*a*c + b**2) - 252
*c**2*d**10*(-4*a*c + b**2)**(5/2)*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2)) - 9*c*
d**10*(b + 2*c*x)**7/(a + b*x + c*x**2) - d**10*(b + 2*c*x)**9/(2*(a + b*x + c*x
**2)**2)

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Mathematica [A]  time = 0.212649, size = 192, normalized size = 1.2 \[ d^{10} \left (128 c^3 x \left (48 a^2 c^2-30 a b^2 c+5 b^4\right )-256 c^5 x^3 \left (4 a c-3 b^2\right )+128 b c^4 x^2 \left (5 b^2-12 a c\right )-252 c^2 \left (4 a c-b^2\right )^{5/2} \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )+\frac{17 c \left (4 a c-b^2\right )^3 (b+2 c x)}{a+x (b+c x)}-\frac{\left (b^2-4 a c\right )^4 (b+2 c x)}{2 (a+x (b+c x))^2}+512 b c^6 x^4+\frac{1024 c^7 x^5}{5}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^10/(a + b*x + c*x^2)^3,x]

[Out]

d^10*(128*c^3*(5*b^4 - 30*a*b^2*c + 48*a^2*c^2)*x + 128*b*c^4*(5*b^2 - 12*a*c)*x
^2 - 256*c^5*(-3*b^2 + 4*a*c)*x^3 + 512*b*c^6*x^4 + (1024*c^7*x^5)/5 - ((b^2 - 4
*a*c)^4*(b + 2*c*x))/(2*(a + x*(b + c*x))^2) + (17*c*(-b^2 + 4*a*c)^3*(b + 2*c*x
))/(a + x*(b + c*x)) - 252*c^2*(-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 = 0.021, size = 751, normalized size = 4.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^10/(c*x^2+b*x+a)^3,x)

[Out]

-2448*d^10/(c*x^2+b*x+a)^2*x^2*a^2*b^3*c^4+12096*d^10*c^4/(4*a*c-b^2)^(1/2)*arct
an((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*b^2-3024*d^10*c^3/(4*a*c-b^2)^(1/2)*arctan((
2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^4+1024/5*d^10*c^7*x^5-1/2*d^10/(c*x^2+b*x+a)^2*b
^9+186*d^10/(c*x^2+b*x+a)^2*x*a*b^6*c^2-1632*d^10/(c*x^2+b*x+a)^2*x^3*a^2*b^2*c^
5+612*d^10/(c*x^2+b*x+a)^2*x^2*a*b^5*c^3-288*d^10/(c*x^2+b*x+a)^2*x*a^3*b^2*c^4-
504*d^10/(c*x^2+b*x+a)^2*x*a^2*b^4*c^3+408*d^10/(c*x^2+b*x+a)^2*x^3*a*b^4*c^4+32
64*d^10/(c*x^2+b*x+a)^2*x^2*a^3*b*c^5-3840*d^10*a*b^2*c^4*x+156*d^10/(c*x^2+b*x+
a)^2*a^2*b^5*c^2-9*d^10/(c*x^2+b*x+a)^2*a*b^7*c+2176*d^10/(c*x^2+b*x+a)^2*x^3*a^
3*c^6-34*d^10/(c*x^2+b*x+a)^2*x^3*b^6*c^3-51*d^10/(c*x^2+b*x+a)^2*x^2*b^7*c^2+19
20*d^10/(c*x^2+b*x+a)^2*x*a^4*c^5+640*d^10*x^2*b^3*c^4+512*d^10*b*c^6*x^4-1024*d
^10*x^3*a*c^6+768*d^10*x^3*b^2*c^5+6144*d^10*a^2*c^5*x+640*d^10*b^4*c^3*x-18*d^1
0/(c*x^2+b*x+a)^2*x*b^8*c+960*d^10/(c*x^2+b*x+a)^2*a^4*b*c^4-688*d^10/(c*x^2+b*x
+a)^2*a^3*b^3*c^3-16128*d^10*c^5/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^
(1/2))*a^3+252*d^10*c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^
6-1536*d^10*x^2*a*b*c^5

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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)^10/(c*x^2 + b*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226396, 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)^10/(c*x^2 + b*x + a)^3,x, algorithm="fricas")

[Out]

[1/10*(2048*c^9*d^10*x^9 + 9216*b*c^8*d^10*x^8 + 1536*(13*b^2*c^7 - 4*a*c^8)*d^1
0*x^7 + 5376*(5*b^3*c^6 - 4*a*b*c^7)*d^10*x^6 + 5376*(5*b^4*c^5 - 10*a*b^2*c^6 +
8*a^2*c^7)*d^10*x^5 + 6400*(3*b^5*c^4 - 10*a*b^3*c^5 + 12*a^2*b*c^6)*d^10*x^4 +
20*(303*b^6*c^3 - 436*a*b^4*c^4 - 2736*a^2*b^2*c^5 + 6720*a^3*c^6)*d^10*x^3 - 1
0*(51*b^7*c^2 - 1892*a*b^5*c^3 + 9488*a^2*b^3*c^4 - 14016*a^3*b*c^5)*d^10*x^2 -
20*(9*b^8*c - 93*a*b^6*c^2 - 68*a^2*b^4*c^3 + 2064*a^3*b^2*c^4 - 4032*a^4*c^5)*d
^10*x - 5*(b^9 + 18*a*b^7*c - 312*a^2*b^5*c^2 + 1376*a^3*b^3*c^3 - 1920*a^4*b*c^
4)*d^10 + 1260*((b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^10*x^4 + 2*(b^5*c^3 - 8*a
*b^3*c^4 + 16*a^2*b*c^5)*d^10*x^3 + (b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*d^10*x^
2 + 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*d^10*x + (a^2*b^4*c^2 - 8*a^3*b
^2*c^3 + 16*a^4*c^4)*d^10)*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), 1/10*(2048*c^9*d^10*x^9 + 9216*b*c^8*d^10*x^
8 + 1536*(13*b^2*c^7 - 4*a*c^8)*d^10*x^7 + 5376*(5*b^3*c^6 - 4*a*b*c^7)*d^10*x^6
+ 5376*(5*b^4*c^5 - 10*a*b^2*c^6 + 8*a^2*c^7)*d^10*x^5 + 6400*(3*b^5*c^4 - 10*a
*b^3*c^5 + 12*a^2*b*c^6)*d^10*x^4 + 20*(303*b^6*c^3 - 436*a*b^4*c^4 - 2736*a^2*b
^2*c^5 + 6720*a^3*c^6)*d^10*x^3 - 10*(51*b^7*c^2 - 1892*a*b^5*c^3 + 9488*a^2*b^3
*c^4 - 14016*a^3*b*c^5)*d^10*x^2 - 20*(9*b^8*c - 93*a*b^6*c^2 - 68*a^2*b^4*c^3 +
2064*a^3*b^2*c^4 - 4032*a^4*c^5)*d^10*x - 5*(b^9 + 18*a*b^7*c - 312*a^2*b^5*c^2
+ 1376*a^3*b^3*c^3 - 1920*a^4*b*c^4)*d^10 - 2520*((b^4*c^4 - 8*a*b^2*c^5 + 16*a
^2*c^6)*d^10*x^4 + 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d^10*x^3 + (b^6*c^2
- 6*a*b^4*c^3 + 32*a^3*c^5)*d^10*x^2 + 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c
^4)*d^10*x + (a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d^10)*sqrt(-b^2 + 4*a*c)
*arctan((2*c*x + b)/sqrt(-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 [A]  time = 38.7825, size = 660, normalized size = 4.12 \[ 512 b c^{6} d^{10} x^{4} + \frac{1024 c^{7} d^{10} x^{5}}{5} + 126 c^{2} d^{10} \sqrt{- \left (4 a c - b^{2}\right )^{5}} \log{\left (x + \frac{2016 a^{2} b c^{4} d^{10} - 1008 a b^{3} c^{3} d^{10} + 126 b^{5} c^{2} d^{10} - 126 c^{2} d^{10} \sqrt{- \left (4 a c - b^{2}\right )^{5}}}{4032 a^{2} c^{5} d^{10} - 2016 a b^{2} c^{4} d^{10} + 252 b^{4} c^{3} d^{10}} \right )} - 126 c^{2} d^{10} \sqrt{- \left (4 a c - b^{2}\right )^{5}} \log{\left (x + \frac{2016 a^{2} b c^{4} d^{10} - 1008 a b^{3} c^{3} d^{10} + 126 b^{5} c^{2} d^{10} + 126 c^{2} d^{10} \sqrt{- \left (4 a c - b^{2}\right )^{5}}}{4032 a^{2} c^{5} d^{10} - 2016 a b^{2} c^{4} d^{10} + 252 b^{4} c^{3} d^{10}} \right )} + x^{3} \left (- 1024 a c^{6} d^{10} + 768 b^{2} c^{5} d^{10}\right ) + x^{2} \left (- 1536 a b c^{5} d^{10} + 640 b^{3} c^{4} d^{10}\right ) + x \left (6144 a^{2} c^{5} d^{10} - 3840 a b^{2} c^{4} d^{10} + 640 b^{4} c^{3} d^{10}\right ) + \frac{1920 a^{4} b c^{4} d^{10} - 1376 a^{3} b^{3} c^{3} d^{10} + 312 a^{2} b^{5} c^{2} d^{10} - 18 a b^{7} c d^{10} - b^{9} d^{10} + x^{3} \left (4352 a^{3} c^{6} d^{10} - 3264 a^{2} b^{2} c^{5} d^{10} + 816 a b^{4} c^{4} d^{10} - 68 b^{6} c^{3} d^{10}\right ) + x^{2} \left (6528 a^{3} b c^{5} d^{10} - 4896 a^{2} b^{3} c^{4} d^{10} + 1224 a b^{5} c^{3} d^{10} - 102 b^{7} c^{2} d^{10}\right ) + x \left (3840 a^{4} c^{5} d^{10} - 576 a^{3} b^{2} c^{4} d^{10} - 1008 a^{2} b^{4} c^{3} d^{10} + 372 a b^{6} c^{2} d^{10} - 36 b^{8} c d^{10}\right )}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \left (4 a c + 2 b^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x+b*d)**10/(c*x**2+b*x+a)**3,x)

[Out]

512*b*c**6*d**10*x**4 + 1024*c**7*d**10*x**5/5 + 126*c**2*d**10*sqrt(-(4*a*c - b
**2)**5)*log(x + (2016*a**2*b*c**4*d**10 - 1008*a*b**3*c**3*d**10 + 126*b**5*c**
2*d**10 - 126*c**2*d**10*sqrt(-(4*a*c - b**2)**5))/(4032*a**2*c**5*d**10 - 2016*
a*b**2*c**4*d**10 + 252*b**4*c**3*d**10)) - 126*c**2*d**10*sqrt(-(4*a*c - b**2)*
*5)*log(x + (2016*a**2*b*c**4*d**10 - 1008*a*b**3*c**3*d**10 + 126*b**5*c**2*d**
10 + 126*c**2*d**10*sqrt(-(4*a*c - b**2)**5))/(4032*a**2*c**5*d**10 - 2016*a*b**
2*c**4*d**10 + 252*b**4*c**3*d**10)) + x**3*(-1024*a*c**6*d**10 + 768*b**2*c**5*
d**10) + x**2*(-1536*a*b*c**5*d**10 + 640*b**3*c**4*d**10) + x*(6144*a**2*c**5*d
**10 - 3840*a*b**2*c**4*d**10 + 640*b**4*c**3*d**10) + (1920*a**4*b*c**4*d**10 -
1376*a**3*b**3*c**3*d**10 + 312*a**2*b**5*c**2*d**10 - 18*a*b**7*c*d**10 - b**9
*d**10 + x**3*(4352*a**3*c**6*d**10 - 3264*a**2*b**2*c**5*d**10 + 816*a*b**4*c**
4*d**10 - 68*b**6*c**3*d**10) + x**2*(6528*a**3*b*c**5*d**10 - 4896*a**2*b**3*c*
*4*d**10 + 1224*a*b**5*c**3*d**10 - 102*b**7*c**2*d**10) + x*(3840*a**4*c**5*d**
10 - 576*a**3*b**2*c**4*d**10 - 1008*a**2*b**4*c**3*d**10 + 372*a*b**6*c**2*d**1
0 - 36*b**8*c*d**10))/(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/XCAS [A]  time = 0.219268, size = 621, normalized size = 3.88 \[ \frac{252 \,{\left (b^{6} c^{2} d^{10} - 12 \, a b^{4} c^{3} d^{10} + 48 \, a^{2} b^{2} c^{4} d^{10} - 64 \, a^{3} c^{5} d^{10}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} - \frac{68 \, b^{6} c^{3} d^{10} x^{3} - 816 \, a b^{4} c^{4} d^{10} x^{3} + 3264 \, a^{2} b^{2} c^{5} d^{10} x^{3} - 4352 \, a^{3} c^{6} d^{10} x^{3} + 102 \, b^{7} c^{2} d^{10} x^{2} - 1224 \, a b^{5} c^{3} d^{10} x^{2} + 4896 \, a^{2} b^{3} c^{4} d^{10} x^{2} - 6528 \, a^{3} b c^{5} d^{10} x^{2} + 36 \, b^{8} c d^{10} x - 372 \, a b^{6} c^{2} d^{10} x + 1008 \, a^{2} b^{4} c^{3} d^{10} x + 576 \, a^{3} b^{2} c^{4} d^{10} x - 3840 \, a^{4} c^{5} d^{10} x + b^{9} d^{10} + 18 \, a b^{7} c d^{10} - 312 \, a^{2} b^{5} c^{2} d^{10} + 1376 \, a^{3} b^{3} c^{3} d^{10} - 1920 \, a^{4} b c^{4} d^{10}}{2 \,{\left (c x^{2} + b x + a\right )}^{2}} + \frac{128 \,{\left (8 \, c^{22} d^{10} x^{5} + 20 \, b c^{21} d^{10} x^{4} + 30 \, b^{2} c^{20} d^{10} x^{3} - 40 \, a c^{21} d^{10} x^{3} + 25 \, b^{3} c^{19} d^{10} x^{2} - 60 \, a b c^{20} d^{10} x^{2} + 25 \, b^{4} c^{18} d^{10} x - 150 \, a b^{2} c^{19} d^{10} x + 240 \, a^{2} c^{20} d^{10} x\right )}}{5 \, c^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^10/(c*x^2 + b*x + a)^3,x, algorithm="giac")

[Out]

252*(b^6*c^2*d^10 - 12*a*b^4*c^3*d^10 + 48*a^2*b^2*c^4*d^10 - 64*a^3*c^5*d^10)*a
rctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/sqrt(-b^2 + 4*a*c) - 1/2*(68*b^6*c^3*d^10*
x^3 - 816*a*b^4*c^4*d^10*x^3 + 3264*a^2*b^2*c^5*d^10*x^3 - 4352*a^3*c^6*d^10*x^3
+ 102*b^7*c^2*d^10*x^2 - 1224*a*b^5*c^3*d^10*x^2 + 4896*a^2*b^3*c^4*d^10*x^2 -
6528*a^3*b*c^5*d^10*x^2 + 36*b^8*c*d^10*x - 372*a*b^6*c^2*d^10*x + 1008*a^2*b^4*
c^3*d^10*x + 576*a^3*b^2*c^4*d^10*x - 3840*a^4*c^5*d^10*x + b^9*d^10 + 18*a*b^7*
c*d^10 - 312*a^2*b^5*c^2*d^10 + 1376*a^3*b^3*c^3*d^10 - 1920*a^4*b*c^4*d^10)/(c*
x^2 + b*x + a)^2 + 128/5*(8*c^22*d^10*x^5 + 20*b*c^21*d^10*x^4 + 30*b^2*c^20*d^1
0*x^3 - 40*a*c^21*d^10*x^3 + 25*b^3*c^19*d^10*x^2 - 60*a*b*c^20*d^10*x^2 + 25*b^
4*c^18*d^10*x - 150*a*b^2*c^19*d^10*x + 240*a^2*c^20*d^10*x)/c^15