3.1157 \(\int \frac{(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=100 \[ 20 c d^6 \left (b^2-4 a c\right ) (b+2 c x)-20 c d^6 \left (b^2-4 a c\right )^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{d^6 (b+2 c x)^5}{a+b x+c x^2}+\frac{20}{3} c d^6 (b+2 c x)^3 \]

[Out]

20*c*(b^2 - 4*a*c)*d^6*(b + 2*c*x) + (20*c*d^6*(b + 2*c*x)^3)/3 - (d^6*(b + 2*c*
x)^5)/(a + b*x + c*x^2) - 20*c*(b^2 - 4*a*c)^(3/2)*d^6*ArcTanh[(b + 2*c*x)/Sqrt[
b^2 - 4*a*c]]

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Rubi [A]  time = 0.21341, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ 20 c d^6 \left (b^2-4 a c\right ) (b+2 c x)-20 c d^6 \left (b^2-4 a c\right )^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{d^6 (b+2 c x)^5}{a+b x+c x^2}+\frac{20}{3} c d^6 (b+2 c x)^3 \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^2,x]

[Out]

20*c*(b^2 - 4*a*c)*d^6*(b + 2*c*x) + (20*c*d^6*(b + 2*c*x)^3)/3 - (d^6*(b + 2*c*
x)^5)/(a + b*x + c*x^2) - 20*c*(b^2 - 4*a*c)^(3/2)*d^6*ArcTanh[(b + 2*c*x)/Sqrt[
b^2 - 4*a*c]]

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Rubi in Sympy [A]  time = 51.2734, size = 112, normalized size = 1.12 \[ 20 b c d^{6} \left (- 4 a c + b^{2}\right ) + 40 c^{2} d^{6} x \left (- 4 a c + b^{2}\right ) + \frac{20 c d^{6} \left (b + 2 c x\right )^{3}}{3} - 20 c d^{6} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )} - \frac{d^{6} \left (b + 2 c x\right )^{5}}{a + b x + c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**6/(c*x**2+b*x+a)**2,x)

[Out]

20*b*c*d**6*(-4*a*c + b**2) + 40*c**2*d**6*x*(-4*a*c + b**2) + 20*c*d**6*(b + 2*
c*x)**3/3 - 20*c*d**6*(-4*a*c + b**2)**(3/2)*atanh((b + 2*c*x)/sqrt(-4*a*c + b**
2)) - d**6*(b + 2*c*x)**5/(a + b*x + c*x**2)

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Mathematica [A]  time = 0.123169, size = 108, normalized size = 1.08 \[ d^6 \left (-16 c^2 x \left (8 a c-3 b^2\right )-\frac{\left (b^2-4 a c\right )^2 (b+2 c x)}{a+x (b+c x)}+20 c \left (4 a c-b^2\right )^{3/2} \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )+32 b c^3 x^2+\frac{64 c^4 x^3}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^2,x]

[Out]

d^6*(-16*c^2*(-3*b^2 + 8*a*c)*x + 32*b*c^3*x^2 + (64*c^4*x^3)/3 - ((b^2 - 4*a*c)
^2*(b + 2*c*x))/(a + x*(b + c*x)) + 20*c*(-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 = 0.013, size = 312, normalized size = 3.1 \[{\frac{64\,{d}^{6}{c}^{4}{x}^{3}}{3}}+32\,{d}^{6}b{c}^{3}{x}^{2}-128\,{d}^{6}xa{c}^{3}+48\,{d}^{6}{b}^{2}{c}^{2}x-32\,{\frac{{d}^{6}{a}^{2}{c}^{3}x}{c{x}^{2}+bx+a}}+16\,{\frac{{d}^{6}a{b}^{2}{c}^{2}x}{c{x}^{2}+bx+a}}-2\,{\frac{{d}^{6}{b}^{4}cx}{c{x}^{2}+bx+a}}-16\,{\frac{{d}^{6}{a}^{2}b{c}^{2}}{c{x}^{2}+bx+a}}+8\,{\frac{{d}^{6}ac{b}^{3}}{c{x}^{2}+bx+a}}-{\frac{{d}^{6}{b}^{5}}{c{x}^{2}+bx+a}}+320\,{\frac{{d}^{6}{a}^{2}{c}^{3}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-160\,{\frac{{d}^{6}a{b}^{2}{c}^{2}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+20\,{\frac{{d}^{6}{b}^{4}c}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^6/(c*x^2+b*x+a)^2,x)

[Out]

64/3*d^6*c^4*x^3+32*d^6*b*c^3*x^2-128*d^6*x*a*c^3+48*d^6*b^2*c^2*x-32*d^6/(c*x^2
+b*x+a)*a^2*c^3*x+16*d^6/(c*x^2+b*x+a)*a*b^2*c^2*x-2*d^6/(c*x^2+b*x+a)*b^4*c*x-1
6*d^6/(c*x^2+b*x+a)*a^2*b*c^2+8*d^6/(c*x^2+b*x+a)*a*c*b^3-d^6/(c*x^2+b*x+a)*b^5+
320*d^6*c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2-160*d^6*c^
2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^2+20*d^6*c/(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]  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)^6/(c*x^2 + b*x + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221547, size = 1, normalized size = 0.01 \[ \left [\frac{64 \, c^{5} d^{6} x^{5} + 160 \, b c^{4} d^{6} x^{4} + 80 \,{\left (3 \, b^{2} c^{3} - 4 \, a c^{4}\right )} d^{6} x^{3} + 144 \,{\left (b^{3} c^{2} - 2 \, a b c^{3}\right )} d^{6} x^{2} - 6 \,{\left (b^{4} c - 32 \, a b^{2} c^{2} + 80 \, a^{2} c^{3}\right )} d^{6} x - 3 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d^{6} - 30 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{6} x^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} d^{6} x +{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{6}\right )} \sqrt{b^{2} - 4 \, a c} \log \left (\frac{2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt{b^{2} - 4 \, a c}{\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right )}{3 \,{\left (c x^{2} + b x + a\right )}}, \frac{64 \, c^{5} d^{6} x^{5} + 160 \, b c^{4} d^{6} x^{4} + 80 \,{\left (3 \, b^{2} c^{3} - 4 \, a c^{4}\right )} d^{6} x^{3} + 144 \,{\left (b^{3} c^{2} - 2 \, a b c^{3}\right )} d^{6} x^{2} - 6 \,{\left (b^{4} c - 32 \, a b^{2} c^{2} + 80 \, a^{2} c^{3}\right )} d^{6} x - 3 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d^{6} - 60 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{6} x^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} d^{6} x +{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{6}\right )} \sqrt{-b^{2} + 4 \, a c} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{3 \,{\left (c x^{2} + b x + a\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^6/(c*x^2 + b*x + a)^2,x, algorithm="fricas")

[Out]

[1/3*(64*c^5*d^6*x^5 + 160*b*c^4*d^6*x^4 + 80*(3*b^2*c^3 - 4*a*c^4)*d^6*x^3 + 14
4*(b^3*c^2 - 2*a*b*c^3)*d^6*x^2 - 6*(b^4*c - 32*a*b^2*c^2 + 80*a^2*c^3)*d^6*x -
3*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^6 - 30*((b^2*c^2 - 4*a*c^3)*d^6*x^2 + (b^3*
c - 4*a*b*c^2)*d^6*x + (a*b^2*c - 4*a^2*c^2)*d^6)*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), 1/3*(64*c^5*d^6*x^5 + 160*b*c^4*d^6*x^4 + 80*(3*b^2*c^3 - 4*a
*c^4)*d^6*x^3 + 144*(b^3*c^2 - 2*a*b*c^3)*d^6*x^2 - 6*(b^4*c - 32*a*b^2*c^2 + 80
*a^2*c^3)*d^6*x - 3*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^6 - 60*((b^2*c^2 - 4*a*c^
3)*d^6*x^2 + (b^3*c - 4*a*b*c^2)*d^6*x + (a*b^2*c - 4*a^2*c^2)*d^6)*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 = 7.40364, size = 313, normalized size = 3.13 \[ 32 b c^{3} d^{6} x^{2} + \frac{64 c^{4} d^{6} x^{3}}{3} - 10 c d^{6} \sqrt{- \left (4 a c - b^{2}\right )^{3}} \log{\left (x + \frac{40 a b c^{2} d^{6} - 10 b^{3} c d^{6} - 10 c d^{6} \sqrt{- \left (4 a c - b^{2}\right )^{3}}}{80 a c^{3} d^{6} - 20 b^{2} c^{2} d^{6}} \right )} + 10 c d^{6} \sqrt{- \left (4 a c - b^{2}\right )^{3}} \log{\left (x + \frac{40 a b c^{2} d^{6} - 10 b^{3} c d^{6} + 10 c d^{6} \sqrt{- \left (4 a c - b^{2}\right )^{3}}}{80 a c^{3} d^{6} - 20 b^{2} c^{2} d^{6}} \right )} + x \left (- 128 a c^{3} d^{6} + 48 b^{2} c^{2} d^{6}\right ) - \frac{16 a^{2} b c^{2} d^{6} - 8 a b^{3} c d^{6} + b^{5} d^{6} + x \left (32 a^{2} c^{3} d^{6} - 16 a b^{2} c^{2} d^{6} + 2 b^{4} c d^{6}\right )}{a + b x + c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x+b*d)**6/(c*x**2+b*x+a)**2,x)

[Out]

32*b*c**3*d**6*x**2 + 64*c**4*d**6*x**3/3 - 10*c*d**6*sqrt(-(4*a*c - b**2)**3)*l
og(x + (40*a*b*c**2*d**6 - 10*b**3*c*d**6 - 10*c*d**6*sqrt(-(4*a*c - b**2)**3))/
(80*a*c**3*d**6 - 20*b**2*c**2*d**6)) + 10*c*d**6*sqrt(-(4*a*c - b**2)**3)*log(x
+ (40*a*b*c**2*d**6 - 10*b**3*c*d**6 + 10*c*d**6*sqrt(-(4*a*c - b**2)**3))/(80*
a*c**3*d**6 - 20*b**2*c**2*d**6)) + x*(-128*a*c**3*d**6 + 48*b**2*c**2*d**6) - (
16*a**2*b*c**2*d**6 - 8*a*b**3*c*d**6 + b**5*d**6 + x*(32*a**2*c**3*d**6 - 16*a*
b**2*c**2*d**6 + 2*b**4*c*d**6))/(a + b*x + c*x**2)

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GIAC/XCAS [A]  time = 0.215613, size = 266, normalized size = 2.66 \[ \frac{20 \,{\left (b^{4} c d^{6} - 8 \, a b^{2} c^{2} d^{6} + 16 \, a^{2} c^{3} d^{6}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} - \frac{2 \, b^{4} c d^{6} x - 16 \, a b^{2} c^{2} d^{6} x + 32 \, a^{2} c^{3} d^{6} x + b^{5} d^{6} - 8 \, a b^{3} c d^{6} + 16 \, a^{2} b c^{2} d^{6}}{c x^{2} + b x + a} + \frac{16 \,{\left (4 \, c^{10} d^{6} x^{3} + 6 \, b c^{9} d^{6} x^{2} + 9 \, b^{2} c^{8} d^{6} x - 24 \, a c^{9} d^{6} x\right )}}{3 \, c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^6/(c*x^2 + b*x + a)^2,x, algorithm="giac")

[Out]

20*(b^4*c*d^6 - 8*a*b^2*c^2*d^6 + 16*a^2*c^3*d^6)*arctan((2*c*x + b)/sqrt(-b^2 +
4*a*c))/sqrt(-b^2 + 4*a*c) - (2*b^4*c*d^6*x - 16*a*b^2*c^2*d^6*x + 32*a^2*c^3*d
^6*x + b^5*d^6 - 8*a*b^3*c*d^6 + 16*a^2*b*c^2*d^6)/(c*x^2 + b*x + a) + 16/3*(4*c
^10*d^6*x^3 + 6*b*c^9*d^6*x^2 + 9*b^2*c^8*d^6*x - 24*a*c^9*d^6*x)/c^6