3.1177 \(\int \frac{1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=140 \[ \frac{60 c^2}{d^2 \left (b^2-4 a c\right )^3 (b+2 c x)}-\frac{60 c^2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{d^2 \left (b^2-4 a c\right )^{7/2}}+\frac{5 c}{d^2 \left (b^2-4 a c\right )^2 (b+2 c x) \left (a+b x+c x^2\right )}-\frac{1}{2 d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^2} \]

[Out]

(60*c^2)/((b^2 - 4*a*c)^3*d^2*(b + 2*c*x)) - 1/(2*(b^2 - 4*a*c)*d^2*(b + 2*c*x)*
(a + b*x + c*x^2)^2) + (5*c)/((b^2 - 4*a*c)^2*d^2*(b + 2*c*x)*(a + b*x + c*x^2))
- (60*c^2*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(7/2)*d^2)

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Rubi [A]  time = 0.267453, antiderivative size = 140, 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 \[ \frac{60 c^2}{d^2 \left (b^2-4 a c\right )^3 (b+2 c x)}-\frac{60 c^2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{d^2 \left (b^2-4 a c\right )^{7/2}}+\frac{5 c}{d^2 \left (b^2-4 a c\right )^2 (b+2 c x) \left (a+b x+c x^2\right )}-\frac{1}{2 d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^3),x]

[Out]

(60*c^2)/((b^2 - 4*a*c)^3*d^2*(b + 2*c*x)) - 1/(2*(b^2 - 4*a*c)*d^2*(b + 2*c*x)*
(a + b*x + c*x^2)^2) + (5*c)/((b^2 - 4*a*c)^2*d^2*(b + 2*c*x)*(a + b*x + c*x^2))
- (60*c^2*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(7/2)*d^2)

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Rubi in Sympy [A]  time = 56.5547, size = 133, normalized size = 0.95 \[ - \frac{60 c^{2} \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{d^{2} \left (- 4 a c + b^{2}\right )^{\frac{7}{2}}} + \frac{60 c^{2}}{d^{2} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{3}} + \frac{5 c}{d^{2} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )} - \frac{1}{2 d^{2} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*c*d*x+b*d)**2/(c*x**2+b*x+a)**3,x)

[Out]

-60*c**2*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(d**2*(-4*a*c + b**2)**(7/2)) +
60*c**2/(d**2*(b + 2*c*x)*(-4*a*c + b**2)**3) + 5*c/(d**2*(b + 2*c*x)*(-4*a*c +
b**2)**2*(a + b*x + c*x**2)) - 1/(2*d**2*(b + 2*c*x)*(-4*a*c + b**2)*(a + b*x +
c*x**2)**2)

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Mathematica [A]  time = 0.272497, size = 119, normalized size = 0.85 \[ \frac{\frac{120 c^2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-\frac{\left (b^2-4 a c\right ) (b+2 c x)}{(a+x (b+c x))^2}+\frac{14 c (b+2 c x)}{a+x (b+c x)}+\frac{64 c^2}{b+2 c x}}{2 d^2 \left (b^2-4 a c\right )^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^3),x]

[Out]

((64*c^2)/(b + 2*c*x) - ((b^2 - 4*a*c)*(b + 2*c*x))/(a + x*(b + c*x))^2 + (14*c*
(b + 2*c*x))/(a + x*(b + c*x)) + (120*c^2*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]]
)/Sqrt[-b^2 + 4*a*c])/(2*(b^2 - 4*a*c)^3*d^2)

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Maple [B]  time = 0.021, size = 273, normalized size = 2. \[ -32\,{\frac{{c}^{2}}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( 2\,cx+b \right ) }}-14\,{\frac{{c}^{3}{x}^{3}}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}-21\,{\frac{b{c}^{2}{x}^{2}}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}-18\,{\frac{a{c}^{2}x}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}-6\,{\frac{{b}^{2}xc}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}-9\,{\frac{abc}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+{\frac{{b}^{3}}{2\,{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}-60\,{\frac{{c}^{2}}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{7/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*c*d*x+b*d)^2/(c*x^2+b*x+a)^3,x)

[Out]

-32/d^2*c^2/(4*a*c-b^2)^3/(2*c*x+b)-14/d^2/(4*a*c-b^2)^3/(c*x^2+b*x+a)^2*c^3*x^3
-21/d^2/(4*a*c-b^2)^3/(c*x^2+b*x+a)^2*b*c^2*x^2-18/d^2/(4*a*c-b^2)^3/(c*x^2+b*x+
a)^2*a*c^2*x-6/d^2/(4*a*c-b^2)^3/(c*x^2+b*x+a)^2*x*b^2*c-9/d^2/(4*a*c-b^2)^3/(c*
x^2+b*x+a)^2*a*b*c+1/2/d^2/(4*a*c-b^2)^3/(c*x^2+b*x+a)^2*b^3-60/d^2/(4*a*c-b^2)^
(7/2)*c^2*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))

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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(1/((2*c*d*x + b*d)^2*(c*x^2 + b*x + a)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231218, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^2*(c*x^2 + b*x + a)^3),x, algorithm="fricas")

[Out]

[-1/2*(60*(2*c^5*x^5 + 5*b*c^4*x^4 + a^2*b*c^2 + 4*(b^2*c^3 + a*c^4)*x^3 + (b^3*
c^2 + 6*a*b*c^3)*x^2 + 2*(a*b^2*c^2 + a^2*c^3)*x)*log((b^3 - 4*a*b*c + 2*(b^2*c
- 4*a*c^2)*x + (2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^2 + b
*x + a)) - (120*c^4*x^4 + 240*b*c^3*x^3 - b^4 + 18*a*b^2*c + 64*a^2*c^2 + 10*(13
*b^2*c^2 + 20*a*c^3)*x^2 + 10*(b^3*c + 20*a*b*c^2)*x)*sqrt(b^2 - 4*a*c))/((2*(b^
6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^2*x^5 + 5*(b^7*c^2 - 12*a*
b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^2*x^4 + 4*(b^8*c - 11*a*b^6*c^2 + 36*
a^2*b^4*c^3 - 16*a^3*b^2*c^4 - 64*a^4*c^5)*d^2*x^3 + (b^9 - 6*a*b^7*c - 24*a^2*b
^5*c^2 + 224*a^3*b^3*c^3 - 384*a^4*b*c^4)*d^2*x^2 + 2*(a*b^8 - 11*a^2*b^6*c + 36
*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 64*a^5*c^4)*d^2*x + (a^2*b^7 - 12*a^3*b^5*c + 48
*a^4*b^3*c^2 - 64*a^5*b*c^3)*d^2)*sqrt(b^2 - 4*a*c)), 1/2*(120*(2*c^5*x^5 + 5*b*
c^4*x^4 + a^2*b*c^2 + 4*(b^2*c^3 + a*c^4)*x^3 + (b^3*c^2 + 6*a*b*c^3)*x^2 + 2*(a
*b^2*c^2 + a^2*c^3)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (
120*c^4*x^4 + 240*b*c^3*x^3 - b^4 + 18*a*b^2*c + 64*a^2*c^2 + 10*(13*b^2*c^2 + 2
0*a*c^3)*x^2 + 10*(b^3*c + 20*a*b*c^2)*x)*sqrt(-b^2 + 4*a*c))/((2*(b^6*c^3 - 12*
a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^2*x^5 + 5*(b^7*c^2 - 12*a*b^5*c^3 + 4
8*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^2*x^4 + 4*(b^8*c - 11*a*b^6*c^2 + 36*a^2*b^4*c^3
- 16*a^3*b^2*c^4 - 64*a^4*c^5)*d^2*x^3 + (b^9 - 6*a*b^7*c - 24*a^2*b^5*c^2 + 22
4*a^3*b^3*c^3 - 384*a^4*b*c^4)*d^2*x^2 + 2*(a*b^8 - 11*a^2*b^6*c + 36*a^3*b^4*c^
2 - 16*a^4*b^2*c^3 - 64*a^5*c^4)*d^2*x + (a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^
2 - 64*a^5*b*c^3)*d^2)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 34.6981, size = 801, normalized size = 5.72 \[ \frac{30 c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} \log{\left (x + \frac{- 7680 a^{4} c^{6} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} + 7680 a^{3} b^{2} c^{5} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} - 2880 a^{2} b^{4} c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} + 480 a b^{6} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} - 30 b^{8} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} + 30 b c^{2}}{60 c^{3}} \right )}}{d^{2}} - \frac{30 c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} \log{\left (x + \frac{7680 a^{4} c^{6} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} - 7680 a^{3} b^{2} c^{5} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} + 2880 a^{2} b^{4} c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} - 480 a b^{6} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} + 30 b^{8} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} + 30 b c^{2}}{60 c^{3}} \right )}}{d^{2}} - \frac{64 a^{2} c^{2} + 18 a b^{2} c - b^{4} + 240 b c^{3} x^{3} + 120 c^{4} x^{4} + x^{2} \left (200 a c^{3} + 130 b^{2} c^{2}\right ) + x \left (200 a b c^{2} + 10 b^{3} c\right )}{128 a^{5} b c^{3} d^{2} - 96 a^{4} b^{3} c^{2} d^{2} + 24 a^{3} b^{5} c d^{2} - 2 a^{2} b^{7} d^{2} + x^{5} \left (256 a^{3} c^{6} d^{2} - 192 a^{2} b^{2} c^{5} d^{2} + 48 a b^{4} c^{4} d^{2} - 4 b^{6} c^{3} d^{2}\right ) + x^{4} \left (640 a^{3} b c^{5} d^{2} - 480 a^{2} b^{3} c^{4} d^{2} + 120 a b^{5} c^{3} d^{2} - 10 b^{7} c^{2} d^{2}\right ) + x^{3} \left (512 a^{4} c^{5} d^{2} + 128 a^{3} b^{2} c^{4} d^{2} - 288 a^{2} b^{4} c^{3} d^{2} + 88 a b^{6} c^{2} d^{2} - 8 b^{8} c d^{2}\right ) + x^{2} \left (768 a^{4} b c^{4} d^{2} - 448 a^{3} b^{3} c^{3} d^{2} + 48 a^{2} b^{5} c^{2} d^{2} + 12 a b^{7} c d^{2} - 2 b^{9} d^{2}\right ) + x \left (256 a^{5} c^{4} d^{2} + 64 a^{4} b^{2} c^{3} d^{2} - 144 a^{3} b^{4} c^{2} d^{2} + 44 a^{2} b^{6} c d^{2} - 4 a b^{8} d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2*c*d*x+b*d)**2/(c*x**2+b*x+a)**3,x)

[Out]

30*c**2*sqrt(-1/(4*a*c - b**2)**7)*log(x + (-7680*a**4*c**6*sqrt(-1/(4*a*c - b**
2)**7) + 7680*a**3*b**2*c**5*sqrt(-1/(4*a*c - b**2)**7) - 2880*a**2*b**4*c**4*sq
rt(-1/(4*a*c - b**2)**7) + 480*a*b**6*c**3*sqrt(-1/(4*a*c - b**2)**7) - 30*b**8*
c**2*sqrt(-1/(4*a*c - b**2)**7) + 30*b*c**2)/(60*c**3))/d**2 - 30*c**2*sqrt(-1/(
4*a*c - b**2)**7)*log(x + (7680*a**4*c**6*sqrt(-1/(4*a*c - b**2)**7) - 7680*a**3
*b**2*c**5*sqrt(-1/(4*a*c - b**2)**7) + 2880*a**2*b**4*c**4*sqrt(-1/(4*a*c - b**
2)**7) - 480*a*b**6*c**3*sqrt(-1/(4*a*c - b**2)**7) + 30*b**8*c**2*sqrt(-1/(4*a*
c - b**2)**7) + 30*b*c**2)/(60*c**3))/d**2 - (64*a**2*c**2 + 18*a*b**2*c - b**4
+ 240*b*c**3*x**3 + 120*c**4*x**4 + x**2*(200*a*c**3 + 130*b**2*c**2) + x*(200*a
*b*c**2 + 10*b**3*c))/(128*a**5*b*c**3*d**2 - 96*a**4*b**3*c**2*d**2 + 24*a**3*b
**5*c*d**2 - 2*a**2*b**7*d**2 + x**5*(256*a**3*c**6*d**2 - 192*a**2*b**2*c**5*d*
*2 + 48*a*b**4*c**4*d**2 - 4*b**6*c**3*d**2) + x**4*(640*a**3*b*c**5*d**2 - 480*
a**2*b**3*c**4*d**2 + 120*a*b**5*c**3*d**2 - 10*b**7*c**2*d**2) + x**3*(512*a**4
*c**5*d**2 + 128*a**3*b**2*c**4*d**2 - 288*a**2*b**4*c**3*d**2 + 88*a*b**6*c**2*
d**2 - 8*b**8*c*d**2) + x**2*(768*a**4*b*c**4*d**2 - 448*a**3*b**3*c**3*d**2 + 4
8*a**2*b**5*c**2*d**2 + 12*a*b**7*c*d**2 - 2*b**9*d**2) + x*(256*a**5*c**4*d**2
+ 64*a**4*b**2*c**3*d**2 - 144*a**3*b**4*c**2*d**2 + 44*a**2*b**6*c*d**2 - 4*a*b
**8*d**2))

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GIAC/XCAS [A]  time = 0.217418, size = 406, normalized size = 2.9 \[ \frac{32 \, c^{8} d^{11}}{{\left (b^{6} c^{6} d^{12} - 12 \, a b^{4} c^{7} d^{12} + 48 \, a^{2} b^{2} c^{8} d^{12} - 64 \, a^{3} c^{9} d^{12}\right )}{\left (2 \, c d x + b d\right )}} + \frac{60 \, c^{2} \arctan \left (\frac{\frac{b^{2} d}{2 \, c d x + b d} - \frac{4 \, a c d}{2 \, c d x + b d}}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt{-b^{2} + 4 \, a c} d^{2}} - \frac{4 \,{\left (\frac{9 \, b^{2} c^{2} d}{{\left (2 \, c d x + b d\right )}^{3}} - \frac{36 \, a c^{3} d}{{\left (2 \, c d x + b d\right )}^{3}} - \frac{7 \, c^{2}}{{\left (2 \, c d x + b d\right )} d}\right )}}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )}{\left (\frac{b^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - \frac{4 \, a c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^2*(c*x^2 + b*x + a)^3),x, algorithm="giac")

[Out]

32*c^8*d^11/((b^6*c^6*d^12 - 12*a*b^4*c^7*d^12 + 48*a^2*b^2*c^8*d^12 - 64*a^3*c^
9*d^12)*(2*c*d*x + b*d)) + 60*c^2*arctan((b^2*d/(2*c*d*x + b*d) - 4*a*c*d/(2*c*d
*x + b*d))/sqrt(-b^2 + 4*a*c))/((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^2) - 4*(9*b^2*c^2*d/(2*c*d*x + b*d)^3 - 36*a*c^3*d/(2*c*d*
x + b*d)^3 - 7*c^2/((2*c*d*x + b*d)*d))/((b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64
*a^3*c^3)*(b^2*d^2/(2*c*d*x + b*d)^2 - 4*a*c*d^2/(2*c*d*x + b*d)^2 - 1)^2)