3.1179 \(\int \frac{1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=168 \[ \frac{140 c^2}{d^4 \left (b^2-4 a c\right )^4 (b+2 c x)}+\frac{140 c^2}{3 d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^3}-\frac{140 c^2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{d^4 \left (b^2-4 a c\right )^{9/2}}+\frac{7 c}{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3 \left (a+b x+c x^2\right )}-\frac{1}{2 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \left (a+b x+c x^2\right )^2} \]

[Out]

(140*c^2)/(3*(b^2 - 4*a*c)^3*d^4*(b + 2*c*x)^3) + (140*c^2)/((b^2 - 4*a*c)^4*d^4
*(b + 2*c*x)) - 1/(2*(b^2 - 4*a*c)*d^4*(b + 2*c*x)^3*(a + b*x + c*x^2)^2) + (7*c
)/((b^2 - 4*a*c)^2*d^4*(b + 2*c*x)^3*(a + b*x + c*x^2)) - (140*c^2*ArcTanh[(b +
2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(9/2)*d^4)

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Rubi [A]  time = 0.364538, antiderivative size = 168, 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 \[ \frac{140 c^2}{d^4 \left (b^2-4 a c\right )^4 (b+2 c x)}+\frac{140 c^2}{3 d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^3}-\frac{140 c^2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{d^4 \left (b^2-4 a c\right )^{9/2}}+\frac{7 c}{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3 \left (a+b x+c x^2\right )}-\frac{1}{2 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^3),x]

[Out]

(140*c^2)/(3*(b^2 - 4*a*c)^3*d^4*(b + 2*c*x)^3) + (140*c^2)/((b^2 - 4*a*c)^4*d^4
*(b + 2*c*x)) - 1/(2*(b^2 - 4*a*c)*d^4*(b + 2*c*x)^3*(a + b*x + c*x^2)^2) + (7*c
)/((b^2 - 4*a*c)^2*d^4*(b + 2*c*x)^3*(a + b*x + c*x^2)) - (140*c^2*ArcTanh[(b +
2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(9/2)*d^4)

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Rubi in Sympy [A]  time = 77.3346, size = 165, normalized size = 0.98 \[ - \frac{140 c^{2} \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{d^{4} \left (- 4 a c + b^{2}\right )^{\frac{9}{2}}} + \frac{140 c^{2}}{d^{4} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{4}} + \frac{140 c^{2}}{3 d^{4} \left (b + 2 c x\right )^{3} \left (- 4 a c + b^{2}\right )^{3}} + \frac{7 c}{d^{4} \left (b + 2 c x\right )^{3} \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )} - \frac{1}{2 d^{4} \left (b + 2 c x\right )^{3} \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*c*d*x+b*d)**4/(c*x**2+b*x+a)**3,x)

[Out]

-140*c**2*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(d**4*(-4*a*c + b**2)**(9/2)) +
140*c**2/(d**4*(b + 2*c*x)*(-4*a*c + b**2)**4) + 140*c**2/(3*d**4*(b + 2*c*x)**
3*(-4*a*c + b**2)**3) + 7*c/(d**4*(b + 2*c*x)**3*(-4*a*c + b**2)**2*(a + b*x + c
*x**2)) - 1/(2*d**4*(b + 2*c*x)**3*(-4*a*c + b**2)*(a + b*x + c*x**2)**2)

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Mathematica [A]  time = 0.909337, size = 140, normalized size = 0.83 \[ \frac{\frac{64 c^2 \left (b^2-4 a c\right )}{(b+2 c x)^3}+\frac{840 c^2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x)}{(a+x (b+c x))^2}+\frac{66 c (b+2 c x)}{a+x (b+c x)}+\frac{576 c^2}{b+2 c x}}{6 d^4 \left (b^2-4 a c\right )^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^3),x]

[Out]

((64*c^2*(b^2 - 4*a*c))/(b + 2*c*x)^3 + (576*c^2)/(b + 2*c*x) - (3*(b^2 - 4*a*c)
*(b + 2*c*x))/(a + x*(b + c*x))^2 + (66*c*(b + 2*c*x))/(a + x*(b + c*x)) + (840*
c^2*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(6*(b^2 - 4*a*c)
^4*d^4)

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Maple [A]  time = 0.025, size = 301, normalized size = 1.8 \[ 96\,{\frac{{c}^{2}}{{d}^{4} \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( 2\,cx+b \right ) }}-{\frac{32\,{c}^{2}}{3\,{d}^{4} \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( 2\,cx+b \right ) ^{3}}}+22\,{\frac{{c}^{3}{x}^{3}}{{d}^{4} \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+33\,{\frac{b{c}^{2}{x}^{2}}{{d}^{4} \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+26\,{\frac{a{c}^{2}x}{{d}^{4} \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+10\,{\frac{{b}^{2}xc}{{d}^{4} \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+13\,{\frac{abc}{{d}^{4} \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) ^{2}}}-{\frac{{b}^{3}}{2\,{d}^{4} \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+140\,{\frac{{c}^{2}}{{d}^{4} \left ( 4\,ac-{b}^{2} \right ) ^{9/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^3,x)

[Out]

96/d^4/(4*a*c-b^2)^4*c^2/(2*c*x+b)-32/3/d^4*c^2/(4*a*c-b^2)^3/(2*c*x+b)^3+22/d^4
/(4*a*c-b^2)^4/(c*x^2+b*x+a)^2*c^3*x^3+33/d^4/(4*a*c-b^2)^4/(c*x^2+b*x+a)^2*b*c^
2*x^2+26/d^4/(4*a*c-b^2)^4/(c*x^2+b*x+a)^2*a*c^2*x+10/d^4/(4*a*c-b^2)^4/(c*x^2+b
*x+a)^2*x*b^2*c+13/d^4/(4*a*c-b^2)^4/(c*x^2+b*x+a)^2*a*b*c-1/2/d^4/(4*a*c-b^2)^4
/(c*x^2+b*x+a)^2*b^3+140/d^4/(4*a*c-b^2)^(9/2)*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)^4*(c*x^2 + b*x + a)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245024, 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)^4*(c*x^2 + b*x + a)^3),x, algorithm="fricas")

[Out]

[1/6*(420*(8*c^7*x^7 + 28*b*c^6*x^6 + a^2*b^3*c^2 + 2*(19*b^2*c^5 + 8*a*c^6)*x^5
+ 5*(5*b^3*c^4 + 8*a*b*c^5)*x^4 + 4*(2*b^4*c^3 + 9*a*b^2*c^4 + 2*a^2*c^5)*x^3 +
(b^5*c^2 + 14*a*b^3*c^3 + 12*a^2*b*c^4)*x^2 + 2*(a*b^4*c^2 + 3*a^2*b^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)) + (3360*c^6*x^6 + 10080*b*c^5*x^5 - 3*b^
6 + 78*a*b^4*c + 640*a^2*b^2*c^2 - 256*a^3*c^3 + 5600*(2*b^2*c^4 + a*c^5)*x^4 +
5600*(b^3*c^3 + 2*a*b*c^4)*x^3 + 14*(83*b^4*c^2 + 536*a*b^2*c^3 + 128*a^2*c^4)*x
^2 + 14*(3*b^5*c + 136*a*b^3*c^2 + 128*a^2*b*c^3)*x)*sqrt(b^2 - 4*a*c))/((8*(b^8
*c^5 - 16*a*b^6*c^6 + 96*a^2*b^4*c^7 - 256*a^3*b^2*c^8 + 256*a^4*c^9)*d^4*x^7 +
28*(b^9*c^4 - 16*a*b^7*c^5 + 96*a^2*b^5*c^6 - 256*a^3*b^3*c^7 + 256*a^4*b*c^8)*d
^4*x^6 + 2*(19*b^10*c^3 - 296*a*b^8*c^4 + 1696*a^2*b^6*c^5 - 4096*a^3*b^4*c^6 +
2816*a^4*b^2*c^7 + 2048*a^5*c^8)*d^4*x^5 + 5*(5*b^11*c^2 - 72*a*b^9*c^3 + 352*a^
2*b^7*c^4 - 512*a^3*b^5*c^5 - 768*a^4*b^3*c^6 + 2048*a^5*b*c^7)*d^4*x^4 + 4*(2*b
^12*c - 23*a*b^10*c^2 + 50*a^2*b^8*c^3 + 320*a^3*b^6*c^4 - 1600*a^4*b^4*c^5 + 17
92*a^5*b^2*c^6 + 512*a^6*c^7)*d^4*x^3 + (b^13 - 2*a*b^11*c - 116*a^2*b^9*c^2 + 8
96*a^3*b^7*c^3 - 2176*a^4*b^5*c^4 + 512*a^5*b^3*c^5 + 3072*a^6*b*c^6)*d^4*x^2 +
2*(a*b^12 - 13*a^2*b^10*c + 48*a^3*b^8*c^2 + 32*a^4*b^6*c^3 - 512*a^5*b^4*c^4 +
768*a^6*b^2*c^5)*d^4*x + (a^2*b^11 - 16*a^3*b^9*c + 96*a^4*b^7*c^2 - 256*a^5*b^5
*c^3 + 256*a^6*b^3*c^4)*d^4)*sqrt(b^2 - 4*a*c)), 1/6*(840*(8*c^7*x^7 + 28*b*c^6*
x^6 + a^2*b^3*c^2 + 2*(19*b^2*c^5 + 8*a*c^6)*x^5 + 5*(5*b^3*c^4 + 8*a*b*c^5)*x^4
+ 4*(2*b^4*c^3 + 9*a*b^2*c^4 + 2*a^2*c^5)*x^3 + (b^5*c^2 + 14*a*b^3*c^3 + 12*a^
2*b*c^4)*x^2 + 2*(a*b^4*c^2 + 3*a^2*b^2*c^3)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*
x + b)/(b^2 - 4*a*c)) + (3360*c^6*x^6 + 10080*b*c^5*x^5 - 3*b^6 + 78*a*b^4*c + 6
40*a^2*b^2*c^2 - 256*a^3*c^3 + 5600*(2*b^2*c^4 + a*c^5)*x^4 + 5600*(b^3*c^3 + 2*
a*b*c^4)*x^3 + 14*(83*b^4*c^2 + 536*a*b^2*c^3 + 128*a^2*c^4)*x^2 + 14*(3*b^5*c +
136*a*b^3*c^2 + 128*a^2*b*c^3)*x)*sqrt(-b^2 + 4*a*c))/((8*(b^8*c^5 - 16*a*b^6*c
^6 + 96*a^2*b^4*c^7 - 256*a^3*b^2*c^8 + 256*a^4*c^9)*d^4*x^7 + 28*(b^9*c^4 - 16*
a*b^7*c^5 + 96*a^2*b^5*c^6 - 256*a^3*b^3*c^7 + 256*a^4*b*c^8)*d^4*x^6 + 2*(19*b^
10*c^3 - 296*a*b^8*c^4 + 1696*a^2*b^6*c^5 - 4096*a^3*b^4*c^6 + 2816*a^4*b^2*c^7
+ 2048*a^5*c^8)*d^4*x^5 + 5*(5*b^11*c^2 - 72*a*b^9*c^3 + 352*a^2*b^7*c^4 - 512*a
^3*b^5*c^5 - 768*a^4*b^3*c^6 + 2048*a^5*b*c^7)*d^4*x^4 + 4*(2*b^12*c - 23*a*b^10
*c^2 + 50*a^2*b^8*c^3 + 320*a^3*b^6*c^4 - 1600*a^4*b^4*c^5 + 1792*a^5*b^2*c^6 +
512*a^6*c^7)*d^4*x^3 + (b^13 - 2*a*b^11*c - 116*a^2*b^9*c^2 + 896*a^3*b^7*c^3 -
2176*a^4*b^5*c^4 + 512*a^5*b^3*c^5 + 3072*a^6*b*c^6)*d^4*x^2 + 2*(a*b^12 - 13*a^
2*b^10*c + 48*a^3*b^8*c^2 + 32*a^4*b^6*c^3 - 512*a^5*b^4*c^4 + 768*a^6*b^2*c^5)*
d^4*x + (a^2*b^11 - 16*a^3*b^9*c + 96*a^4*b^7*c^2 - 256*a^5*b^5*c^3 + 256*a^6*b^
3*c^4)*d^4)*sqrt(-b^2 + 4*a*c))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2*c*d*x+b*d)**4/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220558, size = 420, normalized size = 2.5 \[ \frac{140 \, c^{2} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{8} d^{4} - 16 \, a b^{6} c d^{4} + 96 \, a^{2} b^{4} c^{2} d^{4} - 256 \, a^{3} b^{2} c^{3} d^{4} + 256 \, a^{4} c^{4} d^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{44 \, c^{3} x^{3} + 66 \, b c^{2} x^{2} + 20 \, b^{2} c x + 52 \, a c^{2} x - b^{3} + 26 \, a b c}{2 \,{\left (b^{8} d^{4} - 16 \, a b^{6} c d^{4} + 96 \, a^{2} b^{4} c^{2} d^{4} - 256 \, a^{3} b^{2} c^{3} d^{4} + 256 \, a^{4} c^{4} d^{4}\right )}{\left (c x^{2} + b x + a\right )}^{2}} + \frac{64 \,{\left (18 \, c^{4} x^{2} + 18 \, b c^{3} x + 5 \, b^{2} c^{2} - 2 \, a c^{3}\right )}}{3 \,{\left (b^{8} d^{4} - 16 \, a b^{6} c d^{4} + 96 \, a^{2} b^{4} c^{2} d^{4} - 256 \, a^{3} b^{2} c^{3} d^{4} + 256 \, a^{4} c^{4} d^{4}\right )}{\left (2 \, c x + b\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^4*(c*x^2 + b*x + a)^3),x, algorithm="giac")

[Out]

140*c^2*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^8*d^4 - 16*a*b^6*c*d^4 + 96*a
^2*b^4*c^2*d^4 - 256*a^3*b^2*c^3*d^4 + 256*a^4*c^4*d^4)*sqrt(-b^2 + 4*a*c)) + 1/
2*(44*c^3*x^3 + 66*b*c^2*x^2 + 20*b^2*c*x + 52*a*c^2*x - b^3 + 26*a*b*c)/((b^8*d
^4 - 16*a*b^6*c*d^4 + 96*a^2*b^4*c^2*d^4 - 256*a^3*b^2*c^3*d^4 + 256*a^4*c^4*d^4
)*(c*x^2 + b*x + a)^2) + 64/3*(18*c^4*x^2 + 18*b*c^3*x + 5*b^2*c^2 - 2*a*c^3)/((
b^8*d^4 - 16*a*b^6*c*d^4 + 96*a^2*b^4*c^2*d^4 - 256*a^3*b^2*c^3*d^4 + 256*a^4*c^
4*d^4)*(2*c*x + b)^3)