3.17 \(\int \frac{g+h x}{\left (a+b x+c x^2\right ) \left (a d+b d x+c d x^2\right )^2} \, dx\)

Optimal. Leaf size=140 \[ \frac{3 (b+2 c x) (2 c g-b h)}{2 d^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{-2 a h+x (2 c g-b h)+b g}{2 d^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{6 c (2 c g-b h) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{d^2 \left (b^2-4 a c\right )^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.307764, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.147 \[ \frac{3 (b+2 c x) (2 c g-b h)}{2 d^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{-2 a h+x (2 c g-b h)+b g}{2 d^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{6 c (2 c g-b h) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{d^2 \left (b^2-4 a c\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 49.7295, size = 131, normalized size = 0.94 \[ \frac{6 c \left (b h - 2 c g\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{d^{2} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} - \frac{3 \left (b + 2 c x\right ) \left (\frac{b h}{2} - c g\right )}{d^{2} \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )} + \frac{2 a h - b g + x \left (b h - 2 c g\right )}{2 d^{2} \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.263056, size = 131, normalized size = 0.94 \[ \frac{\frac{\left (b^2-4 a c\right ) (2 a h-b g+b h x-2 c g x)}{(a+x (b+c x))^2}-\frac{12 c (b h-2 c g) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{3 (b+2 c x) (2 c g-b h)}{a+x (b+c x)}}{2 d^2 \left (b^2-4 a c\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(((b^2 - 4*a*c)*(-(b*g) + 2*a*h - 2*c*g*x + b*h*x))/(a + x*(b + c*x))^2 + (3*(2*
c*g - b*h)*(b + 2*c*x))/(a + x*(b + c*x)) - (12*c*(-2*c*g + b*h)*ArcTan[(b + 2*c
*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(2*(b^2 - 4*a*c)^2*d^2)

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Maple [B]  time = 0.007, size = 340, normalized size = 2.4 \[ -{\frac{bxh}{2\,{d}^{2} \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) ^{2}}}+{\frac{cxg}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) ^{2}}}-{\frac{ah}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) ^{2}}}+{\frac{bg}{2\,{d}^{2} \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) ^{2}}}-3\,{\frac{cxbh}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}+6\,{\frac{{c}^{2}xg}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}-{\frac{3\,{b}^{2}h}{2\,{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}+3\,{\frac{bcg}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}-6\,{\frac{bch}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{5/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+12\,{\frac{{c}^{2}g}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{5/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/d^2/(4*a*c-b^2)/(c*x^2+b*x+a)^2*x*b*h+1/d^2/(4*a*c-b^2)/(c*x^2+b*x+a)^2*x*c
*g-1/d^2/(4*a*c-b^2)/(c*x^2+b*x+a)^2*a*h+1/2/d^2/(4*a*c-b^2)/(c*x^2+b*x+a)^2*b*g
-3/d^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)*c*x*b*h+6/d^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)*c^2
*x*g-3/2/d^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)*b^2*h+3/d^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)
*b*c*g-6/d^2/(4*a*c-b^2)^(5/2)*c*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*h+12/d^2/
(4*a*c-b^2)^(5/2)*c^2*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*g

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.305622, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*(6*(2*a^2*c^2*g - a^2*b*c*h + (2*c^4*g - b*c^3*h)*x^4 + 2*(2*b*c^3*g - b^2
*c^2*h)*x^3 + (2*(b^2*c^2 + 2*a*c^3)*g - (b^3*c + 2*a*b*c^2)*h)*x^2 + 2*(2*a*b*c
^2*g - a*b^2*c*h)*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)) - (6*(2*c^3*g - b*c^
2*h)*x^3 + 9*(2*b*c^2*g - b^2*c*h)*x^2 - (b^3 - 10*a*b*c)*g - (a*b^2 + 8*a^2*c)*
h + 2*(2*(b^2*c + 5*a*c^2)*g - (b^3 + 5*a*b*c)*h)*x)*sqrt(b^2 - 4*a*c))/(((b^4*c
^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*
d^2*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^2*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a
^3*b*c^2)*d^2*x + (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*d^2)*sqrt(b^2 - 4*a*c)),
1/2*(12*(2*a^2*c^2*g - a^2*b*c*h + (2*c^4*g - b*c^3*h)*x^4 + 2*(2*b*c^3*g - b^2*
c^2*h)*x^3 + (2*(b^2*c^2 + 2*a*c^3)*g - (b^3*c + 2*a*b*c^2)*h)*x^2 + 2*(2*a*b*c^
2*g - a*b^2*c*h)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (6*(
2*c^3*g - b*c^2*h)*x^3 + 9*(2*b*c^2*g - b^2*c*h)*x^2 - (b^3 - 10*a*b*c)*g - (a*b
^2 + 8*a^2*c)*h + 2*(2*(b^2*c + 5*a*c^2)*g - (b^3 + 5*a*b*c)*h)*x)*sqrt(-b^2 + 4
*a*c))/(((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2*x^4 + 2*(b^5*c - 8*a*b^3*c^2 +
16*a^2*b*c^3)*d^2*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^2*x^2 + 2*(a*b^5 - 8*a
^2*b^3*c + 16*a^3*b*c^2)*d^2*x + (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*d^2)*sqrt(
-b^2 + 4*a*c))]

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Sympy [A]  time = 4.81789, size = 709, normalized size = 5.06 \[ \frac{3 c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (b h - 2 c g\right ) \log{\left (x + \frac{- 192 a^{3} c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (b h - 2 c g\right ) + 144 a^{2} b^{2} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (b h - 2 c g\right ) - 36 a b^{4} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (b h - 2 c g\right ) + 3 b^{6} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (b h - 2 c g\right ) + 3 b^{2} c h - 6 b c^{2} g}{6 b c^{2} h - 12 c^{3} g} \right )}}{d^{2}} - \frac{3 c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (b h - 2 c g\right ) \log{\left (x + \frac{192 a^{3} c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (b h - 2 c g\right ) - 144 a^{2} b^{2} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (b h - 2 c g\right ) + 36 a b^{4} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (b h - 2 c g\right ) - 3 b^{6} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (b h - 2 c g\right ) + 3 b^{2} c h - 6 b c^{2} g}{6 b c^{2} h - 12 c^{3} g} \right )}}{d^{2}} - \frac{8 a^{2} c h + a b^{2} h - 10 a b c g + b^{3} g + x^{3} \left (6 b c^{2} h - 12 c^{3} g\right ) + x^{2} \left (9 b^{2} c h - 18 b c^{2} g\right ) + x \left (10 a b c h - 20 a c^{2} g + 2 b^{3} h - 4 b^{2} c g\right )}{32 a^{4} c^{2} d^{2} - 16 a^{3} b^{2} c d^{2} + 2 a^{2} b^{4} d^{2} + x^{4} \left (32 a^{2} c^{4} d^{2} - 16 a b^{2} c^{3} d^{2} + 2 b^{4} c^{2} d^{2}\right ) + x^{3} \left (64 a^{2} b c^{3} d^{2} - 32 a b^{3} c^{2} d^{2} + 4 b^{5} c d^{2}\right ) + x^{2} \left (64 a^{3} c^{3} d^{2} - 12 a b^{4} c d^{2} + 2 b^{6} d^{2}\right ) + x \left (64 a^{3} b c^{2} d^{2} - 32 a^{2} b^{3} c d^{2} + 4 a b^{5} d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*c*sqrt(-1/(4*a*c - b**2)**5)*(b*h - 2*c*g)*log(x + (-192*a**3*c**4*sqrt(-1/(4*
a*c - b**2)**5)*(b*h - 2*c*g) + 144*a**2*b**2*c**3*sqrt(-1/(4*a*c - b**2)**5)*(b
*h - 2*c*g) - 36*a*b**4*c**2*sqrt(-1/(4*a*c - b**2)**5)*(b*h - 2*c*g) + 3*b**6*c
*sqrt(-1/(4*a*c - b**2)**5)*(b*h - 2*c*g) + 3*b**2*c*h - 6*b*c**2*g)/(6*b*c**2*h
- 12*c**3*g))/d**2 - 3*c*sqrt(-1/(4*a*c - b**2)**5)*(b*h - 2*c*g)*log(x + (192*
a**3*c**4*sqrt(-1/(4*a*c - b**2)**5)*(b*h - 2*c*g) - 144*a**2*b**2*c**3*sqrt(-1/
(4*a*c - b**2)**5)*(b*h - 2*c*g) + 36*a*b**4*c**2*sqrt(-1/(4*a*c - b**2)**5)*(b*
h - 2*c*g) - 3*b**6*c*sqrt(-1/(4*a*c - b**2)**5)*(b*h - 2*c*g) + 3*b**2*c*h - 6*
b*c**2*g)/(6*b*c**2*h - 12*c**3*g))/d**2 - (8*a**2*c*h + a*b**2*h - 10*a*b*c*g +
b**3*g + x**3*(6*b*c**2*h - 12*c**3*g) + x**2*(9*b**2*c*h - 18*b*c**2*g) + x*(1
0*a*b*c*h - 20*a*c**2*g + 2*b**3*h - 4*b**2*c*g))/(32*a**4*c**2*d**2 - 16*a**3*b
**2*c*d**2 + 2*a**2*b**4*d**2 + x**4*(32*a**2*c**4*d**2 - 16*a*b**2*c**3*d**2 +
2*b**4*c**2*d**2) + x**3*(64*a**2*b*c**3*d**2 - 32*a*b**3*c**2*d**2 + 4*b**5*c*d
**2) + x**2*(64*a**3*c**3*d**2 - 12*a*b**4*c*d**2 + 2*b**6*d**2) + x*(64*a**3*b*
c**2*d**2 - 32*a**2*b**3*c*d**2 + 4*a*b**5*d**2))

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GIAC/XCAS [A]  time = 0.277088, size = 296, normalized size = 2.11 \[ \frac{6 \,{\left (2 \, c^{2} g - b c h\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} d^{2} - 8 \, a b^{2} c d^{2} + 16 \, a^{2} c^{2} d^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{12 \, c^{3} g x^{3} - 6 \, b c^{2} h x^{3} + 18 \, b c^{2} g x^{2} - 9 \, b^{2} c h x^{2} + 4 \, b^{2} c g x + 20 \, a c^{2} g x - 2 \, b^{3} h x - 10 \, a b c h x - b^{3} g + 10 \, a b c g - a b^{2} h - 8 \, a^{2} c h}{2 \,{\left (b^{4} d^{2} - 8 \, a b^{2} c d^{2} + 16 \, a^{2} c^{2} d^{2}\right )}{\left (c x^{2} + b x + a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*(2*c^2*g - b*c*h)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^4*d^2 - 8*a*b^2*c
*d^2 + 16*a^2*c^2*d^2)*sqrt(-b^2 + 4*a*c)) + 1/2*(12*c^3*g*x^3 - 6*b*c^2*h*x^3 +
18*b*c^2*g*x^2 - 9*b^2*c*h*x^2 + 4*b^2*c*g*x + 20*a*c^2*g*x - 2*b^3*h*x - 10*a*
b*c*h*x - b^3*g + 10*a*b*c*g - a*b^2*h - 8*a^2*c*h)/((b^4*d^2 - 8*a*b^2*c*d^2 +
16*a^2*c^2*d^2)*(c*x^2 + b*x + a)^2)