3.196 \(\int \frac{\text{b1}+\text{c1} x}{\left (a+2 b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=130 \[ \frac{3 (b+c x) (\text{b1} c-b \text{c1})}{8 \left (b^2-a c\right )^2 \left (a+2 b x+c x^2\right )}-\frac{-a \text{c1}+x (\text{b1} c-b \text{c1})+b \text{b1}}{4 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^2}-\frac{3 c (\text{b1} c-b \text{c1}) \tanh ^{-1}\left (\frac{b+c x}{\sqrt{b^2-a c}}\right )}{8 \left (b^2-a c\right )^{5/2}} \]

[Out]

-(b*b1 - a*c1 + (b1*c - b*c1)*x)/(4*(b^2 - a*c)*(a + 2*b*x + c*x^2)^2) + (3*(b1*
c - b*c1)*(b + c*x))/(8*(b^2 - a*c)^2*(a + 2*b*x + c*x^2)) - (3*c*(b1*c - b*c1)*
ArcTanh[(b + c*x)/Sqrt[b^2 - a*c]])/(8*(b^2 - a*c)^(5/2))

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Rubi [A]  time = 0.164698, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{3 (b+c x) (\text{b1} c-b \text{c1})}{8 \left (b^2-a c\right )^2 \left (a+2 b x+c x^2\right )}-\frac{-a \text{c1}+x (\text{b1} c-b \text{c1})+b \text{b1}}{4 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^2}-\frac{3 c (\text{b1} c-b \text{c1}) \tanh ^{-1}\left (\frac{b+c x}{\sqrt{b^2-a c}}\right )}{8 \left (b^2-a c\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(b1 + c1*x)/(a + 2*b*x + c*x^2)^3,x]

[Out]

-(b*b1 - a*c1 + (b1*c - b*c1)*x)/(4*(b^2 - a*c)*(a + 2*b*x + c*x^2)^2) + (3*(b1*
c - b*c1)*(b + c*x))/(8*(b^2 - a*c)^2*(a + 2*b*x + c*x^2)) - (3*c*(b1*c - b*c1)*
ArcTanh[(b + c*x)/Sqrt[b^2 - a*c]])/(8*(b^2 - a*c)^(5/2))

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Rubi in Sympy [A]  time = 9.28953, size = 121, normalized size = 0.93 \[ \frac{3 c \left (b c_{1} - b_{1} c\right ) \operatorname{atanh}{\left (\frac{b + c x}{\sqrt{- a c + b^{2}}} \right )}}{8 \left (- a c + b^{2}\right )^{\frac{5}{2}}} - \frac{3 \left (2 b + 2 c x\right ) \left (b c_{1} - b_{1} c\right )}{16 \left (- a c + b^{2}\right )^{2} \left (a + 2 b x + c x^{2}\right )} + \frac{2 a c_{1} - 2 b b_{1} + x \left (2 b c_{1} - 2 b_{1} c\right )}{8 \left (- a c + b^{2}\right ) \left (a + 2 b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c1*x+b1)/(c*x**2+2*b*x+a)**3,x)

[Out]

3*c*(b*c1 - b1*c)*atanh((b + c*x)/sqrt(-a*c + b**2))/(8*(-a*c + b**2)**(5/2)) -
3*(2*b + 2*c*x)*(b*c1 - b1*c)/(16*(-a*c + b**2)**2*(a + 2*b*x + c*x**2)) + (2*a*
c1 - 2*b*b1 + x*(2*b*c1 - 2*b1*c))/(8*(-a*c + b**2)*(a + 2*b*x + c*x**2)**2)

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Mathematica [A]  time = 0.228586, size = 127, normalized size = 0.98 \[ \frac{\frac{2 \left (b^2-a c\right ) (a \text{c1}-b \text{b1}+b \text{c1} x-\text{b1} c x)}{(a+x (2 b+c x))^2}+\frac{3 c (\text{b1} c-b \text{c1}) \tan ^{-1}\left (\frac{b+c x}{\sqrt{a c-b^2}}\right )}{\sqrt{a c-b^2}}+\frac{3 (b+c x) (\text{b1} c-b \text{c1})}{a+x (2 b+c x)}}{8 \left (b^2-a c\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(b1 + c1*x)/(a + 2*b*x + c*x^2)^3,x]

[Out]

((2*(b^2 - a*c)*(-(b*b1) + a*c1 - b1*c*x + b*c1*x))/(a + x*(2*b + c*x))^2 + (3*(
b1*c - b*c1)*(b + c*x))/(a + x*(2*b + c*x)) + (3*c*(b1*c - b*c1)*ArcTan[(b + c*x
)/Sqrt[-b^2 + a*c]])/Sqrt[-b^2 + a*c])/(8*(b^2 - a*c)^2)

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Maple [B]  time = 0.007, size = 274, normalized size = 2.1 \[{\frac{ \left ( -2\,b{\it c1}+2\,{\it b1}\,c \right ) x+2\,b{\it b1}-2\,a{\it c1}}{ \left ( 8\,ac-8\,{b}^{2} \right ) \left ( c{x}^{2}+2\,bx+a \right ) ^{2}}}-6\,{\frac{cxb{\it c1}}{ \left ( 4\,ac-4\,{b}^{2} \right ) ^{2} \left ( c{x}^{2}+2\,bx+a \right ) }}+6\,{\frac{x{c}^{2}{\it b1}}{ \left ( 4\,ac-4\,{b}^{2} \right ) ^{2} \left ( c{x}^{2}+2\,bx+a \right ) }}-6\,{\frac{{b}^{2}{\it c1}}{ \left ( 4\,ac-4\,{b}^{2} \right ) ^{2} \left ( c{x}^{2}+2\,bx+a \right ) }}+6\,{\frac{b{\it b1}\,c}{ \left ( 4\,ac-4\,{b}^{2} \right ) ^{2} \left ( c{x}^{2}+2\,bx+a \right ) }}-6\,{\frac{{\it c1}\,bc}{ \left ( 4\,ac-4\,{b}^{2} \right ) ^{2}\sqrt{ac-{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,cx+2\,b}{\sqrt{ac-{b}^{2}}}} \right ) }+6\,{\frac{{\it b1}\,{c}^{2}}{ \left ( 4\,ac-4\,{b}^{2} \right ) ^{2}\sqrt{ac-{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,cx+2\,b}{\sqrt{ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c1*x+b1)/(c*x^2+2*b*x+a)^3,x)

[Out]

1/2*((-2*b*c1+2*b1*c)*x+2*b*b1-2*a*c1)/(4*a*c-4*b^2)/(c*x^2+2*b*x+a)^2-6/(4*a*c-
4*b^2)^2/(c*x^2+2*b*x+a)*x*c*b*c1+6/(4*a*c-4*b^2)^2/(c*x^2+2*b*x+a)*x*c^2*b1-6/(
4*a*c-4*b^2)^2/(c*x^2+2*b*x+a)*b^2*c1+6/(4*a*c-4*b^2)^2/(c*x^2+2*b*x+a)*b*b1*c-6
/(4*a*c-4*b^2)^2*c/(a*c-b^2)^(1/2)*arctan(1/2*(2*c*x+2*b)/(a*c-b^2)^(1/2))*b*c1+
6/(4*a*c-4*b^2)^2*c^2/(a*c-b^2)^(1/2)*arctan(1/2*(2*c*x+2*b)/(a*c-b^2)^(1/2))*b1

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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((c1*x + b1)/(c*x^2 + 2*b*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229314, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c1*x + b1)/(c*x^2 + 2*b*x + a)^3,x, algorithm="fricas")

[Out]

[-1/16*(3*(a^2*b1*c^2 - a^2*b*c*c1 + (b1*c^4 - b*c^3*c1)*x^4 + 4*(b*b1*c^3 - b^2
*c^2*c1)*x^3 + 2*(2*b^2*b1*c^2 + a*b1*c^3 - (2*b^3*c + a*b*c^2)*c1)*x^2 + 4*(a*b
*b1*c^2 - a*b^2*c*c1)*x)*log((2*b^3 - 2*a*b*c + 2*(b^2*c - a*c^2)*x + (c^2*x^2 +
2*b*c*x + 2*b^2 - a*c)*sqrt(b^2 - a*c))/(c*x^2 + 2*b*x + a)) + 2*(2*b^3*b1 - 5*
a*b*b1*c - 3*(b1*c^3 - b*c^2*c1)*x^3 - 9*(b*b1*c^2 - b^2*c*c1)*x^2 + (a*b^2 + 2*
a^2*c)*c1 - (4*b^2*b1*c + 5*a*b1*c^2 - (4*b^3 + 5*a*b*c)*c1)*x)*sqrt(b^2 - a*c))
/((a^2*b^4 - 2*a^3*b^2*c + a^4*c^2 + (b^4*c^2 - 2*a*b^2*c^3 + a^2*c^4)*x^4 + 4*(
b^5*c - 2*a*b^3*c^2 + a^2*b*c^3)*x^3 + 2*(2*b^6 - 3*a*b^4*c + a^3*c^3)*x^2 + 4*(
a*b^5 - 2*a^2*b^3*c + a^3*b*c^2)*x)*sqrt(b^2 - a*c)), 1/8*(3*(a^2*b1*c^2 - a^2*b
*c*c1 + (b1*c^4 - b*c^3*c1)*x^4 + 4*(b*b1*c^3 - b^2*c^2*c1)*x^3 + 2*(2*b^2*b1*c^
2 + a*b1*c^3 - (2*b^3*c + a*b*c^2)*c1)*x^2 + 4*(a*b*b1*c^2 - a*b^2*c*c1)*x)*arct
an(-sqrt(-b^2 + a*c)*(c*x + b)/(b^2 - a*c)) - (2*b^3*b1 - 5*a*b*b1*c - 3*(b1*c^3
- b*c^2*c1)*x^3 - 9*(b*b1*c^2 - b^2*c*c1)*x^2 + (a*b^2 + 2*a^2*c)*c1 - (4*b^2*b
1*c + 5*a*b1*c^2 - (4*b^3 + 5*a*b*c)*c1)*x)*sqrt(-b^2 + a*c))/((a^2*b^4 - 2*a^3*
b^2*c + a^4*c^2 + (b^4*c^2 - 2*a*b^2*c^3 + a^2*c^4)*x^4 + 4*(b^5*c - 2*a*b^3*c^2
+ a^2*b*c^3)*x^3 + 2*(2*b^6 - 3*a*b^4*c + a^3*c^3)*x^2 + 4*(a*b^5 - 2*a^2*b^3*c
+ a^3*b*c^2)*x)*sqrt(-b^2 + a*c))]

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Sympy [A]  time = 3.89344, size = 622, normalized size = 4.78 \[ \frac{3 c \sqrt{- \frac{1}{\left (a c - b^{2}\right )^{5}}} \left (b c_{1} - b_{1} c\right ) \log{\left (x + \frac{- 3 a^{3} c^{4} \sqrt{- \frac{1}{\left (a c - b^{2}\right )^{5}}} \left (b c_{1} - b_{1} c\right ) + 9 a^{2} b^{2} c^{3} \sqrt{- \frac{1}{\left (a c - b^{2}\right )^{5}}} \left (b c_{1} - b_{1} c\right ) - 9 a b^{4} c^{2} \sqrt{- \frac{1}{\left (a c - b^{2}\right )^{5}}} \left (b c_{1} - b_{1} c\right ) + 3 b^{6} c \sqrt{- \frac{1}{\left (a c - b^{2}\right )^{5}}} \left (b c_{1} - b_{1} c\right ) + 3 b^{2} c c_{1} - 3 b b_{1} c^{2}}{3 b c^{2} c_{1} - 3 b_{1} c^{3}} \right )}}{16} - \frac{3 c \sqrt{- \frac{1}{\left (a c - b^{2}\right )^{5}}} \left (b c_{1} - b_{1} c\right ) \log{\left (x + \frac{3 a^{3} c^{4} \sqrt{- \frac{1}{\left (a c - b^{2}\right )^{5}}} \left (b c_{1} - b_{1} c\right ) - 9 a^{2} b^{2} c^{3} \sqrt{- \frac{1}{\left (a c - b^{2}\right )^{5}}} \left (b c_{1} - b_{1} c\right ) + 9 a b^{4} c^{2} \sqrt{- \frac{1}{\left (a c - b^{2}\right )^{5}}} \left (b c_{1} - b_{1} c\right ) - 3 b^{6} c \sqrt{- \frac{1}{\left (a c - b^{2}\right )^{5}}} \left (b c_{1} - b_{1} c\right ) + 3 b^{2} c c_{1} - 3 b b_{1} c^{2}}{3 b c^{2} c_{1} - 3 b_{1} c^{3}} \right )}}{16} - \frac{2 a^{2} c c_{1} + a b^{2} c_{1} - 5 a b b_{1} c + 2 b^{3} b_{1} + x^{3} \left (3 b c^{2} c_{1} - 3 b_{1} c^{3}\right ) + x^{2} \left (9 b^{2} c c_{1} - 9 b b_{1} c^{2}\right ) + x \left (5 a b c c_{1} - 5 a b_{1} c^{2} + 4 b^{3} c_{1} - 4 b^{2} b_{1} c\right )}{8 a^{4} c^{2} - 16 a^{3} b^{2} c + 8 a^{2} b^{4} + x^{4} \left (8 a^{2} c^{4} - 16 a b^{2} c^{3} + 8 b^{4} c^{2}\right ) + x^{3} \left (32 a^{2} b c^{3} - 64 a b^{3} c^{2} + 32 b^{5} c\right ) + x^{2} \left (16 a^{3} c^{3} - 48 a b^{4} c + 32 b^{6}\right ) + x \left (32 a^{3} b c^{2} - 64 a^{2} b^{3} c + 32 a b^{5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c1*x+b1)/(c*x**2+2*b*x+a)**3,x)

[Out]

3*c*sqrt(-1/(a*c - b**2)**5)*(b*c1 - b1*c)*log(x + (-3*a**3*c**4*sqrt(-1/(a*c -
b**2)**5)*(b*c1 - b1*c) + 9*a**2*b**2*c**3*sqrt(-1/(a*c - b**2)**5)*(b*c1 - b1*c
) - 9*a*b**4*c**2*sqrt(-1/(a*c - b**2)**5)*(b*c1 - b1*c) + 3*b**6*c*sqrt(-1/(a*c
- b**2)**5)*(b*c1 - b1*c) + 3*b**2*c*c1 - 3*b*b1*c**2)/(3*b*c**2*c1 - 3*b1*c**3
))/16 - 3*c*sqrt(-1/(a*c - b**2)**5)*(b*c1 - b1*c)*log(x + (3*a**3*c**4*sqrt(-1/
(a*c - b**2)**5)*(b*c1 - b1*c) - 9*a**2*b**2*c**3*sqrt(-1/(a*c - b**2)**5)*(b*c1
- b1*c) + 9*a*b**4*c**2*sqrt(-1/(a*c - b**2)**5)*(b*c1 - b1*c) - 3*b**6*c*sqrt(
-1/(a*c - b**2)**5)*(b*c1 - b1*c) + 3*b**2*c*c1 - 3*b*b1*c**2)/(3*b*c**2*c1 - 3*
b1*c**3))/16 - (2*a**2*c*c1 + a*b**2*c1 - 5*a*b*b1*c + 2*b**3*b1 + x**3*(3*b*c**
2*c1 - 3*b1*c**3) + x**2*(9*b**2*c*c1 - 9*b*b1*c**2) + x*(5*a*b*c*c1 - 5*a*b1*c*
*2 + 4*b**3*c1 - 4*b**2*b1*c))/(8*a**4*c**2 - 16*a**3*b**2*c + 8*a**2*b**4 + x**
4*(8*a**2*c**4 - 16*a*b**2*c**3 + 8*b**4*c**2) + x**3*(32*a**2*b*c**3 - 64*a*b**
3*c**2 + 32*b**5*c) + x**2*(16*a**3*c**3 - 48*a*b**4*c + 32*b**6) + x*(32*a**3*b
*c**2 - 64*a**2*b**3*c + 32*a*b**5))

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GIAC/XCAS [A]  time = 0.204036, size = 262, normalized size = 2.02 \[ \frac{3 \,{\left (b_{1} c^{2} - b c c_{1}\right )} \arctan \left (\frac{c x + b}{\sqrt{-b^{2} + a c}}\right )}{8 \,{\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} \sqrt{-b^{2} + a c}} + \frac{3 \, b_{1} c^{3} x^{3} - 3 \, b c^{2} c_{1} x^{3} + 9 \, b b_{1} c^{2} x^{2} - 9 \, b^{2} c c_{1} x^{2} + 4 \, b^{2} b_{1} c x + 5 \, a b_{1} c^{2} x - 4 \, b^{3} c_{1} x - 5 \, a b c c_{1} x - 2 \, b^{3} b_{1} + 5 \, a b b_{1} c - a b^{2} c_{1} - 2 \, a^{2} c c_{1}}{8 \,{\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )}{\left (c x^{2} + 2 \, b x + a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c1*x + b1)/(c*x^2 + 2*b*x + a)^3,x, algorithm="giac")

[Out]

3/8*(b1*c^2 - b*c*c1)*arctan((c*x + b)/sqrt(-b^2 + a*c))/((b^4 - 2*a*b^2*c + a^2
*c^2)*sqrt(-b^2 + a*c)) + 1/8*(3*b1*c^3*x^3 - 3*b*c^2*c1*x^3 + 9*b*b1*c^2*x^2 -
9*b^2*c*c1*x^2 + 4*b^2*b1*c*x + 5*a*b1*c^2*x - 4*b^3*c1*x - 5*a*b*c*c1*x - 2*b^3
*b1 + 5*a*b*b1*c - a*b^2*c1 - 2*a^2*c*c1)/((b^4 - 2*a*b^2*c + a^2*c^2)*(c*x^2 +
2*b*x + a)^2)