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

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 = 0.072158, 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, Rules used = {638, 614, 618, 206} \[ \frac{3 (b+c x) (\text{b1} c-b \text{c1})}{8 \left (b^2-a c\right )^2 \left (a+2 b x+c x^2\right )}-\frac{-a \text{c1}+x (\text{b1} c-b \text{c1})+b \text{b1}}{4 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^2}-\frac{3 c (\text{b1} c-b \text{c1}) \tanh ^{-1}\left (\frac{b+c x}{\sqrt{b^2-a c}}\right )}{8 \left (b^2-a c\right )^{5/2}} \]

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))

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\text{b1}+\text{c1} x}{\left (a+2 b x+c x^2\right )^3} \, dx &=-\frac{b \text{b1}-a \text{c1}+(\text{b1} c-b \text{c1}) x}{4 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^2}-\frac{(3 (\text{b1} c-b \text{c1})) \int \frac{1}{\left (a+2 b x+c x^2\right )^2} \, dx}{4 \left (b^2-a c\right )}\\ &=-\frac{b \text{b1}-a \text{c1}+(\text{b1} c-b \text{c1}) x}{4 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^2}+\frac{3 (\text{b1} c-b \text{c1}) (b+c x)}{8 \left (b^2-a c\right )^2 \left (a+2 b x+c x^2\right )}+\frac{(3 c (\text{b1} c-b \text{c1})) \int \frac{1}{a+2 b x+c x^2} \, dx}{8 \left (b^2-a c\right )^2}\\ &=-\frac{b \text{b1}-a \text{c1}+(\text{b1} c-b \text{c1}) x}{4 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^2}+\frac{3 (\text{b1} c-b \text{c1}) (b+c x)}{8 \left (b^2-a c\right )^2 \left (a+2 b x+c x^2\right )}-\frac{(3 c (\text{b1} c-b \text{c1})) \operatorname{Subst}\left (\int \frac{1}{4 \left (b^2-a c\right )-x^2} \, dx,x,2 b+2 c x\right )}{4 \left (b^2-a c\right )^2}\\ &=-\frac{b \text{b1}-a \text{c1}+(\text{b1} c-b \text{c1}) x}{4 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^2}+\frac{3 (\text{b1} c-b \text{c1}) (b+c x)}{8 \left (b^2-a c\right )^2 \left (a+2 b x+c x^2\right )}-\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}}\\ \end{align*}

Mathematica [A]  time = 0.138804, 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 = 0.004, size = 274, normalized size = 2.1 \begin{align*}{\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 ) } \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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 [B]  time = 1.92107, size = 2279, normalized size = 17.53 \begin{align*} \text{result too large to display} \end{align*}

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*(4*b^5*b1 - 14*a*b^3*b1*c + 10*a^2*b*b1*c^2 - 6*(b^2*b1*c^3 - a*b1*c^4 - (b^3*c^2 - a*b*c^3)*c1)*x^3 -
18*(b^3*b1*c^2 - a*b*b1*c^3 - (b^4*c - a*b^2*c^2)*c1)*x^2 + 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)*sqrt(b^2 - a*c)*log((c^2*x^2 + 2*b*c*x + 2*b^2 - a*c + 2*sqrt(b^2 - a*c)*(c*x + b))/(c*x^2 +
2*b*x + a)) + 2*(a*b^4 + a^2*b^2*c - 2*a^3*c^2)*c1 - 2*(4*b^4*b1*c + a*b^2*b1*c^2 - 5*a^2*b1*c^3 - (4*b^5 + a*
b^3*c - 5*a^2*b*c^2)*c1)*x)/(a^2*b^6 - 3*a^3*b^4*c + 3*a^4*b^2*c^2 - a^5*c^3 + (b^6*c^2 - 3*a*b^4*c^3 + 3*a^2*
b^2*c^4 - a^3*c^5)*x^4 + 4*(b^7*c - 3*a*b^5*c^2 + 3*a^2*b^3*c^3 - a^3*b*c^4)*x^3 + 2*(2*b^8 - 5*a*b^6*c + 3*a^
2*b^4*c^2 + a^3*b^2*c^3 - a^4*c^4)*x^2 + 4*(a*b^7 - 3*a^2*b^5*c + 3*a^3*b^3*c^2 - a^4*b*c^3)*x), -1/8*(2*b^5*b
1 - 7*a*b^3*b1*c + 5*a^2*b*b1*c^2 - 3*(b^2*b1*c^3 - a*b1*c^4 - (b^3*c^2 - a*b*c^3)*c1)*x^3 - 9*(b^3*b1*c^2 - a
*b*b1*c^3 - (b^4*c - a*b^2*c^2)*c1)*x^2 + 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)*s
qrt(-b^2 + a*c)*arctan(-sqrt(-b^2 + a*c)*(c*x + b)/(b^2 - a*c)) + (a*b^4 + a^2*b^2*c - 2*a^3*c^2)*c1 - (4*b^4*
b1*c + a*b^2*b1*c^2 - 5*a^2*b1*c^3 - (4*b^5 + a*b^3*c - 5*a^2*b*c^2)*c1)*x)/(a^2*b^6 - 3*a^3*b^4*c + 3*a^4*b^2
*c^2 - a^5*c^3 + (b^6*c^2 - 3*a*b^4*c^3 + 3*a^2*b^2*c^4 - a^3*c^5)*x^4 + 4*(b^7*c - 3*a*b^5*c^2 + 3*a^2*b^3*c^
3 - a^3*b*c^4)*x^3 + 2*(2*b^8 - 5*a*b^6*c + 3*a^2*b^4*c^2 + a^3*b^2*c^3 - a^4*c^4)*x^2 + 4*(a*b^7 - 3*a^2*b^5*
c + 3*a^3*b^3*c^2 - a^4*b*c^3)*x)]

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Sympy [B]  time = 2.03427, size = 622, normalized size = 4.78 \begin{align*} \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 )} \end{align*}

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 [A]  time = 1.07355, size = 262, normalized size = 2.02 \begin{align*} \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}} \end{align*}

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)