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

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

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Rubi [A]  time = 0.07, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, 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 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])

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 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]

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*}

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Mathematica [A]  time = 0.14, 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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IntegrateAlgebraic [A]  time = 0.18, size = 195, normalized size = 1.50 \[ \frac {-2 a^2 c \text {c1}-a b^2 \text {c1}+5 a b \text {b1} c-5 a b c \text {c1} x+5 a \text {b1} c^2 x-2 b^3 \text {b1}-4 b^3 \text {c1} x+4 b^2 \text {b1} c x-9 b^2 c \text {c1} x^2+9 b \text {b1} c^2 x^2-3 b c^2 \text {c1} x^3+3 \text {b1} c^3 x^3}{8 \left (b^2-a c\right )^2 \left (a+2 b x+c x^2\right )^2}-\frac {3 \left (b c \text {c1}-\text {b1} c^2\right ) \tan ^{-1}\left (\frac {c x}{\sqrt {a c-b^2}}+\frac {b}{\sqrt {a c-b^2}}\right )}{8 \left (b^2-a c\right )^2 \sqrt {a c-b^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b^3*b1 + 5*a*b*b1*c - a*b^2*c1 - 2*a^2*c*c1 + 4*b^2*b1*c*x + 5*a*b1*c^2*x - 4*b^3*c1*x - 5*a*b*c*c1*x + 9*
b*b1*c^2*x^2 - 9*b^2*c*c1*x^2 + 3*b1*c^3*x^3 - 3*b*c^2*c1*x^3)/(8*(b^2 - a*c)^2*(a + 2*b*x + c*x^2)^2) - (3*(-
(b1*c^2) + b*c*c1)*ArcTan[b/Sqrt[-b^2 + a*c] + (c*x)/Sqrt[-b^2 + a*c]])/(8*(b^2 - a*c)^2*Sqrt[-b^2 + a*c])

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fricas [B]  time = 1.27, size = 1104, normalized size = 8.49 \[ \left [-\frac {4 \, b^{5} b_{1} - 14 \, a b^{3} b_{1} c + 10 \, a^{2} b b_{1} c^{2} - 6 \, {\left (b^{2} b_{1} c^{3} - a b_{1} c^{4} - {\left (b^{3} c^{2} - a b c^{3}\right )} c_{1}\right )} x^{3} - 18 \, {\left (b^{3} b_{1} c^{2} - a b b_{1} c^{3} - {\left (b^{4} c - a b^{2} c^{2}\right )} c_{1}\right )} x^{2} + 3 \, {\left (a^{2} b_{1} c^{2} - a^{2} b c c_{1} + {\left (b_{1} c^{4} - b c^{3} c_{1}\right )} x^{4} + 4 \, {\left (b b_{1} c^{3} - b^{2} c^{2} c_{1}\right )} x^{3} + 2 \, {\left (2 \, b^{2} b_{1} c^{2} + a b_{1} c^{3} - {\left (2 \, b^{3} c + a b c^{2}\right )} c_{1}\right )} x^{2} + 4 \, {\left (a b b_{1} c^{2} - a b^{2} c c_{1}\right )} x\right )} \sqrt {b^{2} - a c} \log \left (\frac {c^{2} x^{2} + 2 \, b c x + 2 \, b^{2} - a c + 2 \, \sqrt {b^{2} - a c} {\left (c x + b\right )}}{c x^{2} + 2 \, b x + a}\right ) + 2 \, {\left (a b^{4} + a^{2} b^{2} c - 2 \, a^{3} c^{2}\right )} c_{1} - 2 \, {\left (4 \, b^{4} b_{1} c + a b^{2} b_{1} c^{2} - 5 \, a^{2} b_{1} c^{3} - {\left (4 \, b^{5} + a b^{3} c - 5 \, a^{2} b c^{2}\right )} c_{1}\right )} x}{16 \, {\left (a^{2} b^{6} - 3 \, a^{3} b^{4} c + 3 \, a^{4} b^{2} c^{2} - a^{5} c^{3} + {\left (b^{6} c^{2} - 3 \, a b^{4} c^{3} + 3 \, a^{2} b^{2} c^{4} - a^{3} c^{5}\right )} x^{4} + 4 \, {\left (b^{7} c - 3 \, a b^{5} c^{2} + 3 \, a^{2} b^{3} c^{3} - a^{3} b c^{4}\right )} x^{3} + 2 \, {\left (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}\right )} x^{2} + 4 \, {\left (a b^{7} - 3 \, a^{2} b^{5} c + 3 \, a^{3} b^{3} c^{2} - a^{4} b c^{3}\right )} x\right )}}, -\frac {2 \, b^{5} b_{1} - 7 \, a b^{3} b_{1} c + 5 \, a^{2} b b_{1} c^{2} - 3 \, {\left (b^{2} b_{1} c^{3} - a b_{1} c^{4} - {\left (b^{3} c^{2} - a b c^{3}\right )} c_{1}\right )} x^{3} - 9 \, {\left (b^{3} b_{1} c^{2} - a b b_{1} c^{3} - {\left (b^{4} c - a b^{2} c^{2}\right )} c_{1}\right )} x^{2} + 3 \, {\left (a^{2} b_{1} c^{2} - a^{2} b c c_{1} + {\left (b_{1} c^{4} - b c^{3} c_{1}\right )} x^{4} + 4 \, {\left (b b_{1} c^{3} - b^{2} c^{2} c_{1}\right )} x^{3} + 2 \, {\left (2 \, b^{2} b_{1} c^{2} + a b_{1} c^{3} - {\left (2 \, b^{3} c + a b c^{2}\right )} c_{1}\right )} x^{2} + 4 \, {\left (a b b_{1} c^{2} - a b^{2} c c_{1}\right )} x\right )} \sqrt {-b^{2} + a c} \arctan \left (-\frac {\sqrt {-b^{2} + a c} {\left (c x + b\right )}}{b^{2} - a c}\right ) + {\left (a b^{4} + a^{2} b^{2} c - 2 \, a^{3} c^{2}\right )} c_{1} - {\left (4 \, b^{4} b_{1} c + a b^{2} b_{1} c^{2} - 5 \, a^{2} b_{1} c^{3} - {\left (4 \, b^{5} + a b^{3} c - 5 \, a^{2} b c^{2}\right )} c_{1}\right )} x}{8 \, {\left (a^{2} b^{6} - 3 \, a^{3} b^{4} c + 3 \, a^{4} b^{2} c^{2} - a^{5} c^{3} + {\left (b^{6} c^{2} - 3 \, a b^{4} c^{3} + 3 \, a^{2} b^{2} c^{4} - a^{3} c^{5}\right )} x^{4} + 4 \, {\left (b^{7} c - 3 \, a b^{5} c^{2} + 3 \, a^{2} b^{3} c^{3} - a^{3} b c^{4}\right )} x^{3} + 2 \, {\left (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}\right )} x^{2} + 4 \, {\left (a b^{7} - 3 \, a^{2} b^{5} c + 3 \, a^{3} b^{3} c^{2} - a^{4} b c^{3}\right )} x\right )}}\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, 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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giac [A]  time = 1.03, size = 194, normalized size = 1.49 \[ \frac {3 \, {\left (b_{1} c^{2} - b c c_{1}\right )} \arctan \left (\frac {c x + b}{\sqrt {-b^{2} + a c}}\right )}{8 \, {\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} \sqrt {-b^{2} + a c}} + \frac {3 \, b_{1} c^{3} x^{3} - 3 \, b c^{2} c_{1} x^{3} + 9 \, b b_{1} c^{2} x^{2} - 9 \, b^{2} c c_{1} x^{2} + 4 \, b^{2} b_{1} c x + 5 \, a b_{1} c^{2} x - 4 \, b^{3} c_{1} x - 5 \, a b c c_{1} x - 2 \, b^{3} b_{1} + 5 \, a b b_{1} c - a b^{2} c_{1} - 2 \, a^{2} c c_{1}}{8 \, {\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} {\left (c x^{2} + 2 \, b x + a\right )}^{2}} \]

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)

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maple [A]  time = 0.35, size = 155, normalized size = 1.19

method	result	size
default	\(\frac {\left (-2 b \mathit {c1} +2 \mathit {b1} c \right ) x +2 b \mathit {b1} -2 a \mathit {c1}}{2 \left (4 a c -4 b^{2}\right ) \left (c \,x^{2}+2 b x +a \right )^{2}}+\frac {3 \left (-2 b \mathit {c1} +2 \mathit {b1} c \right ) \left (\frac {2 c x +2 b}{\left (4 a c -4 b^{2}\right ) \left (c \,x^{2}+2 b x +a \right )}+\frac {2 c \arctan \left (\frac {2 c x +2 b}{2 \sqrt {a c -b^{2}}}\right )}{\left (4 a c -4 b^{2}\right ) \sqrt {a c -b^{2}}}\right )}{2 \left (4 a c -4 b^{2}\right )}\)	\(155\)
risch	\(\frac {-\frac {3 c^{2} \left (b \mathit {c1} -\mathit {b1} c \right ) x^{3}}{8 \left (a^{2} c^{2}-2 a \,b^{2} c +b^{4}\right )}-\frac {9 b c \left (b \mathit {c1} -\mathit {b1} c \right ) x^{2}}{8 \left (a^{2} c^{2}-2 a \,b^{2} c +b^{4}\right )}-\frac {\left (5 a c +4 b^{2}\right ) \left (b \mathit {c1} -\mathit {b1} c \right ) x}{8 \left (a^{2} c^{2}-2 a \,b^{2} c +b^{4}\right )}-\frac {2 a^{2} c \mathit {c1} +a \,b^{2} \mathit {c1} -5 a b \mathit {b1} c +2 b^{3} \mathit {b1}}{8 \left (a^{2} c^{2}-2 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{2}+2 b x +a \right )^{2}}-\frac {3 c \ln \left (\left (a^{2} c^{3}-2 a \,b^{2} c^{2}+b^{4} c \right ) x -\left (-a c +b^{2}\right )^{\frac {5}{2}}+a^{2} b \,c^{2}-2 a \,b^{3} c +b^{5}\right ) b \mathit {c1}}{16 \left (-a c +b^{2}\right )^{\frac {5}{2}}}+\frac {3 c^{2} \ln \left (\left (a^{2} c^{3}-2 a \,b^{2} c^{2}+b^{4} c \right ) x -\left (-a c +b^{2}\right )^{\frac {5}{2}}+a^{2} b \,c^{2}-2 a \,b^{3} c +b^{5}\right ) \mathit {b1}}{16 \left (-a c +b^{2}\right )^{\frac {5}{2}}}+\frac {3 c \ln \left (\left (-a^{2} c^{3}+2 a \,b^{2} c^{2}-b^{4} c \right ) x -\left (-a c +b^{2}\right )^{\frac {5}{2}}-a^{2} b \,c^{2}+2 a \,b^{3} c -b^{5}\right ) b \mathit {c1}}{16 \left (-a c +b^{2}\right )^{\frac {5}{2}}}-\frac {3 c^{2} \ln \left (\left (-a^{2} c^{3}+2 a \,b^{2} c^{2}-b^{4} c \right ) x -\left (-a c +b^{2}\right )^{\frac {5}{2}}-a^{2} b \,c^{2}+2 a \,b^{3} c -b^{5}\right ) \mathit {b1}}{16 \left (-a c +b^{2}\right )^{\frac {5}{2}}}\)	\(473\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c1*x+b1)/(c*x^2+2*b*x+a)^3,x,method=_RETURNVERBOSE)

[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+3/2*(-2*b*c1+2*b1*c)/(4*a*c-4*b^2)*((2*
c*x+2*b)/(4*a*c-4*b^2)/(c*x^2+2*b*x+a)+2*c/(4*a*c-4*b^2)/(a*c-b^2)^(1/2)*arctan(1/2*(2*c*x+2*b)/(a*c-b^2)^(1/2
)))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a*c>0)', see `assume?`
for more details)Is 4*b^2-4*a*c positive or negative?

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mupad [B]  time = 0.50, size = 360, normalized size = 2.77 \[ \frac {3\,c\,\mathrm {atan}\left (\frac {8\,\left (\frac {3\,c^2\,x\,\left (b\,c_{1}-b_{1}\,c\right )}{8\,{\left (a\,c-b^2\right )}^{5/2}}+\frac {3\,c\,\left (b\,c_{1}-b_{1}\,c\right )\,\left (16\,a^2\,b\,c^2-32\,a\,b^3\,c+16\,b^5\right )}{128\,{\left (a\,c-b^2\right )}^{5/2}\,\left (a^2\,c^2-2\,a\,b^2\,c+b^4\right )}\right )\,\left (a^2\,c^2-2\,a\,b^2\,c+b^4\right )}{3\,b_{1}\,c^2-3\,b\,c\,c_{1}}\right )\,\left (b\,c_{1}-b_{1}\,c\right )}{8\,{\left (a\,c-b^2\right )}^{5/2}}-\frac {\frac {2\,c\,c_{1}\,a^2+c_{1}\,a\,b^2-5\,b_{1}\,c\,a\,b+2\,b_{1}\,b^3}{8\,\left (a^2\,c^2-2\,a\,b^2\,c+b^4\right )}+\frac {x\,\left (4\,b^2+5\,a\,c\right )\,\left (b\,c_{1}-b_{1}\,c\right )}{8\,\left (a^2\,c^2-2\,a\,b^2\,c+b^4\right )}+\frac {3\,c^2\,x^3\,\left (b\,c_{1}-b_{1}\,c\right )}{8\,\left (a^2\,c^2-2\,a\,b^2\,c+b^4\right )}+\frac {9\,b\,c\,x^2\,\left (b\,c_{1}-b_{1}\,c\right )}{8\,\left (a^2\,c^2-2\,a\,b^2\,c+b^4\right )}}{a^2+x^2\,\left (4\,b^2+2\,a\,c\right )+c^2\,x^4+4\,a\,b\,x+4\,b\,c\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*c*atan((8*((3*c^2*x*(b*c1 - b1*c))/(8*(a*c - b^2)^(5/2)) + (3*c*(b*c1 - b1*c)*(16*b^5 + 16*a^2*b*c^2 - 32*a
*b^3*c))/(128*(a*c - b^2)^(5/2)*(b^4 + a^2*c^2 - 2*a*b^2*c)))*(b^4 + a^2*c^2 - 2*a*b^2*c))/(3*b1*c^2 - 3*b*c*c
1))*(b*c1 - b1*c))/(8*(a*c - b^2)^(5/2)) - ((2*b^3*b1 + a*b^2*c1 + 2*a^2*c*c1 - 5*a*b*b1*c)/(8*(b^4 + a^2*c^2
- 2*a*b^2*c)) + (x*(5*a*c + 4*b^2)*(b*c1 - b1*c))/(8*(b^4 + a^2*c^2 - 2*a*b^2*c)) + (3*c^2*x^3*(b*c1 - b1*c))/
(8*(b^4 + a^2*c^2 - 2*a*b^2*c)) + (9*b*c*x^2*(b*c1 - b1*c))/(8*(b^4 + a^2*c^2 - 2*a*b^2*c)))/(a^2 + x^2*(2*a*c
+ 4*b^2) + c^2*x^4 + 4*a*b*x + 4*b*c*x^3)

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sympy [B]  time = 1.89, 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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