3.197 \(\int \frac {\text {b1}+\text {c1} x}{(a+2 b x+c x^2)^4} \, dx\)

Optimal. Leaf size=173 \[ \frac {5 c^2 (\text {b1} c-b \text {c1}) \tanh ^{-1}\left (\frac {b+c x}{\sqrt {b^2-a c}}\right )}{16 \left (b^2-a c\right )^{7/2}}-\frac {5 c (b+c x) (\text {b1} c-b \text {c1})}{16 \left (b^2-a c\right )^3 \left (a+2 b x+c x^2\right )}+\frac {5 (b+c x) (\text {b1} c-b \text {c1})}{24 \left (b^2-a c\right )^2 \left (a+2 b x+c x^2\right )^2}-\frac {-a \text {c1}+x (\text {b1} c-b \text {c1})+b \text {b1}}{6 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^3} \]

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Rubi [A]  time = 0.11, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {638, 614, 618, 206} \[ \frac {5 c^2 (\text {b1} c-b \text {c1}) \tanh ^{-1}\left (\frac {b+c x}{\sqrt {b^2-a c}}\right )}{16 \left (b^2-a c\right )^{7/2}}-\frac {5 c (b+c x) (\text {b1} c-b \text {c1})}{16 \left (b^2-a c\right )^3 \left (a+2 b x+c x^2\right )}+\frac {5 (b+c x) (\text {b1} c-b \text {c1})}{24 \left (b^2-a c\right )^2 \left (a+2 b x+c x^2\right )^2}-\frac {-a \text {c1}+x (\text {b1} c-b \text {c1})+b \text {b1}}{6 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(b1 + c1*x)/(a + 2*b*x + c*x^2)^4,x]

[Out]

-(b*b1 - a*c1 + (b1*c - b*c1)*x)/(6*(b^2 - a*c)*(a + 2*b*x + c*x^2)^3) + (5*(b1*c - b*c1)*(b + c*x))/(24*(b^2
- a*c)^2*(a + 2*b*x + c*x^2)^2) - (5*c*(b1*c - b*c1)*(b + c*x))/(16*(b^2 - a*c)^3*(a + 2*b*x + c*x^2)) + (5*c^
2*(b1*c - b*c1)*ArcTanh[(b + c*x)/Sqrt[b^2 - a*c]])/(16*(b^2 - a*c)^(7/2))

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 )^4} \, dx &=-\frac {b \text {b1}-a \text {c1}+(\text {b1} c-b \text {c1}) x}{6 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^3}-\frac {(5 (\text {b1} c-b \text {c1})) \int \frac {1}{\left (a+2 b x+c x^2\right )^3} \, dx}{6 \left (b^2-a c\right )}\\ &=-\frac {b \text {b1}-a \text {c1}+(\text {b1} c-b \text {c1}) x}{6 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^3}+\frac {5 (\text {b1} c-b \text {c1}) (b+c x)}{24 \left (b^2-a c\right )^2 \left (a+2 b x+c x^2\right )^2}+\frac {(5 c (\text {b1} c-b \text {c1})) \int \frac {1}{\left (a+2 b x+c x^2\right )^2} \, dx}{8 \left (b^2-a c\right )^2}\\ &=-\frac {b \text {b1}-a \text {c1}+(\text {b1} c-b \text {c1}) x}{6 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^3}+\frac {5 (\text {b1} c-b \text {c1}) (b+c x)}{24 \left (b^2-a c\right )^2 \left (a+2 b x+c x^2\right )^2}-\frac {5 c (\text {b1} c-b \text {c1}) (b+c x)}{16 \left (b^2-a c\right )^3 \left (a+2 b x+c x^2\right )}-\frac {\left (5 c^2 (\text {b1} c-b \text {c1})\right ) \int \frac {1}{a+2 b x+c x^2} \, dx}{16 \left (b^2-a c\right )^3}\\ &=-\frac {b \text {b1}-a \text {c1}+(\text {b1} c-b \text {c1}) x}{6 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^3}+\frac {5 (\text {b1} c-b \text {c1}) (b+c x)}{24 \left (b^2-a c\right )^2 \left (a+2 b x+c x^2\right )^2}-\frac {5 c (\text {b1} c-b \text {c1}) (b+c x)}{16 \left (b^2-a c\right )^3 \left (a+2 b x+c x^2\right )}+\frac {\left (5 c^2 (\text {b1} c-b \text {c1})\right ) \operatorname {Subst}\left (\int \frac {1}{4 \left (b^2-a c\right )-x^2} \, dx,x,2 b+2 c x\right )}{8 \left (b^2-a c\right )^3}\\ &=-\frac {b \text {b1}-a \text {c1}+(\text {b1} c-b \text {c1}) x}{6 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^3}+\frac {5 (\text {b1} c-b \text {c1}) (b+c x)}{24 \left (b^2-a c\right )^2 \left (a+2 b x+c x^2\right )^2}-\frac {5 c (\text {b1} c-b \text {c1}) (b+c x)}{16 \left (b^2-a c\right )^3 \left (a+2 b x+c x^2\right )}+\frac {5 c^2 (\text {b1} c-b \text {c1}) \tanh ^{-1}\left (\frac {b+c x}{\sqrt {b^2-a c}}\right )}{16 \left (b^2-a c\right )^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 168, normalized size = 0.97 \[ \frac {\frac {15 c^2 (b \text {c1}-\text {b1} c) \tan ^{-1}\left (\frac {b+c x}{\sqrt {a c-b^2}}\right )}{\sqrt {a c-b^2}}-\frac {10 \left (b^2-a c\right ) (b+c x) (b \text {c1}-\text {b1} c)}{(a+x (2 b+c x))^2}+\frac {8 \left (b^2-a c\right )^2 (a \text {c1}-b \text {b1}+b \text {c1} x-\text {b1} c x)}{(a+x (2 b+c x))^3}+\frac {15 c (b+c x) (b \text {c1}-\text {b1} c)}{a+x (2 b+c x)}}{48 \left (b^2-a c\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(b1 + c1*x)/(a + 2*b*x + c*x^2)^4,x]

[Out]

((8*(b^2 - a*c)^2*(-(b*b1) + a*c1 - b1*c*x + b*c1*x))/(a + x*(2*b + c*x))^3 - (10*(b^2 - a*c)*(-(b1*c) + b*c1)
*(b + c*x))/(a + x*(2*b + c*x))^2 + (15*c*(-(b1*c) + b*c1)*(b + c*x))/(a + x*(2*b + c*x)) + (15*c^2*(-(b1*c) +
b*c1)*ArcTan[(b + c*x)/Sqrt[-b^2 + a*c]])/Sqrt[-b^2 + a*c])/(48*(b^2 - a*c)^3)

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IntegrateAlgebraic [A]  time = 0.29, size = 340, normalized size = 1.97 \[ \frac {8 a^3 c^2 \text {c1}+9 a^2 b^2 c \text {c1}-33 a^2 b \text {b1} c^2+33 a^2 b c^2 \text {c1} x-33 a^2 \text {b1} c^3 x-2 a b^4 \text {c1}+26 a b^3 \text {b1} c+54 a b^3 c \text {c1} x-54 a b^2 \text {b1} c^2 x+120 a b^2 c^2 \text {c1} x^2-120 a b \text {b1} c^3 x^2+40 a b c^3 \text {c1} x^3-40 a \text {b1} c^4 x^3-8 b^5 \text {b1}-12 b^5 \text {c1} x+12 b^4 \text {b1} c x+30 b^4 c \text {c1} x^2-30 b^3 \text {b1} c^2 x^2+110 b^3 c^2 \text {c1} x^3-110 b^2 \text {b1} c^3 x^3+75 b^2 c^3 \text {c1} x^4-75 b \text {b1} c^4 x^4+15 b c^4 \text {c1} x^5-15 \text {b1} c^5 x^5}{48 \left (b^2-a c\right )^3 \left (a+2 b x+c x^2\right )^3}+\frac {5 \left (b c^2 \text {c1}-\text {b1} c^3\right ) \tan ^{-1}\left (\frac {c x}{\sqrt {a c-b^2}}+\frac {b}{\sqrt {a c-b^2}}\right )}{16 \left (b^2-a c\right )^3 \sqrt {a c-b^2}} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b1 + c1*x)/(a + 2*b*x + c*x^2)^4,x]

[Out]

(-8*b^5*b1 + 26*a*b^3*b1*c - 33*a^2*b*b1*c^2 - 2*a*b^4*c1 + 9*a^2*b^2*c*c1 + 8*a^3*c^2*c1 + 12*b^4*b1*c*x - 54
*a*b^2*b1*c^2*x - 33*a^2*b1*c^3*x - 12*b^5*c1*x + 54*a*b^3*c*c1*x + 33*a^2*b*c^2*c1*x - 30*b^3*b1*c^2*x^2 - 12
0*a*b*b1*c^3*x^2 + 30*b^4*c*c1*x^2 + 120*a*b^2*c^2*c1*x^2 - 110*b^2*b1*c^3*x^3 - 40*a*b1*c^4*x^3 + 110*b^3*c^2
*c1*x^3 + 40*a*b*c^3*c1*x^3 - 75*b*b1*c^4*x^4 + 75*b^2*c^3*c1*x^4 - 15*b1*c^5*x^5 + 15*b*c^4*c1*x^5)/(48*(b^2
- a*c)^3*(a + 2*b*x + c*x^2)^3) + (5*(-(b1*c^3) + b*c^2*c1)*ArcTan[b/Sqrt[-b^2 + a*c] + (c*x)/Sqrt[-b^2 + a*c]
])/(16*(b^2 - a*c)^3*Sqrt[-b^2 + a*c])

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fricas [B]  time = 1.34, size = 1950, normalized size = 11.27 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c1*x+b1)/(c*x^2+2*b*x+a)^4,x, algorithm="fricas")

[Out]

[-1/96*(16*b^7*b1 - 68*a*b^5*b1*c + 118*a^2*b^3*b1*c^2 - 66*a^3*b*b1*c^3 + 30*(b^2*b1*c^5 - a*b1*c^6 - (b^3*c^
4 - a*b*c^5)*c1)*x^5 + 150*(b^3*b1*c^4 - a*b*b1*c^5 - (b^4*c^3 - a*b^2*c^4)*c1)*x^4 + 20*(11*b^4*b1*c^3 - 7*a*
b^2*b1*c^4 - 4*a^2*b1*c^5 - (11*b^5*c^2 - 7*a*b^3*c^3 - 4*a^2*b*c^4)*c1)*x^3 + 60*(b^5*b1*c^2 + 3*a*b^3*b1*c^3
- 4*a^2*b*b1*c^4 - (b^6*c + 3*a*b^4*c^2 - 4*a^2*b^2*c^3)*c1)*x^2 - 15*(a^3*b1*c^3 - a^3*b*c^2*c1 + (b1*c^6 -
b*c^5*c1)*x^6 + 6*(b*b1*c^5 - b^2*c^4*c1)*x^5 + 3*(4*b^2*b1*c^4 + a*b1*c^5 - (4*b^3*c^3 + a*b*c^4)*c1)*x^4 + 4
*(2*b^3*b1*c^3 + 3*a*b*b1*c^4 - (2*b^4*c^2 + 3*a*b^2*c^3)*c1)*x^3 + 3*(4*a*b^2*b1*c^3 + a^2*b1*c^4 - (4*a*b^3*
c^2 + a^2*b*c^3)*c1)*x^2 + 6*(a^2*b*b1*c^3 - a^2*b^2*c^2*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*(2*a*b^6 - 11*a^2*b^4*c + a^3*b^2*c^2 + 8*a^4*c
^3)*c1 - 6*(4*b^6*b1*c - 22*a*b^4*b1*c^2 + 7*a^2*b^2*b1*c^3 + 11*a^3*b1*c^4 - (4*b^7 - 22*a*b^5*c + 7*a^2*b^3*
c^2 + 11*a^3*b*c^3)*c1)*x)/(a^3*b^8 - 4*a^4*b^6*c + 6*a^5*b^4*c^2 - 4*a^6*b^2*c^3 + a^7*c^4 + (b^8*c^3 - 4*a*b
^6*c^4 + 6*a^2*b^4*c^5 - 4*a^3*b^2*c^6 + a^4*c^7)*x^6 + 6*(b^9*c^2 - 4*a*b^7*c^3 + 6*a^2*b^5*c^4 - 4*a^3*b^3*c
^5 + a^4*b*c^6)*x^5 + 3*(4*b^10*c - 15*a*b^8*c^2 + 20*a^2*b^6*c^3 - 10*a^3*b^4*c^4 + a^5*c^6)*x^4 + 4*(2*b^11
- 5*a*b^9*c + 10*a^3*b^5*c^3 - 10*a^4*b^3*c^4 + 3*a^5*b*c^5)*x^3 + 3*(4*a*b^10 - 15*a^2*b^8*c + 20*a^3*b^6*c^2
- 10*a^4*b^4*c^3 + a^6*c^5)*x^2 + 6*(a^2*b^9 - 4*a^3*b^7*c + 6*a^4*b^5*c^2 - 4*a^5*b^3*c^3 + a^6*b*c^4)*x), -
1/48*(8*b^7*b1 - 34*a*b^5*b1*c + 59*a^2*b^3*b1*c^2 - 33*a^3*b*b1*c^3 + 15*(b^2*b1*c^5 - a*b1*c^6 - (b^3*c^4 -
a*b*c^5)*c1)*x^5 + 75*(b^3*b1*c^4 - a*b*b1*c^5 - (b^4*c^3 - a*b^2*c^4)*c1)*x^4 + 10*(11*b^4*b1*c^3 - 7*a*b^2*b
1*c^4 - 4*a^2*b1*c^5 - (11*b^5*c^2 - 7*a*b^3*c^3 - 4*a^2*b*c^4)*c1)*x^3 + 30*(b^5*b1*c^2 + 3*a*b^3*b1*c^3 - 4*
a^2*b*b1*c^4 - (b^6*c + 3*a*b^4*c^2 - 4*a^2*b^2*c^3)*c1)*x^2 - 15*(a^3*b1*c^3 - a^3*b*c^2*c1 + (b1*c^6 - b*c^5
*c1)*x^6 + 6*(b*b1*c^5 - b^2*c^4*c1)*x^5 + 3*(4*b^2*b1*c^4 + a*b1*c^5 - (4*b^3*c^3 + a*b*c^4)*c1)*x^4 + 4*(2*b
^3*b1*c^3 + 3*a*b*b1*c^4 - (2*b^4*c^2 + 3*a*b^2*c^3)*c1)*x^3 + 3*(4*a*b^2*b1*c^3 + a^2*b1*c^4 - (4*a*b^3*c^2 +
a^2*b*c^3)*c1)*x^2 + 6*(a^2*b*b1*c^3 - a^2*b^2*c^2*c1)*x)*sqrt(-b^2 + a*c)*arctan(-sqrt(-b^2 + a*c)*(c*x + b)
/(b^2 - a*c)) + (2*a*b^6 - 11*a^2*b^4*c + a^3*b^2*c^2 + 8*a^4*c^3)*c1 - 3*(4*b^6*b1*c - 22*a*b^4*b1*c^2 + 7*a^
2*b^2*b1*c^3 + 11*a^3*b1*c^4 - (4*b^7 - 22*a*b^5*c + 7*a^2*b^3*c^2 + 11*a^3*b*c^3)*c1)*x)/(a^3*b^8 - 4*a^4*b^6
*c + 6*a^5*b^4*c^2 - 4*a^6*b^2*c^3 + a^7*c^4 + (b^8*c^3 - 4*a*b^6*c^4 + 6*a^2*b^4*c^5 - 4*a^3*b^2*c^6 + a^4*c^
7)*x^6 + 6*(b^9*c^2 - 4*a*b^7*c^3 + 6*a^2*b^5*c^4 - 4*a^3*b^3*c^5 + a^4*b*c^6)*x^5 + 3*(4*b^10*c - 15*a*b^8*c^
2 + 20*a^2*b^6*c^3 - 10*a^3*b^4*c^4 + a^5*c^6)*x^4 + 4*(2*b^11 - 5*a*b^9*c + 10*a^3*b^5*c^3 - 10*a^4*b^3*c^4 +
3*a^5*b*c^5)*x^3 + 3*(4*a*b^10 - 15*a^2*b^8*c + 20*a^3*b^6*c^2 - 10*a^4*b^4*c^3 + a^6*c^5)*x^2 + 6*(a^2*b^9 -
4*a^3*b^7*c + 6*a^4*b^5*c^2 - 4*a^5*b^3*c^3 + a^6*b*c^4)*x)]

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giac [B]  time = 0.85, size = 363, normalized size = 2.10 \[ -\frac {5 \, {\left (b_{1} c^{3} - b c^{2} c_{1}\right )} \arctan \left (\frac {c x + b}{\sqrt {-b^{2} + a c}}\right )}{16 \, {\left (b^{6} - 3 \, a b^{4} c + 3 \, a^{2} b^{2} c^{2} - a^{3} c^{3}\right )} \sqrt {-b^{2} + a c}} - \frac {15 \, b_{1} c^{5} x^{5} - 15 \, b c^{4} c_{1} x^{5} + 75 \, b b_{1} c^{4} x^{4} - 75 \, b^{2} c^{3} c_{1} x^{4} + 110 \, b^{2} b_{1} c^{3} x^{3} + 40 \, a b_{1} c^{4} x^{3} - 110 \, b^{3} c^{2} c_{1} x^{3} - 40 \, a b c^{3} c_{1} x^{3} + 30 \, b^{3} b_{1} c^{2} x^{2} + 120 \, a b b_{1} c^{3} x^{2} - 30 \, b^{4} c c_{1} x^{2} - 120 \, a b^{2} c^{2} c_{1} x^{2} - 12 \, b^{4} b_{1} c x + 54 \, a b^{2} b_{1} c^{2} x + 33 \, a^{2} b_{1} c^{3} x + 12 \, b^{5} c_{1} x - 54 \, a b^{3} c c_{1} x - 33 \, a^{2} b c^{2} c_{1} x + 8 \, b^{5} b_{1} - 26 \, a b^{3} b_{1} c + 33 \, a^{2} b b_{1} c^{2} + 2 \, a b^{4} c_{1} - 9 \, a^{2} b^{2} c c_{1} - 8 \, a^{3} c^{2} c_{1}}{48 \, {\left (b^{6} - 3 \, a b^{4} c + 3 \, a^{2} b^{2} c^{2} - a^{3} c^{3}\right )} {\left (c x^{2} + 2 \, b x + a\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c1*x+b1)/(c*x^2+2*b*x+a)^4,x, algorithm="giac")

[Out]

-5/16*(b1*c^3 - b*c^2*c1)*arctan((c*x + b)/sqrt(-b^2 + a*c))/((b^6 - 3*a*b^4*c + 3*a^2*b^2*c^2 - a^3*c^3)*sqrt
(-b^2 + a*c)) - 1/48*(15*b1*c^5*x^5 - 15*b*c^4*c1*x^5 + 75*b*b1*c^4*x^4 - 75*b^2*c^3*c1*x^4 + 110*b^2*b1*c^3*x
^3 + 40*a*b1*c^4*x^3 - 110*b^3*c^2*c1*x^3 - 40*a*b*c^3*c1*x^3 + 30*b^3*b1*c^2*x^2 + 120*a*b*b1*c^3*x^2 - 30*b^
4*c*c1*x^2 - 120*a*b^2*c^2*c1*x^2 - 12*b^4*b1*c*x + 54*a*b^2*b1*c^2*x + 33*a^2*b1*c^3*x + 12*b^5*c1*x - 54*a*b
^3*c*c1*x - 33*a^2*b*c^2*c1*x + 8*b^5*b1 - 26*a*b^3*b1*c + 33*a^2*b*b1*c^2 + 2*a*b^4*c1 - 9*a^2*b^2*c*c1 - 8*a
^3*c^2*c1)/((b^6 - 3*a*b^4*c + 3*a^2*b^2*c^2 - a^3*c^3)*(c*x^2 + 2*b*x + a)^3)

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maple [A]  time = 0.38, size = 206, normalized size = 1.19

method	result	size
default	\(\frac {\left (-2 b \mathit {c1} +2 \mathit {b1} c \right ) x +2 b \mathit {b1} -2 a \mathit {c1}}{3 \left (4 a c -4 b^{2}\right ) \left (c \,x^{2}+2 b x +a \right )^{3}}+\frac {5 \left (-2 b \mathit {c1} +2 \mathit {b1} c \right ) \left (\frac {2 c x +2 b}{2 \left (4 a c -4 b^{2}\right ) \left (c \,x^{2}+2 b x +a \right )^{2}}+\frac {3 c \left (\frac {2 c x +2 b}{\left (4 a c -4 b^{2}\right ) \left (c \,x^{2}+2 b x +a \right )}+\frac {2 c \arctan \left (\frac {2 c x +2 b}{2 \sqrt {a c -b^{2}}}\right )}{\left (4 a c -4 b^{2}\right ) \sqrt {a c -b^{2}}}\right )}{4 a c -4 b^{2}}\right )}{3 \left (4 a c -4 b^{2}\right )}\)	\(206\)
risch	\(\frac {-\frac {5 c^{4} \left (b \mathit {c1} -\mathit {b1} c \right ) x^{5}}{16 \left (c^{3} a^{3}-3 b^{2} c^{2} a^{2}+3 b^{4} c a -b^{6}\right )}-\frac {25 c^{3} \left (b \mathit {c1} -\mathit {b1} c \right ) b \,x^{4}}{16 \left (c^{3} a^{3}-3 b^{2} c^{2} a^{2}+3 b^{4} c a -b^{6}\right )}-\frac {5 \left (4 a c +11 b^{2}\right ) c^{2} \left (b \mathit {c1} -\mathit {b1} c \right ) x^{3}}{24 \left (c^{3} a^{3}-3 b^{2} c^{2} a^{2}+3 b^{4} c a -b^{6}\right )}-\frac {5 b \left (4 a c +b^{2}\right ) c \left (b \mathit {c1} -\mathit {b1} c \right ) x^{2}}{8 \left (c^{3} a^{3}-3 b^{2} c^{2} a^{2}+3 b^{4} c a -b^{6}\right )}-\frac {\left (11 a^{2} b \,c^{2} \mathit {c1} -11 a^{2} \mathit {b1} \,c^{3}+18 a \,b^{3} c \mathit {c1} -18 a \,b^{2} \mathit {b1} \,c^{2}-4 b^{5} \mathit {c1} +4 b^{4} \mathit {b1} c \right ) x}{16 \left (c^{3} a^{3}-3 b^{2} c^{2} a^{2}+3 b^{4} c a -b^{6}\right )}-\frac {8 a^{3} c^{2} \mathit {c1} +9 a^{2} b^{2} c \mathit {c1} -33 a^{2} b \mathit {b1} \,c^{2}-2 a \,b^{4} \mathit {c1} +26 a \,b^{3} \mathit {b1} c -8 b^{5} \mathit {b1}}{48 \left (c^{3} a^{3}-3 b^{2} c^{2} a^{2}+3 b^{4} c a -b^{6}\right )}}{\left (c \,x^{2}+2 b x +a \right )^{3}}+\frac {5 c^{2} \ln \left (\left (-a^{3} c^{4}+3 a^{2} b^{2} c^{3}-3 a \,b^{4} c^{2}+b^{6} c \right ) x -\left (-a c +b^{2}\right )^{\frac {7}{2}}-a^{3} b \,c^{3}+3 a^{2} b^{3} c^{2}-3 a \,b^{5} c +b^{7}\right ) b \mathit {c1}}{32 \left (-a c +b^{2}\right )^{\frac {7}{2}}}-\frac {5 c^{3} \ln \left (\left (-a^{3} c^{4}+3 a^{2} b^{2} c^{3}-3 a \,b^{4} c^{2}+b^{6} c \right ) x -\left (-a c +b^{2}\right )^{\frac {7}{2}}-a^{3} b \,c^{3}+3 a^{2} b^{3} c^{2}-3 a \,b^{5} c +b^{7}\right ) \mathit {b1}}{32 \left (-a c +b^{2}\right )^{\frac {7}{2}}}-\frac {5 c^{2} \ln \left (\left (a^{3} c^{4}-3 a^{2} b^{2} c^{3}+3 a \,b^{4} c^{2}-b^{6} c \right ) x -\left (-a c +b^{2}\right )^{\frac {7}{2}}+a^{3} b \,c^{3}-3 a^{2} b^{3} c^{2}+3 a \,b^{5} c -b^{7}\right ) b \mathit {c1}}{32 \left (-a c +b^{2}\right )^{\frac {7}{2}}}+\frac {5 c^{3} \ln \left (\left (a^{3} c^{4}-3 a^{2} b^{2} c^{3}+3 a \,b^{4} c^{2}-b^{6} c \right ) x -\left (-a c +b^{2}\right )^{\frac {7}{2}}+a^{3} b \,c^{3}-3 a^{2} b^{3} c^{2}+3 a \,b^{5} c -b^{7}\right ) \mathit {b1}}{32 \left (-a c +b^{2}\right )^{\frac {7}{2}}}\)	\(792\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c1*x+b1)/(c*x^2+2*b*x+a)^4,x,method=_RETURNVERBOSE)

[Out]

1/3*((-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)^3+5/3*(-2*b*c1+2*b1*c)/(4*a*c-4*b^2)*(1/2
*(2*c*x+2*b)/(4*a*c-4*b^2)/(c*x^2+2*b*x+a)^2+3*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)^4,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 = 0.71, size = 640, normalized size = 3.70 \[ \frac {\frac {5\,c^4\,x^5\,\left (b\,c_{1}-b_{1}\,c\right )}{16\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}-\frac {-8\,c_{1}\,a^3\,c^2-9\,c_{1}\,a^2\,b^2\,c+33\,b_{1}\,a^2\,b\,c^2+2\,c_{1}\,a\,b^4-26\,b_{1}\,a\,b^3\,c+8\,b_{1}\,b^5}{48\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}+\frac {x\,\left (b\,c_{1}-b_{1}\,c\right )\,\left (11\,a^2\,c^2+18\,a\,b^2\,c-4\,b^4\right )}{16\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}+\frac {5\,c\,x^3\,\left (11\,b^2\,c+4\,a\,c^2\right )\,\left (b\,c_{1}-b_{1}\,c\right )}{24\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}+\frac {5\,c\,x^2\,\left (b^3+4\,a\,c\,b\right )\,\left (b\,c_{1}-b_{1}\,c\right )}{8\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}+\frac {25\,b\,c^3\,x^4\,\left (b\,c_{1}-b_{1}\,c\right )}{16\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}}{x^3\,\left (8\,b^3+12\,a\,c\,b\right )+x^2\,\left (3\,c\,a^2+12\,a\,b^2\right )+x^4\,\left (12\,b^2\,c+3\,a\,c^2\right )+a^3+c^3\,x^6+6\,b\,c^2\,x^5+6\,a^2\,b\,x}-\frac {5\,c^2\,\mathrm {atan}\left (\frac {16\,\left (\frac {5\,c^3\,x\,\left (b\,c_{1}-b_{1}\,c\right )}{16\,{\left (a\,c-b^2\right )}^{7/2}}+\frac {5\,c^2\,\left (b\,c_{1}-b_{1}\,c\right )\,\left (-32\,a^3\,b\,c^3+96\,a^2\,b^3\,c^2-96\,a\,b^5\,c+32\,b^7\right )}{512\,{\left (a\,c-b^2\right )}^{7/2}\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}\right )\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}{5\,b_{1}\,c^3-5\,b\,c^2\,c_{1}}\right )\,\left (b\,c_{1}-b_{1}\,c\right )}{16\,{\left (a\,c-b^2\right )}^{7/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b1 + c1*x)/(a + 2*b*x + c*x^2)^4,x)

[Out]

((5*c^4*x^5*(b*c1 - b1*c))/(16*(b^6 - a^3*c^3 + 3*a^2*b^2*c^2 - 3*a*b^4*c)) - (8*b^5*b1 - 8*a^3*c^2*c1 + 2*a*b
^4*c1 - 26*a*b^3*b1*c + 33*a^2*b*b1*c^2 - 9*a^2*b^2*c*c1)/(48*(b^6 - a^3*c^3 + 3*a^2*b^2*c^2 - 3*a*b^4*c)) + (
x*(b*c1 - b1*c)*(11*a^2*c^2 - 4*b^4 + 18*a*b^2*c))/(16*(b^6 - a^3*c^3 + 3*a^2*b^2*c^2 - 3*a*b^4*c)) + (5*c*x^3
*(4*a*c^2 + 11*b^2*c)*(b*c1 - b1*c))/(24*(b^6 - a^3*c^3 + 3*a^2*b^2*c^2 - 3*a*b^4*c)) + (5*c*x^2*(b^3 + 4*a*b*
c)*(b*c1 - b1*c))/(8*(b^6 - a^3*c^3 + 3*a^2*b^2*c^2 - 3*a*b^4*c)) + (25*b*c^3*x^4*(b*c1 - b1*c))/(16*(b^6 - a^
3*c^3 + 3*a^2*b^2*c^2 - 3*a*b^4*c)))/(x^3*(8*b^3 + 12*a*b*c) + x^2*(12*a*b^2 + 3*a^2*c) + x^4*(3*a*c^2 + 12*b^
2*c) + a^3 + c^3*x^6 + 6*b*c^2*x^5 + 6*a^2*b*x) - (5*c^2*atan((16*((5*c^3*x*(b*c1 - b1*c))/(16*(a*c - b^2)^(7/
2)) + (5*c^2*(b*c1 - b1*c)*(32*b^7 - 32*a^3*b*c^3 + 96*a^2*b^3*c^2 - 96*a*b^5*c))/(512*(a*c - b^2)^(7/2)*(b^6
- a^3*c^3 + 3*a^2*b^2*c^2 - 3*a*b^4*c)))*(b^6 - a^3*c^3 + 3*a^2*b^2*c^2 - 3*a*b^4*c))/(5*b1*c^3 - 5*b*c^2*c1))
*(b*c1 - b1*c))/(16*(a*c - b^2)^(7/2))

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sympy [B]  time = 3.06, size = 1027, normalized size = 5.94 \[ \frac {5 c^{2} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) \log {\left (x + \frac {- 5 a^{4} c^{6} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) + 20 a^{3} b^{2} c^{5} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) - 30 a^{2} b^{4} c^{4} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) + 20 a b^{6} c^{3} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) - 5 b^{8} c^{2} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) + 5 b^{2} c^{2} c_{1} - 5 b b_{1} c^{3}}{5 b c^{3} c_{1} - 5 b_{1} c^{4}} \right )}}{32} - \frac {5 c^{2} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) \log {\left (x + \frac {5 a^{4} c^{6} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) - 20 a^{3} b^{2} c^{5} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) + 30 a^{2} b^{4} c^{4} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) - 20 a b^{6} c^{3} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) + 5 b^{8} c^{2} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) + 5 b^{2} c^{2} c_{1} - 5 b b_{1} c^{3}}{5 b c^{3} c_{1} - 5 b_{1} c^{4}} \right )}}{32} + \frac {- 8 a^{3} c^{2} c_{1} - 9 a^{2} b^{2} c c_{1} + 33 a^{2} b b_{1} c^{2} + 2 a b^{4} c_{1} - 26 a b^{3} b_{1} c + 8 b^{5} b_{1} + x^{5} \left (- 15 b c^{4} c_{1} + 15 b_{1} c^{5}\right ) + x^{4} \left (- 75 b^{2} c^{3} c_{1} + 75 b b_{1} c^{4}\right ) + x^{3} \left (- 40 a b c^{3} c_{1} + 40 a b_{1} c^{4} - 110 b^{3} c^{2} c_{1} + 110 b^{2} b_{1} c^{3}\right ) + x^{2} \left (- 120 a b^{2} c^{2} c_{1} + 120 a b b_{1} c^{3} - 30 b^{4} c c_{1} + 30 b^{3} b_{1} c^{2}\right ) + x \left (- 33 a^{2} b c^{2} c_{1} + 33 a^{2} b_{1} c^{3} - 54 a b^{3} c c_{1} + 54 a b^{2} b_{1} c^{2} + 12 b^{5} c_{1} - 12 b^{4} b_{1} c\right )}{48 a^{6} c^{3} - 144 a^{5} b^{2} c^{2} + 144 a^{4} b^{4} c - 48 a^{3} b^{6} + x^{6} \left (48 a^{3} c^{6} - 144 a^{2} b^{2} c^{5} + 144 a b^{4} c^{4} - 48 b^{6} c^{3}\right ) + x^{5} \left (288 a^{3} b c^{5} - 864 a^{2} b^{3} c^{4} + 864 a b^{5} c^{3} - 288 b^{7} c^{2}\right ) + x^{4} \left (144 a^{4} c^{5} + 144 a^{3} b^{2} c^{4} - 1296 a^{2} b^{4} c^{3} + 1584 a b^{6} c^{2} - 576 b^{8} c\right ) + x^{3} \left (576 a^{4} b c^{4} - 1344 a^{3} b^{3} c^{3} + 576 a^{2} b^{5} c^{2} + 576 a b^{7} c - 384 b^{9}\right ) + x^{2} \left (144 a^{5} c^{4} + 144 a^{4} b^{2} c^{3} - 1296 a^{3} b^{4} c^{2} + 1584 a^{2} b^{6} c - 576 a b^{8}\right ) + x \left (288 a^{5} b c^{3} - 864 a^{4} b^{3} c^{2} + 864 a^{3} b^{5} c - 288 a^{2} b^{7}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c1*x+b1)/(c*x**2+2*b*x+a)**4,x)

[Out]

5*c**2*sqrt(-1/(a*c - b**2)**7)*(b*c1 - b1*c)*log(x + (-5*a**4*c**6*sqrt(-1/(a*c - b**2)**7)*(b*c1 - b1*c) + 2
0*a**3*b**2*c**5*sqrt(-1/(a*c - b**2)**7)*(b*c1 - b1*c) - 30*a**2*b**4*c**4*sqrt(-1/(a*c - b**2)**7)*(b*c1 - b
1*c) + 20*a*b**6*c**3*sqrt(-1/(a*c - b**2)**7)*(b*c1 - b1*c) - 5*b**8*c**2*sqrt(-1/(a*c - b**2)**7)*(b*c1 - b1
*c) + 5*b**2*c**2*c1 - 5*b*b1*c**3)/(5*b*c**3*c1 - 5*b1*c**4))/32 - 5*c**2*sqrt(-1/(a*c - b**2)**7)*(b*c1 - b1
*c)*log(x + (5*a**4*c**6*sqrt(-1/(a*c - b**2)**7)*(b*c1 - b1*c) - 20*a**3*b**2*c**5*sqrt(-1/(a*c - b**2)**7)*(
b*c1 - b1*c) + 30*a**2*b**4*c**4*sqrt(-1/(a*c - b**2)**7)*(b*c1 - b1*c) - 20*a*b**6*c**3*sqrt(-1/(a*c - b**2)*
*7)*(b*c1 - b1*c) + 5*b**8*c**2*sqrt(-1/(a*c - b**2)**7)*(b*c1 - b1*c) + 5*b**2*c**2*c1 - 5*b*b1*c**3)/(5*b*c*
*3*c1 - 5*b1*c**4))/32 + (-8*a**3*c**2*c1 - 9*a**2*b**2*c*c1 + 33*a**2*b*b1*c**2 + 2*a*b**4*c1 - 26*a*b**3*b1*
c + 8*b**5*b1 + x**5*(-15*b*c**4*c1 + 15*b1*c**5) + x**4*(-75*b**2*c**3*c1 + 75*b*b1*c**4) + x**3*(-40*a*b*c**
3*c1 + 40*a*b1*c**4 - 110*b**3*c**2*c1 + 110*b**2*b1*c**3) + x**2*(-120*a*b**2*c**2*c1 + 120*a*b*b1*c**3 - 30*
b**4*c*c1 + 30*b**3*b1*c**2) + x*(-33*a**2*b*c**2*c1 + 33*a**2*b1*c**3 - 54*a*b**3*c*c1 + 54*a*b**2*b1*c**2 +
12*b**5*c1 - 12*b**4*b1*c))/(48*a**6*c**3 - 144*a**5*b**2*c**2 + 144*a**4*b**4*c - 48*a**3*b**6 + x**6*(48*a**
3*c**6 - 144*a**2*b**2*c**5 + 144*a*b**4*c**4 - 48*b**6*c**3) + x**5*(288*a**3*b*c**5 - 864*a**2*b**3*c**4 + 8
64*a*b**5*c**3 - 288*b**7*c**2) + x**4*(144*a**4*c**5 + 144*a**3*b**2*c**4 - 1296*a**2*b**4*c**3 + 1584*a*b**6
*c**2 - 576*b**8*c) + x**3*(576*a**4*b*c**4 - 1344*a**3*b**3*c**3 + 576*a**2*b**5*c**2 + 576*a*b**7*c - 384*b*
*9) + x**2*(144*a**5*c**4 + 144*a**4*b**2*c**3 - 1296*a**3*b**4*c**2 + 1584*a**2*b**6*c - 576*a*b**8) + x*(288
*a**5*b*c**3 - 864*a**4*b**3*c**2 + 864*a**3*b**5*c - 288*a**2*b**7))

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