3.2.97 \(\int \frac {\text {b1}+\text {c1} x}{(a+2 b x+c x^2)^4} \, dx\) [197]

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

[Out]

1/6*(-b*b1+a*c1-(-b*c1+b1*c)*x)/(-a*c+b^2)/(c*x^2+2*b*x+a)^3+5/24*(-b*c1+b1*c)*(c*x+b)/(-a*c+b^2)^2/(c*x^2+2*b
*x+a)^2-5/16*c*(-b*c1+b1*c)*(c*x+b)/(-a*c+b^2)^3/(c*x^2+2*b*x+a)+5/16*c^2*(-b*c1+b1*c)*arctanh((c*x+b)/(-a*c+b
^2)^(1/2))/(-a*c+b^2)^(7/2)

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Rubi [A]
time = 0.07, 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 = {652, 628, 632, 212} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(b*b1 - a*c1 + (b1*c - b*c1)*x)/((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 212

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

Rule 628

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 632

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 652

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)/((p + 1)*(b^2 - 4*a*c)))*(a + b*x + c*x^2)^(p + 1), 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 ) \text {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 = 0.14, size = 168, normalized size = 0.97 \begin {gather*} \frac {\frac {8 \left (b^2-a c\right )^2 (-b \text {b1}+a \text {c1}-\text {b1} c x+b \text {c1} x)}{(a+x (2 b+c x))^3}-\frac {10 \left (b^2-a c\right ) (-\text {b1} c+b \text {c1}) (b+c x)}{(a+x (2 b+c x))^2}+\frac {15 c (-\text {b1} c+b \text {c1}) (b+c x)}{a+x (2 b+c x)}+\frac {15 c^2 (-\text {b1} c+b \text {c1}) \tan ^{-1}\left (\frac {b+c x}{\sqrt {-b^2+a c}}\right )}{\sqrt {-b^2+a c}}}{48 \left (b^2-a c\right )^3} \end {gather*}

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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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(1239\) vs. \(2(173)=346\).
time = 33.00, size = 1215, normalized size = 7.02 \begin {gather*} \frac {-16 a^3 c^2 \text {c1}-18 a^2 b^2 c \text {c1}+66 a^2 b \text {b1} c^2+4 a b^4 \text {c1}-52 a b^3 \text {b1} c+16 b^5 \text {b1}-6 x \left (11 a^2 b c^2 \text {c1}-11 a^2 \text {b1} c^3+18 a b^3 c \text {c1}-18 a b^2 \text {b1} c^2-4 b^5 \text {c1}+4 b^4 \text {b1} c\right )-60 b c x^2 \left (4 a b c \text {c1}-4 a \text {b1} c^2+b^3 \text {c1}-b^2 \text {b1} c\right )+15 c^2 \left (a^6 c^3-3 a^5 b^2 c^2+3 a^4 b^4 c-a^3 b^6+6 a^2 b x \left (a^3 c^3-3 a^2 b^2 c^2+3 a b^4 c-b^6\right )+3 a x^2 \left (a^4 c^4+a^3 b^2 c^3-9 a^2 b^4 c^2+11 a b^6 c-4 b^8\right )+4 b x^3 \left (3 a^4 c^4-7 a^3 b^2 c^3+3 a^2 b^4 c^2+3 a b^6 c-2 b^8\right )+3 c x^4 \left (a^4 c^4+a^3 b^2 c^3-9 a^2 b^4 c^2+11 a b^6 c-4 b^8\right )+6 b c^2 x^5 \left (a^3 c^3-3 a^2 b^2 c^2+3 a b^4 c-b^6\right )+c^3 x^6 \left (a^3 c^3-3 a^2 b^2 c^2+3 a b^4 c-b^6\right )\right ) \left (b \text {c1}-\text {b1} c\right ) \left (\text {Log}\left [\frac {a^4 c^4 \left (-b \text {c1}+\text {b1} c\right ) \sqrt {-\frac {1}{{\left (a c-b^2\right )}^7}}+4 a^3 b^2 c^3 \left (b \text {c1}-\text {b1} c\right ) \sqrt {-\frac {1}{{\left (a c-b^2\right )}^7}}+6 a^2 b^4 c^2 \left (-b \text {c1}+\text {b1} c\right ) \sqrt {-\frac {1}{{\left (a c-b^2\right )}^7}}+4 a b^6 c \left (b \text {c1}-\text {b1} c\right ) \sqrt {-\frac {1}{{\left (a c-b^2\right )}^7}}+b^8 \left (-b \text {c1}+\text {b1} c\right ) \sqrt {-\frac {1}{{\left (a c-b^2\right )}^7}}+b^2 \text {c1}-b \text {b1} c+c x \left (b \text {c1}-\text {b1} c\right )}{c \left (b \text {c1}-\text {b1} c\right )}\right ]-\text {Log}\left [\frac {a^4 c^4 \left (b \text {c1}-\text {b1} c\right ) \sqrt {-\frac {1}{{\left (a c-b^2\right )}^7}}+4 a^3 b^2 c^3 \left (-b \text {c1}+\text {b1} c\right ) \sqrt {-\frac {1}{{\left (a c-b^2\right )}^7}}+6 a^2 b^4 c^2 \left (b \text {c1}-\text {b1} c\right ) \sqrt {-\frac {1}{{\left (a c-b^2\right )}^7}}-4 a b^6 c \left (b \text {c1}-\text {b1} c\right ) \sqrt {-\frac {1}{{\left (a c-b^2\right )}^7}}+b^8 \left (b \text {c1}-\text {b1} c\right ) \sqrt {-\frac {1}{{\left (a c-b^2\right )}^7}}+b^2 \text {c1}-b \text {b1} c+c x \left (b \text {c1}-\text {b1} c\right )}{c \left (b \text {c1}-\text {b1} c\right )}\right ]\right ) \sqrt {-\frac {1}{{\left (a c-b^2\right )}^7}}+20 c^2 x^3 \left (-4 a b c \text {c1}+4 a \text {b1} c^2-11 b^3 \text {c1}+11 b^2 \text {b1} c\right )-150 b c^3 x^4 \left (b \text {c1}-\text {b1} c\right )+30 c^4 x^5 \left (-b \text {c1}+\text {b1} c\right )}{96 a^6 c^3-288 a^5 b^2 c^2+288 a^4 b^4 c-96 a^3 b^6+576 a^2 b x \left (a^3 c^3-3 a^2 b^2 c^2+3 a b^4 c-b^6\right )+288 a x^2 \left (a^4 c^4+a^3 b^2 c^3-9 a^2 b^4 c^2+11 a b^6 c-4 b^8\right )+384 b x^3 \left (3 a^4 c^4-7 a^3 b^2 c^3+3 a^2 b^4 c^2+3 a b^6 c-2 b^8\right )+288 c x^4 \left (a^4 c^4+a^3 b^2 c^3-9 a^2 b^4 c^2+11 a b^6 c-4 b^8\right )+576 b c^2 x^5 \left (a^3 c^3-3 a^2 b^2 c^2+3 a b^4 c-b^6\right )+96 c^3 x^6 \left (a^3 c^3-3 a^2 b^2 c^2+3 a b^4 c-b^6\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-16 a ^ 3 c ^ 2 c1 - 18 a ^ 2 b ^ 2 c c1 + 66 a ^ 2 b b1 c ^ 2 + 4 a b ^ 4 c1 - 52 a b ^ 3 b1 c + 16 b ^ 5 b1
- 6 x (11 a ^ 2 b c ^ 2 c1 - 11 a ^ 2 b1 c ^ 3 + 18 a b ^ 3 c c1 - 18 a b ^ 2 b1 c ^ 2 - 4 b ^ 5 c1 + 4 b ^ 4
b1 c) - 60 b c x ^ 2 (4 a b c c1 - 4 a b1 c ^ 2 + b ^ 3 c1 - b ^ 2 b1 c) + 15 c ^ 2 (a ^ 6 c ^ 3 - 3 a ^ 5 b
^ 2 c ^ 2 + 3 a ^ 4 b ^ 4 c - a ^ 3 b ^ 6 + 6 a ^ 2 b x (a ^ 3 c ^ 3 - 3 a ^ 2 b ^ 2 c ^ 2 + 3 a b ^ 4 c - b ^
6) + 3 a x ^ 2 (a ^ 4 c ^ 4 + a ^ 3 b ^ 2 c ^ 3 - 9 a ^ 2 b ^ 4 c ^ 2 + 11 a b ^ 6 c - 4 b ^ 8) + 4 b x ^ 3 (
3 a ^ 4 c ^ 4 - 7 a ^ 3 b ^ 2 c ^ 3 + 3 a ^ 2 b ^ 4 c ^ 2 + 3 a b ^ 6 c - 2 b ^ 8) + 3 c x ^ 4 (a ^ 4 c ^ 4 +
a ^ 3 b ^ 2 c ^ 3 - 9 a ^ 2 b ^ 4 c ^ 2 + 11 a b ^ 6 c - 4 b ^ 8) + 6 b c ^ 2 x ^ 5 (a ^ 3 c ^ 3 - 3 a ^ 2 b ^
2 c ^ 2 + 3 a b ^ 4 c - b ^ 6) + c ^ 3 x ^ 6 (a ^ 3 c ^ 3 - 3 a ^ 2 b ^ 2 c ^ 2 + 3 a b ^ 4 c - b ^ 6)) (b c1
- b1 c) (Log[(a ^ 4 c ^ 4 (-b c1 + b1 c) Sqrt[-1 / (a c - b ^ 2) ^ 7] + 4 a ^ 3 b ^ 2 c ^ 3 (b c1 - b1 c) Sqr
t[-1 / (a c - b ^ 2) ^ 7] + 6 a ^ 2 b ^ 4 c ^ 2 (-b c1 + b1 c) Sqrt[-1 / (a c - b ^ 2) ^ 7] + 4 a b ^ 6 c (b c
1 - b1 c) Sqrt[-1 / (a c - b ^ 2) ^ 7] + b ^ 8 (-b c1 + b1 c) Sqrt[-1 / (a c - b ^ 2) ^ 7] + b ^ 2 c1 - b b1 c
+ c x (b c1 - b1 c)) / (c (b c1 - b1 c))] - Log[(a ^ 4 c ^ 4 (b c1 - b1 c) Sqrt[-1 / (a c - b ^ 2) ^ 7] + 4 a
^ 3 b ^ 2 c ^ 3 (-b c1 + b1 c) Sqrt[-1 / (a c - b ^ 2) ^ 7] + 6 a ^ 2 b ^ 4 c ^ 2 (b c1 - b1 c) Sqrt[-1 / (a
c - b ^ 2) ^ 7] - 4 a b ^ 6 c (b c1 - b1 c) Sqrt[-1 / (a c - b ^ 2) ^ 7] + b ^ 8 (b c1 - b1 c) Sqrt[-1 / (a c
- b ^ 2) ^ 7] + b ^ 2 c1 - b b1 c + c x (b c1 - b1 c)) / (c (b c1 - b1 c))]) Sqrt[-1 / (a c - b ^ 2) ^ 7] + 20
c ^ 2 x ^ 3 (-4 a b c c1 + 4 a b1 c ^ 2 - 11 b ^ 3 c1 + 11 b ^ 2 b1 c) - 150 b c ^ 3 x ^ 4 (b c1 - b1 c) + 30
c ^ 4 x ^ 5 (-b c1 + b1 c)) / (96 (a ^ 6 c ^ 3 - 3 a ^ 5 b ^ 2 c ^ 2 + 3 a ^ 4 b ^ 4 c - a ^ 3 b ^ 6 + 6 a ^
2 b x (a ^ 3 c ^ 3 - 3 a ^ 2 b ^ 2 c ^ 2 + 3 a b ^ 4 c - b ^ 6) + 3 a x ^ 2 (a ^ 4 c ^ 4 + a ^ 3 b ^ 2 c ^ 3 -
9 a ^ 2 b ^ 4 c ^ 2 + 11 a b ^ 6 c - 4 b ^ 8) + 4 b x ^ 3 (3 a ^ 4 c ^ 4 - 7 a ^ 3 b ^ 2 c ^ 3 + 3 a ^ 2 b ^
4 c ^ 2 + 3 a b ^ 6 c - 2 b ^ 8) + 3 c x ^ 4 (a ^ 4 c ^ 4 + a ^ 3 b ^ 2 c ^ 3 - 9 a ^ 2 b ^ 4 c ^ 2 + 11 a b ^
6 c - 4 b ^ 8) + 6 b c ^ 2 x ^ 5 (a ^ 3 c ^ 3 - 3 a ^ 2 b ^ 2 c ^ 2 + 3 a b ^ 4 c - b ^ 6) + c ^ 3 x ^ 6 (a ^
3 c ^ 3 - 3 a ^ 2 b ^ 2 c ^ 2 + 3 a b ^ 4 c - b ^ 6)))

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Maple [A]
time = 0.12, 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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 962 vs. \(2 (161) = 322\).
time = 0.35, size = 1950, normalized size = 11.27

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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1027 vs. \(2 (158) = 316\)
time = 1.71, size = 1027, normalized size = 5.94 \begin {gather*} \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} \cdot \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} \cdot \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} \cdot \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} \cdot \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} \cdot \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 )} \end {gather*}

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 363 vs. \(2 (161) = 322\).
time = 0.00, size = 387, normalized size = 2.24 \begin {gather*} \frac {-15 x^{5} c_{1} c^{4} b+15 x^{5} b_{1} c^{5}-75 x^{4} c_{1} c^{3} b^{2}+75 x^{4} b_{1} c^{4} b-40 x^{3} c_{1} c^{3} b a-110 x^{3} c_{1} c^{2} b^{3}+40 x^{3} b_{1} c^{4} a+110 x^{3} b_{1} c^{3} b^{2}-120 x^{2} c_{1} c^{2} b^{2} a-30 x^{2} c_{1} c b^{4}+120 x^{2} b_{1} c^{3} b a+30 x^{2} b_{1} c^{2} b^{3}-33 x c_{1} c^{2} b a^{2}-54 x c_{1} c b^{3} a+12 x c_{1} b^{5}+33 x b_{1} c^{3} a^{2}+54 x b_{1} c^{2} b^{2} a-12 x b_{1} c b^{4}-8 c_{1} c^{2} a^{3}-9 c_{1} c b^{2} a^{2}+2 c_{1} b^{4} a+33 b_{1} c^{2} b a^{2}-26 b_{1} c b^{3} a+8 b_{1} b^{5}}{\left (48 c^{3} a^{3}-144 c^{2} b^{2} a^{2}+144 c b^{4} a-48 b^{6}\right ) \left (x^{2} c+2 x b+a\right )^{3}}+\frac {\left (5 c_{1} c^{2} b-5 b_{1} c^{3}\right ) \arctan \left (\frac {b+c x}{\sqrt {-b^{2}+a c}}\right )}{2 \left (-8 c^{3} a^{3}+24 c^{2} b^{2} a^{2}-24 c b^{4} a+8 b^{6}\right ) \sqrt {-b^{2}+a c}} \end {gather*}

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]

-5/16*(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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Mupad [B]
time = 0.71, size = 640, normalized size = 3.70 \begin {gather*} \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}} \end {gather*}

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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