3.4 \(\int \frac {1}{\sqrt {a+b x+c x^2} (d+b x+c x^2)^2} \, dx\)

Optimal. Leaf size=129 \[ \frac {\left (4 c (a-2 d)+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-d} (b+2 c x)}{\sqrt {b^2-4 c d} \sqrt {a+b x+c x^2}}\right )}{(a-d)^{3/2} \left (b^2-4 c d\right )^{3/2}}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )} \]

[Out]

(b^2+4*c*(a-2*d))*arctanh((2*c*x+b)*(a-d)^(1/2)/(b^2-4*c*d)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a-d)^(3/2)/(b^2-4*c*d)
^(3/2)-(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(a-d)/(b^2-4*c*d)/(c*x^2+b*x+d)

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Rubi [A]  time = 0.17, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {974, 12, 982, 208} \[ \frac {\left (4 c (a-2 d)+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-d} (b+2 c x)}{\sqrt {b^2-4 c d} \sqrt {a+b x+c x^2}}\right )}{(a-d)^{3/2} \left (b^2-4 c d\right )^{3/2}}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b*x + c*x^2]*(d + b*x + c*x^2)^2),x]

[Out]

-(((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/((a - d)*(b^2 - 4*c*d)*(d + b*x + c*x^2))) + ((b^2 + 4*c*(a - 2*d))*ArcT
anh[(Sqrt[a - d]*(b + 2*c*x))/(Sqrt[b^2 - 4*c*d]*Sqrt[a + b*x + c*x^2])])/((a - d)^(3/2)*(b^2 - 4*c*d)^(3/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 208

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

Rule 974

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[((2*a
*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a*f) + c*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*x)*(a + b*x + c*x^2)^(p +
1)*(d + e*x + f*x^2)^(q + 1))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), x] - Dist[1/
((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*
x^2)^q*Simp[2*c*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(a*f*(
p + 1) - c*d*(p + 2)) - e*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f))*(p + q + 2) + (2*f*(2*a*c^2*e - b^
2*c*e + b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(b*f*(p + 1) - c*e*(2*p +
q + 4)))*x + c*f*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e,
f, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e
- b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]

Rule 982

Int[1/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-2*e, Su
bst[Int[1/(e*(b*e - 4*a*f) - (b*d - a*e)*x^2), x], x, (e + 2*f*x)/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[c*e - b*f, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^2} \, dx &=-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )}+\frac {\int -\frac {c^2 \left (b^2+4 c (a-2 d)\right ) (a-d)}{2 \sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )} \, dx}{c^2 (a-d)^2 \left (b^2-4 c d\right )}\\ &=-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )}-\frac {\left (b^2+4 c (a-2 d)\right ) \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )} \, dx}{2 (a-d) \left (b^2-4 c d\right )}\\ &=-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )}+\frac {\left (b \left (b^2+4 c (a-2 d)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b \left (b^2-4 c d\right )-(a b-b d) x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{(a-d) \left (b^2-4 c d\right )}\\ &=-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )}+\frac {\left (b^2+4 c (a-2 d)\right ) \tanh ^{-1}\left (\frac {\sqrt {a-d} (b+2 c x)}{\sqrt {b^2-4 c d} \sqrt {a+b x+c x^2}}\right )}{(a-d)^{3/2} \left (b^2-4 c d\right )^{3/2}}\\ \end {align*}

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Mathematica [B]  time = 0.92, size = 296, normalized size = 2.29 \[ \frac {1}{2} \left (-\frac {8 c (b+2 c x) \sqrt {a+x (b+c x)}}{(a-d) \left (4 c d-b^2\right ) \left (\sqrt {b^2-4 c d}-b-2 c x\right ) \left (\sqrt {b^2-4 c d}+b+2 c x\right )}-\frac {\left (4 c (a-2 d)+b^2\right ) \tanh ^{-1}\left (\frac {4 a c-2 c x \sqrt {b^2-4 c d}-b \left (\sqrt {b^2-4 c d}+b\right )}{4 c \sqrt {a-d} \sqrt {a+x (b+c x)}}\right )}{(a-d)^{3/2} \left (b^2-4 c d\right )^{3/2}}-\frac {\left (4 c (a-2 d)+b^2\right ) \tanh ^{-1}\left (\frac {-2 c \left (2 a+x \sqrt {b^2-4 c d}\right )-b \sqrt {b^2-4 c d}+b^2}{4 c \sqrt {a-d} \sqrt {a+x (b+c x)}}\right )}{(a-d)^{3/2} \left (b^2-4 c d\right )^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b*x + c*x^2]*(d + b*x + c*x^2)^2),x]

[Out]

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

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fricas [B]  time = 1.86, size = 1544, normalized size = 11.97 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x+d)^2/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[1/4*(sqrt(a*b^2 + 4*c*d^2 - (b^2 + 4*a*c)*d)*(8*c*d^2 - (b^2*c + 4*a*c^2 - 8*c^2*d)*x^2 - (b^2 + 4*a*c)*d - (
b^3 + 4*a*b*c - 8*b*c*d)*x)*log((8*a^2*b^4 + (b^4*c^2 + 24*a*b^2*c^3 + 16*a^2*c^4 + 128*c^4*d^2 - 32*(b^2*c^3
+ 4*a*c^4)*d)*x^4 + 2*(b^5*c + 24*a*b^3*c^2 + 16*a^2*b*c^3 + 128*b*c^3*d^2 - 32*(b^3*c^2 + 4*a*b*c^3)*d)*x^3 +
(b^4 + 24*a*b^2*c + 16*a^2*c^2)*d^2 + (b^6 + 32*a*b^4*c + 48*a^2*b^2*c^2 + 32*(5*b^2*c^2 + 4*a*c^3)*d^2 - 2*(
19*b^4*c + 104*a*b^2*c^2 + 48*a^2*c^3)*d)*x^2 - 4*(2*a*b^3 + 2*(b^2*c^2 + 4*a*c^3 - 8*c^3*d)*x^3 + 3*(b^3*c +
4*a*b*c^2 - 8*b*c^2*d)*x^2 - (b^3 + 4*a*b*c)*d + (b^4 + 8*a*b^2*c - 2*(5*b^2*c + 4*a*c^2)*d)*x)*sqrt(a*b^2 + 4
*c*d^2 - (b^2 + 4*a*c)*d)*sqrt(c*x^2 + b*x + a) - 8*(a*b^4 + 4*a^2*b^2*c)*d + 2*(4*a*b^5 + 16*a^2*b^3*c + 16*(
b^3*c + 4*a*b*c^2)*d^2 - (3*b^5 + 40*a*b^3*c + 48*a^2*b*c^2)*d)*x)/(c^2*x^4 + 2*b*c*x^3 + 2*b*d*x + (b^2 + 2*c
*d)*x^2 + d^2)) - 4*(a*b^3 + 4*b*c*d^2 - (b^3 + 4*a*b*c)*d + 2*(a*b^2*c + 4*c^2*d^2 - (b^2*c + 4*a*c^2)*d)*x)*
sqrt(c*x^2 + b*x + a))/(a^2*b^4*d + 16*c^2*d^5 - 8*(b^2*c + 4*a*c^2)*d^4 + (b^4 + 16*a*b^2*c + 16*a^2*c^2)*d^3
- 2*(a*b^4 + 4*a^2*b^2*c)*d^2 + (a^2*b^4*c + 16*c^3*d^4 - 8*(b^2*c^2 + 4*a*c^3)*d^3 + (b^4*c + 16*a*b^2*c^2 +
16*a^2*c^3)*d^2 - 2*(a*b^4*c + 4*a^2*b^2*c^2)*d)*x^2 + (a^2*b^5 + 16*b*c^2*d^4 - 8*(b^3*c + 4*a*b*c^2)*d^3 +
(b^5 + 16*a*b^3*c + 16*a^2*b*c^2)*d^2 - 2*(a*b^5 + 4*a^2*b^3*c)*d)*x), -1/2*(sqrt(-a*b^2 - 4*c*d^2 + (b^2 + 4*
a*c)*d)*(8*c*d^2 - (b^2*c + 4*a*c^2 - 8*c^2*d)*x^2 - (b^2 + 4*a*c)*d - (b^3 + 4*a*b*c - 8*b*c*d)*x)*arctan(-1/
2*(2*a*b^2 + (b^2*c + 4*a*c^2 - 8*c^2*d)*x^2 - (b^2 + 4*a*c)*d + (b^3 + 4*a*b*c - 8*b*c*d)*x)*sqrt(-a*b^2 - 4*
c*d^2 + (b^2 + 4*a*c)*d)*sqrt(c*x^2 + b*x + a)/(a^2*b^3 + 4*a*b*c*d^2 + 2*(a*b^2*c^2 + 4*c^3*d^2 - (b^2*c^2 +
4*a*c^3)*d)*x^3 + 3*(a*b^3*c + 4*b*c^2*d^2 - (b^3*c + 4*a*b*c^2)*d)*x^2 - (a*b^3 + 4*a^2*b*c)*d + (a*b^4 + 2*a
^2*b^2*c + 4*(b^2*c + 2*a*c^2)*d^2 - (b^4 + 6*a*b^2*c + 8*a^2*c^2)*d)*x)) + 2*(a*b^3 + 4*b*c*d^2 - (b^3 + 4*a*
b*c)*d + 2*(a*b^2*c + 4*c^2*d^2 - (b^2*c + 4*a*c^2)*d)*x)*sqrt(c*x^2 + b*x + a))/(a^2*b^4*d + 16*c^2*d^5 - 8*(
b^2*c + 4*a*c^2)*d^4 + (b^4 + 16*a*b^2*c + 16*a^2*c^2)*d^3 - 2*(a*b^4 + 4*a^2*b^2*c)*d^2 + (a^2*b^4*c + 16*c^3
*d^4 - 8*(b^2*c^2 + 4*a*c^3)*d^3 + (b^4*c + 16*a*b^2*c^2 + 16*a^2*c^3)*d^2 - 2*(a*b^4*c + 4*a^2*b^2*c^2)*d)*x^
2 + (a^2*b^5 + 16*b*c^2*d^4 - 8*(b^3*c + 4*a*b*c^2)*d^3 + (b^5 + 16*a*b^3*c + 16*a^2*b*c^2)*d^2 - 2*(a*b^5 + 4
*a^2*b^3*c)*d)*x)]

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giac [B]  time = 2.27, size = 1166, normalized size = 9.04 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x+d)^2/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

-1/2*((b^2 + 4*a*c - 8*c*d)*log(abs((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*c + 4*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^2*a*c^2 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^2*d + (sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*sq
rt(c) + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c^(3/2) - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c^(3/2)*d
+ 3*a*b^2*c + 4*sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2)*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(3/2) - 4*a^2*
c^2 - 2*b^2*c*d + 4*sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2)*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c + sqrt(a*b
^2 - b^2*d - 4*a*c*d + 4*c*d^2)*b^2*sqrt(c)))/sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2) - (b^2 + 4*a*c - 8*c*d)*
log(abs((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*c + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^2 - 8*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^2*c^2*d + (sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*sqrt(c) + 4*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))*a*b*c^(3/2) - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c^(3/2)*d + 3*a*b^2*c - 4*sqrt(a*b^2 -
b^2*d - 4*a*c*d + 4*c*d^2)*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(3/2) - 4*a^2*c^2 - 2*b^2*c*d - 4*sqrt(a*b
^2 - b^2*d - 4*a*c*d + 4*c*d^2)*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c - sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d
^2)*b^2*sqrt(c)))/sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2))/(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2) + ((sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^2*b^2*sqrt(c) + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^(3/2) - 8*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^2*c^(3/2)*d + (sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*a*b*c - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c*d + 3*a*b^2*sqrt(c) - 4*a^2*c^(3/2) - 2*b^2*sqrt(c)*d)/(
((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*c + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*sqrt(c) + (sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^2*b^2 - 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^2*c*d - 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*sqrt(c) + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*sqrt(c)
*d - a*b^2 + a^2*c + b^2*d)*(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2))

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maple [B]  time = 0.03, size = 829, normalized size = 6.43 \[ \frac {2 c \ln \left (\frac {2 a -2 d +\sqrt {b^{2}-4 c d}\, \left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )+2 \sqrt {a -d}\, \sqrt {a +\left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )^{2} c -d +\sqrt {b^{2}-4 c d}\, \left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )}}{x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}}\right )}{\left (b^{2}-4 c d \right )^{\frac {3}{2}} \sqrt {a -d}}-\frac {2 c \ln \left (\frac {2 a -2 d -\sqrt {b^{2}-4 c d}\, \left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )+2 \sqrt {a -d}\, \sqrt {a +\left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )^{2} c -d -\sqrt {b^{2}-4 c d}\, \left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )}}{x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}}\right )}{\left (b^{2}-4 c d \right )^{\frac {3}{2}} \sqrt {a -d}}+\frac {\ln \left (\frac {2 a -2 d +\sqrt {b^{2}-4 c d}\, \left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )+2 \sqrt {a -d}\, \sqrt {a +\left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )^{2} c -d +\sqrt {b^{2}-4 c d}\, \left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )}}{x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}}\right )}{2 \sqrt {b^{2}-4 c d}\, \left (a -d \right )^{\frac {3}{2}}}-\frac {\ln \left (\frac {2 a -2 d -\sqrt {b^{2}-4 c d}\, \left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )+2 \sqrt {a -d}\, \sqrt {a +\left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )^{2} c -d -\sqrt {b^{2}-4 c d}\, \left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )}}{x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}}\right )}{2 \sqrt {b^{2}-4 c d}\, \left (a -d \right )^{\frac {3}{2}}}-\frac {\sqrt {a +\left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )^{2} c -d +\sqrt {b^{2}-4 c d}\, \left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )}}{\left (b^{2}-4 c d \right ) \left (a -d \right ) \left (x +\frac {b}{2 c}-\frac {\sqrt {b^{2}-4 c d}}{2 c}\right )}-\frac {\sqrt {a +\left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )^{2} c -d -\sqrt {b^{2}-4 c d}\, \left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )}}{\left (b^{2}-4 c d \right ) \left (a -d \right ) \left (x +\frac {b}{2 c}+\frac {\sqrt {b^{2}-4 c d}}{2 c}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x^2+b*x+d)^2/(c*x^2+b*x+a)^(1/2),x)

[Out]

-2/(b^2-4*c*d)^(3/2)*c/(a-d)^(1/2)*ln((2*a-2*d-(b^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)+2*(a-d)^(1/2)
*(a+(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)^2*c-d-(b^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c))^(1/2))/(x+1/2*(b+
(b^2-4*c*d)^(1/2))/c))+2/(b^2-4*c*d)^(3/2)*c/(a-d)^(1/2)*ln((2*a-2*d+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^
(1/2))/c)+2*(a-d)^(1/2)*(a+(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c-d+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/
2))/c))^(1/2))/(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))-1/(b^2-4*c*d)/(a-d)/(x-1/2/c*(b^2-4*c*d)^(1/2)+1/2/c*b)*(a+(x
-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c-d+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))^(1/2)+1/2/(b^2-4*c*d)
^(1/2)/(a-d)^(3/2)*ln((2*a-2*d+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+2*(a-d)^(1/2)*(a+(x-1/2*(-b+
(b^2-4*c*d)^(1/2))/c)^2*c-d+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))^(1/2))/(x-1/2*(-b+(b^2-4*c*d)^
(1/2))/c))-1/(b^2-4*c*d)/(a-d)/(x+1/2/c*(b^2-4*c*d)^(1/2)+1/2/c*b)*(a+(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)^2*c-d-(b
^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c))^(1/2)-1/2/(b^2-4*c*d)^(1/2)/(a-d)^(3/2)*ln((2*a-2*d-(b^2-4*c*
d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)+2*(a-d)^(1/2)*(a+(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)^2*c-d-(b^2-4*c*d)^(1
/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c))^(1/2))/(x+1/2*(b+(b^2-4*c*d)^(1/2))/c))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x^{2} + b x + a} {\left (c x^{2} + b x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x+d)^2/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(c*x^2 + b*x + a)*(c*x^2 + b*x + d)^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {c\,x^2+b\,x+a}\,{\left (c\,x^2+b\,x+d\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x + c*x^2)^(1/2)*(d + b*x + c*x^2)^2),x)

[Out]

int(1/((a + b*x + c*x^2)^(1/2)*(d + b*x + c*x^2)^2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + b x + c x^{2}} \left (b x + c x^{2} + d\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x**2+b*x+d)**2/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral(1/(sqrt(a + b*x + c*x**2)*(b*x + c*x**2 + d)**2), x)

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