3.1.4 \(\int \frac {1}{\sqrt {a+b x+c x^2} (d+b x+c x^2)^2} \, dx\) [4]

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

[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 = 0.11, 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 = {988, 12, 996, 214} \begin {gather*} \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 )} \end {gather*}

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 214

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

Rule 988

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 996

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 ) \text {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 [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.69, size = 215, normalized size = 1.67 \begin {gather*} \frac {\frac {2 (b+2 c x) \sqrt {a+x (b+c x)}}{d+x (b+c x)}+\left (b^2+4 c (a-2 d)\right ) \text {RootSum}\left [-a b^2+a^2 c+b^2 d+2 a b \sqrt {c} \text {$\#$1}-4 b \sqrt {c} d \text {$\#$1}+b^2 \text {$\#$1}^2-2 a c \text {$\#$1}^2+4 c d \text {$\#$1}^2-2 b \sqrt {c} \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {\log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )}{a \sqrt {c}-2 \sqrt {c} d+b \text {$\#$1}-\sqrt {c} \text {$\#$1}^2}\&\right ]}{2 (a-d) \left (-b^2+4 c d\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*(b + 2*c*x)*Sqrt[a + x*(b + c*x)])/(d + x*(b + c*x)) + (b^2 + 4*c*(a - 2*d))*RootSum[-(a*b^2) + a^2*c + b^
2*d + 2*a*b*Sqrt[c]*#1 - 4*b*Sqrt[c]*d*#1 + b^2*#1^2 - 2*a*c*#1^2 + 4*c*d*#1^2 - 2*b*Sqrt[c]*#1^3 + c*#1^4 & ,
Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]/(a*Sqrt[c] - 2*Sqrt[c]*d + b*#1 - Sqrt[c]*#1^2) & ])/(2*(a - d
)*(-b^2 + 4*c*d))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(826\) vs. \(2(117)=234\).
time = 0.17, size = 827, normalized size = 6.41

method	result	size
default	\(-\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 {\left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )^{2} c -\sqrt {b^{2}-4 c d}\, \left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )+a -d}}{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 {-\frac {\sqrt {\left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )^{2} c +\sqrt {b^{2}-4 c d}\, \left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )+a -d}}{\left (a -d \right ) \left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )}+\frac {\sqrt {b^{2}-4 c d}\, \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 {\left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )^{2} c +\sqrt {b^{2}-4 c d}\, \left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )+a -d}}{x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}}\right )}{2 \left (a -d \right )^{\frac {3}{2}}}}{b^{2}-4 c 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 {\left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )^{2} c +\sqrt {b^{2}-4 c d}\, \left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )+a -d}}{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 {-\frac {\sqrt {\left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )^{2} c -\sqrt {b^{2}-4 c d}\, \left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )+a -d}}{\left (a -d \right ) \left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )}-\frac {\sqrt {b^{2}-4 c d}\, \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 {\left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )^{2} c -\sqrt {b^{2}-4 c d}\, \left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )+a -d}}{x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}}\right )}{2 \left (a -d \right )^{\frac {3}{2}}}}{b^{2}-4 c d}\)	\(827\)

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,method=_RETURNVERBOSE)

[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)
*((x+1/2*(b+(b^2-4*c*d)^(1/2))/c)^2*c-(b^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)+a-d)^(1/2))/(x+1/2*(b+
(b^2-4*c*d)^(1/2))/c))+1/(b^2-4*c*d)*(-1/(a-d)/(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)*((x-1/2*(-b+(b^2-4*c*d)^(1/2))
/c)^2*c+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+a-d)^(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)*((x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c+(
b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+a-d)^(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)*((x-1/2*(-b+
(b^2-4*c*d)^(1/2))/c)^2*c+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+a-d)^(1/2))/(x-1/2*(-b+(b^2-4*c*d
)^(1/2))/c))+1/(b^2-4*c*d)*(-1/(a-d)/(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)*((x+1/2*(b+(b^2-4*c*d)^(1/2))/c)^2*c-(b^2
-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)+a-d)^(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)*((x+1/2*(b+(b^2-4*c*d)^(1/2))/c)^2*c-(b^2-4*c*d)^(1/2
)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)+a-d)^(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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 667 vs. \(2 (117) = 234\).
time = 3.59, size = 1544, normalized size = 11.97 \begin {gather*} \left [\frac {\sqrt {a b^{2} + 4 \, c d^{2} - {\left (b^{2} + 4 \, a c\right )} d} {\left (8 \, c d^{2} - {\left (b^{2} c + 4 \, a c^{2} - 8 \, c^{2} d\right )} x^{2} - {\left (b^{2} + 4 \, a c\right )} d - {\left (b^{3} + 4 \, a b c - 8 \, b c d\right )} x\right )} \log \left (\frac {8 \, a^{2} b^{4} + {\left (b^{4} c^{2} + 24 \, a b^{2} c^{3} + 16 \, a^{2} c^{4} + 128 \, c^{4} d^{2} - 32 \, {\left (b^{2} c^{3} + 4 \, a c^{4}\right )} d\right )} x^{4} + 2 \, {\left (b^{5} c + 24 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3} + 128 \, b c^{3} d^{2} - 32 \, {\left (b^{3} c^{2} + 4 \, a b c^{3}\right )} d\right )} x^{3} + {\left (b^{4} + 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{2} + {\left (b^{6} + 32 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} + 32 \, {\left (5 \, b^{2} c^{2} + 4 \, a c^{3}\right )} d^{2} - 2 \, {\left (19 \, b^{4} c + 104 \, a b^{2} c^{2} + 48 \, a^{2} c^{3}\right )} d\right )} x^{2} - 4 \, {\left (2 \, a b^{3} + 2 \, {\left (b^{2} c^{2} + 4 \, a c^{3} - 8 \, c^{3} d\right )} x^{3} + 3 \, {\left (b^{3} c + 4 \, a b c^{2} - 8 \, b c^{2} d\right )} x^{2} - {\left (b^{3} + 4 \, a b c\right )} d + {\left (b^{4} + 8 \, a b^{2} c - 2 \, {\left (5 \, b^{2} c + 4 \, a c^{2}\right )} d\right )} x\right )} \sqrt {a b^{2} + 4 \, c d^{2} - {\left (b^{2} + 4 \, a c\right )} d} \sqrt {c x^{2} + b x + a} - 8 \, {\left (a b^{4} + 4 \, a^{2} b^{2} c\right )} d + 2 \, {\left (4 \, a b^{5} + 16 \, a^{2} b^{3} c + 16 \, {\left (b^{3} c + 4 \, a b c^{2}\right )} d^{2} - {\left (3 \, b^{5} + 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} d\right )} x}{c^{2} x^{4} + 2 \, b c x^{3} + 2 \, b d x + {\left (b^{2} + 2 \, c d\right )} x^{2} + d^{2}}\right ) - 4 \, {\left (a b^{3} + 4 \, b c d^{2} - {\left (b^{3} + 4 \, a b c\right )} d + 2 \, {\left (a b^{2} c + 4 \, c^{2} d^{2} - {\left (b^{2} c + 4 \, a c^{2}\right )} d\right )} x\right )} \sqrt {c x^{2} + b x + a}}{4 \, {\left (a^{2} b^{4} d + 16 \, c^{2} d^{5} - 8 \, {\left (b^{2} c + 4 \, a c^{2}\right )} d^{4} + {\left (b^{4} + 16 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{3} - 2 \, {\left (a b^{4} + 4 \, a^{2} b^{2} c\right )} d^{2} + {\left (a^{2} b^{4} c + 16 \, c^{3} d^{4} - 8 \, {\left (b^{2} c^{2} + 4 \, a c^{3}\right )} d^{3} + {\left (b^{4} c + 16 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{2} - 2 \, {\left (a b^{4} c + 4 \, a^{2} b^{2} c^{2}\right )} d\right )} x^{2} + {\left (a^{2} b^{5} + 16 \, b c^{2} d^{4} - 8 \, {\left (b^{3} c + 4 \, a b c^{2}\right )} d^{3} + {\left (b^{5} + 16 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d^{2} - 2 \, {\left (a b^{5} + 4 \, a^{2} b^{3} c\right )} d\right )} x\right )}}, -\frac {\sqrt {-a b^{2} - 4 \, c d^{2} + {\left (b^{2} + 4 \, a c\right )} d} {\left (8 \, c d^{2} - {\left (b^{2} c + 4 \, a c^{2} - 8 \, c^{2} d\right )} x^{2} - {\left (b^{2} + 4 \, a c\right )} d - {\left (b^{3} + 4 \, a b c - 8 \, b c d\right )} x\right )} \arctan \left (-\frac {{\left (2 \, a b^{2} + {\left (b^{2} c + 4 \, a c^{2} - 8 \, c^{2} d\right )} x^{2} - {\left (b^{2} + 4 \, a c\right )} d + {\left (b^{3} + 4 \, a b c - 8 \, b c d\right )} x\right )} \sqrt {-a b^{2} - 4 \, c d^{2} + {\left (b^{2} + 4 \, a c\right )} d} \sqrt {c x^{2} + b x + a}}{2 \, {\left (a^{2} b^{3} + 4 \, a b c d^{2} + 2 \, {\left (a b^{2} c^{2} + 4 \, c^{3} d^{2} - {\left (b^{2} c^{2} + 4 \, a c^{3}\right )} d\right )} x^{3} + 3 \, {\left (a b^{3} c + 4 \, b c^{2} d^{2} - {\left (b^{3} c + 4 \, a b c^{2}\right )} d\right )} x^{2} - {\left (a b^{3} + 4 \, a^{2} b c\right )} d + {\left (a b^{4} + 2 \, a^{2} b^{2} c + 4 \, {\left (b^{2} c + 2 \, a c^{2}\right )} d^{2} - {\left (b^{4} + 6 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} d\right )} x\right )}}\right ) + 2 \, {\left (a b^{3} + 4 \, b c d^{2} - {\left (b^{3} + 4 \, a b c\right )} d + 2 \, {\left (a b^{2} c + 4 \, c^{2} d^{2} - {\left (b^{2} c + 4 \, a c^{2}\right )} d\right )} x\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left (a^{2} b^{4} d + 16 \, c^{2} d^{5} - 8 \, {\left (b^{2} c + 4 \, a c^{2}\right )} d^{4} + {\left (b^{4} + 16 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{3} - 2 \, {\left (a b^{4} + 4 \, a^{2} b^{2} c\right )} d^{2} + {\left (a^{2} b^{4} c + 16 \, c^{3} d^{4} - 8 \, {\left (b^{2} c^{2} + 4 \, a c^{3}\right )} d^{3} + {\left (b^{4} c + 16 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{2} - 2 \, {\left (a b^{4} c + 4 \, a^{2} b^{2} c^{2}\right )} d\right )} x^{2} + {\left (a^{2} b^{5} + 16 \, b c^{2} d^{4} - 8 \, {\left (b^{3} c + 4 \, a b c^{2}\right )} d^{3} + {\left (b^{5} + 16 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d^{2} - 2 \, {\left (a b^{5} + 4 \, a^{2} b^{3} c\right )} d\right )} x\right )}}\right ] \end {gather*}

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

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1166 vs. \(2 (117) = 234\).
time = 5.01, size = 1166, normalized size = 9.04 \begin {gather*} -\frac {\frac {{\left (b^{2} + 4 \, a c - 8 \, c d\right )} \log \left ({\left | {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{2} c + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a c^{2} - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c^{2} d + {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{3} \sqrt {c} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b c^{\frac {3}{2}} - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b c^{\frac {3}{2}} d + 3 \, a b^{2} c + 4 \, \sqrt {a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c^{\frac {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}} {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b c + \sqrt {a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} b^{2} \sqrt {c} \right |}\right )}{\sqrt {a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}}} - \frac {{\left (b^{2} + 4 \, a c - 8 \, c d\right )} \log \left ({\left | {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{2} c + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a c^{2} - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c^{2} d + {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{3} \sqrt {c} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b c^{\frac {3}{2}} - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b c^{\frac {3}{2}} d + 3 \, a b^{2} c - 4 \, \sqrt {a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c^{\frac {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}} {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b c - \sqrt {a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} b^{2} \sqrt {c} \right |}\right )}{\sqrt {a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}}}}{2 \, {\left (a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}\right )}} + \frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{2} \sqrt {c} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a c^{\frac {3}{2}} - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c^{\frac {3}{2}} d + {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{3} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b c - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b c d + 3 \, a b^{2} \sqrt {c} - 4 \, a^{2} c^{\frac {3}{2}} - 2 \, b^{2} \sqrt {c} d}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} c + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b \sqrt {c} + {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{2} - 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a c + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c d - 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b \sqrt {c} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b \sqrt {c} d - a b^{2} + a^{2} c + b^{2} d\right )} {\left (a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}\right )}} \end {gather*}

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

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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