3.1.5 \(\int \frac {1}{\sqrt {a+b x+c x^2} (d+b x+c x^2)^3} \, dx\) [5]

Optimal. Leaf size=224 \[ -\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{2 (a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )^2}+\frac {3 \left (b^2+4 c (a-2 d)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{4 (a-d)^2 \left (b^2-4 c d\right )^2 \left (d+b x+c x^2\right )}-\frac {\left (3 b^4+8 b^2 c (a-4 d)+16 c^2 \left (3 a^2-8 a d+8 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {a-d} (b+2 c x)}{\sqrt {b^2-4 c d} \sqrt {a+b x+c x^2}}\right )}{4 (a-d)^{5/2} \left (b^2-4 c d\right )^{5/2}} \]

[Out]

-1/4*(3*b^4+8*b^2*c*(a-4*d)+16*c^2*(3*a^2-8*a*d+8*d^2))*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)^(5/2)/(b^2-4*c*d)^(5/2)-1/2*(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)
^2+3/4*(b^2+4*c*(a-2*d))*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(a-d)^2/(b^2-4*c*d)^2/(c*x^2+b*x+d)

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Rubi [A]
time = 0.27, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {988, 1074, 12, 996, 214} \begin {gather*} -\frac {\left (16 c^2 \left (3 a^2-8 a d+8 d^2\right )+8 b^2 c (a-4 d)+3 b^4\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 )}{4 (a-d)^{5/2} \left (b^2-4 c d\right )^{5/2}}+\frac {3 (b+2 c x) \left (4 c (a-2 d)+b^2\right ) \sqrt {a+b x+c x^2}}{4 (a-d)^2 \left (b^2-4 c d\right )^2 \left (b x+c x^2+d\right )}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{2 (a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b*x + c*x^2]*(d + b*x + c*x^2)^3),x]

[Out]

-1/2*((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/((a - d)*(b^2 - 4*c*d)*(d + b*x + c*x^2)^2) + (3*(b^2 + 4*c*(a - 2*d)
)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*(a - d)^2*(b^2 - 4*c*d)^2*(d + b*x + c*x^2)) - ((3*b^4 + 8*b^2*c*(a -
4*d) + 16*c^2*(3*a^2 - 8*a*d + 8*d^2))*ArcTanh[(Sqrt[a - d]*(b + 2*c*x))/(Sqrt[b^2 - 4*c*d]*Sqrt[a + b*x + c*x
^2])])/(4*(a - d)^(5/2)*(b^2 - 4*c*d)^(5/2))

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]

Rule 1074

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_
)^2)^(q_), x_Symbol] :> Simp[(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)))*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f -
c*(b*e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b*c*d - 2*a*c*e + a*b*f) + C*(b^2*d - a*b*e
- 2*a*(c*d - a*f)))*x), 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[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f)
)*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C
*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f -
c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c
*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(
c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*
e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b,
c, d, e, f, A, B, C, 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]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^3} \, dx &=-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{2 (a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )^2}+\frac {\int \frac {-\frac {1}{2} c^2 (a-d) \left (3 b^2+12 a c-16 c d\right )-4 b c^3 (a-d) x-4 c^4 (a-d) x^2}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^2} \, dx}{2 c^2 (a-d)^2 \left (b^2-4 c d\right )}\\ &=-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{2 (a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )^2}+\frac {3 \left (b^2+4 c (a-2 d)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{4 (a-d)^2 \left (b^2-4 c d\right )^2 \left (d+b x+c x^2\right )}-\frac {\int -\frac {c^4 (a-d)^2 \left (3 b^4+8 b^2 c (a-4 d)+16 c^2 \left (3 a^2-8 a d+8 d^2\right )\right )}{4 \sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )} \, dx}{2 c^4 (a-d)^4 \left (b^2-4 c d\right )^2}\\ &=-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{2 (a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )^2}+\frac {3 \left (b^2+4 c (a-2 d)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{4 (a-d)^2 \left (b^2-4 c d\right )^2 \left (d+b x+c x^2\right )}+\frac {\left (3 b^4+8 b^2 c (a-4 d)+16 c^2 \left (3 a^2-8 a d+8 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )} \, dx}{8 (a-d)^2 \left (b^2-4 c d\right )^2}\\ &=-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{2 (a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )^2}+\frac {3 \left (b^2+4 c (a-2 d)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{4 (a-d)^2 \left (b^2-4 c d\right )^2 \left (d+b x+c x^2\right )}-\frac {\left (b \left (3 b^4+8 b^2 c (a-4 d)+16 c^2 \left (3 a^2-8 a d+8 d^2\right )\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 )}{4 (a-d)^2 \left (b^2-4 c d\right )^2}\\ &=-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{2 (a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )^2}+\frac {3 \left (b^2+4 c (a-2 d)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{4 (a-d)^2 \left (b^2-4 c d\right )^2 \left (d+b x+c x^2\right )}-\frac {\left (3 b^4+8 b^2 c (a-4 d)+16 c^2 \left (3 a^2-8 a d+8 d^2\right )\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 )}{4 (a-d)^{5/2} \left (b^2-4 c d\right )^{5/2}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(486\) vs. \(2(224)=448\).
time = 10.44, size = 486, normalized size = 2.17 \begin {gather*} \frac {-2 \sqrt {a-d} \sqrt {b^2-4 c d} (b+2 c x) \sqrt {a+x (b+c x)} \left (2 (a-d) \left (b^2-4 c d\right )-3 \left (b^2+4 c (a-2 d)\right ) (d+x (b+c x))\right )+\left (3 b^4+8 b^2 c (a-4 d)+16 c^2 \left (3 a^2-8 a d+8 d^2\right )\right ) (d+x (b+c x))^2 \log \left (b-\sqrt {b^2-4 c d}+2 c x\right )-\left (3 b^4+8 b^2 c (a-4 d)+16 c^2 \left (3 a^2-8 a d+8 d^2\right )\right ) (d+x (b+c x))^2 \log \left (b+\sqrt {b^2-4 c d}+2 c x\right )+\left (3 b^4+8 b^2 c (a-4 d)+16 c^2 \left (3 a^2-8 a d+8 d^2\right )\right ) (d+x (b+c x))^2 \log \left (b^2+b \sqrt {b^2-4 c d}+2 c \left (-2 a+\sqrt {b^2-4 c d} x-2 \sqrt {a-d} \sqrt {a+x (b+c x)}\right )\right )-\left (3 b^4+8 b^2 c (a-4 d)+16 c^2 \left (3 a^2-8 a d+8 d^2\right )\right ) (d+x (b+c x))^2 \log \left (-b^2+b \sqrt {b^2-4 c d}+2 c \left (2 a+\sqrt {b^2-4 c d} x+2 \sqrt {a-d} \sqrt {a+x (b+c x)}\right )\right )}{8 (a-d)^{5/2} \left (b^2-4 c d\right )^{5/2} (d+x (b+c x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b*x + c*x^2]*(d + b*x + c*x^2)^3),x]

[Out]

(-2*Sqrt[a - d]*Sqrt[b^2 - 4*c*d]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(2*(a - d)*(b^2 - 4*c*d) - 3*(b^2 + 4*c*(a
- 2*d))*(d + x*(b + c*x))) + (3*b^4 + 8*b^2*c*(a - 4*d) + 16*c^2*(3*a^2 - 8*a*d + 8*d^2))*(d + x*(b + c*x))^2
*Log[b - Sqrt[b^2 - 4*c*d] + 2*c*x] - (3*b^4 + 8*b^2*c*(a - 4*d) + 16*c^2*(3*a^2 - 8*a*d + 8*d^2))*(d + x*(b +
c*x))^2*Log[b + Sqrt[b^2 - 4*c*d] + 2*c*x] + (3*b^4 + 8*b^2*c*(a - 4*d) + 16*c^2*(3*a^2 - 8*a*d + 8*d^2))*(d
+ x*(b + c*x))^2*Log[b^2 + b*Sqrt[b^2 - 4*c*d] + 2*c*(-2*a + Sqrt[b^2 - 4*c*d]*x - 2*Sqrt[a - d]*Sqrt[a + x*(b
+ c*x)])] - (3*b^4 + 8*b^2*c*(a - 4*d) + 16*c^2*(3*a^2 - 8*a*d + 8*d^2))*(d + x*(b + c*x))^2*Log[-b^2 + b*Sqr
t[b^2 - 4*c*d] + 2*c*(2*a + Sqrt[b^2 - 4*c*d]*x + 2*Sqrt[a - d]*Sqrt[a + x*(b + c*x)])])/(8*(a - d)^(5/2)*(b^2
- 4*c*d)^(5/2)*(d + x*(b + c*x))^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1867\) vs. \(2(204)=408\).
time = 0.15, size = 1868, normalized size = 8.34

method	result	size
default	\(\text {Expression too large to display}\)	\(1868\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x^2+b*x+d)^3/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/(b^2-4*c*d)^(3/2)*(-1/2/(a-d)/(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^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)-3/4*(b^2-4*c*d)^(1/2)/(a-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)))+1/2*c/(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)))-1/(b^2-4*c*d)^(3/2)*(-1/2/(a-d)/(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)^
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)+3/4*(b^2-4
*c*d)^(1/2)/(a-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)))+1/2*c/(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)))+6*c^2/(b^2-4*c*d)^(5/2)/(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))-3/(b^2-4
*c*d)^2*c*(-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)))-6*c^2/(b^2-4*c*d)^(5/2)/(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))-3/(b^2-4*c*d
)^2*c*(-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)^3/(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)^3), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1804 vs. \(2 (204) = 408\).
time = 6.55, size = 3818, normalized size = 17.04 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x+d)^3/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[1/16*((128*c^2*d^4 + (3*b^4*c^2 + 8*a*b^2*c^3 + 48*a^2*c^4 + 128*c^4*d^2 - 32*(b^2*c^3 + 4*a*c^4)*d)*x^4 - 32
*(b^2*c + 4*a*c^2)*d^3 + 2*(3*b^5*c + 8*a*b^3*c^2 + 48*a^2*b*c^3 + 128*b*c^3*d^2 - 32*(b^3*c^2 + 4*a*b*c^3)*d)
*x^3 + (3*b^4 + 8*a*b^2*c + 48*a^2*c^2)*d^2 + (3*b^6 + 8*a*b^4*c + 48*a^2*b^2*c^2 + 256*c^3*d^3 + 64*(b^2*c^2
- 4*a*c^3)*d^2 - 2*(13*b^4*c + 56*a*b^2*c^2 - 48*a^2*c^3)*d)*x^2 + 2*(128*b*c^2*d^3 - 32*(b^3*c + 4*a*b*c^2)*d
^2 + (3*b^5 + 8*a*b^3*c + 48*a^2*b*c^2)*d)*x)*sqrt(a*b^2 + 4*c*d^2 - (b^2 + 4*a*c)*d)*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*(2*a^2*b^5 + 128*b*c^2*d^4 - 52*
(b^3*c + 4*a*b*c^2)*d^3 - 6*(a*b^4*c^2 + 4*a^2*b^2*c^3 - 32*c^4*d^3 + 12*(b^2*c^3 + 4*a*c^4)*d^2 - (b^4*c^2 +
16*a*b^2*c^3 + 16*a^2*c^4)*d)*x^3 + 5*(b^5 + 16*a*b^3*c + 16*a^2*b*c^2)*d^2 - 9*(a*b^5*c + 4*a^2*b^3*c^2 - 32*
b*c^3*d^3 + 12*(b^3*c^2 + 4*a*b*c^3)*d^2 - (b^5*c + 16*a*b^3*c^2 + 16*a^2*b*c^3)*d)*x^2 - 7*(a*b^5 + 4*a^2*b^3
*c)*d - (3*a*b^6 + 8*a^2*b^4*c - 256*c^3*d^4 + 8*(b^2*c^2 + 52*a*c^3)*d^3 + 2*(13*b^4*c - 8*a*b^2*c^2 - 80*a^2
*c^3)*d^2 - (3*b^6 + 34*a*b^4*c - 8*a^2*b^2*c^2)*d)*x)*sqrt(c*x^2 + b*x + a))/(a^3*b^6*d^2 + 64*c^3*d^8 - 48*(
b^2*c^2 + 4*a*c^3)*d^7 + 12*(b^4*c + 12*a*b^2*c^2 + 16*a^2*c^3)*d^6 - (b^6 + 36*a*b^4*c + 144*a^2*b^2*c^2 + 64
*a^3*c^3)*d^5 + 3*(a*b^6 + 12*a^2*b^4*c + 16*a^3*b^2*c^2)*d^4 + (a^3*b^6*c^2 + 64*c^5*d^6 - 48*(b^2*c^4 + 4*a*
c^5)*d^5 + 12*(b^4*c^3 + 12*a*b^2*c^4 + 16*a^2*c^5)*d^4 - (b^6*c^2 + 36*a*b^4*c^3 + 144*a^2*b^2*c^4 + 64*a^3*c
^5)*d^3 + 3*(a*b^6*c^2 + 12*a^2*b^4*c^3 + 16*a^3*b^2*c^4)*d^2 - 3*(a^2*b^6*c^2 + 4*a^3*b^4*c^3)*d)*x^4 - 3*(a^
2*b^6 + 4*a^3*b^4*c)*d^3 + 2*(a^3*b^7*c + 64*b*c^4*d^6 - 48*(b^3*c^3 + 4*a*b*c^4)*d^5 + 12*(b^5*c^2 + 12*a*b^3
*c^3 + 16*a^2*b*c^4)*d^4 - (b^7*c + 36*a*b^5*c^2 + 144*a^2*b^3*c^3 + 64*a^3*b*c^4)*d^3 + 3*(a*b^7*c + 12*a^2*b
^5*c^2 + 16*a^3*b^3*c^3)*d^2 - 3*(a^2*b^7*c + 4*a^3*b^5*c^2)*d)*x^3 + (a^3*b^8 + 128*c^4*d^7 - 32*(b^2*c^3 + 1
2*a*c^4)*d^6 - 24*(b^4*c^2 - 4*a*b^2*c^3 - 16*a^2*c^4)*d^5 + 2*(5*b^6*c + 36*a*b^4*c^2 - 48*a^2*b^2*c^3 - 64*a
^3*c^4)*d^4 - (b^8 + 30*a*b^6*c + 72*a^2*b^4*c^2 - 32*a^3*b^2*c^3)*d^3 + 3*(a*b^8 + 10*a^2*b^6*c + 8*a^3*b^4*c
^2)*d^2 - (3*a^2*b^8 + 10*a^3*b^6*c)*d)*x^2 + 2*(a^3*b^7*d + 64*b*c^3*d^7 - 48*(b^3*c^2 + 4*a*b*c^3)*d^6 + 12*
(b^5*c + 12*a*b^3*c^2 + 16*a^2*b*c^3)*d^5 - (b^7 + 36*a*b^5*c + 144*a^2*b^3*c^2 + 64*a^3*b*c^3)*d^4 + 3*(a*b^7
+ 12*a^2*b^5*c + 16*a^3*b^3*c^2)*d^3 - 3*(a^2*b^7 + 4*a^3*b^5*c)*d^2)*x), -1/8*((128*c^2*d^4 + (3*b^4*c^2 + 8
*a*b^2*c^3 + 48*a^2*c^4 + 128*c^4*d^2 - 32*(b^2*c^3 + 4*a*c^4)*d)*x^4 - 32*(b^2*c + 4*a*c^2)*d^3 + 2*(3*b^5*c
+ 8*a*b^3*c^2 + 48*a^2*b*c^3 + 128*b*c^3*d^2 - 32*(b^3*c^2 + 4*a*b*c^3)*d)*x^3 + (3*b^4 + 8*a*b^2*c + 48*a^2*c
^2)*d^2 + (3*b^6 + 8*a*b^4*c + 48*a^2*b^2*c^2 + 256*c^3*d^3 + 64*(b^2*c^2 - 4*a*c^3)*d^2 - 2*(13*b^4*c + 56*a*
b^2*c^2 - 48*a^2*c^3)*d)*x^2 + 2*(128*b*c^2*d^3 - 32*(b^3*c + 4*a*b*c^2)*d^2 + (3*b^5 + 8*a*b^3*c + 48*a^2*b*c
^2)*d)*x)*sqrt(-a*b^2 - 4*c*d^2 + (b^2 + 4*a*c)*d)*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*(2*a^2*b^5 + 128*b*c^2*d^4 - 52*(b^3*c + 4*a*b*c^2)*d^3 - 6*(a*b^4*c^2 + 4*a^2*b^
2*c^3 - 32*c^4*d^3 + 12*(b^2*c^3 + 4*a*c^4)*d^2 - (b^4*c^2 + 16*a*b^2*c^3 + 16*a^2*c^4)*d)*x^3 + 5*(b^5 + 16*a
*b^3*c + 16*a^2*b*c^2)*d^2 - 9*(a*b^5*c + 4*a^2*b^3*c^2 - 32*b*c^3*d^3 + 12*(b^3*c^2 + 4*a*b*c^3)*d^2 - (b^5*c
+ 16*a*b^3*c^2 + 16*a^2*b*c^3)*d)*x^2 - 7*(a*b^5 + 4*a^2*b^3*c)*d - (3*a*b^6 + 8*a^2*b^4*c - 256*c^3*d^4 + 8*
(b^2*c^2 + 52*a*c^3)*d^3 + 2*(13*b^4*c - 8*a*b^2*c^2 - 80*a^2*c^3)*d^2 - (3*b^6 + 34*a*b^4*c - 8*a^2*b^2*c^2)*
d)*x)*sqrt(c*x^2 + b*x + a))/(a^3*b^6*d^2 + 64*c^3*d^8 - 48*(b^2*c^2 + 4*a*c^3)*d^7 + 12*(b^4*c + 12*a*b^2*c^2
+ 16*a^2*c^3)*d^6 - (b^6 + 36*a*b^4*c + 144*a^2*b^2*c^2 + 64*a^3*c^3)*d^5 + 3*(a*b^6 + 12*a^2*b^4*c + 16*a^3*
b^2*c^2)*d^4 + (a^3*b^6*c^2 + 64*c^5*d^6 - 48*(b^2*c^4 + 4*a*c^5)*d^5 + 12*(b^4*c^3 + 12*a*b^2*c^4 + 16*a^2*c^
5)*d^4 - (b^6*c^2 + 36*a*b^4*c^3 + 144*a^2*b^2*...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x**2+b*x+d)**3/(c*x**2+b*x+a)**(1/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2986 vs. \(2 (204) = 408\).
time = 4.94, size = 2986, normalized size = 13.33 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x+d)^3/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

-1/8*((3*b^4 + 8*a*b^2*c + 48*a^2*c^2 - 32*b^2*c*d - 128*a*c^2*d + 128*c^2*d^2)*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) - (3*b^4 + 8*a*b^2*c + 48*a^2*c^2 - 32*b^2*c*d - 128*a*c^2*d + 128*c^2*d^2)*log(a
bs(-(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^2*b^4 - 2*a*b^4*d - 8*a^2*b^2*c*d + b^4*d^2 + 16*a*b
^2*c*d^2 + 16*a^2*c^2*d^2 - 8*b^2*c*d^3 - 32*a*c^2*d^3 + 16*c^2*d^4) - 1/4*(3*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^6*b^4*c^(3/2) + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^2*c^(5/2) + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^6*a^2*c^(7/2) - 32*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^2*c^(5/2)*d - 128*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^6*a*c^(7/2)*d + 128*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*c^(7/2)*d^2 + 9*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^5*b^5*c + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^3*c^2 + 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^5*a^2*b*c^3 - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^3*c^2*d - 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*
a*b*c^3*d + 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b*c^3*d^2 + 9*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^6*
sqrt(c) + 15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^4*c^(3/2) + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a
^2*b^2*c^(5/2) - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*c^(7/2) - 78*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^4*b^4*c^(3/2)*d - 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^2*c^(5/2)*d + 672*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^4*a^2*c^(7/2)*d + 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^(5/2)*d^2 - 1152*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^4*a*c^(7/2)*d^2 + 768*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*c^(7/2)*d^3 + 3*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^3*b^7 - 10*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^5*c - 288*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^3*a^3*b*c^3 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^5*c*d + 160*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^3*a*b^3*c^2*d + 1344*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b*c^3*d - 256*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^3*b^3*c^2*d^2 - 2304*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c^3*d^2 + 1536*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^3*b*c^3*d^3 - 14*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^6*sqrt(c) - 71*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^2*a^2*b^4*c^(3/2) - 200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^2*c^(5/2) + 144*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^2*a^4*c^(7/2) + 23*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^6*sqrt(c)*d + 280*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^2*a*b^4*c^(3/2)*d + 1168*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^2*c^(5/2)*d - 640
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*c^(7/2)*d - 272*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^4*c^(3/2)*d
^2 - 2048*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^2*c^(5/2)*d^2 + 640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*
a^2*c^(7/2)*d^2 + 1152*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*c^(5/2)*d^3 - 5*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))*a*b^7 - 47*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^5*c - 56*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b
^3*c^2 + 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b*c^3 + 5*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^7*d + 136
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^5*c*d + 496*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^3*c^2*d - 640*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b*c^3*d - 80*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^5*c*d^2 - 896*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))*a*b^3*c^2*d^2 + 640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b*c^3*d^2 + 384*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))*b^3*c^2*d^3 - 11*a^2*b^6*sqrt(c) - 11*a^3*b^4*c^(3/2) + 72*a^4*b^2*c^(5/2) - 4
8*a^5*c^(7/2) + 17*a*b^6*sqrt(c)*d + 118*a^2*b^4*c^(3/2)*d - 256*a^3*b^2*c^(5/2)*d + 96*a^4*c^(7/2)*d - 6*b^6*
sqrt(c)*d^2 - 152*a*b^4*c^(3/2)*d^2 + 160*a^2*b^2*c^(5/2)*d^2 + 48*b^4*c^(3/2)*d^3)/((a^2*b^4 - 2*a*b^4*d - 8*
a^2*b^2*c*d + b^4*d^2 + 16*a*b^2*c*d^2 + 16*a^2*c^2*d^2 - 8*b^2*c*d^3 - 32*a*c^2*d^3 + 16*c^2*d^4)*((sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^4*c + 2*(sqrt(c)*x - ...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{\sqrt {c\,x^2+b\,x+a}\,{\left (c\,x^2+b\,x+d\right )}^3} \,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)^3),x)

[Out]

int(1/((a + b*x + c*x^2)^(1/2)*(d + b*x + c*x^2)^3), x)

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