### 3.2381 $$\int \frac{1}{(d+e x)^4 \sqrt{a+b x+c x^2}} \, dx$$

Optimal. Leaf size=293 $-\frac{e \sqrt{a+b x+c x^2} \left (-4 c e (4 a e+11 b d)+15 b^2 e^2+44 c^2 d^2\right )}{24 (d+e x) \left (a e^2-b d e+c d^2\right )^3}+\frac{(2 c d-b e) \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac{5 e \sqrt{a+b x+c x^2} (2 c d-b e)}{12 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}-\frac{e \sqrt{a+b x+c x^2}}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}$

[Out]

-(e*Sqrt[a + b*x + c*x^2])/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) - (5*e*(2*c*d - b*e)*Sqrt[a + b*x + c*x^2])
/(12*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) - (e*(44*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(11*b*d + 4*a*e))*Sqrt[a + b
*x + c*x^2])/(24*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) + ((2*c*d - b*e)*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d +
3*a*e))*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(16*(
c*d^2 - b*d*e + a*e^2)^(7/2))

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Rubi [A]  time = 0.359475, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.227, Rules used = {744, 834, 806, 724, 206} $-\frac{e \sqrt{a+b x+c x^2} \left (-4 c e (4 a e+11 b d)+15 b^2 e^2+44 c^2 d^2\right )}{24 (d+e x) \left (a e^2-b d e+c d^2\right )^3}+\frac{(2 c d-b e) \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac{5 e \sqrt{a+b x+c x^2} (2 c d-b e)}{12 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}-\frac{e \sqrt{a+b x+c x^2}}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)^4*Sqrt[a + b*x + c*x^2]),x]

[Out]

-(e*Sqrt[a + b*x + c*x^2])/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) - (5*e*(2*c*d - b*e)*Sqrt[a + b*x + c*x^2])
/(12*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) - (e*(44*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(11*b*d + 4*a*e))*Sqrt[a + b
*x + c*x^2])/(24*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) + ((2*c*d - b*e)*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d +
3*a*e))*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(16*(
c*d^2 - b*d*e + a*e^2)^(7/2))

Rule 744

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)
*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)),
Int[(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x], x]
/; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e
, 0] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ
[p]) || ILtQ[Simplify[m + 2*p + 3], 0])

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 724

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

Rule 206

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

Rubi steps

\begin{align*} \int \frac{1}{(d+e x)^4 \sqrt{a+b x+c x^2}} \, dx &=-\frac{e \sqrt{a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac{\int \frac{\frac{1}{2} (-6 c d+5 b e)+2 c e x}{(d+e x)^3 \sqrt{a+b x+c x^2}} \, dx}{3 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{e \sqrt{a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac{5 e (2 c d-b e) \sqrt{a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{\int \frac{\frac{1}{4} \left (24 c^2 d^2+15 b^2 e^2-2 c e (17 b d+8 a e)\right )-\frac{5}{2} c e (2 c d-b e) x}{(d+e x)^2 \sqrt{a+b x+c x^2}} \, dx}{6 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{e \sqrt{a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac{5 e (2 c d-b e) \sqrt{a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac{e \left (44 c^2 d^2+15 b^2 e^2-4 c e (11 b d+4 a e)\right ) \sqrt{a+b x+c x^2}}{24 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac{\left ((2 c d-b e) \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right )\right ) \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{16 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{e \sqrt{a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac{5 e (2 c d-b e) \sqrt{a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac{e \left (44 c^2 d^2+15 b^2 e^2-4 c e (11 b d+4 a e)\right ) \sqrt{a+b x+c x^2}}{24 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac{\left ((2 c d-b e) \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac{-b d+2 a e-(2 c d-b e) x}{\sqrt{a+b x+c x^2}}\right )}{8 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{e \sqrt{a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac{5 e (2 c d-b e) \sqrt{a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac{e \left (44 c^2 d^2+15 b^2 e^2-4 c e (11 b d+4 a e)\right ) \sqrt{a+b x+c x^2}}{24 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac{(2 c d-b e) \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b d e+a e^2} \sqrt{a+b x+c x^2}}\right )}{16 \left (c d^2-b d e+a e^2\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.673365, size = 288, normalized size = 0.98 $-\frac{\frac{e \sqrt{a+x (b+c x)} \left (-4 c e (4 a e+11 b d)+15 b^2 e^2+44 c^2 d^2\right )}{4 (d+e x) \left (e (a e-b d)+c d^2\right )}+\frac{3 (2 c d-b e) \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{8 \left (e (a e-b d)+c d^2\right )^{3/2}}+\frac{2 e \sqrt{a+x (b+c x)} \left (e (a e-b d)+c d^2\right )}{(d+e x)^3}+\frac{5 e \sqrt{a+x (b+c x)} (2 c d-b e)}{2 (d+e x)^2}}{6 \left (e (a e-b d)+c d^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)^4*Sqrt[a + b*x + c*x^2]),x]

[Out]

-((2*e*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c*x)])/(d + e*x)^3 + (5*e*(2*c*d - b*e)*Sqrt[a + x*(b + c*x)
])/(2*(d + e*x)^2) + (e*(44*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(11*b*d + 4*a*e))*Sqrt[a + x*(b + c*x)])/(4*(c*d^2 +
e*(-(b*d) + a*e))*(d + e*x)) + (3*(2*c*d - b*e)*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*ArcTanh[(-(b*d
) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(8*(c*d^2 + e*(-(b*d)
+ a*e))^(3/2)))/(6*(c*d^2 + e*(-(b*d) + a*e))^2)

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Maple [B]  time = 0.236, size = 1665, normalized size = 5.7 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x)

[Out]

-1/3/e^2/(a*e^2-b*d*e+c*d^2)/(d/e+x)^3*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+5/12/
(a*e^2-b*d*e+c*d^2)^2/(d/e+x)^2*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b-5/6/e/(a*e
^2-b*d*e+c*d^2)^2/(d/e+x)^2*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c*d-5/8*e^2/(a*e
^2-b*d*e+c*d^2)^3/(d/e+x)*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2+5/2*e/(a*e^2-b
*d*e+c*d^2)^3/(d/e+x)*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*c*d-5/2/(a*e^2-b*d*e
+c*d^2)^3/(d/e+x)*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c^2*d^2+5/16*e^2/(a*e^2-b*
d*e+c*d^2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d
*e+c*d^2)/e^2)^(1/2)*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b^3-15/8*e/(a
*e^2-b*d*e+c*d^2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*
e^2-b*d*e+c*d^2)/e^2)^(1/2)*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b^2*c*
d+15/4/(a*e^2-b*d*e+c*d^2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+
x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x
))*b*c^2*d^2-5/2/e/(a*e^2-b*d*e+c*d^2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*
c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(
1/2))/(d/e+x))*c^3*d^3-3/4/(a*e^2-b*d*e+c*d^2)^2*c/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e
^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d
^2)/e^2)^(1/2))/(d/e+x))*b+3/2/e/(a*e^2-b*d*e+c*d^2)^2*c^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+
c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b
*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*d+2/3*c/(a*e^2-b*d*e+c*d^2)^2/(d/e+x)*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e
^2-b*d*e+c*d^2)/e^2)^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 97.8904, size = 5632, normalized size = 19.22 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[1/96*(3*(16*c^3*d^6 - 24*b*c^2*d^5*e + 6*(3*b^2*c - 4*a*c^2)*d^4*e^2 - (5*b^3 - 12*a*b*c)*d^3*e^3 + (16*c^3*d
^3*e^3 - 24*b*c^2*d^2*e^4 + 6*(3*b^2*c - 4*a*c^2)*d*e^5 - (5*b^3 - 12*a*b*c)*e^6)*x^3 + 3*(16*c^3*d^4*e^2 - 24
*b*c^2*d^3*e^3 + 6*(3*b^2*c - 4*a*c^2)*d^2*e^4 - (5*b^3 - 12*a*b*c)*d*e^5)*x^2 + 3*(16*c^3*d^5*e - 24*b*c^2*d^
4*e^2 + 6*(3*b^2*c - 4*a*c^2)*d^3*e^3 - (5*b^3 - 12*a*b*c)*d^2*e^4)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*
d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 4*sqrt(c*d^2 - b*d*e +
a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e
)*x)/(e^2*x^2 + 2*d*e*x + d^2)) - 4*(72*c^3*d^6*e - 162*b*c^2*d^5*e^2 - 34*a^2*b*d*e^6 + 8*a^3*e^7 + (123*b^2*
c + 92*a*c^2)*d^4*e^3 - (33*b^3 + 136*a*b*c)*d^3*e^4 + (59*a*b^2 + 28*a^2*c)*d^2*e^5 + (44*c^3*d^4*e^3 - 88*b*
c^2*d^3*e^4 + (59*b^2*c + 28*a*c^2)*d^2*e^5 - (15*b^3 + 28*a*b*c)*d*e^6 + (15*a*b^2 - 16*a^2*c)*e^7)*x^2 + 2*(
54*c^3*d^5*e^2 - 113*b*c^2*d^4*e^3 - 5*a^2*b*e^7 + (79*b^2*c + 48*a*c^2)*d^3*e^4 - 2*(10*b^3 + 29*a*b*c)*d^2*e
^5 + (25*a*b^2 - 6*a^2*c)*d*e^6)*x)*sqrt(c*x^2 + b*x + a))/(c^4*d^11 - 4*b*c^3*d^10*e - 4*a^3*b*d^4*e^7 + a^4*
d^3*e^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^9*e^2 - 4*(b^3*c + 3*a*b*c^2)*d^8*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^7
*e^4 - 4*(a*b^3 + 3*a^2*b*c)*d^6*e^5 + 2*(3*a^2*b^2 + 2*a^3*c)*d^5*e^6 + (c^4*d^8*e^3 - 4*b*c^3*d^7*e^4 - 4*a^
3*b*d*e^10 + a^4*e^11 + 2*(3*b^2*c^2 + 2*a*c^3)*d^6*e^5 - 4*(b^3*c + 3*a*b*c^2)*d^5*e^6 + (b^4 + 12*a*b^2*c +
6*a^2*c^2)*d^4*e^7 - 4*(a*b^3 + 3*a^2*b*c)*d^3*e^8 + 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^9)*x^3 + 3*(c^4*d^9*e^2 - 4
*b*c^3*d^8*e^3 - 4*a^3*b*d^2*e^9 + a^4*d*e^10 + 2*(3*b^2*c^2 + 2*a*c^3)*d^7*e^4 - 4*(b^3*c + 3*a*b*c^2)*d^6*e^
5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^5*e^6 - 4*(a*b^3 + 3*a^2*b*c)*d^4*e^7 + 2*(3*a^2*b^2 + 2*a^3*c)*d^3*e^8)*
x^2 + 3*(c^4*d^10*e - 4*b*c^3*d^9*e^2 - 4*a^3*b*d^3*e^8 + a^4*d^2*e^9 + 2*(3*b^2*c^2 + 2*a*c^3)*d^8*e^3 - 4*(b
^3*c + 3*a*b*c^2)*d^7*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^6*e^5 - 4*(a*b^3 + 3*a^2*b*c)*d^5*e^6 + 2*(3*a^2*
b^2 + 2*a^3*c)*d^4*e^7)*x), 1/48*(3*(16*c^3*d^6 - 24*b*c^2*d^5*e + 6*(3*b^2*c - 4*a*c^2)*d^4*e^2 - (5*b^3 - 12
*a*b*c)*d^3*e^3 + (16*c^3*d^3*e^3 - 24*b*c^2*d^2*e^4 + 6*(3*b^2*c - 4*a*c^2)*d*e^5 - (5*b^3 - 12*a*b*c)*e^6)*x
^3 + 3*(16*c^3*d^4*e^2 - 24*b*c^2*d^3*e^3 + 6*(3*b^2*c - 4*a*c^2)*d^2*e^4 - (5*b^3 - 12*a*b*c)*d*e^5)*x^2 + 3*
(16*c^3*d^5*e - 24*b*c^2*d^4*e^2 + 6*(3*b^2*c - 4*a*c^2)*d^3*e^3 - (5*b^3 - 12*a*b*c)*d^2*e^4)*x)*sqrt(-c*d^2
+ b*d*e - a*e^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x
)/(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*x)) - 2*(72
*c^3*d^6*e - 162*b*c^2*d^5*e^2 - 34*a^2*b*d*e^6 + 8*a^3*e^7 + (123*b^2*c + 92*a*c^2)*d^4*e^3 - (33*b^3 + 136*a
*b*c)*d^3*e^4 + (59*a*b^2 + 28*a^2*c)*d^2*e^5 + (44*c^3*d^4*e^3 - 88*b*c^2*d^3*e^4 + (59*b^2*c + 28*a*c^2)*d^2
*e^5 - (15*b^3 + 28*a*b*c)*d*e^6 + (15*a*b^2 - 16*a^2*c)*e^7)*x^2 + 2*(54*c^3*d^5*e^2 - 113*b*c^2*d^4*e^3 - 5*
a^2*b*e^7 + (79*b^2*c + 48*a*c^2)*d^3*e^4 - 2*(10*b^3 + 29*a*b*c)*d^2*e^5 + (25*a*b^2 - 6*a^2*c)*d*e^6)*x)*sqr
t(c*x^2 + b*x + a))/(c^4*d^11 - 4*b*c^3*d^10*e - 4*a^3*b*d^4*e^7 + a^4*d^3*e^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^9*e
^2 - 4*(b^3*c + 3*a*b*c^2)*d^8*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^7*e^4 - 4*(a*b^3 + 3*a^2*b*c)*d^6*e^5 +
2*(3*a^2*b^2 + 2*a^3*c)*d^5*e^6 + (c^4*d^8*e^3 - 4*b*c^3*d^7*e^4 - 4*a^3*b*d*e^10 + a^4*e^11 + 2*(3*b^2*c^2 +
2*a*c^3)*d^6*e^5 - 4*(b^3*c + 3*a*b*c^2)*d^5*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^7 - 4*(a*b^3 + 3*a^2*b
*c)*d^3*e^8 + 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^9)*x^3 + 3*(c^4*d^9*e^2 - 4*b*c^3*d^8*e^3 - 4*a^3*b*d^2*e^9 + a^4*
d*e^10 + 2*(3*b^2*c^2 + 2*a*c^3)*d^7*e^4 - 4*(b^3*c + 3*a*b*c^2)*d^6*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^5*
e^6 - 4*(a*b^3 + 3*a^2*b*c)*d^4*e^7 + 2*(3*a^2*b^2 + 2*a^3*c)*d^3*e^8)*x^2 + 3*(c^4*d^10*e - 4*b*c^3*d^9*e^2 -
4*a^3*b*d^3*e^8 + a^4*d^2*e^9 + 2*(3*b^2*c^2 + 2*a*c^3)*d^8*e^3 - 4*(b^3*c + 3*a*b*c^2)*d^7*e^4 + (b^4 + 12*a
*b^2*c + 6*a^2*c^2)*d^6*e^5 - 4*(a*b^3 + 3*a^2*b*c)*d^5*e^6 + 2*(3*a^2*b^2 + 2*a^3*c)*d^4*e^7)*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d + e x\right )^{4} \sqrt{a + b x + c x^{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**4/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral(1/((d + e*x)**4*sqrt(a + b*x + c*x**2)), x)

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Giac [B]  time = 1.33898, size = 2807, normalized size = 9.58 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

1/8*(16*c^3*d^3 - 24*b*c^2*d^2*e + 18*b^2*c*d*e^2 - 24*a*c^2*d*e^2 - 5*b^3*e^3 + 12*a*b*c*e^3)*arctan(-((sqrt(
c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c
*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5
+ a^3*e^6)*sqrt(-c*d^2 + b*d*e - a*e^2)) - 1/24*(240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*c^(7/2)*d^4*e + 35
2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^4*d^5 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*c^3*d^3*e^2 - 400*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c^3*d^4*e + 528*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*c^(7/2)*d^5 - 3
60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b*c^(5/2)*d^3*e^2 - 756*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*c^(
5/2)*d^4*e - 816*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^(7/2)*d^4*e + 264*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))*b^2*c^3*d^5 - 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b*c^2*d^2*e^3 + 204*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^3*b^2*c^2*d^3*e^2 - 656*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c^3*d^3*e^2 - 336*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))*b^3*c^2*d^4*e - 816*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c^3*d^4*e + 44*b^3*c^(5/2)*d^5 + 270*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^(3/2)*d^2*e^3 - 360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*c^(5/2)
*d^2*e^3 + 498*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^3*c^(3/2)*d^3*e^2 + 648*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^2*a*b*c^(5/2)*d^3*e^2 - 44*b^4*c^(3/2)*d^4*e - 204*a*b^2*c^(5/2)*d^4*e + 54*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^5*b^2*c*d*e^4 - 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*c^2*d*e^4 + 34*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^3*b^3*c*d^2*e^3 + 264*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c^2*d^2*e^3 + 180*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))*b^4*c*d^3*e^2 + 900*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^2*c^2*d^3*e^2 + 480*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))*a^2*c^3*d^3*e^2 - 75*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^3*sqrt(c)*d*e^4 + 180*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^4*a*b*c^(3/2)*d*e^4 - 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^4*sqrt(c)*d^2*
e^3 - 432*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^2*c^(3/2)*d^2*e^3 + 288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^2*a^2*c^(5/2)*d^2*e^3 + 15*b^5*sqrt(c)*d^3*e^2 + 206*a*b^3*c^(3/2)*d^3*e^2 + 240*a^2*b*c^(5/2)*d^3*e^2 - 15*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^3*e^5 + 36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b*c*e^5 - 40*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^3*b^4*d*e^4 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^2*c*d*e^4 + 192*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*c^2*d*e^4 - 33*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^5*d^2*e^3 - 450*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))*a*b^3*c*d^2*e^3 - 432*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b*c^2*d^2*e^3 +
120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^3*sqrt(c)*d*e^4 - 78*a*b^4*sqrt(c)*d^2*e^3 - 222*a^2*b^2*c^(3/2)
*d^2*e^3 - 88*a^3*c^(5/2)*d^2*e^3 + 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^3*e^5 - 96*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^3*a^2*b*c*e^5 + 66*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^4*d*e^4 + 306*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))*a^2*b^2*c*d*e^4 - 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*c^2*d*e^4 - 96*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^2*a^3*c^(3/2)*e^5 + 111*a^2*b^3*sqrt(c)*d*e^4 + 28*a^3*b*c^(3/2)*d*e^4 - 33*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))*a^2*b^3*e^5 - 36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b*c*e^5 - 48*a^3*b^2*sqrt(c)*e^
5 + 32*a^4*c^(3/2)*e^5)/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*
d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6)*((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*
e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c)*d + b*d - a*e)^3)