### 3.2305 $$\int \frac{\sqrt{d+e x}}{(a+b x+c x^2)^3} \, dx$$

Optimal. Leaf size=634 $-\frac{\sqrt{d+e x} \left (-c x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-4 b c \left (2 a e^2+3 c d^2\right )-4 a c^2 d e+13 b^2 c d e+b^3 \left (-e^2\right )\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{c} \left (-8 c^2 d e \left (-3 d \sqrt{b^2-4 a c}-13 a e+18 b d\right )+2 c e^2 \left (-2 b \left (6 d \sqrt{b^2-4 a c}+13 a e\right )+10 a e \sqrt{b^2-4 a c}+23 b^2 d\right )+b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+96 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{c} \left (-8 c^2 d e \left (3 d \sqrt{b^2-4 a c}-13 a e+18 b d\right )+2 c e^2 \left (12 b d \sqrt{b^2-4 a c}-10 a e \sqrt{b^2-4 a c}-26 a b e+23 b^2 d\right )+b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+96 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac{(b+2 c x) \sqrt{d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}$

[Out]

-((b + 2*c*x)*Sqrt[d + e*x])/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (Sqrt[d + e*x]*(13*b^2*c*d*e - 4*a*c^2*d*
e - b^3*e^2 - 4*b*c*(3*c*d^2 + 2*a*e^2) - c*(24*c^2*d^2 + b^2*e^2 - 4*c*e*(6*b*d - 5*a*e))*x))/(4*(b^2 - 4*a*c
)^2*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)) - (Sqrt[c]*(96*c^3*d^3 + b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 8*c^
2*d*e*(18*b*d - 3*Sqrt[b^2 - 4*a*c]*d - 13*a*e) + 2*c*e^2*(23*b^2*d + 10*a*Sqrt[b^2 - 4*a*c]*e - 2*b*(6*Sqrt[b
^2 - 4*a*c]*d + 13*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(4
*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2)) + (Sqrt[c]*(96*c
^3*d^3 + b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2*d*e*(18*b*d + 3*Sqrt[b^2 - 4*a*c]*d - 13*a*e) + 2*c*e^2*(23*b
^2*d + 12*b*Sqrt[b^2 - 4*a*c]*d - 26*a*b*e - 10*a*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x]
)/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c]
)*e]*(c*d^2 - b*d*e + a*e^2))

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Rubi [A]  time = 4.97375, antiderivative size = 634, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.227, Rules used = {736, 822, 826, 1166, 208} $-\frac{\sqrt{d+e x} \left (-c x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-4 b c \left (2 a e^2+3 c d^2\right )-4 a c^2 d e+13 b^2 c d e+b^3 \left (-e^2\right )\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{c} \left (-8 c^2 d e \left (-3 d \sqrt{b^2-4 a c}-13 a e+18 b d\right )+2 c e^2 \left (-2 b \left (6 d \sqrt{b^2-4 a c}+13 a e\right )+10 a e \sqrt{b^2-4 a c}+23 b^2 d\right )+b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+96 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{c} \left (-8 c^2 d e \left (3 d \sqrt{b^2-4 a c}-13 a e+18 b d\right )+2 c e^2 \left (12 b d \sqrt{b^2-4 a c}-10 a e \sqrt{b^2-4 a c}-26 a b e+23 b^2 d\right )+b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+96 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac{(b+2 c x) \sqrt{d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((b + 2*c*x)*Sqrt[d + e*x])/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (Sqrt[d + e*x]*(13*b^2*c*d*e - 4*a*c^2*d*
e - b^3*e^2 - 4*b*c*(3*c*d^2 + 2*a*e^2) - c*(24*c^2*d^2 + b^2*e^2 - 4*c*e*(6*b*d - 5*a*e))*x))/(4*(b^2 - 4*a*c
)^2*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)) - (Sqrt[c]*(96*c^3*d^3 + b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 8*c^
2*d*e*(18*b*d - 3*Sqrt[b^2 - 4*a*c]*d - 13*a*e) + 2*c*e^2*(23*b^2*d + 10*a*Sqrt[b^2 - 4*a*c]*e - 2*b*(6*Sqrt[b
^2 - 4*a*c]*d + 13*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(4
*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2)) + (Sqrt[c]*(96*c
^3*d^3 + b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2*d*e*(18*b*d + 3*Sqrt[b^2 - 4*a*c]*d - 13*a*e) + 2*c*e^2*(23*b
^2*d + 12*b*Sqrt[b^2 - 4*a*c]*d - 26*a*b*e - 10*a*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x]
)/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c]
)*e]*(c*d^2 - b*d*e + a*e^2))

Rule 736

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

Rule 822

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

Rule 826

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

Rule 1166

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

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]

Rubi steps

\begin{align*} \int \frac{\sqrt{d+e x}}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac{(b+2 c x) \sqrt{d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{\int \frac{-6 c d+\frac{b e}{2}-5 c e x}{\sqrt{d+e x} \left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=-\frac{(b+2 c x) \sqrt{d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{\sqrt{d+e x} \left (13 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (3 c d^2+2 a e^2\right )-c \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac{\int \frac{\frac{1}{4} \left (-48 c^3 d^3-b^3 e^3-b c e^2 (11 b d-16 a e)+4 c^2 d e (15 b d-13 a e)\right )-\frac{1}{4} c e \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x}{\sqrt{d+e x} \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{(b+2 c x) \sqrt{d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{\sqrt{d+e x} \left (13 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (3 c d^2+2 a e^2\right )-c \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{4} e \left (-48 c^3 d^3-b^3 e^3-b c e^2 (11 b d-16 a e)+4 c^2 d e (15 b d-13 a e)\right )+\frac{1}{4} c d e \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right )-\frac{1}{4} c e \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt{d+e x}\right )}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{(b+2 c x) \sqrt{d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{\sqrt{d+e x} \left (13 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (3 c d^2+2 a e^2\right )-c \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac{\left (c \left (96 c^3 d^3+b^2 \left (b-\sqrt{b^2-4 a c}\right ) e^3-8 c^2 d e \left (18 b d+3 \sqrt{b^2-4 a c} d-13 a e\right )+2 c e^2 \left (23 b^2 d+12 b \sqrt{b^2-4 a c} d-26 a b e-10 a \sqrt{b^2-4 a c} e\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{2} \sqrt{b^2-4 a c} e+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{8 \left (b^2-4 a c\right )^{5/2} \left (c d^2-b d e+a e^2\right )}+\frac{\left (c \left (96 c^3 d^3+b^2 \left (b+\sqrt{b^2-4 a c}\right ) e^3-8 c^2 d e \left (18 b d-3 \sqrt{b^2-4 a c} d-13 a e\right )+2 c e^2 \left (23 b^2 d+10 a \sqrt{b^2-4 a c} e-2 b \left (6 \sqrt{b^2-4 a c} d+13 a e\right )\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2} \sqrt{b^2-4 a c} e+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{8 \left (b^2-4 a c\right )^{5/2} \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{(b+2 c x) \sqrt{d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{\sqrt{d+e x} \left (13 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (3 c d^2+2 a e^2\right )-c \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac{\sqrt{c} \left (96 c^3 d^3+b^2 \left (b+\sqrt{b^2-4 a c}\right ) e^3-8 c^2 d e \left (18 b d-3 \sqrt{b^2-4 a c} d-13 a e\right )+2 c e^2 \left (23 b^2 d+10 a \sqrt{b^2-4 a c} e-2 b \left (6 \sqrt{b^2-4 a c} d+13 a e\right )\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}+\frac{\sqrt{c} \left (96 c^3 d^3+b^2 \left (b-\sqrt{b^2-4 a c}\right ) e^3-8 c^2 d e \left (18 b d+3 \sqrt{b^2-4 a c} d-13 a e\right )+2 c e^2 \left (23 b^2 d+12 b \sqrt{b^2-4 a c} d-26 a b e-10 a \sqrt{b^2-4 a c} e\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}\\ \end{align*}

Mathematica [A]  time = 4.29806, size = 580, normalized size = 0.91 $-\frac{\sqrt{d+e x} \left (4 b c \left (2 a e^2+3 c d (d-2 e x)\right )+4 c^2 \left (a e (d+5 e x)+6 c d^2 x\right )+b^2 c e (e x-13 d)+b^3 e^2\right )}{4 \left (b^2-4 a c\right )^2 (a+x (b+c x)) \left (e (b d-a e)-c d^2\right )}-\frac{\sqrt{c} \left (\frac{\left (8 c^2 d e \left (3 d \sqrt{b^2-4 a c}+13 a e-18 b d\right )+2 c e^2 \left (-2 b \left (6 d \sqrt{b^2-4 a c}+13 a e\right )+10 a e \sqrt{b^2-4 a c}+23 b^2 d\right )+b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+96 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}}-\frac{\left (-8 c^2 d e \left (3 d \sqrt{b^2-4 a c}-13 a e+18 b d\right )+2 c e^2 \left (12 b d \sqrt{b^2-4 a c}-10 a e \sqrt{b^2-4 a c}-26 a b e+23 b^2 d\right )+b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+96 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \left (e (a e-b d)+c d^2\right )}-\frac{(b+2 c x) \sqrt{d+e x}}{2 \left (b^2-4 a c\right ) (a+x (b+c x))^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((b + 2*c*x)*Sqrt[d + e*x])/(2*(b^2 - 4*a*c)*(a + x*(b + c*x))^2) - (Sqrt[d + e*x]*(b^3*e^2 + b^2*c*e*(-13*d
+ e*x) + 4*b*c*(2*a*e^2 + 3*c*d*(d - 2*e*x)) + 4*c^2*(6*c*d^2*x + a*e*(d + 5*e*x))))/(4*(b^2 - 4*a*c)^2*(-(c*d
^2) + e*(b*d - a*e))*(a + x*(b + c*x))) - (Sqrt[c]*(((96*c^3*d^3 + b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 + 8*c^2*d*e
*(-18*b*d + 3*Sqrt[b^2 - 4*a*c]*d + 13*a*e) + 2*c*e^2*(23*b^2*d + 10*a*Sqrt[b^2 - 4*a*c]*e - 2*b*(6*Sqrt[b^2 -
4*a*c]*d + 13*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/Sqrt[2
*c*d + (-b + Sqrt[b^2 - 4*a*c])*e] - ((96*c^3*d^3 + b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2*d*e*(18*b*d + 3*Sq
rt[b^2 - 4*a*c]*d - 13*a*e) + 2*c*e^2*(23*b^2*d + 12*b*Sqrt[b^2 - 4*a*c]*d - 26*a*b*e - 10*a*Sqrt[b^2 - 4*a*c]
*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/Sqrt[2*c*d - (b + Sqrt[
b^2 - 4*a*c])*e]))/(4*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*(c*d^2 + e*(-(b*d) + a*e)))

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Maple [B]  time = 0.352, size = 3360, normalized size = 5.3 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-40*e^3*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(4*b*e-8*c*d+4*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((b*e-2*
c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)
^(1/2))*a-3*e^2*c/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2*(-4*
a*c*e^2+b^2*e^2)^(1/2)/(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(3/2)*b+6*e*c^2/(-e^2*(4*a*c-b^2))^(1/2
)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2*(-4*a*c*e^2+b^2*e^2)^(1/2)/(-b*e+2*c*d+(-4*a*
c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(3/2)*d+40*e^3*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(-4*b*e+8*c*d+4*(-4*a*
c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/
((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a-34*e^3*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(-4*b*e+8
*c*d+4*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/
2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^2-96*e*c^4/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2
)^2/(-4*b*e+8*c*d+4*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan
h((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*d^2+96*e*c^4/(-e^2*(4*a*c-b^2))^(1/
2)/(4*a*c-b^2)^2/(4*b*e-8*c*d+4*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(
1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*d^2+34*e^3*c^2/(-e^2*(4*a*
c-b^2))^(1/2)/(4*a*c-b^2)^2/(4*b*e-8*c*d+4*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^
(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^2+96*e^2*c^3/
(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(-4*b*e+8*c*d+4*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((-b*e+2*c*d+(-e^2*
(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*
b*d+36*e^2*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(-4*b*e+8*c*d+4*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((-b
*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1
/2))*c)^(1/2))*(-4*a*c*e^2+b^2*e^2)^(1/2)*b-72*e*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(-4*b*e+8*c*d+4*(-
4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1
/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*(-4*a*c*e^2+b^2*e^2)^(1/2)*d-96*e^2*c^3/(-e^2*(4*a*c-b^2)
)^(1/2)/(4*a*c-b^2)^2/(4*b*e-8*c*d+4*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))
*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b*d+36*e^2*c^2/(-e^2*
(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(4*b*e-8*c*d+4*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b
^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*(-4*a*c*e^
2+b^2*e^2)^(1/2)*b-3*e^2*c/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2
))^2*(-4*a*c*e^2+b^2*e^2)^(1/2)/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(3/2)*b+3/2*e^2/(-e^2*(4*a*c-b
^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2*(-4*a*c*e^2+b^2*e^2)^(1/2)*(e*x+d)^(
1/2)*b+7*e^3*c/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2*(e*x+d)
^(1/2)*a-7*e^3*c/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2*(e*x+
d)^(1/2)*a+3/2*e^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2*(-4
*a*c*e^2+b^2*e^2)^(1/2)*(e*x+d)^(1/2)*b-3*e*c/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c+1/2/c*(e^2
*(-4*a*c+b^2))^(1/2))^2*(-4*a*c*e^2+b^2*e^2)^(1/2)*(e*x+d)^(1/2)*d+10*e^3*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-
b^2)^2/(e*x+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(3/2)*
a-5/2*e^3*c/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-b*e+2*c*
d-(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(3/2)*b^2-3*e*c/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c-1/
2/c*(e^2*(-4*a*c+b^2))^(1/2))^2*(-4*a*c*e^2+b^2*e^2)^(1/2)*(e*x+d)^(1/2)*d-10*e^3*c^2/(-e^2*(4*a*c-b^2))^(1/2)
/(4*a*c-b^2)^2/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d
)^(3/2)*a+5/2*e^3*c/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-
b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(3/2)*b^2-72*e*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(4*b*e
-8*c*d+4*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/
2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*(-4*a*c*e^2+b^2*e^2)^(1/2)*d+6*e*c^2/(-e^2*(4*a*c
-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2*(-4*a*c*e^2+b^2*e^2)^(1/2)/(-b*e+2
*c*d-(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(3/2)*d-7/4*e^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c
-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2*(e*x+d)^(1/2)*b^2+7/4*e^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b
*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2*(e*x+d)^(1/2)*b^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x + d}}{{\left (c x^{2} + b x + a\right )}^{3}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(e*x + d)/(c*x^2 + b*x + a)^3, x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out