### 3.646 $$\int \frac{\sqrt{d+e x}}{(a+c x^2)^3} \, dx$$

Optimal. Leaf size=849 $\frac{\sqrt{d+e x} x}{4 a \left (c x^2+a\right )^2}+\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\sqrt{d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (c x^2+a\right )}$

[Out]

(x*Sqrt[d + e*x])/(4*a*(a + c*x^2)^2) + (Sqrt[d + e*x]*(a*d*e + (6*c*d^2 + 5*a*e^2)*x))/(16*a^2*(c*d^2 + a*e^2
)*(a + c*x^2)) + (e*(6*c^(3/2)*d^3 + 8*a*Sqrt[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(6*c*d^2 + 5*a*e^2))*ArcTanh[(Sqr
t[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] - Sqrt[2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(3
2*Sqrt[2]*a^2*c^(3/4)*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(6*c^(3/2)*d^3 + 8*a*S
qrt[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(6*c*d^2 + 5*a*e^2))*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] + Sqrt[
2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(32*Sqrt[2]*a^2*c^(3/4)*(c*d^2 + a*e^2)^(3/2
)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(6*c^(3/2)*d^3 + 8*a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(6*c*d^
2 + 5*a*e^2))*Log[Sqrt[c*d^2 + a*e^2] - Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] +
Sqrt[c]*(d + e*x)])/(64*Sqrt[2]*a^2*c^(3/4)*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*
(6*c^(3/2)*d^3 + 8*a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(6*c*d^2 + 5*a*e^2))*Log[Sqrt[c*d^2 + a*e^2] + Sqrt[2
]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(64*Sqrt[2]*a^2*c^(3/4)*(c
*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

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Rubi [A]  time = 2.88348, antiderivative size = 849, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.421, Rules used = {737, 823, 827, 1169, 634, 618, 206, 628} $\frac{\sqrt{d+e x} x}{4 a \left (c x^2+a\right )^2}+\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\sqrt{d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (c x^2+a\right )}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[d + e*x]/(a + c*x^2)^3,x]

[Out]

(x*Sqrt[d + e*x])/(4*a*(a + c*x^2)^2) + (Sqrt[d + e*x]*(a*d*e + (6*c*d^2 + 5*a*e^2)*x))/(16*a^2*(c*d^2 + a*e^2
)*(a + c*x^2)) + (e*(6*c^(3/2)*d^3 + 8*a*Sqrt[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(6*c*d^2 + 5*a*e^2))*ArcTanh[(Sqr
t[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] - Sqrt[2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(3
2*Sqrt[2]*a^2*c^(3/4)*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(6*c^(3/2)*d^3 + 8*a*S
qrt[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(6*c*d^2 + 5*a*e^2))*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] + Sqrt[
2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(32*Sqrt[2]*a^2*c^(3/4)*(c*d^2 + a*e^2)^(3/2
)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(6*c^(3/2)*d^3 + 8*a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(6*c*d^
2 + 5*a*e^2))*Log[Sqrt[c*d^2 + a*e^2] - Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] +
Sqrt[c]*(d + e*x)])/(64*Sqrt[2]*a^2*c^(3/4)*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*
(6*c^(3/2)*d^3 + 8*a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(6*c*d^2 + 5*a*e^2))*Log[Sqrt[c*d^2 + a*e^2] + Sqrt[2
]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(64*Sqrt[2]*a^2*c^(3/4)*(c
*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

Rule 737

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

Rule 823

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

Rule 827

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

Rule 1169

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

Rule 634

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

Rule 618

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

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{d+e x}}{\left (a+c x^2\right )^3} \, dx &=\frac{x \sqrt{d+e x}}{4 a \left (a+c x^2\right )^2}-\frac{\int \frac{-3 d-\frac{5 e x}{2}}{\sqrt{d+e x} \left (a+c x^2\right )^2} \, dx}{4 a}\\ &=\frac{x \sqrt{d+e x}}{4 a \left (a+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac{\int \frac{\frac{1}{4} c d \left (12 c d^2+13 a e^2\right )+\frac{1}{4} c e \left (6 c d^2+5 a e^2\right ) x}{\sqrt{d+e x} \left (a+c x^2\right )} \, dx}{8 a^2 c \left (c d^2+a e^2\right )}\\ &=\frac{x \sqrt{d+e x}}{4 a \left (a+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{4} c d e \left (6 c d^2+5 a e^2\right )+\frac{1}{4} c d e \left (12 c d^2+13 a e^2\right )+\frac{1}{4} c e \left (6 c d^2+5 a e^2\right ) x^2}{c d^2+a e^2-2 c d x^2+c x^4} \, dx,x,\sqrt{d+e x}\right )}{4 a^2 c \left (c d^2+a e^2\right )}\\ &=\frac{x \sqrt{d+e x}}{4 a \left (a+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \left (-\frac{1}{4} c d e \left (6 c d^2+5 a e^2\right )+\frac{1}{4} c d e \left (12 c d^2+13 a e^2\right )\right )}{\sqrt [4]{c}}-\left (-\frac{1}{4} c d e \left (6 c d^2+5 a e^2\right )-\frac{1}{4} \sqrt{c} e \sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )+\frac{1}{4} c d e \left (12 c d^2+13 a e^2\right )\right ) x}{\frac{\sqrt{c d^2+a e^2}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{d+e x}\right )}{8 \sqrt{2} a^2 c^{5/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \left (-\frac{1}{4} c d e \left (6 c d^2+5 a e^2\right )+\frac{1}{4} c d e \left (12 c d^2+13 a e^2\right )\right )}{\sqrt [4]{c}}+\left (-\frac{1}{4} c d e \left (6 c d^2+5 a e^2\right )-\frac{1}{4} \sqrt{c} e \sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )+\frac{1}{4} c d e \left (12 c d^2+13 a e^2\right )\right ) x}{\frac{\sqrt{c d^2+a e^2}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{d+e x}\right )}{8 \sqrt{2} a^2 c^{5/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}\\ &=\frac{x \sqrt{d+e x}}{4 a \left (a+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac{\left (e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{-\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac{\sqrt{c d^2+a e^2}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{d+e x}\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\left (e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac{\sqrt{c d^2+a e^2}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{d+e x}\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\left (e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{c d^2+a e^2}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{d+e x}\right )}{64 a^2 c \left (c d^2+a e^2\right )^{3/2}}+\frac{\left (e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{c d^2+a e^2}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{d+e x}\right )}{64 a^2 c \left (c d^2+a e^2\right )^{3/2}}\\ &=\frac{x \sqrt{d+e x}}{4 a \left (a+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt{c d^2+a e^2}-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c} (d+e x)\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt{c d^2+a e^2}+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c} (d+e x)\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}-\frac{\left (e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{2 \left (d-\frac{\sqrt{c d^2+a e^2}}{\sqrt{c}}\right )-x^2} \, dx,x,-\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt{d+e x}\right )}{32 a^2 c \left (c d^2+a e^2\right )^{3/2}}-\frac{\left (e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{2 \left (d-\frac{\sqrt{c d^2+a e^2}}{\sqrt{c}}\right )-x^2} \, dx,x,\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt{d+e x}\right )}{32 a^2 c \left (c d^2+a e^2\right )^{3/2}}\\ &=\frac{x \sqrt{d+e x}}{4 a \left (a+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}{\sqrt [4]{c}}-\sqrt{2} \sqrt{d+e x}\right )}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}{\sqrt [4]{c}}+\sqrt{2} \sqrt{d+e x}\right )}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt{c d^2+a e^2}-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c} (d+e x)\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt{c d^2+a e^2}+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c} (d+e x)\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}\\ \end{align*}

Mathematica [A]  time = 0.870625, size = 412, normalized size = 0.49 $\frac{\frac{2 (d+e x)^{3/2} \left (5 a^2 e^3+a c d e (3 d+8 e x)+6 c^2 d^3 x\right )}{a+c x^2}+\frac{\sqrt{\sqrt{c} d-\sqrt{-a} e} \left (5 a^2 e^4+6 \sqrt{-a} c^{3/2} d^3 e+19 a c d^2 e^2+8 \sqrt{-a} a \sqrt{c} d e^3+12 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )-\sqrt{\sqrt{-a} e+\sqrt{c} d} \left (5 a^2 e^4-6 \sqrt{-a} c^{3/2} d^3 e+19 a c d^2 e^2+8 (-a)^{3/2} \sqrt{c} d e^3+12 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )-4 \sqrt{-a} c^{3/4} d e \sqrt{d+e x} \left (4 a e^2+3 c d^2\right )}{\sqrt{-a} c^{3/4}}+\frac{8 a (d+e x)^{3/2} \left (a e^2+c d^2\right ) (a e+c d x)}{\left (a+c x^2\right )^2}}{32 a^2 \left (a e^2+c d^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((8*a*(c*d^2 + a*e^2)*(a*e + c*d*x)*(d + e*x)^(3/2))/(a + c*x^2)^2 + (2*(d + e*x)^(3/2)*(5*a^2*e^3 + 6*c^2*d^3
*x + a*c*d*e*(3*d + 8*e*x)))/(a + c*x^2) + (-4*Sqrt[-a]*c^(3/4)*d*e*(3*c*d^2 + 4*a*e^2)*Sqrt[d + e*x] + Sqrt[S
qrt[c]*d - Sqrt[-a]*e]*(12*c^2*d^4 + 6*Sqrt[-a]*c^(3/2)*d^3*e + 19*a*c*d^2*e^2 + 8*Sqrt[-a]*a*Sqrt[c]*d*e^3 +
5*a^2*e^4)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]] - Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(12*c^
2*d^4 - 6*Sqrt[-a]*c^(3/2)*d^3*e + 19*a*c*d^2*e^2 + 8*(-a)^(3/2)*Sqrt[c]*d*e^3 + 5*a^2*e^4)*ArcTanh[(c^(1/4)*S
qrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]])/(Sqrt[-a]*c^(3/4)))/(32*a^2*(c*d^2 + a*e^2)^2)

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{3}}\sqrt{ex+d}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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} + 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+a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 4.63693, size = 7772, normalized size = 9.15 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*((a^4*c*d^2 + a^5*e^2 + (a^2*c^3*d^2 + a^3*c^2*e^2)*x^4 + 2*(a^3*c^2*d^2 + a^4*c*e^2)*x^2)*sqrt(-(144*c^3
*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 + (a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^
2*e^4 + a^8*c*e^6)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*
e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a
^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6))*log((3024*c^3*d^6*e^5 + 7884*a*c^2*d^4*e^7 +
5625*a^2*c*d^2*e^9 + 625*a^3*e^11)*sqrt(e*x + d) + (126*a^3*c^3*d^5*e^6 + 318*a^4*c^2*d^3*e^8 + 200*a^5*c*d*e^
10 - (12*a^5*c^7*d^10 + 55*a^6*c^6*d^8*e^2 + 98*a^7*c^5*d^6*e^4 + 84*a^8*c^4*d^4*e^6 + 34*a^9*c^3*d^2*e^8 + 5*
a^10*c^2*e^10)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2
+ 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))*sqrt(-
(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 + (a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^
7*c^2*d^2*e^4 + a^8*c*e^6)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c
^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^
12)))/(a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6))) - (a^4*c*d^2 + a^5*e^2 + (a^2*c^3*d^
2 + a^3*c^2*e^2)*x^4 + 2*(a^3*c^2*d^2 + a^4*c*e^2)*x^2)*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3
*e^4 + 105*a^3*d*e^6 + (a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6)*sqrt(-(441*c^2*d^4*e^
10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^
6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c
^2*d^2*e^4 + a^8*c*e^6))*log((3024*c^3*d^6*e^5 + 7884*a*c^2*d^4*e^7 + 5625*a^2*c*d^2*e^9 + 625*a^3*e^11)*sqrt(
e*x + d) - (126*a^3*c^3*d^5*e^6 + 318*a^4*c^2*d^3*e^8 + 200*a^5*c*d*e^10 - (12*a^5*c^7*d^10 + 55*a^6*c^6*d^8*e
^2 + 98*a^7*c^5*d^6*e^4 + 84*a^8*c^4*d^4*e^6 + 34*a^9*c^3*d^2*e^8 + 5*a^10*c^2*e^10)*sqrt(-(441*c^2*d^4*e^10 +
1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^
6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a
^2*c*d^3*e^4 + 105*a^3*d*e^6 + (a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6)*sqrt(-(441*c^
2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^
8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 +
3*a^7*c^2*d^2*e^4 + a^8*c*e^6))) + (a^4*c*d^2 + a^5*e^2 + (a^2*c^3*d^2 + a^3*c^2*e^2)*x^4 + 2*(a^3*c^2*d^2 +
a^4*c*e^2)*x^2)*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - (a^5*c^4*d^6 + 3*
a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(
a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*
d^2*e^10 + a^11*c^3*e^12)))/(a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6))*log((3024*c^3*d
^6*e^5 + 7884*a*c^2*d^4*e^7 + 5625*a^2*c*d^2*e^9 + 625*a^3*e^11)*sqrt(e*x + d) + (126*a^3*c^3*d^5*e^6 + 318*a^
4*c^2*d^3*e^8 + 200*a^5*c*d*e^10 + (12*a^5*c^7*d^10 + 55*a^6*c^6*d^8*e^2 + 98*a^7*c^5*d^6*e^4 + 84*a^8*c^4*d^4
*e^6 + 34*a^9*c^3*d^2*e^8 + 5*a^10*c^2*e^10)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*
c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*
e^10 + a^11*c^3*e^12)))*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - (a^5*c^4*
d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2
*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a
^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6))) - (a^4
*c*d^2 + a^5*e^2 + (a^2*c^3*d^2 + a^3*c^2*e^2)*x^4 + 2*(a^3*c^2*d^2 + a^4*c*e^2)*x^2)*sqrt(-(144*c^3*d^7 + 420
*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - (a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^
8*c*e^6)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a
^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5*c^4*d^6
+ 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6))*log((3024*c^3*d^6*e^5 + 7884*a*c^2*d^4*e^7 + 5625*a^2*c
*d^2*e^9 + 625*a^3*e^11)*sqrt(e*x + d) - (126*a^3*c^3*d^5*e^6 + 318*a^4*c^2*d^3*e^8 + 200*a^5*c*d*e^10 + (12*a
^5*c^7*d^10 + 55*a^6*c^6*d^8*e^2 + 98*a^7*c^5*d^6*e^4 + 84*a^8*c^4*d^4*e^6 + 34*a^9*c^3*d^2*e^8 + 5*a^10*c^2*e
^10)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a^7*c
^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))*sqrt(-(144*c^3*d
^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - (a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*
e^4 + a^8*c*e^6)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^
2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5
*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6))) + 4*(a*c*d*e*x^2 + a^2*d*e + (6*c^2*d^2 + 5*a*
c*e^2)*x^3 + (10*a*c*d^2 + 9*a^2*e^2)*x)*sqrt(e*x + d))/(a^4*c*d^2 + a^5*e^2 + (a^2*c^3*d^2 + a^3*c^2*e^2)*x^4
+ 2*(a^3*c^2*d^2 + a^4*c*e^2)*x^2)

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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+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+a)^3,x, algorithm="giac")

[Out]

Timed out