3.634 \(\int \frac{\sqrt{d+e x}}{(a+c x^2)^2} \, dx\)

Optimal. Leaf size=675 \[ \frac{e \left (\sqrt{a e^2+c d^2}+\sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a c^{3/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \left (\sqrt{a e^2+c d^2}+\sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a c^{3/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \left (d-\frac{\sqrt{a e^2+c d^2}}{\sqrt{c}}\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a \sqrt [4]{c} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left (d-\frac{\sqrt{a e^2+c d^2}}{\sqrt{c}}\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a \sqrt [4]{c} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{x \sqrt{d+e x}}{2 a \left (a+c x^2\right )} \]

[Out]

(x*Sqrt[d + e*x])/(2*a*(a + c*x^2)) + (e*(Sqrt[c]*d + Sqrt[c*d^2 + 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]]])/(4*Sqrt[2]*a*c^(3/4)*Sqrt
[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(Sqrt[c]*d + Sqrt[c*d^2 + a*e^2])*ArcTanh[(Sqrt[Sq
rt[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]]])/(4*Sqr
t[2]*a*c^(3/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(d - Sqrt[c*d^2 + a*e^2]/Sqrt[c
])*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)])/(8*Sqrt[2]*a*c^(1/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*(d - Sqrt[c*d^2
+ a*e^2]/Sqrt[c])*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)])/(8*Sqrt[2]*a*c^(1/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

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Rubi [A]  time = 1.01755, antiderivative size = 675, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {737, 827, 1169, 634, 618, 206, 628} \[ \frac{e \left (\sqrt{a e^2+c d^2}+\sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a c^{3/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \left (\sqrt{a e^2+c d^2}+\sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a c^{3/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \left (d-\frac{\sqrt{a e^2+c d^2}}{\sqrt{c}}\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a \sqrt [4]{c} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left (d-\frac{\sqrt{a e^2+c d^2}}{\sqrt{c}}\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a \sqrt [4]{c} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{x \sqrt{d+e x}}{2 a \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]/(a + c*x^2)^2,x]

[Out]

(x*Sqrt[d + e*x])/(2*a*(a + c*x^2)) + (e*(Sqrt[c]*d + Sqrt[c*d^2 + 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]]])/(4*Sqrt[2]*a*c^(3/4)*Sqrt
[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(Sqrt[c]*d + Sqrt[c*d^2 + a*e^2])*ArcTanh[(Sqrt[Sq
rt[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]]])/(4*Sqr
t[2]*a*c^(3/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(d - Sqrt[c*d^2 + a*e^2]/Sqrt[c
])*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)])/(8*Sqrt[2]*a*c^(1/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*(d - Sqrt[c*d^2
+ a*e^2]/Sqrt[c])*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)])/(8*Sqrt[2]*a*c^(1/4)*Sqrt[c*d^2 + a*e^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 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 )^2} \, dx &=\frac{x \sqrt{d+e x}}{2 a \left (a+c x^2\right )}-\frac{\int \frac{-d-\frac{e x}{2}}{\sqrt{d+e x} \left (a+c x^2\right )} \, dx}{2 a}\\ &=\frac{x \sqrt{d+e x}}{2 a \left (a+c x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{d e}{2}-\frac{e x^2}{2}}{c d^2+a e^2-2 c d x^2+c x^4} \, dx,x,\sqrt{d+e x}\right )}{a}\\ &=\frac{x \sqrt{d+e x}}{2 a \left (a+c x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{d e \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}{\sqrt{2} \sqrt [4]{c}}-\left (-\frac{d e}{2}+\frac{e \sqrt{c d^2+a e^2}}{2 \sqrt{c}}\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 )}{2 \sqrt{2} a \sqrt [4]{c} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{d e \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}{\sqrt{2} \sqrt [4]{c}}+\left (-\frac{d e}{2}+\frac{e \sqrt{c d^2+a e^2}}{2 \sqrt{c}}\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 )}{2 \sqrt{2} a \sqrt [4]{c} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}\\ &=\frac{x \sqrt{d+e x}}{2 a \left (a+c x^2\right )}+\frac{\left (e \left (1+\frac{\sqrt{c} d}{\sqrt{c d^2+a e^2}}\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 )}{8 a c}+\frac{\left (e \left (1+\frac{\sqrt{c} d}{\sqrt{c d^2+a e^2}}\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 )}{8 a c}-\frac{\left (e \left (d-\frac{\sqrt{c d^2+a e^2}}{\sqrt{c}}\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 )}{8 \sqrt{2} a \sqrt [4]{c} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\left (e \left (d-\frac{\sqrt{c d^2+a e^2}}{\sqrt{c}}\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 )}{8 \sqrt{2} a \sqrt [4]{c} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}\\ &=\frac{x \sqrt{d+e x}}{2 a \left (a+c x^2\right )}-\frac{e \left (d-\frac{\sqrt{c d^2+a e^2}}{\sqrt{c}}\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 )}{8 \sqrt{2} a \sqrt [4]{c} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (d-\frac{\sqrt{c d^2+a e^2}}{\sqrt{c}}\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 )}{8 \sqrt{2} a \sqrt [4]{c} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}-\frac{\left (e \left (1+\frac{\sqrt{c} d}{\sqrt{c d^2+a e^2}}\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 )}{4 a c}-\frac{\left (e \left (1+\frac{\sqrt{c} d}{\sqrt{c d^2+a e^2}}\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 )}{4 a c}\\ &=\frac{x \sqrt{d+e x}}{2 a \left (a+c x^2\right )}+\frac{e \left (1+\frac{\sqrt{c} d}{\sqrt{c d^2+a e^2}}\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 )}{4 \sqrt{2} a c^{3/4} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (1+\frac{\sqrt{c} d}{\sqrt{c d^2+a e^2}}\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 )}{4 \sqrt{2} a c^{3/4} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (d-\frac{\sqrt{c d^2+a e^2}}{\sqrt{c}}\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 )}{8 \sqrt{2} a \sqrt [4]{c} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (d-\frac{\sqrt{c d^2+a e^2}}{\sqrt{c}}\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 )}{8 \sqrt{2} a \sqrt [4]{c} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}\\ \end{align*}

Mathematica [A]  time = 0.588015, size = 265, normalized size = 0.39 \[ \frac{-\frac{\sqrt{\sqrt{c} d-\sqrt{-a} e} \left (2 \sqrt{-a} c d^2-a \sqrt{c} d e+\sqrt{-a} a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{2 a c^{3/4}}+\frac{\sqrt{\sqrt{-a} e+\sqrt{c} d} \left (2 \sqrt{-a} c d^2+a \sqrt{c} d e+\sqrt{-a} a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{2 a c^{3/4}}+\frac{x \sqrt{d+e x} \left (a e^2+c d^2\right )}{a+c x^2}}{2 a \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((c*d^2 + a*e^2)*x*Sqrt[d + e*x])/(a + c*x^2) - (Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(2*Sqrt[-a]*c*d^2 - a*Sqrt[c]*d
*e + Sqrt[-a]*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]])/(2*a*c^(3/4)) + (Sqrt[Sqrt
[c]*d + Sqrt[-a]*e]*(2*Sqrt[-a]*c*d^2 + a*Sqrt[c]*d*e + Sqrt[-a]*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[S
qrt[c]*d + Sqrt[-a]*e]])/(2*a*c^(3/4)))/(2*a*(c*d^2 + a*e^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 ) ^{2}}\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)^2,x)

[Out]

int((e*x+d)^(1/2)/(c*x^2+a)^2,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 )}^{2}}\,{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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.46788, size = 2593, normalized size = 3.84 \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)^2,x, algorithm="fricas")

[Out]

1/8*((a*c*x^2 + a^2)*sqrt(-(4*c*d^3 + 3*a*d*e^2 + (a^3*c^2*d^2 + a^4*c*e^2)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4
*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 + a^4*c*e^2))*log((4*c*d^2*e^3 + a*e^5)*sqrt(e*x + d) + (a^2*c*d*e^4 -
(2*a^3*c^4*d^4 + 3*a^4*c^3*d^2*e^2 + a^5*c^2*e^4)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))*
sqrt(-(4*c*d^3 + 3*a*d*e^2 + (a^3*c^2*d^2 + a^4*c*e^2)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^
4)))/(a^3*c^2*d^2 + a^4*c*e^2))) - (a*c*x^2 + a^2)*sqrt(-(4*c*d^3 + 3*a*d*e^2 + (a^3*c^2*d^2 + a^4*c*e^2)*sqrt
(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 + a^4*c*e^2))*log((4*c*d^2*e^3 + a*e^5)*s
qrt(e*x + d) - (a^2*c*d*e^4 - (2*a^3*c^4*d^4 + 3*a^4*c^3*d^2*e^2 + a^5*c^2*e^4)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4
*c^4*d^2*e^2 + a^5*c^3*e^4)))*sqrt(-(4*c*d^3 + 3*a*d*e^2 + (a^3*c^2*d^2 + a^4*c*e^2)*sqrt(-e^6/(a^3*c^5*d^4 +
2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 + a^4*c*e^2))) + (a*c*x^2 + a^2)*sqrt(-(4*c*d^3 + 3*a*d*e^2 -
(a^3*c^2*d^2 + a^4*c*e^2)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 + a^4*c*e^2
))*log((4*c*d^2*e^3 + a*e^5)*sqrt(e*x + d) + (a^2*c*d*e^4 + (2*a^3*c^4*d^4 + 3*a^4*c^3*d^2*e^2 + a^5*c^2*e^4)*
sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))*sqrt(-(4*c*d^3 + 3*a*d*e^2 - (a^3*c^2*d^2 + a^4*c*
e^2)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 + a^4*c*e^2))) - (a*c*x^2 + a^2)
*sqrt(-(4*c*d^3 + 3*a*d*e^2 - (a^3*c^2*d^2 + a^4*c*e^2)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e
^4)))/(a^3*c^2*d^2 + a^4*c*e^2))*log((4*c*d^2*e^3 + a*e^5)*sqrt(e*x + d) - (a^2*c*d*e^4 + (2*a^3*c^4*d^4 + 3*a
^4*c^3*d^2*e^2 + a^5*c^2*e^4)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))*sqrt(-(4*c*d^3 + 3*a
*d*e^2 - (a^3*c^2*d^2 + a^4*c*e^2)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 +
a^4*c*e^2))) + 4*sqrt(e*x + d)*x)/(a*c*x^2 + a^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)**2,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)^2,x, algorithm="giac")

[Out]

Timed out