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

Optimal. Leaf size=478 \[ \frac{e \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} c^{3/4} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}-\frac{e \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} c^{3/4} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} c^{3/4} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} c^{3/4} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}} \]

[Out]

(e*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]]])/(Sqrt[2]*c^(3/4)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*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]]])/(Sqrt[2]*c^(3/4)*Sqrt[Sqrt
[c]*d - Sqrt[c*d^2 + a*e^2]]) + (e*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)])/(2*Sqrt[2]*c^(3/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) - (e*Log[Sq
rt[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)])/
(2*Sqrt[2]*c^(3/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

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Rubi [A]  time = 0.404863, antiderivative size = 478, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {700, 1129, 634, 618, 206, 628} \[ \frac{e \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} c^{3/4} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}-\frac{e \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} c^{3/4} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} c^{3/4} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} c^{3/4} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]/(a + c*x^2),x]

[Out]

(e*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]]])/(Sqrt[2]*c^(3/4)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*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]]])/(Sqrt[2]*c^(3/4)*Sqrt[Sqrt
[c]*d - Sqrt[c*d^2 + a*e^2]]) + (e*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)])/(2*Sqrt[2]*c^(3/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) - (e*Log[Sq
rt[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)])/
(2*Sqrt[2]*c^(3/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

Rule 700

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

Rule 1129

Int[(x_)^(m_.)/((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*r), Int[x^(m - 1)/(q - r*x + x^2), x], x] - Dist[1/(2*c*r), Int[x^(m - 1)/(q + r*x + x^2),
x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 1] && LtQ[m, 3] && 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}}{a+c x^2} \, dx &=(2 e) \operatorname{Subst}\left (\int \frac{x^2}{c d^2+a e^2-2 c d x^2+c x^4} \, dx,x,\sqrt{d+e x}\right )\\ &=\frac{e \operatorname{Subst}\left (\int \frac{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 )}{\sqrt{2} c^{3/4} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}-\frac{e \operatorname{Subst}\left (\int \frac{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 )}{\sqrt{2} c^{3/4} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}\\ &=\frac{e \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 )}{2 c}+\frac{e \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 )}{2 c}+\frac{e \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 )}{2 \sqrt{2} c^{3/4} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}-\frac{e \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 )}{2 \sqrt{2} c^{3/4} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}\\ &=\frac{e \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 )}{2 \sqrt{2} c^{3/4} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}-\frac{e \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 )}{2 \sqrt{2} c^{3/4} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}-\frac{e \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 )}{c}-\frac{e \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 )}{c}\\ &=\frac{e \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 )}{\sqrt{2} c^{3/4} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \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 )}{\sqrt{2} c^{3/4} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}+\frac{e \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 )}{2 \sqrt{2} c^{3/4} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}-\frac{e \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 )}{2 \sqrt{2} c^{3/4} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}\\ \end{align*}

Mathematica [A]  time = 0.0923689, size = 135, normalized size = 0.28 \[ \frac{\sqrt{\sqrt{c} d-\sqrt{-a} e} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )-\sqrt{\sqrt{-a} e+\sqrt{c} d} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} c^{3/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]] - Sqrt[Sqrt[c]*d +
Sqrt[-a]*e]*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]])/(Sqrt[-a]*c^(3/4))

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Maple [B]  time = 0.245, size = 1176, normalized size = 2.5 \begin{align*} \text{result 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+a),x)

[Out]

-1/4*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)/a/e/c^(1/2)*ln((e*x+d)*c^(1/2)-(e*x+d)^(1/2)*(2*(c*(a*e^2+c*d^2))
^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))*d+1/4*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)/a/c^(3/2)/e*ln((e*x+d)*
c^(1/2)-(e*x+d)^(1/2)*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(a*c*e^2+c^2*d^2)^(1/2)-1/2
*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)/a/e/c^(1/2)*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)/(4*(a*e^2+c*d^2)^
(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)-(2*(c*(a*e^2+c*d^2))^(1/2
)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*d+1/2*(2*(a*c*e^2+c^2*d
^2)^(1/2)+2*c*d)^(1/2)/a/c^(3/2)/e*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c
*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)-(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4
*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(a*c*e^2+c^2*d^2)^(1/2)+1/4*(2*(a*c*e^2+c
^2*d^2)^(1/2)+2*c*d)^(1/2)/a/e/c^(1/2)*ln((e*x+d)*c^(1/2)+(e*x+d)^(1/2)*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2
)+(a*e^2+c*d^2)^(1/2))*d-1/4*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)/a/c^(3/2)/e*ln((e*x+d)*c^(1/2)+(e*x+d)^(1
/2)*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(a*c*e^2+c^2*d^2)^(1/2)-1/2*(2*(a*c*e^2+c^2*d
^2)^(1/2)+2*c*d)^(1/2)/a/e/c^(1/2)*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c
*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4
*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*d+1/2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(
1/2)/a/c^(3/2)/e*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1
/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/
2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(a*c*e^2+c^2*d^2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x + d}}{c x^{2} + a}\,{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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.9146, size = 743, normalized size = 1.55 \begin{align*} -\frac{1}{2} \, \sqrt{-\frac{a c \sqrt{-\frac{e^{2}}{a c^{3}}} + d}{a c}} \log \left (a c^{2} \sqrt{-\frac{a c \sqrt{-\frac{e^{2}}{a c^{3}}} + d}{a c}} \sqrt{-\frac{e^{2}}{a c^{3}}} + \sqrt{e x + d} e\right ) + \frac{1}{2} \, \sqrt{-\frac{a c \sqrt{-\frac{e^{2}}{a c^{3}}} + d}{a c}} \log \left (-a c^{2} \sqrt{-\frac{a c \sqrt{-\frac{e^{2}}{a c^{3}}} + d}{a c}} \sqrt{-\frac{e^{2}}{a c^{3}}} + \sqrt{e x + d} e\right ) + \frac{1}{2} \, \sqrt{\frac{a c \sqrt{-\frac{e^{2}}{a c^{3}}} - d}{a c}} \log \left (a c^{2} \sqrt{\frac{a c \sqrt{-\frac{e^{2}}{a c^{3}}} - d}{a c}} \sqrt{-\frac{e^{2}}{a c^{3}}} + \sqrt{e x + d} e\right ) - \frac{1}{2} \, \sqrt{\frac{a c \sqrt{-\frac{e^{2}}{a c^{3}}} - d}{a c}} \log \left (-a c^{2} \sqrt{\frac{a c \sqrt{-\frac{e^{2}}{a c^{3}}} - d}{a c}} \sqrt{-\frac{e^{2}}{a c^{3}}} + \sqrt{e x + d} e\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-(a*c*sqrt(-e^2/(a*c^3)) + d)/(a*c))*log(a*c^2*sqrt(-(a*c*sqrt(-e^2/(a*c^3)) + d)/(a*c))*sqrt(-e^2/(
a*c^3)) + sqrt(e*x + d)*e) + 1/2*sqrt(-(a*c*sqrt(-e^2/(a*c^3)) + d)/(a*c))*log(-a*c^2*sqrt(-(a*c*sqrt(-e^2/(a*
c^3)) + d)/(a*c))*sqrt(-e^2/(a*c^3)) + sqrt(e*x + d)*e) + 1/2*sqrt((a*c*sqrt(-e^2/(a*c^3)) - d)/(a*c))*log(a*c
^2*sqrt((a*c*sqrt(-e^2/(a*c^3)) - d)/(a*c))*sqrt(-e^2/(a*c^3)) + sqrt(e*x + d)*e) - 1/2*sqrt((a*c*sqrt(-e^2/(a
*c^3)) - d)/(a*c))*log(-a*c^2*sqrt((a*c*sqrt(-e^2/(a*c^3)) - d)/(a*c))*sqrt(-e^2/(a*c^3)) + sqrt(e*x + d)*e)

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Sympy [A]  time = 5.04361, size = 75, normalized size = 0.16 \begin{align*} 2 e \operatorname{RootSum}{\left (256 t^{4} a^{2} c^{3} e^{4} + 32 t^{2} a c^{2} d e^{2} + a e^{2} + c d^{2}, \left ( t \mapsto t \log{\left (64 t^{3} a c^{2} e^{2} + 4 t c d + \sqrt{d + e x} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e*RootSum(256*_t**4*a**2*c**3*e**4 + 32*_t**2*a*c**2*d*e**2 + a*e**2 + c*d**2, Lambda(_t, _t*log(64*_t**3*a*
c**2*e**2 + 4*_t*c*d + sqrt(d + e*x))))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError