### 3.615 $$\int \frac{1}{\sqrt{d+e x} (a-c x^2)} \, dx$$

Optimal. Leaf size=134 $\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} \sqrt [4]{c} \sqrt{\sqrt{a} e+\sqrt{c} d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \sqrt [4]{c} \sqrt{\sqrt{c} d-\sqrt{a} e}}$

[Out]

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

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Rubi [A]  time = 0.113946, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.15, Rules used = {708, 1093, 208} $\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} \sqrt [4]{c} \sqrt{\sqrt{a} e+\sqrt{c} d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \sqrt [4]{c} \sqrt{\sqrt{c} d-\sqrt{a} e}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(Sqrt[d + e*x]*(a - c*x^2)),x]

[Out]

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

Rule 708

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(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 1093

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/
2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 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{1}{\sqrt{d+e x} \left (a-c x^2\right )} \, dx &=(2 e) \operatorname{Subst}\left (\int \frac{1}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt{d+e x}\right )\\ &=-\frac{\sqrt{c} \operatorname{Subst}\left (\int \frac{1}{c d-\sqrt{a} \sqrt{c} e-c x^2} \, dx,x,\sqrt{d+e x}\right )}{\sqrt{a}}+\frac{\sqrt{c} \operatorname{Subst}\left (\int \frac{1}{c d+\sqrt{a} \sqrt{c} e-c x^2} \, dx,x,\sqrt{d+e x}\right )}{\sqrt{a}}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \sqrt [4]{c} \sqrt{\sqrt{c} d-\sqrt{a} e}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d+\sqrt{a} e}}\right )}{\sqrt{a} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{a} e}}\\ \end{align*}

Mathematica [A]  time = 0.0868462, size = 125, normalized size = 0.93 $\frac{\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{\sqrt{a} e+\sqrt{c} d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{\sqrt{c} d-\sqrt{a} e}}}{\sqrt{a} \sqrt [4]{c}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.205, size = 110, normalized size = 0.8 \begin{align*}{ce\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{ce{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*e/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))+
c*e/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.94096, size = 1791, normalized size = 13.37 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: NotImplementedError