### 3.616 $$\int \frac{1}{(d+e x)^{3/2} (a-c x^2)} \, dx$$

Optimal. Leaf size=160 $\frac{2 e}{\sqrt{d+e x} \left (c d^2-a e^2\right )}-\frac{\sqrt [4]{c} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \left (\sqrt{c} d-\sqrt{a} e\right )^{3/2}}+\frac{\sqrt [4]{c} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} \left (\sqrt{a} e+\sqrt{c} d\right )^{3/2}}$

[Out]

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

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Rubi [A]  time = 0.208049, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {710, 827, 1166, 208} $\frac{2 e}{\sqrt{d+e x} \left (c d^2-a e^2\right )}-\frac{\sqrt [4]{c} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \left (\sqrt{c} d-\sqrt{a} e\right )^{3/2}}+\frac{\sqrt [4]{c} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} \left (\sqrt{a} e+\sqrt{c} d\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)^(3/2)*(a - c*x^2)),x]

[Out]

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

Rule 710

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

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

Mathematica [C]  time = 0.0861443, size = 136, normalized size = 0.85 $\frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{a} e}\right )+\left (\sqrt{a} e-\sqrt{c} d\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{a} e}\right )}{\sqrt{a} \sqrt{d+e x} \left (c d^2-a e^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)^(3/2)*(a - c*x^2)),x]

[Out]

((Sqrt[c]*d + Sqrt[a]*e)*Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[a]*e)] + (-(Sqr
t[c]*d) + Sqrt[a]*e)*Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)])/(Sqrt[a]*(c
*d^2 - a*e^2)*Sqrt[d + e*x])

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Maple [B]  time = 0.209, size = 291, normalized size = 1.8 \begin{align*} -{\frac{{c}^{2}ed}{a{e}^{2}-c{d}^{2}}\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}}}}-{\frac{ce}{a{e}^{2}-c{d}^{2}}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}-{\frac{{c}^{2}ed}{a{e}^{2}-c{d}^{2}}{\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}}}}+{\frac{ce}{a{e}^{2}-c{d}^{2}}{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}-2\,{\frac{e}{ \left ( a{e}^{2}-c{d}^{2} \right ) \sqrt{ex+d}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e*c^2/(a*e^2-c*d^2)/(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))*d-e*c/(a*e^2-c*d^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))-e*c^2/(a*e^2-c*d^2)/(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))*d+e*c/(a*e^2-c*d^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))-2*e/(a*e^2-c*d^2)/(e*x+d)^(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 )}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.52326, size = 5617, normalized size = 35.11 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)*sqrt((c^2*d^3 + 3*a*c*d*e^2 + (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*
a^3*c*d^2*e^4 - a^4*e^6)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 +
15*a^3*c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 - 3*
a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))*log((3*c^2*d^2*e + a*c*e^3)*sqrt(e*x + d) + (6*a*c^2*d^3*e^2 + 2
*a^2*c*d*e^4 - (a*c^4*d^8 - 2*a^2*c^3*d^6*e^2 + 2*a^4*c*d^2*e^6 - a^5*e^8)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e
^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^
8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))*sqrt((c^2*d^3 + 3*a*c*d*e^2 + (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*
e^4 - a^4*e^6)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^
4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^
4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))) - (c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)*sqrt((c^2*d^3 + 3*a*c*d*e^2 + (
a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/
(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*
e^10 + a^7*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))*log((3*c^2*d^2*e + a*c*e^3)*sq
rt(e*x + d) - (6*a*c^2*d^3*e^2 + 2*a^2*c*d*e^4 - (a*c^4*d^8 - 2*a^2*c^3*d^6*e^2 + 2*a^4*c*d^2*e^6 - a^5*e^8)*s
qrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 - 20*a
^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))*sqrt((c^2*d^3 + 3*a*c*d*e^2 + (a*c^3*d^6
- 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^1
2 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7
*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))) + (c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*
x)*sqrt((c^2*d^3 + 3*a*c*d*e^2 - (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((9*c^3*d^4*e
^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 +
15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^
6))*log((3*c^2*d^2*e + a*c*e^3)*sqrt(e*x + d) + (6*a*c^2*d^3*e^2 + 2*a^2*c*d*e^4 + (a*c^4*d^8 - 2*a^2*c^3*d^6*
e^2 + 2*a^4*c*d^2*e^6 - a^5*e^8)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^
10*e^2 + 15*a^3*c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))*sqrt((c
^2*d^3 + 3*a*c*d*e^2 - (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((9*c^3*d^4*e^2 + 6*a*c
^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^
2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))) - (c*
d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)*sqrt((c^2*d^3 + 3*a*c*d*e^2 - (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^
2*e^4 - a^4*e^6)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*
c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*
d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))*log((3*c^2*d^2*e + a*c*e^3)*sqrt(e*x + d) - (6*a*c^2*d^3*e^2 + 2*a^2*c*d
*e^4 + (a*c^4*d^8 - 2*a^2*c^3*d^6*e^2 + 2*a^4*c*d^2*e^6 - a^5*e^8)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2
*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^
6*c*d^2*e^10 + a^7*e^12)))*sqrt((c^2*d^3 + 3*a*c*d*e^2 - (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^
4*e^6)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^
4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 +
3*a^3*c*d^2*e^4 - a^4*e^6))) + 4*sqrt(e*x + d)*e)/(c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{- a d \sqrt{d + e x} - a e x \sqrt{d + e x} + c d x^{2} \sqrt{d + e x} + c e x^{3} \sqrt{d + e x}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(1/(-a*d*sqrt(d + e*x) - a*e*x*sqrt(d + e*x) + c*d*x**2*sqrt(d + e*x) + c*e*x**3*sqrt(d + e*x)), x)

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

[Out]

Timed out