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

Optimal. Leaf size=265 \[ -\frac{\sqrt [4]{c} \left (2 \sqrt{c} d-5 \sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2}}+\frac{\sqrt [4]{c} \left (5 \sqrt{a} e+2 \sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2}}-\frac{a e-c d x}{2 a \left (a-c x^2\right ) \sqrt{d+e x} \left (c d^2-a e^2\right )}-\frac{e \left (5 a e^2+c d^2\right )}{2 a \sqrt{d+e x} \left (c d^2-a e^2\right )^2} \]

[Out]

-(e*(c*d^2 + 5*a*e^2))/(2*a*(c*d^2 - a*e^2)^2*Sqrt[d + e*x]) - (a*e - c*d*x)/(2*a*(c*d^2 - a*e^2)*Sqrt[d + e*x
]*(a - c*x^2)) - (c^(1/4)*(2*Sqrt[c]*d - 5*Sqrt[a]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]
*e]])/(4*a^(3/2)*(Sqrt[c]*d - Sqrt[a]*e)^(5/2)) + (c^(1/4)*(2*Sqrt[c]*d + 5*Sqrt[a]*e)*ArcTanh[(c^(1/4)*Sqrt[d
+ e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2)*(Sqrt[c]*d + Sqrt[a]*e)^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.479425, antiderivative size = 265, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {741, 829, 827, 1166, 208} \[ -\frac{\sqrt [4]{c} \left (2 \sqrt{c} d-5 \sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2}}+\frac{\sqrt [4]{c} \left (5 \sqrt{a} e+2 \sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2}}-\frac{a e-c d x}{2 a \left (a-c x^2\right ) \sqrt{d+e x} \left (c d^2-a e^2\right )}-\frac{e \left (5 a e^2+c d^2\right )}{2 a \sqrt{d+e x} \left (c d^2-a e^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e*(c*d^2 + 5*a*e^2))/(2*a*(c*d^2 - a*e^2)^2*Sqrt[d + e*x]) - (a*e - c*d*x)/(2*a*(c*d^2 - a*e^2)*Sqrt[d + e*x
]*(a - c*x^2)) - (c^(1/4)*(2*Sqrt[c]*d - 5*Sqrt[a]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]
*e]])/(4*a^(3/2)*(Sqrt[c]*d - Sqrt[a]*e)^(5/2)) + (c^(1/4)*(2*Sqrt[c]*d + 5*Sqrt[a]*e)*ArcTanh[(c^(1/4)*Sqrt[d
+ e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2)*(Sqrt[c]*d + Sqrt[a]*e)^(5/2))

Rule 741

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(a*e + c*d*x)*(
a + c*x^2)^(p + 1))/(2*a*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*
Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a
, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 829

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((e*f - d*g)*(d
+ e*x)^(m + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(c*d^2 + a*e^2), Int[((d + e*x)^(m + 1)*Simp[c*d*f + a*
e*g - c*(e*f - d*g)*x, x])/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] &&
FractionQ[m] && 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 )^2} \, dx &=-\frac{a e-c d x}{2 a \left (c d^2-a e^2\right ) \sqrt{d+e x} \left (a-c x^2\right )}+\frac{\int \frac{\frac{1}{2} \left (2 c d^2-5 a e^2\right )+\frac{3}{2} c d e x}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx}{2 a \left (c d^2-a e^2\right )}\\ &=-\frac{e \left (c d^2+5 a e^2\right )}{2 a \left (c d^2-a e^2\right )^2 \sqrt{d+e x}}-\frac{a e-c d x}{2 a \left (c d^2-a e^2\right ) \sqrt{d+e x} \left (a-c x^2\right )}-\frac{\int \frac{-c d \left (c d^2-4 a e^2\right )-\frac{1}{2} c e \left (c d^2+5 a e^2\right ) x}{\sqrt{d+e x} \left (a-c x^2\right )} \, dx}{2 a \left (c d^2-a e^2\right )^2}\\ &=-\frac{e \left (c d^2+5 a e^2\right )}{2 a \left (c d^2-a e^2\right )^2 \sqrt{d+e x}}-\frac{a e-c d x}{2 a \left (c d^2-a e^2\right ) \sqrt{d+e x} \left (a-c x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{-c d e \left (c d^2-4 a e^2\right )+\frac{1}{2} c d e \left (c d^2+5 a e^2\right )-\frac{1}{2} c e \left (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 )}{a \left (c d^2-a e^2\right )^2}\\ &=-\frac{e \left (c d^2+5 a e^2\right )}{2 a \left (c d^2-a e^2\right )^2 \sqrt{d+e x}}-\frac{a e-c d x}{2 a \left (c d^2-a e^2\right ) \sqrt{d+e x} \left (a-c x^2\right )}-\frac{\left (c \left (2 \sqrt{c} d-5 \sqrt{a} e\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c d-\sqrt{a} \sqrt{c} e-c x^2} \, dx,x,\sqrt{d+e x}\right )}{4 a^{3/2} \left (\sqrt{c} d-\sqrt{a} e\right )^2}+\frac{\left (c \left (2 \sqrt{c} d+5 \sqrt{a} e\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c d+\sqrt{a} \sqrt{c} e-c x^2} \, dx,x,\sqrt{d+e x}\right )}{4 a^{3/2} \left (\sqrt{c} d+\sqrt{a} e\right )^2}\\ &=-\frac{e \left (c d^2+5 a e^2\right )}{2 a \left (c d^2-a e^2\right )^2 \sqrt{d+e x}}-\frac{a e-c d x}{2 a \left (c d^2-a e^2\right ) \sqrt{d+e x} \left (a-c x^2\right )}-\frac{\sqrt [4]{c} \left (2 \sqrt{c} d-5 \sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2}}+\frac{\sqrt [4]{c} \left (2 \sqrt{c} d+5 \sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d+\sqrt{a} e}}\right )}{4 a^{3/2} \left (\sqrt{c} d+\sqrt{a} e\right )^{5/2}}\\ \end{align*}

Mathematica [C]  time = 0.451095, size = 331, normalized size = 1.25 \[ \frac{\frac{3 c^{3/4} d \left (\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{\sqrt{c} d-\sqrt{a} e}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{2 \sqrt{a}}+\frac{\left (5 a e^2+c d^2\right ) \left (\left (\sqrt{a} e+\sqrt{c} d\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{a} e}\right )+\left (\sqrt{a} e-\sqrt{c} d\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{a} e}\right )\right )}{2 \sqrt{a} \sqrt{d+e x} \left (c d^2-a e^2\right )}+\frac{a e-c d x}{\left (a-c x^2\right ) \sqrt{d+e x}}}{2 a \left (a e^2-c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a*e - c*d*x)/(Sqrt[d + e*x]*(a - c*x^2)) + (3*c^(3/4)*d*(ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sq
rt[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]))/(2*Sqrt[a]) + ((c*d^2 + 5*a*e^2)*((Sqrt[c]*d + Sqrt[a]*e)*Hypergeometric2F1[-1/2, 1, 1/2,
(Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[a]*e)] + (-(Sqrt[c]*d) + Sqrt[a]*e)*Hypergeometric2F1[-1/2, 1, 1/2, (Sq
rt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]))/(2*Sqrt[a]*(c*d^2 - a*e^2)*Sqrt[d + e*x]))/(2*a*(-(c*d^2) + a*e^2)
)

________________________________________________________________________________________

Maple [B]  time = 0.249, size = 783, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*e^3/(a*e^2-c*d^2)^2/(e*x+d)^(1/2)-1/2*e^3*c/(a*e^2-c*d^2)^2/(c*e^2*x^2-a*e^2)*(e*x+d)^(3/2)-1/2*e*c^2/(a*e^
2-c*d^2)^2/(c*e^2*x^2-a*e^2)/a*(e*x+d)^(3/2)*d^2+3/2*e^3*c/(a*e^2-c*d^2)^2/(c*e^2*x^2-a*e^2)*d*(e*x+d)^(1/2)+1
/2*e*c^2/(a*e^2-c*d^2)^2/(c*e^2*x^2-a*e^2)*d^3/a*(e*x+d)^(1/2)-2*e^3*c^2/(a*e^2-c*d^2)^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+1/2*e*c^3/(a*e^2-c*d^2)
^2/a/(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^3-5/4*e^3*c/(a*e^2-c*d^2)^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))-1/4*e*c^2/(a*e^2-c*d^2)^2/a/((-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^2-2*e^3*c^2/(a*e^2-c*d^2)^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+1/2*e*c^3/(a*e^2-c*d^2)^2/a/(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^3+5/4*e^3*c/(a*e^2-c*d^2)^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))+1/4*e*c^2/(a*e^2-c*d^2)^2/a/((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^2

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} - a\right )}^{2}{\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)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 6.31496, size = 12070, normalized size = 45.55 \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)^2,x, algorithm="fricas")

[Out]

1/8*((a^2*c^2*d^5 - 2*a^3*c*d^3*e^2 + a^4*d*e^4 - (a*c^3*d^4*e - 2*a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^3 - (a*c^3*d
^5 - 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*x^2 + (a^2*c^2*d^4*e - 2*a^3*c*d^2*e^3 + a^4*e^5)*x)*sqrt((4*c^4*d^7 - 3
5*a*c^3*d^5*e^2 + 70*a^2*c^2*d^3*e^4 + 105*a^3*c*d*e^6 + (a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^
4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10)*sqrt((1225*c^5*d^8*e^6 - 10780*a*c^4*d^6*e^8 + 21966*a^2*
c^3*d^4*e^10 + 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 - 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*
e^4 - 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 - 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 - 120*a^10*c^
3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 - 10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a
^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10))*log((140*c^4*d^6*e^3 - 1491*a*c^3*d^4*e^5 +
3750*a^2*c^2*d^2*e^7 + 625*a^3*c*e^9)*sqrt(e*x + d) + (35*a^2*c^4*d^7*e^4 - 609*a^3*c^3*d^5*e^6 + 1977*a^4*c^
2*d^3*e^8 + 325*a^5*c*d*e^10 + (2*a^3*c^7*d^14 - 19*a^4*c^6*d^12*e^2 + 60*a^5*c^5*d^10*e^4 - 85*a^6*c^4*d^8*e^
6 + 50*a^7*c^3*d^6*e^8 + 3*a^8*c^2*d^4*e^10 - 16*a^9*c*d^2*e^12 + 5*a^10*e^14)*sqrt((1225*c^5*d^8*e^6 - 10780*
a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 + 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 - 10*a^4*c^9*d
^18*e^2 + 45*a^5*c^8*d^16*e^4 - 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 - 252*a^8*c^5*d^10*e^10 + 210*a^9*
c^4*d^8*e^12 - 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 - 10*a^12*c*d^2*e^18 + a^13*e^20)))*sqrt((4*c^4*d^
7 - 35*a*c^3*d^5*e^2 + 70*a^2*c^2*d^3*e^4 + 105*a^3*c*d*e^6 + (a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d
^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10)*sqrt((1225*c^5*d^8*e^6 - 10780*a*c^4*d^6*e^8 + 21966
*a^2*c^3*d^4*e^10 + 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 - 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*
d^16*e^4 - 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 - 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 - 120*a^
10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 - 10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 +
10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10))) - (a^2*c^2*d^5 - 2*a^3*c*d^3*e^2 + a^
4*d*e^4 - (a*c^3*d^4*e - 2*a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^3 - (a*c^3*d^5 - 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*x^
2 + (a^2*c^2*d^4*e - 2*a^3*c*d^2*e^3 + a^4*e^5)*x)*sqrt((4*c^4*d^7 - 35*a*c^3*d^5*e^2 + 70*a^2*c^2*d^3*e^4 + 1
05*a^3*c*d*e^6 + (a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8
- a^8*e^10)*sqrt((1225*c^5*d^8*e^6 - 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 + 7700*a^3*c^2*d^2*e^12 + 6
25*a^4*c*e^14)/(a^3*c^10*d^20 - 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 - 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6
*d^12*e^8 - 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 - 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 - 10*a
^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*
a^7*c*d^2*e^8 - a^8*e^10))*log((140*c^4*d^6*e^3 - 1491*a*c^3*d^4*e^5 + 3750*a^2*c^2*d^2*e^7 + 625*a^3*c*e^9)*s
qrt(e*x + d) - (35*a^2*c^4*d^7*e^4 - 609*a^3*c^3*d^5*e^6 + 1977*a^4*c^2*d^3*e^8 + 325*a^5*c*d*e^10 + (2*a^3*c^
7*d^14 - 19*a^4*c^6*d^12*e^2 + 60*a^5*c^5*d^10*e^4 - 85*a^6*c^4*d^8*e^6 + 50*a^7*c^3*d^6*e^8 + 3*a^8*c^2*d^4*e
^10 - 16*a^9*c*d^2*e^12 + 5*a^10*e^14)*sqrt((1225*c^5*d^8*e^6 - 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 +
7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 - 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 - 120*a^6*
c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 - 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 - 120*a^10*c^3*d^6*e^14 + 4
5*a^11*c^2*d^4*e^16 - 10*a^12*c*d^2*e^18 + a^13*e^20)))*sqrt((4*c^4*d^7 - 35*a*c^3*d^5*e^2 + 70*a^2*c^2*d^3*e^
4 + 105*a^3*c*d*e^6 + (a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^
2*e^8 - a^8*e^10)*sqrt((1225*c^5*d^8*e^6 - 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 + 7700*a^3*c^2*d^2*e^1
2 + 625*a^4*c*e^14)/(a^3*c^10*d^20 - 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 - 120*a^6*c^7*d^14*e^6 + 210*a^
7*c^6*d^12*e^8 - 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 - 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 -
10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6
+ 5*a^7*c*d^2*e^8 - a^8*e^10))) + (a^2*c^2*d^5 - 2*a^3*c*d^3*e^2 + a^4*d*e^4 - (a*c^3*d^4*e - 2*a^2*c^2*d^2*e
^3 + a^3*c*e^5)*x^3 - (a*c^3*d^5 - 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*x^2 + (a^2*c^2*d^4*e - 2*a^3*c*d^2*e^3 + a
^4*e^5)*x)*sqrt((4*c^4*d^7 - 35*a*c^3*d^5*e^2 + 70*a^2*c^2*d^3*e^4 + 105*a^3*c*d*e^6 - (a^3*c^5*d^10 - 5*a^4*c
^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10)*sqrt((1225*c^5*d^8*e^6 - 10
780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 + 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 - 10*a^4*c
^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 - 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 - 252*a^8*c^5*d^10*e^10 + 210*
a^9*c^4*d^8*e^12 - 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 - 10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d
^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10))*log((140*c^4*
d^6*e^3 - 1491*a*c^3*d^4*e^5 + 3750*a^2*c^2*d^2*e^7 + 625*a^3*c*e^9)*sqrt(e*x + d) + (35*a^2*c^4*d^7*e^4 - 609
*a^3*c^3*d^5*e^6 + 1977*a^4*c^2*d^3*e^8 + 325*a^5*c*d*e^10 - (2*a^3*c^7*d^14 - 19*a^4*c^6*d^12*e^2 + 60*a^5*c^
5*d^10*e^4 - 85*a^6*c^4*d^8*e^6 + 50*a^7*c^3*d^6*e^8 + 3*a^8*c^2*d^4*e^10 - 16*a^9*c*d^2*e^12 + 5*a^10*e^14)*s
qrt((1225*c^5*d^8*e^6 - 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 + 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)
/(a^3*c^10*d^20 - 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 - 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 - 25
2*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 - 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 - 10*a^12*c*d^2*e^18
+ a^13*e^20)))*sqrt((4*c^4*d^7 - 35*a*c^3*d^5*e^2 + 70*a^2*c^2*d^3*e^4 + 105*a^3*c*d*e^6 - (a^3*c^5*d^10 - 5*
a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10)*sqrt((1225*c^5*d^8*e^6
- 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 + 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 - 10*
a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 - 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 - 252*a^8*c^5*d^10*e^10 +
210*a^9*c^4*d^8*e^12 - 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 - 10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*
c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10))) - (a^2*
c^2*d^5 - 2*a^3*c*d^3*e^2 + a^4*d*e^4 - (a*c^3*d^4*e - 2*a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^3 - (a*c^3*d^5 - 2*a^2
*c^2*d^3*e^2 + a^3*c*d*e^4)*x^2 + (a^2*c^2*d^4*e - 2*a^3*c*d^2*e^3 + a^4*e^5)*x)*sqrt((4*c^4*d^7 - 35*a*c^3*d^
5*e^2 + 70*a^2*c^2*d^3*e^4 + 105*a^3*c*d*e^6 - (a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6
*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10)*sqrt((1225*c^5*d^8*e^6 - 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^
10 + 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 - 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 - 120*
a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 - 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 - 120*a^10*c^3*d^6*e^14
+ 45*a^11*c^2*d^4*e^16 - 10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6
*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10))*log((140*c^4*d^6*e^3 - 1491*a*c^3*d^4*e^5 + 3750*a^2*
c^2*d^2*e^7 + 625*a^3*c*e^9)*sqrt(e*x + d) - (35*a^2*c^4*d^7*e^4 - 609*a^3*c^3*d^5*e^6 + 1977*a^4*c^2*d^3*e^8
+ 325*a^5*c*d*e^10 - (2*a^3*c^7*d^14 - 19*a^4*c^6*d^12*e^2 + 60*a^5*c^5*d^10*e^4 - 85*a^6*c^4*d^8*e^6 + 50*a^7
*c^3*d^6*e^8 + 3*a^8*c^2*d^4*e^10 - 16*a^9*c*d^2*e^12 + 5*a^10*e^14)*sqrt((1225*c^5*d^8*e^6 - 10780*a*c^4*d^6*
e^8 + 21966*a^2*c^3*d^4*e^10 + 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 - 10*a^4*c^9*d^18*e^2 +
45*a^5*c^8*d^16*e^4 - 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 - 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^
12 - 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 - 10*a^12*c*d^2*e^18 + a^13*e^20)))*sqrt((4*c^4*d^7 - 35*a*c
^3*d^5*e^2 + 70*a^2*c^2*d^3*e^4 + 105*a^3*c*d*e^6 - (a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 1
0*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10)*sqrt((1225*c^5*d^8*e^6 - 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d
^4*e^10 + 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 - 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 -
120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 - 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 - 120*a^10*c^3*d^6
*e^14 + 45*a^11*c^2*d^4*e^16 - 10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^
3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10))) - 4*(2*a*c*d^2*e + 4*a^2*e^3 - (c^2*d^2*e + 5*a
*c*e^3)*x^2 - (c^2*d^3 - a*c*d*e^2)*x)*sqrt(e*x + d))/(a^2*c^2*d^5 - 2*a^3*c*d^3*e^2 + a^4*d*e^4 - (a*c^3*d^4*
e - 2*a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^3 - (a*c^3*d^5 - 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*x^2 + (a^2*c^2*d^4*e -
2*a^3*c*d^2*e^3 + a^4*e^5)*x)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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)^2,x, algorithm="giac")

[Out]

Timed out