### 3.642 $$\int \frac{1}{\sqrt{d+e x} (a-c x^2)^3} \, dx$$

Optimal. Leaf size=315 $-\frac{\sqrt{d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d x \left (c d^2-2 a e^2\right )\right )}{16 a^2 \left (a-c x^2\right ) \left (c d^2-a e^2\right )^2}-\frac{3 \left (-10 \sqrt{a} \sqrt{c} d e+7 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2}}+\frac{3 \left (10 \sqrt{a} \sqrt{c} d e+7 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2}}-\frac{\sqrt{d+e x} (a e-c d x)}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )}$

[Out]

-((a*e - c*d*x)*Sqrt[d + e*x])/(4*a*(c*d^2 - a*e^2)*(a - c*x^2)^2) - (Sqrt[d + e*x]*(a*e*(c*d^2 - 7*a*e^2) - 6
*c*d*(c*d^2 - 2*a*e^2)*x))/(16*a^2*(c*d^2 - a*e^2)^2*(a - c*x^2)) - (3*(4*c*d^2 - 10*Sqrt[a]*Sqrt[c]*d*e + 7*a
*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(32*a^(5/2)*c^(1/4)*(Sqrt[c]*d - Sqrt[a]*e
)^(5/2)) + (3*(4*c*d^2 + 10*Sqrt[a]*Sqrt[c]*d*e + 7*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sq
rt[a]*e]])/(32*a^(5/2)*c^(1/4)*(Sqrt[c]*d + Sqrt[a]*e)^(5/2))

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Rubi [A]  time = 0.634021, antiderivative size = 315, 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, 823, 827, 1166, 208} $-\frac{\sqrt{d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d x \left (c d^2-2 a e^2\right )\right )}{16 a^2 \left (a-c x^2\right ) \left (c d^2-a e^2\right )^2}-\frac{3 \left (-10 \sqrt{a} \sqrt{c} d e+7 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2}}+\frac{3 \left (10 \sqrt{a} \sqrt{c} d e+7 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2}}-\frac{\sqrt{d+e x} (a e-c d x)}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((a*e - c*d*x)*Sqrt[d + e*x])/(4*a*(c*d^2 - a*e^2)*(a - c*x^2)^2) - (Sqrt[d + e*x]*(a*e*(c*d^2 - 7*a*e^2) - 6
*c*d*(c*d^2 - 2*a*e^2)*x))/(16*a^2*(c*d^2 - a*e^2)^2*(a - c*x^2)) - (3*(4*c*d^2 - 10*Sqrt[a]*Sqrt[c]*d*e + 7*a
*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(32*a^(5/2)*c^(1/4)*(Sqrt[c]*d - Sqrt[a]*e
)^(5/2)) + (3*(4*c*d^2 + 10*Sqrt[a]*Sqrt[c]*d*e + 7*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sq
rt[a]*e]])/(32*a^(5/2)*c^(1/4)*(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 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

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}{\sqrt{d+e x} \left (a-c x^2\right )^3} \, dx &=-\frac{(a e-c d x) \sqrt{d+e x}}{4 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )^2}+\frac{\int \frac{\frac{1}{2} \left (6 c d^2-7 a e^2\right )+\frac{5}{2} c d e x}{\sqrt{d+e x} \left (a-c x^2\right )^2} \, dx}{4 a \left (c d^2-a e^2\right )}\\ &=-\frac{(a e-c d x) \sqrt{d+e x}}{4 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )^2}-\frac{\sqrt{d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d \left (c d^2-2 a e^2\right ) x\right )}{16 a^2 \left (c d^2-a e^2\right )^2 \left (a-c x^2\right )}-\frac{\int \frac{-\frac{3}{4} c \left (4 c^2 d^4-9 a c d^2 e^2+7 a^2 e^4\right )-\frac{3}{2} c^2 d e \left (c d^2-2 a e^2\right ) x}{\sqrt{d+e x} \left (a-c x^2\right )} \, dx}{8 a^2 c \left (c d^2-a e^2\right )^2}\\ &=-\frac{(a e-c d x) \sqrt{d+e x}}{4 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )^2}-\frac{\sqrt{d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d \left (c d^2-2 a e^2\right ) x\right )}{16 a^2 \left (c d^2-a e^2\right )^2 \left (a-c x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{\frac{3}{2} c^2 d^2 e \left (c d^2-2 a e^2\right )-\frac{3}{4} c e \left (4 c^2 d^4-9 a c d^2 e^2+7 a^2 e^4\right )-\frac{3}{2} c^2 d e \left (c d^2-2 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 )}{4 a^2 c \left (c d^2-a e^2\right )^2}\\ &=-\frac{(a e-c d x) \sqrt{d+e x}}{4 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )^2}-\frac{\sqrt{d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d \left (c d^2-2 a e^2\right ) x\right )}{16 a^2 \left (c d^2-a e^2\right )^2 \left (a-c x^2\right )}-\frac{\left (3 \sqrt{c} \left (4 c d^2-10 \sqrt{a} \sqrt{c} d e+7 a e^2\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 )}{32 a^{5/2} \left (\sqrt{c} d-\sqrt{a} e\right )^2}+\frac{\left (3 \sqrt{c} \left (4 c d^2+10 \sqrt{a} \sqrt{c} d e+7 a e^2\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 )}{32 a^{5/2} \left (\sqrt{c} d+\sqrt{a} e\right )^2}\\ &=-\frac{(a e-c d x) \sqrt{d+e x}}{4 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )^2}-\frac{\sqrt{d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d \left (c d^2-2 a e^2\right ) x\right )}{16 a^2 \left (c d^2-a e^2\right )^2 \left (a-c x^2\right )}-\frac{3 \left (4 c d^2-10 \sqrt{a} \sqrt{c} d e+7 a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2}}+\frac{3 \left (4 c d^2+10 \sqrt{a} \sqrt{c} d e+7 a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d+\sqrt{a} e}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt{c} d+\sqrt{a} e\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.70031, size = 335, normalized size = 1.06 $\frac{\frac{2 \sqrt{d+e x} \left (7 a^2 e^3-a c d e (d+12 e x)+6 c^2 d^3 x\right )}{a-c x^2}+\frac{8 a \sqrt{d+e x} \left (c d^2-a e^2\right ) (c d x-a e)}{\left (a-c x^2\right )^2}+\frac{3 \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2} \left (10 \sqrt{a} \sqrt{c} d e+7 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )-3 \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2} \left (-10 \sqrt{a} \sqrt{c} d e+7 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \sqrt [4]{c} \sqrt{\sqrt{c} d-\sqrt{a} e} \sqrt{\sqrt{a} e+\sqrt{c} d}}}{32 a^2 \left (c d^2-a e^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((8*a*(c*d^2 - a*e^2)*(-(a*e) + c*d*x)*Sqrt[d + e*x])/(a - c*x^2)^2 + (2*Sqrt[d + e*x]*(7*a^2*e^3 + 6*c^2*d^3*
x - a*c*d*e*(d + 12*e*x)))/(a - c*x^2) + (-3*(Sqrt[c]*d + Sqrt[a]*e)^(5/2)*(4*c*d^2 - 10*Sqrt[a]*Sqrt[c]*d*e +
7*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] + 3*(Sqrt[c]*d - Sqrt[a]*e)^(5/2)*(4*c*
d^2 + 10*Sqrt[a]*Sqrt[c]*d*e + 7*a*e^2)*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]*Sqrt[Sqrt[c]*d + Sqrt[a]*e]))/(32*a^2*(c*d^2 - a*e^2)^2)

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Maple [B]  time = 0.267, size = 956, 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/(-c*x^2+a)^3/(e*x+d)^(1/2),x)

[Out]

-3/16*e/a^2/(e*x+(a*c*e^2)^(1/2)/c)^2/(a*e^2+c*d^2-2*(a*c*e^2)^(1/2)*d)*(e*x+d)^(3/2)*d+9/32*e/c*(a*c*e^2)^(1/
2)/a^2/(e*x+(a*c*e^2)^(1/2)/c)^2/(a*e^2+c*d^2-2*(a*c*e^2)^(1/2)*d)*(e*x+d)^(3/2)+3/16*e/a^2/(e*x+(a*c*e^2)^(1/
2)/c)^2/(c*d-(a*c*e^2)^(1/2))*(e*x+d)^(1/2)*d-11/32*e/c*(a*c*e^2)^(1/2)/a^2/(e*x+(a*c*e^2)^(1/2)/c)^2/(c*d-(a*
c*e^2)^(1/2))*(e*x+d)^(1/2)-21/32*e^3*c/(a*c*e^2)^(1/2)/a/(-a*e^2-c*d^2+2*(a*c*e^2)^(1/2)*d)/((-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))-3/8*e*c^2/(a*c*e^2)^(1/2)/a^2/(-a*e^2
-c*d^2+2*(a*c*e^2)^(1/2)*d)/((-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+15/16*e*c/a^2/(-a*e^2-c*d^2+2*(a*c*e^2)^(1/2)*d)/((-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-3/16*e/a^2/(e*x-(a*c*e^2)^(1/2)/c)^2/(a*e^2+c*d^2+2*(a*c*e^2)^(1/2)
*d)*(e*x+d)^(3/2)*d-9/32*e/c*(a*c*e^2)^(1/2)/a^2/(e*x-(a*c*e^2)^(1/2)/c)^2/(a*e^2+c*d^2+2*(a*c*e^2)^(1/2)*d)*(
e*x+d)^(3/2)+3/16*e/a^2/(e*x-(a*c*e^2)^(1/2)/c)^2/(c*d+(a*c*e^2)^(1/2))*(e*x+d)^(1/2)*d+11/32*e/c*(a*c*e^2)^(1
/2)/a^2/(e*x-(a*c*e^2)^(1/2)/c)^2/(c*d+(a*c*e^2)^(1/2))*(e*x+d)^(1/2)+21/32*e^3*c/(a*c*e^2)^(1/2)/a/(a*e^2+c*d
^2+2*(a*c*e^2)^(1/2)*d)/((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
))+3/8*e*c^2/(a*c*e^2)^(1/2)/a^2/(a*e^2+c*d^2+2*(a*c*e^2)^(1/2)*d)/((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+15/16*e*c/a^2/(a*e^2+c*d^2+2*(a*c*e^2)^(1/2)*d)/((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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{{\left (c x^{2} - a\right )}^{3} \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)^3/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 13.2157, size = 12446, normalized size = 39.51 \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)^3/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

1/64*(3*(a^4*c^2*d^4 - 2*a^5*c*d^2*e^2 + a^6*e^4 + (a^2*c^4*d^4 - 2*a^3*c^3*d^2*e^2 + a^4*c^2*e^4)*x^4 - 2*(a^
3*c^3*d^4 - 2*a^4*c^2*d^2*e^2 + a^5*c*e^4)*x^2)*sqrt((16*c^4*d^9 - 84*a*c^3*d^7*e^2 + 189*a^2*c^2*d^5*e^4 - 21
0*a^3*c*d^3*e^6 + 105*a^4*d*e^8 + (a^5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6
+ 5*a^9*c*d^2*e^8 - a^10*e^10)*sqrt((441*c^4*d^8*e^10 - 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 - 5292*a^3
*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 - 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 - 120*a^8*c^8*d^14*e^
6 + 210*a^9*c^7*d^12*e^8 - 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 - 120*a^12*c^4*d^6*e^14 + 45*a^13*c^
3*d^4*e^16 - 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10
*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e^10))*log(27*(336*c^4*d^8*e^5 - 1788*a*c^3*d^6*e^7 + 4189*a^2*c^2*d
^4*e^9 - 4802*a^3*c*d^2*e^11 + 2401*a^4*e^13)*sqrt(e*x + d) + 27*(42*a^3*c^4*d^8*e^6 - 213*a^4*c^3*d^6*e^8 + 5
15*a^5*c^2*d^4*e^10 - 623*a^6*c*d^2*e^12 + 343*a^7*e^14 - (4*a^5*c^8*d^15 - 31*a^6*c^7*d^13*e^2 + 106*a^7*c^6*
d^11*e^4 - 205*a^8*c^5*d^9*e^6 + 240*a^9*c^4*d^7*e^8 - 169*a^10*c^3*d^5*e^10 + 66*a^11*c^2*d^3*e^12 - 11*a^12*
c*d*e^14)*sqrt((441*c^4*d^8*e^10 - 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 - 5292*a^3*c*d^2*e^16 + 2401*a^
4*e^18)/(a^5*c^11*d^20 - 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 - 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*
e^8 - 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 - 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 - 10*a^14*
c^2*d^2*e^18 + a^15*c*e^20)))*sqrt((16*c^4*d^9 - 84*a*c^3*d^7*e^2 + 189*a^2*c^2*d^5*e^4 - 210*a^3*c*d^3*e^6 +
105*a^4*d*e^8 + (a^5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8
- a^10*e^10)*sqrt((441*c^4*d^8*e^10 - 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 - 5292*a^3*c*d^2*e^16 + 2401
*a^4*e^18)/(a^5*c^11*d^20 - 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 - 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^
12*e^8 - 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 - 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 - 10*a^
14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6 +
5*a^9*c*d^2*e^8 - a^10*e^10))) - 3*(a^4*c^2*d^4 - 2*a^5*c*d^2*e^2 + a^6*e^4 + (a^2*c^4*d^4 - 2*a^3*c^3*d^2*e^
2 + a^4*c^2*e^4)*x^4 - 2*(a^3*c^3*d^4 - 2*a^4*c^2*d^2*e^2 + a^5*c*e^4)*x^2)*sqrt((16*c^4*d^9 - 84*a*c^3*d^7*e^
2 + 189*a^2*c^2*d^5*e^4 - 210*a^3*c*d^3*e^6 + 105*a^4*d*e^8 + (a^5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d
^6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e^10)*sqrt((441*c^4*d^8*e^10 - 2268*a*c^3*d^6*e^12 + 4974
*a^2*c^2*d^4*e^14 - 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 - 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^
16*e^4 - 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 - 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 - 120*a^
12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 - 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 - 5*a^6*c^4*d^8*e
^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e^10))*log(27*(336*c^4*d^8*e^5 - 1788*a*
c^3*d^6*e^7 + 4189*a^2*c^2*d^4*e^9 - 4802*a^3*c*d^2*e^11 + 2401*a^4*e^13)*sqrt(e*x + d) - 27*(42*a^3*c^4*d^8*e
^6 - 213*a^4*c^3*d^6*e^8 + 515*a^5*c^2*d^4*e^10 - 623*a^6*c*d^2*e^12 + 343*a^7*e^14 - (4*a^5*c^8*d^15 - 31*a^6
*c^7*d^13*e^2 + 106*a^7*c^6*d^11*e^4 - 205*a^8*c^5*d^9*e^6 + 240*a^9*c^4*d^7*e^8 - 169*a^10*c^3*d^5*e^10 + 66*
a^11*c^2*d^3*e^12 - 11*a^12*c*d*e^14)*sqrt((441*c^4*d^8*e^10 - 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 - 5
292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 - 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 - 120*a^8*c^8*
d^14*e^6 + 210*a^9*c^7*d^12*e^8 - 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 - 120*a^12*c^4*d^6*e^14 + 45*
a^13*c^3*d^4*e^16 - 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))*sqrt((16*c^4*d^9 - 84*a*c^3*d^7*e^2 + 189*a^2*c^2*d^
5*e^4 - 210*a^3*c*d^3*e^6 + 105*a^4*d*e^8 + (a^5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^
2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e^10)*sqrt((441*c^4*d^8*e^10 - 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14
- 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 - 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 - 120*a^8*c
^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 - 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 - 120*a^12*c^4*d^6*e^14 +
45*a^13*c^3*d^4*e^16 - 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^
6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e^10))) + 3*(a^4*c^2*d^4 - 2*a^5*c*d^2*e^2 + a^6*e^4 + (a^
2*c^4*d^4 - 2*a^3*c^3*d^2*e^2 + a^4*c^2*e^4)*x^4 - 2*(a^3*c^3*d^4 - 2*a^4*c^2*d^2*e^2 + a^5*c*e^4)*x^2)*sqrt((
16*c^4*d^9 - 84*a*c^3*d^7*e^2 + 189*a^2*c^2*d^5*e^4 - 210*a^3*c*d^3*e^6 + 105*a^4*d*e^8 - (a^5*c^5*d^10 - 5*a^
6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e^10)*sqrt((441*c^4*d^8*e^10
- 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 - 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 - 10*a^6*c
^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 - 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 - 252*a^10*c^6*d^10*e^10 + 21
0*a^11*c^5*d^8*e^12 - 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 - 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^
5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e^10))*log(2
7*(336*c^4*d^8*e^5 - 1788*a*c^3*d^6*e^7 + 4189*a^2*c^2*d^4*e^9 - 4802*a^3*c*d^2*e^11 + 2401*a^4*e^13)*sqrt(e*x
+ d) + 27*(42*a^3*c^4*d^8*e^6 - 213*a^4*c^3*d^6*e^8 + 515*a^5*c^2*d^4*e^10 - 623*a^6*c*d^2*e^12 + 343*a^7*e^1
4 + (4*a^5*c^8*d^15 - 31*a^6*c^7*d^13*e^2 + 106*a^7*c^6*d^11*e^4 - 205*a^8*c^5*d^9*e^6 + 240*a^9*c^4*d^7*e^8 -
169*a^10*c^3*d^5*e^10 + 66*a^11*c^2*d^3*e^12 - 11*a^12*c*d*e^14)*sqrt((441*c^4*d^8*e^10 - 2268*a*c^3*d^6*e^12
+ 4974*a^2*c^2*d^4*e^14 - 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 - 10*a^6*c^10*d^18*e^2 + 45*a^7
*c^9*d^16*e^4 - 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 - 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 -
120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 - 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))*sqrt((16*c^4*d^9 - 84*a*
c^3*d^7*e^2 + 189*a^2*c^2*d^5*e^4 - 210*a^3*c*d^3*e^6 + 105*a^4*d*e^8 - (a^5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10
*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e^10)*sqrt((441*c^4*d^8*e^10 - 2268*a*c^3*d^6*e
^12 + 4974*a^2*c^2*d^4*e^14 - 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 - 10*a^6*c^10*d^18*e^2 + 45*
a^7*c^9*d^16*e^4 - 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 - 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^1
2 - 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 - 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 - 5*a^6
*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e^10))) - 3*(a^4*c^2*d^4 - 2*a
^5*c*d^2*e^2 + a^6*e^4 + (a^2*c^4*d^4 - 2*a^3*c^3*d^2*e^2 + a^4*c^2*e^4)*x^4 - 2*(a^3*c^3*d^4 - 2*a^4*c^2*d^2*
e^2 + a^5*c*e^4)*x^2)*sqrt((16*c^4*d^9 - 84*a*c^3*d^7*e^2 + 189*a^2*c^2*d^5*e^4 - 210*a^3*c*d^3*e^6 + 105*a^4*
d*e^8 - (a^5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e
^10)*sqrt((441*c^4*d^8*e^10 - 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 - 5292*a^3*c*d^2*e^16 + 2401*a^4*e^1
8)/(a^5*c^11*d^20 - 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 - 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 -
252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 - 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 - 10*a^14*c^2*d
^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c
*d^2*e^8 - a^10*e^10))*log(27*(336*c^4*d^8*e^5 - 1788*a*c^3*d^6*e^7 + 4189*a^2*c^2*d^4*e^9 - 4802*a^3*c*d^2*e^
11 + 2401*a^4*e^13)*sqrt(e*x + d) - 27*(42*a^3*c^4*d^8*e^6 - 213*a^4*c^3*d^6*e^8 + 515*a^5*c^2*d^4*e^10 - 623*
a^6*c*d^2*e^12 + 343*a^7*e^14 + (4*a^5*c^8*d^15 - 31*a^6*c^7*d^13*e^2 + 106*a^7*c^6*d^11*e^4 - 205*a^8*c^5*d^9
*e^6 + 240*a^9*c^4*d^7*e^8 - 169*a^10*c^3*d^5*e^10 + 66*a^11*c^2*d^3*e^12 - 11*a^12*c*d*e^14)*sqrt((441*c^4*d^
8*e^10 - 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 - 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 - 1
0*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 - 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 - 252*a^10*c^6*d^10*e^
10 + 210*a^11*c^5*d^8*e^12 - 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 - 10*a^14*c^2*d^2*e^18 + a^15*c*e^20
)))*sqrt((16*c^4*d^9 - 84*a*c^3*d^7*e^2 + 189*a^2*c^2*d^5*e^4 - 210*a^3*c*d^3*e^6 + 105*a^4*d*e^8 - (a^5*c^5*d
^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e^10)*sqrt((441*c^4
*d^8*e^10 - 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 - 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20
- 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 - 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 - 252*a^10*c^6*d^10
*e^10 + 210*a^11*c^5*d^8*e^12 - 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 - 10*a^14*c^2*d^2*e^18 + a^15*c*e
^20)))/(a^5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e^
10))) - 4*(5*a^2*c*d^2*e - 11*a^3*e^3 + 6*(c^3*d^3 - 2*a*c^2*d*e^2)*x^3 - (a*c^2*d^2*e - 7*a^2*c*e^3)*x^2 - 2*
(5*a*c^2*d^3 - 8*a^2*c*d*e^2)*x)*sqrt(e*x + d))/(a^4*c^2*d^4 - 2*a^5*c*d^2*e^2 + a^6*e^4 + (a^2*c^4*d^4 - 2*a^
3*c^3*d^2*e^2 + a^4*c^2*e^4)*x^4 - 2*(a^3*c^3*d^4 - 2*a^4*c^2*d^2*e^2 + a^5*c*e^4)*x^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(1/(-c*x**2+a)**3/(e*x+d)**(1/2),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(1/(-c*x^2+a)^3/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

Timed out