3.617 \(\int \frac{1}{(d+e x)^{5/2} (a-c x^2)} \, dx\)

Optimal. Leaf size=190 \[ -\frac{c^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2}}+\frac{c^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2}}+\frac{4 c d e}{\sqrt{d+e x} \left (c d^2-a e^2\right )^2}+\frac{2 e}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.379885, antiderivative size = 190, 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 = {710, 829, 827, 1166, 208} \[ -\frac{c^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2}}+\frac{c^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2}}+\frac{4 c d e}{\sqrt{d+e x} \left (c d^2-a e^2\right )^2}+\frac{2 e}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(5/2)*(a - c*x^2)),x]

[Out]

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

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 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)^{5/2} \left (a-c x^2\right )} \, dx &=\frac{2 e}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac{c \int \frac{d-e x}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx}{c d^2-a e^2}\\ &=\frac{2 e}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac{4 c d e}{\left (c d^2-a e^2\right )^2 \sqrt{d+e x}}-\frac{c \int \frac{-c d^2-a e^2+2 c d e x}{\sqrt{d+e x} \left (a-c x^2\right )} \, dx}{\left (c d^2-a e^2\right )^2}\\ &=\frac{2 e}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac{4 c d e}{\left (c d^2-a e^2\right )^2 \sqrt{d+e x}}-\frac{(2 c) \operatorname{Subst}\left (\int \frac{-2 c d^2 e+e \left (-c d^2-a e^2\right )+2 c d e x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt{d+e x}\right )}{\left (c d^2-a e^2\right )^2}\\ &=\frac{2 e}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac{4 c d e}{\left (c d^2-a e^2\right )^2 \sqrt{d+e x}}-\frac{c^{3/2} \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 )^2}+\frac{c^{3/2} \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 )^2}\\ &=\frac{2 e}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac{4 c d e}{\left (c d^2-a e^2\right )^2 \sqrt{d+e x}}-\frac{c^{3/4} \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 )^{5/2}}+\frac{c^{3/4} \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 )^{5/2}}\\ \end{align*}

Mathematica [C]  time = 0.0593212, size = 130, normalized size = 0.68 \[ \frac{\frac{\, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{a} e}\right )}{\sqrt{c} d-\sqrt{a} e}-\frac{\, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{a} e}\right )}{\sqrt{a} e+\sqrt{c} d}}{3 \sqrt{a} (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(5/2)*(a - c*x^2)),x]

[Out]

(Hypergeometric2F1[-3/2, 1, -1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[a]*e)]/(Sqrt[c]*d - Sqrt[a]*e) - Hyper
geometric2F1[-3/2, 1, -1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]/(Sqrt[c]*d + Sqrt[a]*e))/(3*Sqrt[a]*(
d + e*x)^(3/2))

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Maple [B]  time = 0.211, size = 472, normalized size = 2.5 \begin{align*} -{\frac{2\,e}{3\,a{e}^{2}-3\,c{d}^{2}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+4\,{\frac{ced}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}\sqrt{ex+d}}}+{\frac{{c}^{2}a{e}^{3}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{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{e{c}^{3}{d}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{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}}}}+2\,{\frac{{c}^{2}ed}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}} \right ) }+{\frac{{c}^{2}a{e}^{3}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{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{e{c}^{3}{d}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{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}}}}-2\,{\frac{{c}^{2}ed}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}{\it Artanh} \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(5/2)/(-c*x^2+a),x)

[Out]

-2/3*e/(a*e^2-c*d^2)/(e*x+d)^(3/2)+4*e*c*d/(a*e^2-c*d^2)^2/(e*x+d)^(1/2)+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))*a*e^3+e*c^3/(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^2+2*e*c^2/(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))*d+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))*a*e^3+e*c^3/(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
h((e*x+d)^(1/2)*c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*d^2-2*e*c^2/(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))*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 )}{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(-c*x^2+a),x, algorithm="maxima")

[Out]

-integrate(1/((c*x^2 - a)*(e*x + d)^(5/2)), x)

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Fricas [B]  time = 3.03937, size = 10656, normalized size = 56.08 \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)^(5/2)/(-c*x^2+a),x, algorithm="fricas")

[Out]

1/6*(3*(c^2*d^6 - 2*a*c*d^4*e^2 + a^2*d^2*e^4 + (c^2*d^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e - 2
*a*c*d^3*e^3 + a^2*d*e^5)*x)*sqrt((c^4*d^5 + 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 + (a*c^5*d^10 - 5*a^2*c^4*d^8*
e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d
^6*e^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*
c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 12
0*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 +
10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))*log((5*c^4*d^4*e + 10*a*c^3*d^2*e^3 +
a^2*c^2*e^5)*sqrt(e*x + d) + (15*a*c^4*d^6*e^2 + 35*a^2*c^3*d^4*e^4 + 13*a^3*c^2*d^2*e^6 + a^4*c*e^8 - (a*c^6*
d^13 - 2*a^2*c^5*d^11*e^2 - 5*a^3*c^4*d^9*e^4 + 20*a^4*c^3*d^7*e^6 - 25*a^5*c^2*d^5*e^8 + 14*a^6*c*d^3*e^10 -
3*a^7*d*e^12)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^
10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 2
52*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18
+ a^11*e^20)))*sqrt((c^4*d^5 + 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 + (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c
^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4 + 110*a
^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 -
120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*
e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^
6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))) - 3*(c^2*d^6 - 2*a*c*d^4*e^2 + a^2*d^2*e^4 + (c^2*d
^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e - 2*a*c*d^3*e^3 + a^2*d*e^5)*x)*sqrt((c^4*d^5 + 10*a*c^3*
d^3*e^2 + 5*a^2*c^2*d*e^4 + (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*
c*d^2*e^8 - a^6*e^10)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^
4*c^3*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12
*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d
^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2
*e^8 - a^6*e^10))*log((5*c^4*d^4*e + 10*a*c^3*d^2*e^3 + a^2*c^2*e^5)*sqrt(e*x + d) - (15*a*c^4*d^6*e^2 + 35*a^
2*c^3*d^4*e^4 + 13*a^3*c^2*d^2*e^6 + a^4*c*e^8 - (a*c^6*d^13 - 2*a^2*c^5*d^11*e^2 - 5*a^3*c^4*d^9*e^4 + 20*a^4
*c^3*d^7*e^6 - 25*a^5*c^2*d^5*e^8 + 14*a^6*c*d^3*e^10 - 3*a^7*d*e^12)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4
+ 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^
16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*
c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))*sqrt((c^4*d^5 + 10*a*c^3*d^3*e^2 + 5*a^
2*c^2*d*e^4 + (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^
6*e^10)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a
*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6
*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11
*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^1
0))) + 3*(c^2*d^6 - 2*a*c*d^4*e^2 + a^2*d^2*e^4 + (c^2*d^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e -
2*a*c*d^3*e^3 + a^2*d*e^5)*x)*sqrt((c^4*d^5 + 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 - (a*c^5*d^10 - 5*a^2*c^4*d^
8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6
*d^6*e^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^
3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 -
120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2
+ 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))*log((5*c^4*d^4*e + 10*a*c^3*d^2*e^3
+ a^2*c^2*e^5)*sqrt(e*x + d) + (15*a*c^4*d^6*e^2 + 35*a^2*c^3*d^4*e^4 + 13*a^3*c^2*d^2*e^6 + a^4*c*e^8 + (a*c^
6*d^13 - 2*a^2*c^5*d^11*e^2 - 5*a^3*c^4*d^9*e^4 + 20*a^4*c^3*d^7*e^6 - 25*a^5*c^2*d^5*e^8 + 14*a^6*c*d^3*e^10
- 3*a^7*d*e^12)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*
e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 -
252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^1
8 + a^11*e^20)))*sqrt((c^4*d^5 + 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 - (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3
*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4 + 110
*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4
- 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^
6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*
d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))) - 3*(c^2*d^6 - 2*a*c*d^4*e^2 + a^2*d^2*e^4 + (c^2
*d^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e - 2*a*c*d^3*e^3 + a^2*d*e^5)*x)*sqrt((c^4*d^5 + 10*a*c^
3*d^3*e^2 + 5*a^2*c^2*d*e^4 - (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^
5*c*d^2*e^8 - a^6*e^10)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 +
a^4*c^3*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^
12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c
*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d
^2*e^8 - a^6*e^10))*log((5*c^4*d^4*e + 10*a*c^3*d^2*e^3 + a^2*c^2*e^5)*sqrt(e*x + d) - (15*a*c^4*d^6*e^2 + 35*
a^2*c^3*d^4*e^4 + 13*a^3*c^2*d^2*e^6 + a^4*c*e^8 + (a*c^6*d^13 - 2*a^2*c^5*d^11*e^2 - 5*a^3*c^4*d^9*e^4 + 20*a
^4*c^3*d^7*e^6 - 25*a^5*c^2*d^5*e^8 + 14*a^6*c*d^3*e^10 - 3*a^7*d*e^12)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e
^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*
d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^
8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))*sqrt((c^4*d^5 + 10*a*c^3*d^3*e^2 + 5*
a^2*c^2*d*e^4 - (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 -
a^6*e^10)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/
(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a
^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^
11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e
^10))) + 4*(6*c*d*e^2*x + 7*c*d^2*e - a*e^3)*sqrt(e*x + d))/(c^2*d^6 - 2*a*c*d^4*e^2 + a^2*d^2*e^4 + (c^2*d^4*
e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e - 2*a*c*d^3*e^3 + a^2*d*e^5)*x)

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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/(e*x+d)**(5/2)/(-c*x**2+a),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/(e*x+d)^(5/2)/(-c*x^2+a),x, algorithm="giac")

[Out]

Timed out