### 3.697 $$\int \frac{1}{\sqrt{d+e x} (a+c x^2)^{5/2}} \, dx$$

Optimal. Leaf size=450 $-\frac{\sqrt{\frac{c x^2}{a}+1} \left (5 a e^2+4 c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{6 (-a)^{3/2} \sqrt{c} \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )}+\frac{\sqrt{d+e x} \left (4 c d x \left (2 a e^2+c d^2\right )+a e \left (5 a e^2+c d^2\right )\right )}{6 a^2 \sqrt{a+c x^2} \left (a e^2+c d^2\right )^2}+\frac{\sqrt{d+e x} (a e+c d x)}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}+\frac{2 \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (2 a e^2+c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 (-a)^{3/2} \sqrt{a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}$

[Out]

((a*e + c*d*x)*Sqrt[d + e*x])/(3*a*(c*d^2 + a*e^2)*(a + c*x^2)^(3/2)) + (Sqrt[d + e*x]*(a*e*(c*d^2 + 5*a*e^2)
+ 4*c*d*(c*d^2 + 2*a*e^2)*x))/(6*a^2*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^2]) + (2*Sqrt[c]*d*(c*d^2 + 2*a*e^2)*Sqrt[
d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt
[c]*d - a*e)])/(3*(-a)^(3/2)*(c*d^2 + a*e^2)^2*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x
^2]) - ((4*c*d^2 + 5*a*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[A
rcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(6*(-a)^(3/2)*Sqrt[c]*(c*
d^2 + a*e^2)*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.384817, antiderivative size = 450, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.286, Rules used = {741, 823, 844, 719, 424, 419} $\frac{\sqrt{d+e x} \left (4 c d x \left (2 a e^2+c d^2\right )+a e \left (5 a e^2+c d^2\right )\right )}{6 a^2 \sqrt{a+c x^2} \left (a e^2+c d^2\right )^2}+\frac{\sqrt{d+e x} (a e+c d x)}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}-\frac{\sqrt{\frac{c x^2}{a}+1} \left (5 a e^2+4 c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{6 (-a)^{3/2} \sqrt{c} \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )}+\frac{2 \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (2 a e^2+c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 (-a)^{3/2} \sqrt{a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((a*e + c*d*x)*Sqrt[d + e*x])/(3*a*(c*d^2 + a*e^2)*(a + c*x^2)^(3/2)) + (Sqrt[d + e*x]*(a*e*(c*d^2 + 5*a*e^2)
+ 4*c*d*(c*d^2 + 2*a*e^2)*x))/(6*a^2*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^2]) + (2*Sqrt[c]*d*(c*d^2 + 2*a*e^2)*Sqrt[
d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt
[c]*d - a*e)])/(3*(-a)^(3/2)*(c*d^2 + a*e^2)^2*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x
^2]) - ((4*c*d^2 + 5*a*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[A
rcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(6*(-a)^(3/2)*Sqrt[c]*(c*
d^2 + a*e^2)*Sqrt[d + e*x]*Sqrt[a + c*x^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 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 719

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(2*a*Rt[-(c/a), 2]*(d + e*x)^m*Sqrt[
1 + (c*x^2)/a])/(c*Sqrt[a + c*x^2]*((c*(d + e*x))/(c*d - a*e*Rt[-(c/a), 2]))^m), Subst[Int[(1 + (2*a*e*Rt[-(c/
a), 2]*x^2)/(c*d - a*e*Rt[-(c/a), 2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(1 - Rt[-(c/a), 2]*x)/2]], x] /; FreeQ[{a,
c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m^2, 1/4]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{d+e x} \left (a+c x^2\right )^{5/2}} \, dx &=\frac{(a e+c d x) \sqrt{d+e x}}{3 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^{3/2}}-\frac{\int \frac{\frac{1}{2} \left (-4 c d^2-5 a e^2\right )-\frac{3}{2} c d e x}{\sqrt{d+e x} \left (a+c x^2\right )^{3/2}} \, dx}{3 a \left (c d^2+a e^2\right )}\\ &=\frac{(a e+c d x) \sqrt{d+e x}}{3 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^{3/2}}+\frac{\sqrt{d+e x} \left (a e \left (c d^2+5 a e^2\right )+4 c d \left (c d^2+2 a e^2\right ) x\right )}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt{a+c x^2}}+\frac{\int \frac{\frac{1}{4} a c e^2 \left (c d^2+5 a e^2\right )-c^2 d e \left (c d^2+2 a e^2\right ) x}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{3 a^2 c \left (c d^2+a e^2\right )^2}\\ &=\frac{(a e+c d x) \sqrt{d+e x}}{3 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^{3/2}}+\frac{\sqrt{d+e x} \left (a e \left (c d^2+5 a e^2\right )+4 c d \left (c d^2+2 a e^2\right ) x\right )}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt{a+c x^2}}-\frac{\left (c d \left (c d^2+2 a e^2\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+c x^2}} \, dx}{3 a^2 \left (c d^2+a e^2\right )^2}+\frac{\left (4 c d^2+5 a e^2\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{12 a^2 \left (c d^2+a e^2\right )}\\ &=\frac{(a e+c d x) \sqrt{d+e x}}{3 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^{3/2}}+\frac{\sqrt{d+e x} \left (a e \left (c d^2+5 a e^2\right )+4 c d \left (c d^2+2 a e^2\right ) x\right )}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt{a+c x^2}}-\frac{\left (2 \sqrt{c} d \left (c d^2+2 a e^2\right ) \sqrt{d+e x} \sqrt{1+\frac{c x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 a \sqrt{c} e x^2}{\sqrt{-a} \left (c d-\frac{a \sqrt{c} e}{\sqrt{-a}}\right )}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )}{3 \sqrt{-a} a \left (c d^2+a e^2\right )^2 \sqrt{\frac{c (d+e x)}{c d-\frac{a \sqrt{c} e}{\sqrt{-a}}}} \sqrt{a+c x^2}}+\frac{\left (\left (4 c d^2+5 a e^2\right ) \sqrt{\frac{c (d+e x)}{c d-\frac{a \sqrt{c} e}{\sqrt{-a}}}} \sqrt{1+\frac{c x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 a \sqrt{c} e x^2}{\sqrt{-a} \left (c d-\frac{a \sqrt{c} e}{\sqrt{-a}}\right )}}} \, dx,x,\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )}{6 \sqrt{-a} a \sqrt{c} \left (c d^2+a e^2\right ) \sqrt{d+e x} \sqrt{a+c x^2}}\\ &=\frac{(a e+c d x) \sqrt{d+e x}}{3 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^{3/2}}+\frac{\sqrt{d+e x} \left (a e \left (c d^2+5 a e^2\right )+4 c d \left (c d^2+2 a e^2\right ) x\right )}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt{a+c x^2}}+\frac{2 \sqrt{c} d \left (c d^2+2 a e^2\right ) \sqrt{d+e x} \sqrt{1+\frac{c x^2}{a}} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 (-a)^{3/2} \left (c d^2+a e^2\right )^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{a+c x^2}}-\frac{\left (4 c d^2+5 a e^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{1+\frac{c x^2}{a}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{6 (-a)^{3/2} \sqrt{c} \left (c d^2+a e^2\right ) \sqrt{d+e x} \sqrt{a+c x^2}}\\ \end{align*}

Mathematica [C]  time = 2.0383, size = 570, normalized size = 1.27 $\frac{\sqrt{d+e x} \left (\frac{\sqrt{a} \sqrt{d+e x} \left (5 i a^{3/2} e^3+i \sqrt{a} c d^2 e+8 a \sqrt{c} d e^2+4 c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right ),\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}-\frac{4 d e \left (2 a^2 e^2+a c \left (d^2+2 e^2 x^2\right )+c^2 d^2 x^2\right )}{d+e x}+5 a^2 e^3+\frac{2 a \left (a e^2+c d^2\right ) (a e+c d x)}{a+c x^2}-\frac{4 i c d \sqrt{d+e x} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (2 a e^2+c d^2\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{e}+a c d^2 e+8 a c d e^2 x+4 c^2 d^3 x\right )}{6 a^2 \sqrt{a+c x^2} \left (a e^2+c d^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[d + e*x]*(a*c*d^2*e + 5*a^2*e^3 + 4*c^2*d^3*x + 8*a*c*d*e^2*x + (2*a*(c*d^2 + a*e^2)*(a*e + c*d*x))/(a +
c*x^2) - (4*d*e*(2*a^2*e^2 + c^2*d^2*x^2 + a*c*(d^2 + 2*e^2*x^2)))/(d + e*x) - ((4*I)*c*d*Sqrt[-d - (I*Sqrt[a
]*e)/Sqrt[c]]*(c*d^2 + 2*a*e^2)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] -
e*x)/(d + e*x))]*Sqrt[d + e*x]*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d
- I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/e + (Sqrt[a]*(4*c^(3/2)*d^3 + I*Sqrt[a]*c*d^2*e + 8*a*Sqrt[c]*d*e^
2 + (5*I)*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d +
e*x))]*Sqrt[d + e*x]*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt
[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]))/(6*a^2*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^2]
)

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Maple [B]  time = 0.299, size = 2673, normalized size = 5.9 \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)^(5/2)/(e*x+d)^(1/2),x)

[Out]

-1/6*(3*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*
a^3*c*d*e^4*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(
-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)-9*x^3*a*c^3*d^2*e^3-13*x*a^2*c^2*d^2*e^3-6*x*a*c^3*d^4*e-7*x^2*a*c^
3*d^3*e^2-15*x^2*a^2*c^2*d*e^4-8*x^4*a*c^3*d*e^4+9*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)
^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*x^2*a*c^2*d^2*e^3*(-a*c)^(1/2)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1
/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)-12*El
lipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^2*c^2*d^
3*e^2*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^
(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)-4*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d
)/((-a*c)^(1/2)*e+c*d))^(1/2))*x^2*c^4*d^5*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a
*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+5*EllipticF((-(e*x+d)*c/((-a*c)^(1/2
)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^3*e^5*(-a*c)^(1/2)*(-(e*x+d)*c/((-a*c)^(
1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*
d))^(1/2)-4*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/
2))*a*c^3*d^5*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x
+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)-4*x^4*c^4*d^3*e^2-5*x^3*a^2*c^2*e^5-4*x^3*c^4*d^4*e-7*x*a^3*c*e^5
-7*a^3*c*d*e^4-3*a^2*c^2*d^3*e^2+4*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((
-a*c)^(1/2)*e+c*d))^(1/2))*x^2*c^3*d^4*e*(-a*c)^(1/2)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1
/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)-8*EllipticE((-(e*x+d)*c/(
(-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*x^2*a^2*c^2*d*e^4*(-(e*x+d)*c/(
(-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1
/2)*e-c*d))^(1/2)-12*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+
c*d))^(1/2))*x^2*a*c^3*d^3*e^2*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+
c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+9*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1
/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^2*c*d^2*e^3*(-a*c)^(1/2)*(-(e*x+d)*c/((-a*c)^(1/2)*e
-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1
/2)+4*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a*
c^2*d^4*e*(-a*c)^(1/2)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1
/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+3*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((
-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*x^2*a^2*c^2*d*e^4*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c
*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+5*EllipticF((
-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*x^2*a^2*c*e^5*(-a*c
)^(1/2)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c
)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+3*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c
*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*x^2*a*c^3*d^3*e^2*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2)
)*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+3*EllipticF((-(e*x+d)*c/((-a
*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^2*c^2*d^3*e^2*(-(e*x+d)*c/((-a*c
)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e
-c*d))^(1/2)-8*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^
(1/2))*a^3*c*d*e^4*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*
((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2))/(e*x+d)^(1/2)/(a*e^2+c*d^2)^2/a^2/(c*x^2+a)^(3/2)/c/e

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + a\right )}^{\frac{5}{2}} \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)^(5/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{2} + a} \sqrt{e x + d}}{c^{3} e x^{7} + c^{3} d x^{6} + 3 \, a c^{2} e x^{5} + 3 \, a c^{2} d x^{4} + 3 \, a^{2} c e x^{3} + 3 \, a^{2} c d x^{2} + a^{3} e x + a^{3} d}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*x^2 + a)*sqrt(e*x + d)/(c^3*e*x^7 + c^3*d*x^6 + 3*a*c^2*e*x^5 + 3*a*c^2*d*x^4 + 3*a^2*c*e*x^3
+ 3*a^2*c*d*x^2 + a^3*e*x + a^3*d), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + c x^{2}\right )^{\frac{5}{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)**(5/2)/(e*x+d)**(1/2),x)

[Out]

Integral(1/((a + c*x**2)**(5/2)*sqrt(d + e*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + a\right )}^{\frac{5}{2}} \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)^(5/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

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