### 3.691 $$\int \frac{1}{(d+e x)^{5/2} (a+c x^2)^{3/2}} \, dx$$

Optimal. Leaf size=485 $\frac{\sqrt{c} \sqrt{\frac{c x^2}{a}+1} \left (3 c d^2-5 a e^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 \sqrt{-a} \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )^2}-\frac{c^{3/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (3 c d^2-29 a e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 \sqrt{-a} \sqrt{a+c x^2} \left (a e^2+c d^2\right )^3 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{c d e \sqrt{a+c x^2} \left (3 c d^2-29 a e^2\right )}{3 a \sqrt{d+e x} \left (a e^2+c d^2\right )^3}+\frac{e \sqrt{a+c x^2} \left (3 c d^2-5 a e^2\right )}{3 a (d+e x)^{3/2} \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{a \sqrt{a+c x^2} (d+e x)^{3/2} \left (a e^2+c d^2\right )}$

[Out]

(a*e + c*d*x)/(a*(c*d^2 + a*e^2)*(d + e*x)^(3/2)*Sqrt[a + c*x^2]) + (e*(3*c*d^2 - 5*a*e^2)*Sqrt[a + c*x^2])/(3
*a*(c*d^2 + a*e^2)^2*(d + e*x)^(3/2)) + (c*d*e*(3*c*d^2 - 29*a*e^2)*Sqrt[a + c*x^2])/(3*a*(c*d^2 + a*e^2)^3*Sq
rt[d + e*x]) - (c^(3/2)*d*(3*c*d^2 - 29*a*e^2)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sq
rt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(3*Sqrt[-a]*(c*d^2 + a*e^2)^3*Sqrt[(Sqrt[c]
*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (Sqrt[c]*(3*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[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a
*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(3*Sqrt[-a]*(c*d^2 + a*e^2)^2*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.494563, antiderivative size = 485, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.286, Rules used = {741, 835, 844, 719, 424, 419} $-\frac{c^{3/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (3 c d^2-29 a e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 \sqrt{-a} \sqrt{a+c x^2} \left (a e^2+c d^2\right )^3 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{c d e \sqrt{a+c x^2} \left (3 c d^2-29 a e^2\right )}{3 a \sqrt{d+e x} \left (a e^2+c d^2\right )^3}+\frac{e \sqrt{a+c x^2} \left (3 c d^2-5 a e^2\right )}{3 a (d+e x)^{3/2} \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{a \sqrt{a+c x^2} (d+e x)^{3/2} \left (a e^2+c d^2\right )}+\frac{\sqrt{c} \sqrt{\frac{c x^2}{a}+1} \left (3 c d^2-5 a e^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 \sqrt{-a} \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)^(5/2)*(a + c*x^2)^(3/2)),x]

[Out]

(a*e + c*d*x)/(a*(c*d^2 + a*e^2)*(d + e*x)^(3/2)*Sqrt[a + c*x^2]) + (e*(3*c*d^2 - 5*a*e^2)*Sqrt[a + c*x^2])/(3
*a*(c*d^2 + a*e^2)^2*(d + e*x)^(3/2)) + (c*d*e*(3*c*d^2 - 29*a*e^2)*Sqrt[a + c*x^2])/(3*a*(c*d^2 + a*e^2)^3*Sq
rt[d + e*x]) - (c^(3/2)*d*(3*c*d^2 - 29*a*e^2)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sq
rt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(3*Sqrt[-a]*(c*d^2 + a*e^2)^3*Sqrt[(Sqrt[c]
*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (Sqrt[c]*(3*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[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a
*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(3*Sqrt[-a]*(c*d^2 + a*e^2)^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 835

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((e*f - d*g)
*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[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}{(d+e x)^{5/2} \left (a+c x^2\right )^{3/2}} \, dx &=\frac{a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \sqrt{a+c x^2}}-\frac{\int \frac{-\frac{5 a e^2}{2}-\frac{3}{2} c d e x}{(d+e x)^{5/2} \sqrt{a+c x^2}} \, dx}{a \left (c d^2+a e^2\right )}\\ &=\frac{a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \sqrt{a+c x^2}}+\frac{e \left (3 c d^2-5 a e^2\right ) \sqrt{a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac{2 \int \frac{6 a c d e^2+\frac{1}{4} c e \left (3 c d^2-5 a e^2\right ) x}{(d+e x)^{3/2} \sqrt{a+c x^2}} \, dx}{3 a \left (c d^2+a e^2\right )^2}\\ &=\frac{a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \sqrt{a+c x^2}}+\frac{e \left (3 c d^2-5 a e^2\right ) \sqrt{a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac{c d e \left (3 c d^2-29 a e^2\right ) \sqrt{a+c x^2}}{3 a \left (c d^2+a e^2\right )^3 \sqrt{d+e x}}-\frac{4 \int \frac{-\frac{1}{8} a c e^2 \left (27 c d^2-5 a e^2\right )+\frac{1}{8} c^2 d e \left (3 c d^2-29 a e^2\right ) x}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{3 a \left (c d^2+a e^2\right )^3}\\ &=\frac{a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \sqrt{a+c x^2}}+\frac{e \left (3 c d^2-5 a e^2\right ) \sqrt{a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac{c d e \left (3 c d^2-29 a e^2\right ) \sqrt{a+c x^2}}{3 a \left (c d^2+a e^2\right )^3 \sqrt{d+e x}}-\frac{\left (c^2 d \left (3 c d^2-29 a e^2\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+c x^2}} \, dx}{6 a \left (c d^2+a e^2\right )^3}+\frac{\left (c \left (3 c d^2-5 a e^2\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{6 a \left (c d^2+a e^2\right )^2}\\ &=\frac{a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \sqrt{a+c x^2}}+\frac{e \left (3 c d^2-5 a e^2\right ) \sqrt{a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac{c d e \left (3 c d^2-29 a e^2\right ) \sqrt{a+c x^2}}{3 a \left (c d^2+a e^2\right )^3 \sqrt{d+e x}}-\frac{\left (c^{3/2} d \left (3 c d^2-29 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} \left (c d^2+a e^2\right )^3 \sqrt{\frac{c (d+e x)}{c d-\frac{a \sqrt{c} e}{\sqrt{-a}}}} \sqrt{a+c x^2}}+\frac{\left (\sqrt{c} \left (3 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 )}{3 \sqrt{-a} \left (c d^2+a e^2\right )^2 \sqrt{d+e x} \sqrt{a+c x^2}}\\ &=\frac{a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \sqrt{a+c x^2}}+\frac{e \left (3 c d^2-5 a e^2\right ) \sqrt{a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac{c d e \left (3 c d^2-29 a e^2\right ) \sqrt{a+c x^2}}{3 a \left (c d^2+a e^2\right )^3 \sqrt{d+e x}}-\frac{c^{3/2} d \left (3 c d^2-29 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 \sqrt{-a} \left (c d^2+a e^2\right )^3 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{a+c x^2}}+\frac{\sqrt{c} \left (3 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 )}{3 \sqrt{-a} \left (c d^2+a e^2\right )^2 \sqrt{d+e x} \sqrt{a+c x^2}}\\ \end{align*}

Mathematica [C]  time = 3.62985, size = 634, normalized size = 1.31 $\frac{\frac{c (d+e x) \left (\sqrt{a} e (d+e x)^{3/2} \left (-5 i a^{3/2} e^3+27 i \sqrt{a} c d^2 e-29 a \sqrt{c} d e^2+3 c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right ),\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )-d e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (-29 a^2 e^2+a c \left (3 d^2-29 e^2 x^2\right )+3 c^2 d^2 x^2\right )+\sqrt{c} d (d+e x)^{3/2} \left (29 a^{3/2} e^3-3 \sqrt{a} c d^2 e-29 i a \sqrt{c} d e^2+3 i c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{e \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}+3 c (d+e x)^2 \left (-a^2 e^3+3 a c d e (d-e x)+c^2 d^3 x\right )-2 a e^3 \left (a+c x^2\right ) \left (a e^2+c d^2\right )-20 a c d e^3 \left (a+c x^2\right ) (d+e x)}{3 a \sqrt{a+c x^2} (d+e x)^{3/2} \left (a e^2+c d^2\right )^3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)^(5/2)*(a + c*x^2)^(3/2)),x]

[Out]

(-2*a*e^3*(c*d^2 + a*e^2)*(a + c*x^2) - 20*a*c*d*e^3*(d + e*x)*(a + c*x^2) + 3*c*(d + e*x)^2*(-(a^2*e^3) + c^2
*d^3*x + 3*a*c*d*e*(d - e*x)) + (c*(d + e*x)*(-(d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(-29*a^2*e^2 + 3*c^2*d^
2*x^2 + a*c*(3*d^2 - 29*e^2*x^2))) + Sqrt[c]*d*((3*I)*c^(3/2)*d^3 - 3*Sqrt[a]*c*d^2*e - (29*I)*a*Sqrt[c]*d*e^2
+ 29*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
))]*(d + e*x)^(3/2)*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)] + Sqrt[a]*e*(3*c^(3/2)*d^3 + (27*I)*Sqrt[a]*c*d^2*e - 29*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))]
*(d + e*x)^(3/2)*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)]))/(e*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]))/(3*a*(c*d^2 + a*e^2)^3*(d + e*x)^(3/2)*Sq
rt[a + c*x^2])

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Maple [B]  time = 0.305, size = 2623, normalized size = 5.4 \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)^(5/2)/(c*x^2+a)^(3/2),x)

[Out]

1/3*(3*x^3*c^3*d^3*e^3-5*x^2*a^2*c*e^6+6*x^2*c^3*d^4*e^2-31*x^2*a*c^2*d^2*e^4-26*x*a^2*c*d*e^5+9*x*a*c^2*d^3*e
^3+3*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))*c^3
*d^6*(-(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))*c^2*d^5*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)-29*x^3*a*c^2*d*e^5+24*Elli
pticF((-(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*a^2*c*d*e^
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)+24*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*a*c^2*d^3*e^3*(-(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*c^2*d^4*e^2*(-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)-29*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*a^2*c*d*e^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)-26*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*a*c^2*d^3*e^3*(-(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)+2*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*d^3*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)-29*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^2*c*d^2*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)-26*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^2*d^4*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*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*c^3*d^5*e*(-(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)+24*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^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)+24*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^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*x*c^3*d^5*e+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*a^2*e^6*(-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)+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)
*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)^
(1/2)*a^2*d*e^5+2*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*a*c*d^2*e^4*(-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)-2*a^3*e^6-25*a^2*c*d^2*e^4+9*a*c^2*d^4*e^2
)/(c*x^2+a)^(1/2)/(a*e^2+c*d^2)^3/a/(e*x+d)^(3/2)/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{3}{2}}{\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)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + a)^(3/2)*(e*x + d)^(5/2)), 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^{2} e^{3} x^{7} + 3 \, c^{2} d e^{2} x^{6} + 3 \, a^{2} d^{2} e x +{\left (3 \, c^{2} d^{2} e + 2 \, a c e^{3}\right )} x^{5} + a^{2} d^{3} +{\left (c^{2} d^{3} + 6 \, a c d e^{2}\right )} x^{4} +{\left (6 \, a c d^{2} e + a^{2} e^{3}\right )} x^{3} +{\left (2 \, a c d^{3} + 3 \, a^{2} d e^{2}\right )} x^{2}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(c*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^2 + a)*sqrt(e*x + d)/(c^2*e^3*x^7 + 3*c^2*d*e^2*x^6 + 3*a^2*d^2*e*x + (3*c^2*d^2*e + 2*a*c*e
^3)*x^5 + a^2*d^3 + (c^2*d^3 + 6*a*c*d*e^2)*x^4 + (6*a*c*d^2*e + a^2*e^3)*x^3 + (2*a*c*d^3 + 3*a^2*d*e^2)*x^2)
, 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{3}{2}} \left (d + e x\right )^{\frac{5}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(5/2)/(c*x**2+a)**(3/2),x)

[Out]

Integral(1/((a + c*x**2)**(3/2)*(d + e*x)**(5/2)), 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{3}{2}}{\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)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((c*x^2 + a)^(3/2)*(e*x + d)^(5/2)), x)