### 3.684 $$\int \frac{1}{(d+e x)^{7/2} \sqrt{a+c x^2}} \, dx$$

Optimal. Leaf size=447 $\frac{16 \sqrt{-a} c^{3/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )^2}-\frac{2 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (23 c d^2-9 a e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 \sqrt{a+c x^2} \left (a e^2+c d^2\right )^3 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{2 c e \sqrt{a+c x^2} \left (23 c d^2-9 a e^2\right )}{15 \sqrt{d+e x} \left (a e^2+c d^2\right )^3}-\frac{16 c d e \sqrt{a+c x^2}}{15 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2}-\frac{2 e \sqrt{a+c x^2}}{5 (d+e x)^{5/2} \left (a e^2+c d^2\right )}$

[Out]

(-2*e*Sqrt[a + c*x^2])/(5*(c*d^2 + a*e^2)*(d + e*x)^(5/2)) - (16*c*d*e*Sqrt[a + c*x^2])/(15*(c*d^2 + a*e^2)^2*
(d + e*x)^(3/2)) - (2*c*e*(23*c*d^2 - 9*a*e^2)*Sqrt[a + c*x^2])/(15*(c*d^2 + a*e^2)^3*Sqrt[d + e*x]) - (2*Sqrt
[-a]*c^(3/2)*(23*c*d^2 - 9*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)])/(15*(c*d^2 + a*e^2)^3*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*
d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (16*Sqrt[-a]*c^(3/2)*d*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)])/(15*(c*d^2 + a*e^2)^2*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.408262, antiderivative size = 447, 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 = {745, 835, 844, 719, 424, 419} $\frac{16 \sqrt{-a} c^{3/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )^2}-\frac{2 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (23 c d^2-9 a e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 \sqrt{a+c x^2} \left (a e^2+c d^2\right )^3 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{2 c e \sqrt{a+c x^2} \left (23 c d^2-9 a e^2\right )}{15 \sqrt{d+e x} \left (a e^2+c d^2\right )^3}-\frac{16 c d e \sqrt{a+c x^2}}{15 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2}-\frac{2 e \sqrt{a+c x^2}}{5 (d+e x)^{5/2} \left (a e^2+c d^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)^(7/2)*Sqrt[a + c*x^2]),x]

[Out]

(-2*e*Sqrt[a + c*x^2])/(5*(c*d^2 + a*e^2)*(d + e*x)^(5/2)) - (16*c*d*e*Sqrt[a + c*x^2])/(15*(c*d^2 + a*e^2)^2*
(d + e*x)^(3/2)) - (2*c*e*(23*c*d^2 - 9*a*e^2)*Sqrt[a + c*x^2])/(15*(c*d^2 + a*e^2)^3*Sqrt[d + e*x]) - (2*Sqrt
[-a]*c^(3/2)*(23*c*d^2 - 9*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)])/(15*(c*d^2 + a*e^2)^3*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*
d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (16*Sqrt[-a]*c^(3/2)*d*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)])/(15*(c*d^2 + a*e^2)^2*Sqrt[d + e*x]*Sqrt[a + c*x^2])

Rule 745

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

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)^{7/2} \sqrt{a+c x^2}} \, dx &=-\frac{2 e \sqrt{a+c x^2}}{5 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac{(2 c) \int \frac{-\frac{5 d}{2}+\frac{3 e x}{2}}{(d+e x)^{5/2} \sqrt{a+c x^2}} \, dx}{5 \left (c d^2+a e^2\right )}\\ &=-\frac{2 e \sqrt{a+c x^2}}{5 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac{16 c d e \sqrt{a+c x^2}}{15 \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac{(4 c) \int \frac{\frac{3}{4} \left (5 c d^2-3 a e^2\right )-2 c d e x}{(d+e x)^{3/2} \sqrt{a+c x^2}} \, dx}{15 \left (c d^2+a e^2\right )^2}\\ &=-\frac{2 e \sqrt{a+c x^2}}{5 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac{16 c d e \sqrt{a+c x^2}}{15 \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 c e \left (23 c d^2-9 a e^2\right ) \sqrt{a+c x^2}}{15 \left (c d^2+a e^2\right )^3 \sqrt{d+e x}}-\frac{(8 c) \int \frac{-\frac{1}{8} c d \left (15 c d^2-17 a e^2\right )-\frac{1}{8} c e \left (23 c d^2-9 a e^2\right ) x}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{15 \left (c d^2+a e^2\right )^3}\\ &=-\frac{2 e \sqrt{a+c x^2}}{5 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac{16 c d e \sqrt{a+c x^2}}{15 \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 c e \left (23 c d^2-9 a e^2\right ) \sqrt{a+c x^2}}{15 \left (c d^2+a e^2\right )^3 \sqrt{d+e x}}+\frac{\left (c^2 \left (23 c d^2-9 a e^2\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+c x^2}} \, dx}{15 \left (c d^2+a e^2\right )^3}-\frac{\left (8 c^2 d\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{15 \left (c d^2+a e^2\right )^2}\\ &=-\frac{2 e \sqrt{a+c x^2}}{5 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac{16 c d e \sqrt{a+c x^2}}{15 \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 c e \left (23 c d^2-9 a e^2\right ) \sqrt{a+c x^2}}{15 \left (c d^2+a e^2\right )^3 \sqrt{d+e x}}+\frac{\left (2 a c^{3/2} \left (23 c d^2-9 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 )}{15 \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 (16 a c^{3/2} d \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 )}{15 \sqrt{-a} \left (c d^2+a e^2\right )^2 \sqrt{d+e x} \sqrt{a+c x^2}}\\ &=-\frac{2 e \sqrt{a+c x^2}}{5 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac{16 c d e \sqrt{a+c x^2}}{15 \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 c e \left (23 c d^2-9 a e^2\right ) \sqrt{a+c x^2}}{15 \left (c d^2+a e^2\right )^3 \sqrt{d+e x}}-\frac{2 \sqrt{-a} c^{3/2} \left (23 c d^2-9 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 )}{15 \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{16 \sqrt{-a} c^{3/2} d \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 )}{15 \left (c d^2+a e^2\right )^2 \sqrt{d+e x} \sqrt{a+c x^2}}\\ \end{align*}

Mathematica [C]  time = 3.25857, size = 618, normalized size = 1.38 $\frac{2 \left (e^2 \left (-\left (a+c x^2\right )\right ) \left (8 c d (d+e x) \left (a e^2+c d^2\right )+c (d+e x)^2 \left (23 c d^2-9 a e^2\right )+3 \left (a e^2+c d^2\right )^2\right )-\frac{c (d+e x)^2 \left (\sqrt{c} (d+e x)^{3/2} \left (-9 a^{3/2} e^3+23 \sqrt{a} c d^2 e+17 i a \sqrt{c} d e^2-15 i c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right ),\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+e^2 \left (-\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}\right ) \left (-9 a^2 e^2+a c \left (23 d^2-9 e^2 x^2\right )+23 c^2 d^2 x^2\right )+\sqrt{c} (d+e x)^{3/2} \left (9 a^{3/2} e^3-23 \sqrt{a} c d^2 e-9 i a \sqrt{c} d e^2+23 i c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{15 e \sqrt{a+c x^2} (d+e x)^{5/2} \left (a e^2+c d^2\right )^3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)^(7/2)*Sqrt[a + c*x^2]),x]

[Out]

(2*(-(e^2*(a + c*x^2)*(3*(c*d^2 + a*e^2)^2 + 8*c*d*(c*d^2 + a*e^2)*(d + e*x) + c*(23*c*d^2 - 9*a*e^2)*(d + e*x
)^2)) - (c*(d + e*x)^2*(-(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(-9*a^2*e^2 + 23*c^2*d^2*x^2 + a*c*(23*d^2 - 9*
e^2*x^2))) + Sqrt[c]*((23*I)*c^(3/2)*d^3 - 23*Sqrt[a]*c*d^2*e - (9*I)*a*Sqrt[c]*d*e^2 + 9*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)*Ellipt
icE[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[c]*((-15*I)*c^(3/2)*d^3 + 23*Sqrt[a]*c*d^2*e + (17*I)*a*Sqrt[c]*d*e^2 - 9*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
)]))/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]))/(15*e*(c*d^2 + a*e^2)^3*(d + e*x)^(5/2)*Sqrt[a + c*x^2])

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Maple [B]  time = 0.307, size = 3863, normalized size = 8.6 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(7/2)/(c*x^2+a)^(1/2),x)

[Out]

-2/15*(23*x^4*c^3*d^2*e^4+54*x^3*c^3*d^3*e^3-6*x^2*a^2*c*e^6+34*x^2*c^3*d^4*e^2-9*x^4*a*c^2*e^6-15*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^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)+28*x^2*a*c^2*d^2*e^4-10*x*a^2*c*d*e^5+54*x*a*c^2*d^3*e^3+23*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)-8*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)-10*x^3*a*c^2*d*e^5+18*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*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)-12*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)-16*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)-18*Ellipt
icE((-(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)+28*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)-8*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)-6*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^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)-8*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^2*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)+
14*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^2*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)-9*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)+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^2*c*e^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)+14*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)-9*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*e^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)+23*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^3*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)+46*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)+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^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)-6*Elliptic
F((-(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)-30*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^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)-8*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*d*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)-16*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)+3*a^3*e^6-15*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^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)+5*a^
2*c*d^2*e^4+34*a*c^2*d^4*e^2)/(c*x^2+a)^(1/2)/(a*e^2+c*d^2)^3/(e*x+d)^(5/2)/e

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{2} + a}{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(c*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(c*x^2 + a)*(e*x + d)^(7/2)), 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 e^{4} x^{6} + 4 \, c d e^{3} x^{5} + 4 \, a d^{3} e x + a d^{4} +{\left (6 \, c d^{2} e^{2} + a e^{4}\right )} x^{4} + 4 \,{\left (c d^{3} e + a d e^{3}\right )} x^{3} +{\left (c d^{4} + 6 \, a d^{2} 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)^(7/2)/(c*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + c x^{2}} \left (d + e x\right )^{\frac{7}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(7/2)/(c*x**2+a)**(1/2),x)

[Out]

Integral(1/(sqrt(a + c*x**2)*(d + e*x)**(7/2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{2} + a}{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(c*x^2+a)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(c*x^2 + a)*(e*x + d)^(7/2)), x)