3.657 \(\int (d+e x)^{3/2} \sqrt{a+c x^2} \, dx\)

Optimal. Leaf size=398 \[ -\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (3 c d^2-5 a e^2\right ) \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{105 c^{3/2} e^2 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{a+c x^2} \sqrt{d+e x} \left (-5 a e^2+3 c d^2+24 c d e x\right )}{105 c e}+\frac{4 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (3 c d^2-29 a e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{105 \sqrt{c} e^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 e \left (a+c x^2\right )^{3/2} \sqrt{d+e x}}{7 c} \]

[Out]

(2*Sqrt[d + e*x]*(3*c*d^2 - 5*a*e^2 + 24*c*d*e*x)*Sqrt[a + c*x^2])/(105*c*e) + (2*e*Sqrt[d + e*x]*(a + c*x^2)^
(3/2))/(7*c) + (4*Sqrt[-a]*d*(3*c*d^2 - 29*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)])/(105*Sqrt[c]*e^2*Sqrt[(Sqrt[c]*(d + e*x)
)/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (4*Sqrt[-a]*(3*c*d^2 - 5*a*e^2)*(c*d^2 + 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)])/(105*c^(3/2)*e^2*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.441283, antiderivative size = 398, 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 = {743, 815, 844, 719, 424, 419} \[ -\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (3 c d^2-5 a e^2\right ) \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{105 c^{3/2} e^2 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{a+c x^2} \sqrt{d+e x} \left (-5 a e^2+3 c d^2+24 c d e x\right )}{105 c e}+\frac{4 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (3 c d^2-29 a e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{105 \sqrt{c} e^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 e \left (a+c x^2\right )^{3/2} \sqrt{d+e x}}{7 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(3*c*d^2 - 5*a*e^2 + 24*c*d*e*x)*Sqrt[a + c*x^2])/(105*c*e) + (2*e*Sqrt[d + e*x]*(a + c*x^2)^
(3/2))/(7*c) + (4*Sqrt[-a]*d*(3*c*d^2 - 29*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)])/(105*Sqrt[c]*e^2*Sqrt[(Sqrt[c]*(d + e*x)
)/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (4*Sqrt[-a]*(3*c*d^2 - 5*a*e^2)*(c*d^2 + 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)])/(105*c^(3/2)*e^2*Sqrt[d + e*x]*Sqrt[a + c*x^2])

Rule 743

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))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - a*e^
2*(m - 1) + 2*c*d*e*(m + p)*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] && If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m
, p, x]

Rule 815

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*(a + c*x^2)^p)/(c*e^2*(m + 2*p + 1)*(m
+ 2*p + 2)), x] + Dist[(2*p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a + c*x^2)^(p - 1)*Simp[f*a
*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))
*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !R
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (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 (d+e x)^{3/2} \sqrt{a+c x^2} \, dx &=\frac{2 e \sqrt{d+e x} \left (a+c x^2\right )^{3/2}}{7 c}+\frac{2 \int \frac{\left (\frac{1}{2} \left (7 c d^2-a e^2\right )+4 c d e x\right ) \sqrt{a+c x^2}}{\sqrt{d+e x}} \, dx}{7 c}\\ &=\frac{2 \sqrt{d+e x} \left (3 c d^2-5 a e^2+24 c d e x\right ) \sqrt{a+c x^2}}{105 c e}+\frac{2 e \sqrt{d+e x} \left (a+c x^2\right )^{3/2}}{7 c}+\frac{8 \int \frac{\frac{1}{4} a c e^2 \left (27 c d^2-5 a e^2\right )-\frac{1}{4} 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}{105 c^2 e^2}\\ &=\frac{2 \sqrt{d+e x} \left (3 c d^2-5 a e^2+24 c d e x\right ) \sqrt{a+c x^2}}{105 c e}+\frac{2 e \sqrt{d+e x} \left (a+c x^2\right )^{3/2}}{7 c}+\frac{1}{105} \left (2 d \left (29 a-\frac{3 c d^2}{e^2}\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+c x^2}} \, dx+\frac{\left (2 \left (3 c d^2-5 a e^2\right ) \left (c d^2+a e^2\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{105 c e^2}\\ &=\frac{2 \sqrt{d+e x} \left (3 c d^2-5 a e^2+24 c d e x\right ) \sqrt{a+c x^2}}{105 c e}+\frac{2 e \sqrt{d+e x} \left (a+c x^2\right )^{3/2}}{7 c}+\frac{\left (4 a d \left (29 a-\frac{3 c d^2}{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 )}{105 \sqrt{-a} \sqrt{c} \sqrt{\frac{c (d+e x)}{c d-\frac{a \sqrt{c} e}{\sqrt{-a}}}} \sqrt{a+c x^2}}+\frac{\left (4 a \left (3 c d^2-5 a e^2\right ) \left (c d^2+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 )}{105 \sqrt{-a} c^{3/2} e^2 \sqrt{d+e x} \sqrt{a+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (3 c d^2-5 a e^2+24 c d e x\right ) \sqrt{a+c x^2}}{105 c e}+\frac{2 e \sqrt{d+e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac{4 \sqrt{-a} d \left (29 a-\frac{3 c d^2}{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 )}{105 \sqrt{c} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{a+c x^2}}-\frac{4 \sqrt{-a} \left (3 c d^2-5 a e^2\right ) \left (c d^2+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 )}{105 c^{3/2} e^2 \sqrt{d+e x} \sqrt{a+c x^2}}\\ \end{align*}

Mathematica [C]  time = 3.41283, size = 582, normalized size = 1.46 \[ \frac{\sqrt{d+e x} \left (\frac{2 \left (a+c x^2\right ) \left (10 a e^2+3 c \left (d^2+8 d e x+5 e^2 x^2\right )\right )}{c e}+\frac{4 \left (\sqrt{a} e (d+e x)^{3/2} \left (-5 i a^{3/2} e^3+27 i \sqrt{a} c d^2 e-29 a \sqrt{c} d e^2+3 c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right ),\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )-d e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (-29 a^2 e^2+a c \left (3 d^2-29 e^2 x^2\right )+3 c^2 d^2 x^2\right )+\sqrt{c} d (d+e x)^{3/2} \left (29 a^{3/2} e^3-3 \sqrt{a} c d^2 e-29 i a \sqrt{c} d e^2+3 i c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{c e^3 (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{105 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x]*((2*(a + c*x^2)*(10*a*e^2 + 3*c*(d^2 + 8*d*e*x + 5*e^2*x^2)))/(c*e) + (4*(-(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)]*Sq
rt[-(((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)/Sqr
t[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)]))/(c*e^3*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[
c]]*(d + e*x))))/(105*Sqrt[a + c*x^2])

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Maple [B]  time = 0.394, size = 1386, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)*(15*x^5*c^3*e^5+10*(-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)*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*e^5+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)*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^2*e^3-6*(-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)*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^4*e+48*a^2*c*(-(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))*d*e^4+48*a*c^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)*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))*d^3*e^2-58*a^2*c*(-(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)*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))*d*e^4-52*a*c^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)*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))*d^3*e^2+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)*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^5+39*x^4*c^3*d*e^4+25*x^3*a*c^2*e^5+27*x^3*c^3*d^2*e^3+4
9*x^2*a*c^2*d*e^4+3*x^2*c^3*d^3*e^2+10*x*a^2*c*e^5+27*x*a*c^2*d^2*e^3+10*a^2*c*d*e^4+3*a*c^2*d^3*e^2)/e^3/(c*e
*x^3+c*d*x^2+a*e*x+a*d)/c^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + c x^{2}} \left (d + e x\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + c*x**2)*(d + e*x)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c x^{2} + a}{\left (e x + d\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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