### 3.664 $$\int \sqrt{d+e x} (a+c x^2)^{3/2} \, dx$$

Optimal. Leaf size=448 $-\frac{32 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (3 a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{315 \sqrt{c} e^4 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{8 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{315 \sqrt{c} e^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{4 \sqrt{a+c x^2} \sqrt{d+e x} \left (4 d \left (3 a e^2+c d^2\right )-3 e x \left (c d^2-7 a e^2\right )\right )}{315 e^3}+\frac{2 \left (a+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}-\frac{4 d \left (a+c x^2\right )^{3/2} \sqrt{d+e x}}{21 e}$

[Out]

(4*Sqrt[d + e*x]*(4*d*(c*d^2 + 3*a*e^2) - 3*e*(c*d^2 - 7*a*e^2)*x)*Sqrt[a + c*x^2])/(315*e^3) - (4*d*Sqrt[d +
e*x]*(a + c*x^2)^(3/2))/(21*e) + (2*(d + e*x)^(3/2)*(a + c*x^2)^(3/2))/(9*e) + (8*Sqrt[-a]*(4*c^2*d^4 + 15*a*c
*d^2*e^2 - 21*a^2*e^4)*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)])/(315*Sqrt[c]*e^4*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]
*Sqrt[a + c*x^2]) - (32*Sqrt[-a]*d*(c*d^2 + a*e^2)*(c*d^2 + 3*a*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqr
t[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqr
t[c]*d - a*e)])/(315*Sqrt[c]*e^4*Sqrt[d + e*x]*Sqrt[a + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.467111, antiderivative size = 448, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.333, Rules used = {735, 833, 815, 844, 719, 424, 419} $\frac{8 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{315 \sqrt{c} e^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{4 \sqrt{a+c x^2} \sqrt{d+e x} \left (4 d \left (3 a e^2+c d^2\right )-3 e x \left (c d^2-7 a e^2\right )\right )}{315 e^3}-\frac{32 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (3 a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{315 \sqrt{c} e^4 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{2 \left (a+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}-\frac{4 d \left (a+c x^2\right )^{3/2} \sqrt{d+e x}}{21 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(4*Sqrt[d + e*x]*(4*d*(c*d^2 + 3*a*e^2) - 3*e*(c*d^2 - 7*a*e^2)*x)*Sqrt[a + c*x^2])/(315*e^3) - (4*d*Sqrt[d +
e*x]*(a + c*x^2)^(3/2))/(21*e) + (2*(d + e*x)^(3/2)*(a + c*x^2)^(3/2))/(9*e) + (8*Sqrt[-a]*(4*c^2*d^4 + 15*a*c
*d^2*e^2 - 21*a^2*e^4)*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)])/(315*Sqrt[c]*e^4*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]
*Sqrt[a + c*x^2]) - (32*Sqrt[-a]*d*(c*d^2 + a*e^2)*(c*d^2 + 3*a*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqr
t[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqr
t[c]*d - a*e)])/(315*Sqrt[c]*e^4*Sqrt[d + e*x]*Sqrt[a + c*x^2])

Rule 735

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a + c*x^2)^p)/(
e*(m + 2*p + 1)), x] + Dist[(2*p)/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[a*e - c*d*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] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !Ration
alQ[m] || LtQ[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 833

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

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 \sqrt{d+e x} \left (a+c x^2\right )^{3/2} \, dx &=\frac{2 (d+e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 e}+\frac{2 \int (a e-c d x) \sqrt{d+e x} \sqrt{a+c x^2} \, dx}{3 e}\\ &=-\frac{4 d \sqrt{d+e x} \left (a+c x^2\right )^{3/2}}{21 e}+\frac{2 (d+e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 e}+\frac{4 \int \frac{\left (4 a c d e-\frac{1}{2} c \left (c d^2-7 a e^2\right ) x\right ) \sqrt{a+c x^2}}{\sqrt{d+e x}} \, dx}{21 c e}\\ &=\frac{4 \sqrt{d+e x} \left (4 d \left (c d^2+3 a e^2\right )-3 e \left (c d^2-7 a e^2\right ) x\right ) \sqrt{a+c x^2}}{315 e^3}-\frac{4 d \sqrt{d+e x} \left (a+c x^2\right )^{3/2}}{21 e}+\frac{2 (d+e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 e}+\frac{16 \int \frac{\frac{1}{4} a c^2 d e \left (c d^2+33 a e^2\right )-\frac{1}{4} c^2 \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) x}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{315 c^2 e^3}\\ &=\frac{4 \sqrt{d+e x} \left (4 d \left (c d^2+3 a e^2\right )-3 e \left (c d^2-7 a e^2\right ) x\right ) \sqrt{a+c x^2}}{315 e^3}-\frac{4 d \sqrt{d+e x} \left (a+c x^2\right )^{3/2}}{21 e}+\frac{2 (d+e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 e}+\frac{\left (16 d \left (c d^2+a e^2\right ) \left (c d^2+3 a e^2\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{315 e^4}-\frac{\left (4 \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+c x^2}} \, dx}{315 e^4}\\ &=\frac{4 \sqrt{d+e x} \left (4 d \left (c d^2+3 a e^2\right )-3 e \left (c d^2-7 a e^2\right ) x\right ) \sqrt{a+c x^2}}{315 e^3}-\frac{4 d \sqrt{d+e x} \left (a+c x^2\right )^{3/2}}{21 e}+\frac{2 (d+e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 e}-\frac{\left (8 a \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\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 )}{315 \sqrt{-a} \sqrt{c} e^4 \sqrt{\frac{c (d+e x)}{c d-\frac{a \sqrt{c} e}{\sqrt{-a}}}} \sqrt{a+c x^2}}+\frac{\left (32 a d \left (c d^2+a e^2\right ) \left (c d^2+3 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 )}{315 \sqrt{-a} \sqrt{c} e^4 \sqrt{d+e x} \sqrt{a+c x^2}}\\ &=\frac{4 \sqrt{d+e x} \left (4 d \left (c d^2+3 a e^2\right )-3 e \left (c d^2-7 a e^2\right ) x\right ) \sqrt{a+c x^2}}{315 e^3}-\frac{4 d \sqrt{d+e x} \left (a+c x^2\right )^{3/2}}{21 e}+\frac{2 (d+e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 e}+\frac{8 \sqrt{-a} \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\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 )}{315 \sqrt{c} e^4 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{a+c x^2}}-\frac{32 \sqrt{-a} d \left (c d^2+a e^2\right ) \left (c d^2+3 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 )}{315 \sqrt{c} e^4 \sqrt{d+e x} \sqrt{a+c x^2}}\\ \end{align*}

Mathematica [C]  time = 4.16675, size = 646, normalized size = 1.44 $\frac{\sqrt{d+e x} \left (\frac{2 \left (a+c x^2\right ) \left (a e^2 (29 d+77 e x)+c \left (-6 d^2 e x+8 d^3+5 d e^2 x^2+35 e^3 x^3\right )\right )}{e^3}+\frac{8 \left (\sqrt{a} \sqrt{c} e (d+e x)^{3/2} \left (33 i a^{3/2} \sqrt{c} d e^3-21 a^2 e^4+i \sqrt{a} c^{3/2} d^3 e+15 a c d^2 e^2+4 c^2 d^4\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right ),\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+e^2 \left (a+c x^2\right ) \left (-\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}\right ) \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right )+\sqrt{c} (d+e x)^{3/2} \left (-15 a^{3/2} c d^2 e^3-21 i a^2 \sqrt{c} d e^4+21 a^{5/2} e^5+15 i a c^{3/2} d^3 e^2-4 \sqrt{a} c^2 d^4 e+4 i c^{5/2} d^5\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{c e^5 (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{315 \sqrt{a+c x^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[d + e*x]*((2*(a + c*x^2)*(a*e^2*(29*d + 77*e*x) + c*(8*d^3 - 6*d^2*e*x + 5*d*e^2*x^2 + 35*e^3*x^3)))/e^3
+ (8*(-(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(4*c^2*d^4 + 15*a*c*d^2*e^2 - 21*a^2*e^4)*(a + c*x^2)) + Sqrt[c]
*((4*I)*c^(5/2)*d^5 - 4*Sqrt[a]*c^2*d^4*e + (15*I)*a*c^(3/2)*d^3*e^2 - 15*a^(3/2)*c*d^2*e^3 - (21*I)*a^2*Sqrt[
c]*d*e^4 + 21*a^(5/2)*e^5)*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]*Sqrt[c]*e*(4*c^2*d^4 + I*Sqrt[a]*c^(3/2)*d^3*e + 15*a*c*d^2*
e^2 + (33*I)*a^(3/2)*Sqrt[c]*d*e^3 - 21*a^2*e^4)*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^5*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e
*x))))/(315*Sqrt[a + c*x^2])

________________________________________________________________________________________

Maple [B]  time = 0.25, size = 1731, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(c*x^2+a)^(1/2)*(e*x+d)^(1/2)*(35*x^6*c^3*e^6+40*x^5*c^3*d*e^5+84*(-(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)*Ellipti
cF((-(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^6+72*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))*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)-48*(-(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^5-12*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))*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)-64*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)-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))*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
)-84*(-(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))*a^3*e^6-24*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)+76*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)+16*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)+112*x^4*a*c^2*e^6-x^4*c^3*d^2*e^4+146*x^3*a*c^2*d*e^5+2*x^3*c^3*d^3*e^
3+77*x^2*a^2*c*e^6+28*x^2*a*c^2*d^2*e^4+8*x^2*c^3*d^4*e^2+106*x*a^2*c*d*e^5+2*x*a*c^2*d^3*e^3+29*a^2*c*d^2*e^4
+8*a*c^2*d^4*e^2)/c/(c*e*x^3+c*d*x^2+a*e*x+a*d)/e^5

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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