### 3.663 $$\int (d+e x)^{3/2} (a+c x^2)^{3/2} \, dx$$

Optimal. Leaf size=497 $-\frac{8 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (-15 a^2 e^4+21 a c d^2 e^2+4 c^2 d^4\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{1155 c^{3/2} e^4 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{4 \sqrt{a+c x^2} \sqrt{d+e x} \left (-15 a^2 e^4-3 c d e x \left (c d^2-31 a e^2\right )+21 a c d^2 e^2+4 c^2 d^4\right )}{1155 c e^3}+\frac{2 \left (a+c x^2\right )^{3/2} \sqrt{d+e x} \left (-3 a e^2+c d^2+28 c d e x\right )}{231 c e}+\frac{32 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (c d^2-3 a e^2\right ) \left (9 a e^2+c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{1155 \sqrt{c} e^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 e \left (a+c x^2\right )^{5/2} \sqrt{d+e x}}{11 c}$

[Out]

(4*Sqrt[d + e*x]*(4*c^2*d^4 + 21*a*c*d^2*e^2 - 15*a^2*e^4 - 3*c*d*e*(c*d^2 - 31*a*e^2)*x)*Sqrt[a + c*x^2])/(11
55*c*e^3) + (2*Sqrt[d + e*x]*(c*d^2 - 3*a*e^2 + 28*c*d*e*x)*(a + c*x^2)^(3/2))/(231*c*e) + (2*e*Sqrt[d + e*x]*
(a + c*x^2)^(5/2))/(11*c) + (32*Sqrt[-a]*d*(c*d^2 - 3*a*e^2)*(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)])/(1155*Sqrt[
c]*e^4*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (8*Sqrt[-a]*(c*d^2 + a*e^2)*(4*c^
2*d^4 + 21*a*c*d^2*e^2 - 15*a^2*e^4)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*El
lipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(1155*c^(3/2)*e^
4*Sqrt[d + e*x]*Sqrt[a + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.559344, antiderivative size = 497, 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 = {743, 815, 844, 719, 424, 419} $\frac{4 \sqrt{a+c x^2} \sqrt{d+e x} \left (-15 a^2 e^4-3 c d e x \left (c d^2-31 a e^2\right )+21 a c d^2 e^2+4 c^2 d^4\right )}{1155 c e^3}-\frac{8 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (-15 a^2 e^4+21 a c d^2 e^2+4 c^2 d^4\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{1155 c^{3/2} e^4 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{2 \left (a+c x^2\right )^{3/2} \sqrt{d+e x} \left (-3 a e^2+c d^2+28 c d e x\right )}{231 c e}+\frac{32 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (c d^2-3 a e^2\right ) \left (9 a e^2+c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{1155 \sqrt{c} e^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 e \left (a+c x^2\right )^{5/2} \sqrt{d+e x}}{11 c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(4*Sqrt[d + e*x]*(4*c^2*d^4 + 21*a*c*d^2*e^2 - 15*a^2*e^4 - 3*c*d*e*(c*d^2 - 31*a*e^2)*x)*Sqrt[a + c*x^2])/(11
55*c*e^3) + (2*Sqrt[d + e*x]*(c*d^2 - 3*a*e^2 + 28*c*d*e*x)*(a + c*x^2)^(3/2))/(231*c*e) + (2*e*Sqrt[d + e*x]*
(a + c*x^2)^(5/2))/(11*c) + (32*Sqrt[-a]*d*(c*d^2 - 3*a*e^2)*(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)])/(1155*Sqrt[
c]*e^4*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (8*Sqrt[-a]*(c*d^2 + a*e^2)*(4*c^
2*d^4 + 21*a*c*d^2*e^2 - 15*a^2*e^4)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*El
lipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(1155*c^(3/2)*e^
4*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} \left (a+c x^2\right )^{3/2} \, dx &=\frac{2 e \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac{2 \int \frac{\left (\frac{1}{2} \left (11 c d^2-a e^2\right )+6 c d e x\right ) \left (a+c x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx}{11 c}\\ &=\frac{2 \sqrt{d+e x} \left (c d^2-3 a e^2+28 c d e x\right ) \left (a+c x^2\right )^{3/2}}{231 c e}+\frac{2 e \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac{8 \int \frac{\left (\frac{3}{4} a c e^2 \left (29 c d^2-3 a e^2\right )-\frac{3}{4} c^2 d e \left (c d^2-31 a e^2\right ) x\right ) \sqrt{a+c x^2}}{\sqrt{d+e x}} \, dx}{231 c^2 e^2}\\ &=\frac{4 \sqrt{d+e x} \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4-3 c d e \left (c d^2-31 a e^2\right ) x\right ) \sqrt{a+c x^2}}{1155 c e^3}+\frac{2 \sqrt{d+e x} \left (c d^2-3 a e^2+28 c d e x\right ) \left (a+c x^2\right )^{3/2}}{231 c e}+\frac{2 e \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac{32 \int \frac{\frac{3}{8} a c^2 e^2 \left (c^2 d^4+114 a c d^2 e^2-15 a^2 e^4\right )-\frac{3}{2} c^3 d e \left (c d^2-3 a e^2\right ) \left (c d^2+9 a e^2\right ) x}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{3465 c^3 e^4}\\ &=\frac{4 \sqrt{d+e x} \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4-3 c d e \left (c d^2-31 a e^2\right ) x\right ) \sqrt{a+c x^2}}{1155 c e^3}+\frac{2 \sqrt{d+e x} \left (c d^2-3 a e^2+28 c d e x\right ) \left (a+c x^2\right )^{3/2}}{231 c e}+\frac{2 e \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{11 c}-\frac{\left (16 d \left (c d^2-3 a e^2\right ) \left (c d^2+9 a e^2\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+c x^2}} \, dx}{1155 e^4}+\frac{\left (4 \left (c d^2+a e^2\right ) \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{1155 c e^4}\\ &=\frac{4 \sqrt{d+e x} \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4-3 c d e \left (c d^2-31 a e^2\right ) x\right ) \sqrt{a+c x^2}}{1155 c e^3}+\frac{2 \sqrt{d+e x} \left (c d^2-3 a e^2+28 c d e x\right ) \left (a+c x^2\right )^{3/2}}{231 c e}+\frac{2 e \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{11 c}-\frac{\left (32 a d \left (c d^2-3 a e^2\right ) \left (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 )}{1155 \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 (8 a \left (c d^2+a e^2\right ) \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4\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 )}{1155 \sqrt{-a} c^{3/2} e^4 \sqrt{d+e x} \sqrt{a+c x^2}}\\ &=\frac{4 \sqrt{d+e x} \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4-3 c d e \left (c d^2-31 a e^2\right ) x\right ) \sqrt{a+c x^2}}{1155 c e^3}+\frac{2 \sqrt{d+e x} \left (c d^2-3 a e^2+28 c d e x\right ) \left (a+c x^2\right )^{3/2}}{231 c e}+\frac{2 e \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac{32 \sqrt{-a} d \left (c d^2-3 a e^2\right ) \left (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 )}{1155 \sqrt{c} e^4 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{a+c x^2}}-\frac{8 \sqrt{-a} \left (c d^2+a e^2\right ) \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4\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 )}{1155 c^{3/2} e^4 \sqrt{d+e x} \sqrt{a+c x^2}}\\ \end{align*}

Mathematica [C]  time = 3.98236, size = 695, normalized size = 1.4 $\frac{2 \sqrt{d+e x} \left (e^2 \left (a+c x^2\right ) \left (60 a^2 e^4+a c e^2 \left (47 d^2+326 d e x+195 e^2 x^2\right )+c^2 \left (5 d^2 e^2 x^2-6 d^3 e x+8 d^4+140 d e^3 x^3+105 e^4 x^4\right )\right )+\frac{4 \left (\sqrt{a} e (d+e x)^{3/2} \left (114 i a^{3/2} c d^2 e^3-108 a^2 \sqrt{c} d e^4-15 i a^{5/2} e^5+24 a c^{3/2} d^3 e^2+i \sqrt{a} c^2 d^4 e+4 c^{5/2} d^5\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right ),\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )-4 d e^2 \left (a+c x^2\right ) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (-27 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )+4 \sqrt{c} d (d+e x)^{3/2} \left (-6 a^{3/2} c d^2 e^3-27 i a^2 \sqrt{c} d e^4+27 a^{5/2} e^5+6 i a c^{3/2} d^3 e^2-\sqrt{a} c^2 d^4 e+i c^{5/2} d^5\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{(d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{1155 c e^5 \sqrt{a+c x^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(e^2*(a + c*x^2)*(60*a^2*e^4 + a*c*e^2*(47*d^2 + 326*d*e*x + 195*e^2*x^2) + c^2*(8*d^4 - 6*d^
3*e*x + 5*d^2*e^2*x^2 + 140*d*e^3*x^3 + 105*e^4*x^4)) + (4*(-4*d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(c^2*d^4
+ 6*a*c*d^2*e^2 - 27*a^2*e^4)*(a + c*x^2) + 4*Sqrt[c]*d*(I*c^(5/2)*d^5 - Sqrt[a]*c^2*d^4*e + (6*I)*a*c^(3/2)*
d^3*e^2 - 6*a^(3/2)*c*d^2*e^3 - (27*I)*a^2*Sqrt[c]*d*e^4 + 27*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*S
qrt[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*(4*c^(5/2)
*d^5 + I*Sqrt[a]*c^2*d^4*e + 24*a*c^(3/2)*d^3*e^2 + (114*I)*a^(3/2)*c*d^2*e^3 - 108*a^2*Sqrt[c]*d*e^4 - (15*I)
*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)*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]]*(d + e*x))))/(1155*c*e^5*Sqrt[a + c*x^2])

________________________________________________________________________________________

Maple [B]  time = 0.274, size = 1970, normalized size = 4. \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)^(3/2),x)

[Out]

2/1155*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)*(8*a*c^3*d^5*e^2-24*(-(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*c*d^2*e^5-100
*(-(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*c^2*d^4*e^3+518*x^3*a*c^3*d^2*e^5+581*x^2*a^2*c^2*d*e^6+46*x^2*a*c^3*d^
3*e^4+373*x*a^2*c^2*d^2*e^5+2*x*a*c^3*d^4*e^3+766*x^4*a*c^3*d*e^6+60*(-(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^3*e
^7+245*x^6*c^4*d*e^6+300*x^5*a*c^3*e^7+145*x^5*c^4*d^2*e^5-x^4*c^4*d^3*e^4+255*x^3*a^2*c^2*e^7+2*x^3*c^4*d^4*e
^3+8*x^2*c^4*d^5*e^2+60*x*a^3*c*e^7+105*x^7*c^4*e^7+112*(-(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*c^3*d^5*e^2-432*(-(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*c*d*e^6-336*(-(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^2*c^2*d^3*e^4-16*(-(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)*c^3*d^
6*e-12*(-(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^3*d^5*e^2+372*(-(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^3*c*d*e^6+360*(-(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*c^2*d^3*e^4+16*(-(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^4*d^7+60*a^3*c*d*e^6+47*a^2*c^2*d^3*e^4)/c^2/e^5/(c*e*x^3+c*d*x^2+
a*e*x+a*d)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + a\right )}^{\frac{3}{2}}{\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)^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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

[Out]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + a\right )}^{\frac{3}{2}}{\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)^(3/2),x, algorithm="giac")

[Out]

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