3.670 \(\int \sqrt{d+e x} (a+c x^2)^{5/2} \, dx\)

Optimal. Leaf size=566 \[ -\frac{16 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (177 a^2 e^4+113 a c d^2 e^2+32 c^2 d^4\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{9009 \sqrt{c} e^6 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{8 \sqrt{a+c x^2} \sqrt{d+e x} \left (d \left (177 a^2 e^4+113 a c d^2 e^2+32 c^2 d^4\right )-3 e x \left (-77 a^2 e^4+27 a c d^2 e^2+8 c^2 d^4\right )\right )}{9009 e^5}+\frac{16 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (258 a^2 c d^2 e^4-231 a^3 e^6+137 a c^2 d^4 e^2+32 c^3 d^6\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{9009 \sqrt{c} e^6 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{20 \left (a+c x^2\right )^{3/2} \sqrt{d+e x} \left (4 d \left (5 a e^2+2 c d^2\right )-7 e x \left (c d^2-11 a e^2\right )\right )}{9009 e^3}+\frac{2 \left (a+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac{20 d \left (a+c x^2\right )^{5/2} \sqrt{d+e x}}{143 e} \]

[Out]

(8*Sqrt[d + e*x]*(d*(32*c^2*d^4 + 113*a*c*d^2*e^2 + 177*a^2*e^4) - 3*e*(8*c^2*d^4 + 27*a*c*d^2*e^2 - 77*a^2*e^
4)*x)*Sqrt[a + c*x^2])/(9009*e^5) + (20*Sqrt[d + e*x]*(4*d*(2*c*d^2 + 5*a*e^2) - 7*e*(c*d^2 - 11*a*e^2)*x)*(a
+ c*x^2)^(3/2))/(9009*e^3) - (20*d*Sqrt[d + e*x]*(a + c*x^2)^(5/2))/(143*e) + (2*(d + e*x)^(3/2)*(a + c*x^2)^(
5/2))/(13*e) + (16*Sqrt[-a]*(32*c^3*d^6 + 137*a*c^2*d^4*e^2 + 258*a^2*c*d^2*e^4 - 231*a^3*e^6)*Sqrt[d + e*x]*S
qrt[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)])/(9009*Sqrt[c]*e^6*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (16*Sqrt[-a]*d*(c
*d^2 + a*e^2)*(32*c^2*d^4 + 113*a*c*d^2*e^2 + 177*a^2*e^4)*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)])/(9009*Sqrt[c]*e^6*Sqrt[d + e*x]*Sqrt[a + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.665535, antiderivative size = 566, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{8 \sqrt{a+c x^2} \sqrt{d+e x} \left (d \left (177 a^2 e^4+113 a c d^2 e^2+32 c^2 d^4\right )-3 e x \left (-77 a^2 e^4+27 a c d^2 e^2+8 c^2 d^4\right )\right )}{9009 e^5}-\frac{16 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (177 a^2 e^4+113 a c d^2 e^2+32 c^2 d^4\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{9009 \sqrt{c} e^6 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{16 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (258 a^2 c d^2 e^4-231 a^3 e^6+137 a c^2 d^4 e^2+32 c^3 d^6\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{9009 \sqrt{c} e^6 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{20 \left (a+c x^2\right )^{3/2} \sqrt{d+e x} \left (4 d \left (5 a e^2+2 c d^2\right )-7 e x \left (c d^2-11 a e^2\right )\right )}{9009 e^3}+\frac{2 \left (a+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac{20 d \left (a+c x^2\right )^{5/2} \sqrt{d+e x}}{143 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*Sqrt[d + e*x]*(d*(32*c^2*d^4 + 113*a*c*d^2*e^2 + 177*a^2*e^4) - 3*e*(8*c^2*d^4 + 27*a*c*d^2*e^2 - 77*a^2*e^
4)*x)*Sqrt[a + c*x^2])/(9009*e^5) + (20*Sqrt[d + e*x]*(4*d*(2*c*d^2 + 5*a*e^2) - 7*e*(c*d^2 - 11*a*e^2)*x)*(a
+ c*x^2)^(3/2))/(9009*e^3) - (20*d*Sqrt[d + e*x]*(a + c*x^2)^(5/2))/(143*e) + (2*(d + e*x)^(3/2)*(a + c*x^2)^(
5/2))/(13*e) + (16*Sqrt[-a]*(32*c^3*d^6 + 137*a*c^2*d^4*e^2 + 258*a^2*c*d^2*e^4 - 231*a^3*e^6)*Sqrt[d + e*x]*S
qrt[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)])/(9009*Sqrt[c]*e^6*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (16*Sqrt[-a]*d*(c
*d^2 + a*e^2)*(32*c^2*d^4 + 113*a*c*d^2*e^2 + 177*a^2*e^4)*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)])/(9009*Sqrt[c]*e^6*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 )^{5/2} \, dx &=\frac{2 (d+e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 e}+\frac{10 \int (a e-c d x) \sqrt{d+e x} \left (a+c x^2\right )^{3/2} \, dx}{13 e}\\ &=-\frac{20 d \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{143 e}+\frac{2 (d+e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 e}+\frac{20 \int \frac{\left (6 a c d e-\frac{1}{2} c \left (c d^2-11 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx}{143 c e}\\ &=\frac{20 \sqrt{d+e x} \left (4 d \left (2 c d^2+5 a e^2\right )-7 e \left (c d^2-11 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{9009 e^3}-\frac{20 d \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{143 e}+\frac{2 (d+e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 e}+\frac{80 \int \frac{\left (\frac{1}{4} a c^2 d e \left (c d^2+97 a e^2\right )-\frac{1}{4} c^2 \left (8 c^2 d^4+27 a c d^2 e^2-77 a^2 e^4\right ) x\right ) \sqrt{a+c x^2}}{\sqrt{d+e x}} \, dx}{3003 c^2 e^3}\\ &=\frac{8 \sqrt{d+e x} \left (d \left (32 c^2 d^4+113 a c d^2 e^2+177 a^2 e^4\right )-3 e \left (8 c^2 d^4+27 a c d^2 e^2-77 a^2 e^4\right ) x\right ) \sqrt{a+c x^2}}{9009 e^5}+\frac{20 \sqrt{d+e x} \left (4 d \left (2 c d^2+5 a e^2\right )-7 e \left (c d^2-11 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{9009 e^3}-\frac{20 d \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{143 e}+\frac{2 (d+e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 e}+\frac{64 \int \frac{a c^3 d e \left (c^2 d^4+4 a c d^2 e^2+51 a^2 e^4\right )-\frac{1}{8} c^3 \left (32 c^3 d^6+137 a c^2 d^4 e^2+258 a^2 c d^2 e^4-231 a^3 e^6\right ) x}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{9009 c^3 e^5}\\ &=\frac{8 \sqrt{d+e x} \left (d \left (32 c^2 d^4+113 a c d^2 e^2+177 a^2 e^4\right )-3 e \left (8 c^2 d^4+27 a c d^2 e^2-77 a^2 e^4\right ) x\right ) \sqrt{a+c x^2}}{9009 e^5}+\frac{20 \sqrt{d+e x} \left (4 d \left (2 c d^2+5 a e^2\right )-7 e \left (c d^2-11 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{9009 e^3}-\frac{20 d \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{143 e}+\frac{2 (d+e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 e}+\frac{\left (8 d \left (c d^2+a e^2\right ) \left (32 c^2 d^4+113 a c d^2 e^2+177 a^2 e^4\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{9009 e^6}-\frac{\left (8 \left (32 c^3 d^6+137 a c^2 d^4 e^2+258 a^2 c d^2 e^4-231 a^3 e^6\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+c x^2}} \, dx}{9009 e^6}\\ &=\frac{8 \sqrt{d+e x} \left (d \left (32 c^2 d^4+113 a c d^2 e^2+177 a^2 e^4\right )-3 e \left (8 c^2 d^4+27 a c d^2 e^2-77 a^2 e^4\right ) x\right ) \sqrt{a+c x^2}}{9009 e^5}+\frac{20 \sqrt{d+e x} \left (4 d \left (2 c d^2+5 a e^2\right )-7 e \left (c d^2-11 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{9009 e^3}-\frac{20 d \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{143 e}+\frac{2 (d+e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 e}-\frac{\left (16 a \left (32 c^3 d^6+137 a c^2 d^4 e^2+258 a^2 c d^2 e^4-231 a^3 e^6\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 )}{9009 \sqrt{-a} \sqrt{c} e^6 \sqrt{\frac{c (d+e x)}{c d-\frac{a \sqrt{c} e}{\sqrt{-a}}}} \sqrt{a+c x^2}}+\frac{\left (16 a d \left (c d^2+a e^2\right ) \left (32 c^2 d^4+113 a c d^2 e^2+177 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 )}{9009 \sqrt{-a} \sqrt{c} e^6 \sqrt{d+e x} \sqrt{a+c x^2}}\\ &=\frac{8 \sqrt{d+e x} \left (d \left (32 c^2 d^4+113 a c d^2 e^2+177 a^2 e^4\right )-3 e \left (8 c^2 d^4+27 a c d^2 e^2-77 a^2 e^4\right ) x\right ) \sqrt{a+c x^2}}{9009 e^5}+\frac{20 \sqrt{d+e x} \left (4 d \left (2 c d^2+5 a e^2\right )-7 e \left (c d^2-11 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{9009 e^3}-\frac{20 d \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{143 e}+\frac{2 (d+e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 e}+\frac{16 \sqrt{-a} \left (32 c^3 d^6+137 a c^2 d^4 e^2+258 a^2 c d^2 e^4-231 a^3 e^6\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 )}{9009 \sqrt{c} e^6 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{a+c x^2}}-\frac{16 \sqrt{-a} d \left (c d^2+a e^2\right ) \left (32 c^2 d^4+113 a c d^2 e^2+177 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 )}{9009 \sqrt{c} e^6 \sqrt{d+e x} \sqrt{a+c x^2}}\\ \end{align*}

Mathematica [C]  time = 4.92649, size = 790, normalized size = 1.4 \[ \frac{2 \sqrt{d+e x} \left (e^2 \left (a+c x^2\right ) \left (a^2 e^4 (971 d+2387 e x)+2 a c e^2 \left (-197 d^2 e x+266 d^3+163 d e^2 x^2+1078 e^3 x^3\right )+c^2 \left (80 d^3 e^2 x^2-70 d^2 e^3 x^3-96 d^4 e x+128 d^5+63 d e^4 x^4+693 e^5 x^5\right )\right )+\frac{8 \left (\sqrt{a} \sqrt{c} e (d+e x)^{3/2} \left (32 i a^{3/2} c^{3/2} d^3 e^3+258 a^2 c d^2 e^4+408 i a^{5/2} \sqrt{c} d e^5-231 a^3 e^6+137 a c^2 d^4 e^2+8 i \sqrt{a} c^{5/2} d^5 e+32 c^3 d^6\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right ),\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+e^2 \left (a+c x^2\right ) \left (-\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}\right ) \left (258 a^2 c d^2 e^4-231 a^3 e^6+137 a c^2 d^4 e^2+32 c^3 d^6\right )+\sqrt{c} (d+e x)^{3/2} \left (-137 a^{3/2} c^2 d^4 e^3+258 i a^2 c^{3/2} d^3 e^4-258 a^{5/2} c d^2 e^5-231 i a^3 \sqrt{c} d e^6+231 a^{7/2} e^7+137 i a c^{5/2} d^5 e^2-32 \sqrt{a} c^3 d^6 e+32 i c^{7/2} d^7\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{c (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{9009 e^7 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(e^2*(a + c*x^2)*(a^2*e^4*(971*d + 2387*e*x) + 2*a*c*e^2*(266*d^3 - 197*d^2*e*x + 163*d*e^2*x
^2 + 1078*e^3*x^3) + c^2*(128*d^5 - 96*d^4*e*x + 80*d^3*e^2*x^2 - 70*d^2*e^3*x^3 + 63*d*e^4*x^4 + 693*e^5*x^5)
) + (8*(-(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(32*c^3*d^6 + 137*a*c^2*d^4*e^2 + 258*a^2*c*d^2*e^4 - 231*a^3*e
^6)*(a + c*x^2)) + Sqrt[c]*((32*I)*c^(7/2)*d^7 - 32*Sqrt[a]*c^3*d^6*e + (137*I)*a*c^(5/2)*d^5*e^2 - 137*a^(3/2
)*c^2*d^4*e^3 + (258*I)*a^2*c^(3/2)*d^3*e^4 - 258*a^(5/2)*c*d^2*e^5 - (231*I)*a^3*Sqrt[c]*d*e^6 + 231*a^(7/2)*
e^7)*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*(32*c^3*d^6 + (8*I)*Sqrt[a]*c^(5/2)*d^5*e + 137*a*c^2*d^4*e^2 + (32*I)*a
^(3/2)*c^(3/2)*d^3*e^3 + 258*a^2*c*d^2*e^4 + (408*I)*a^(5/2)*Sqrt[c]*d*e^5 - 231*a^3*e^6)*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*Sq
rt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(9009*e^7*Sqrt[a + c*x^2])

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Maple [B]  time = 0.268, size = 2332, normalized size = 4.1 \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)^(5/2)*(e*x+d)^(1/2),x)

[Out]

2/9009*(c*x^2+a)^(1/2)*(e*x+d)^(1/2)*(-1416*(-(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*d*e^7+256*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^8*((-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)*(-(e*x+d)*c/((-a*c)^(1/2)
*e-c*d))^(1/2)+1848*(-(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^4*e^8-1848*(-(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^4*e^8+3238*x^5*a*c^3*d*e^7-84
0*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^4*e^4*((-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)
*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)-192*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^6*e^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)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)-256*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^7*e*(-a*c)^(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)*(-(e*x+d)*c/((-a*c
)^(1/2)*e-c*d))^(1/2)+1352*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^6*e^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)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)+3160*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^4*e^4*((-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)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)+12
00*(-(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^2*e^6+216*(-(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^2*e^6+756*x^7*c^4*d*e^7+2849*x^6*a*
c^3*e^8-7*x^6*c^4*d^2*e^6+693*x^8*c^4*e^8+148*x^3*a*c^3*d^3*e^5+903*x^2*a^2*c^2*d^2*e^6+516*x^2*a*c^3*d^4*e^4+
3358*x*a^3*c*d*e^7+138*x*a^2*c^2*d^3*e^5+32*x*a*c^3*d^5*e^3+128*x^2*c^4*d^6*e^2+10*x^5*c^4*d^3*e^5+4543*x^4*a^
2*c^2*e^8-16*x^4*c^4*d^4*e^4+32*x^3*c^4*d^5*e^3+2387*x^2*a^3*c*e^8-75*x^4*a*c^3*d^2*e^6+5840*x^3*a^2*c^2*d*e^7
+971*a^3*c*d^2*e^6+532*a^2*c^2*d^4*e^4+128*a*c^3*d^6*e^2-2320*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^3*e^5*(-a*c)^(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)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2
)-1160*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^5*e^3*(-a*c)^(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)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2))/c/(c*e*x^3+c*d*x^2+a*e*x+a*d)/e^7

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + a\right )}^{\frac{5}{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)^(5/2)*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (c^{2} x^{4} + 2 \, a c x^{2} + a^{2}\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((c*x^2+a)^(5/2)*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

integral((c^2*x^4 + 2*a*c*x^2 + a^2)*sqrt(c*x^2 + a)*sqrt(e*x + d), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + c x^{2}\right )^{\frac{5}{2}} \sqrt{d + e x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + c*x**2)**(5/2)*sqrt(d + e*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + a\right )}^{\frac{5}{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)^(5/2)*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

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