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

Optimal. Leaf size=494 \[ -\frac{16 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (45 a^2 e^4+69 a c d^2 e^2+32 c^2 d^4\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{693 \sqrt{c} e^6 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{8 \sqrt{a+c x^2} \sqrt{d+e x} \left (45 a^2 e^4-24 c d e x \left (2 a e^2+c d^2\right )+69 a c d^2 e^2+32 c^2 d^4\right )}{693 e^5}+\frac{16 \sqrt{-a} \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (93 a^2 e^4+93 a c d^2 e^2+32 c^2 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{693 e^6 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{20 \left (a+c x^2\right )^{3/2} \sqrt{d+e x} \left (9 a e^2+8 c d^2-7 c d e x\right )}{693 e^3}+\frac{2 \left (a+c x^2\right )^{5/2} \sqrt{d+e x}}{11 e} \]

[Out]

(8*Sqrt[d + e*x]*(32*c^2*d^4 + 69*a*c*d^2*e^2 + 45*a^2*e^4 - 24*c*d*e*(c*d^2 + 2*a*e^2)*x)*Sqrt[a + c*x^2])/(6
93*e^5) + (20*Sqrt[d + e*x]*(8*c*d^2 + 9*a*e^2 - 7*c*d*e*x)*(a + c*x^2)^(3/2))/(693*e^3) + (2*Sqrt[d + e*x]*(a
+ c*x^2)^(5/2))/(11*e) + (16*Sqrt[-a]*Sqrt[c]*d*(32*c^2*d^4 + 93*a*c*d^2*e^2 + 93*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)]
)/(693*e^6*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (16*Sqrt[-a]*(c*d^2 + a*e^2)*
(32*c^2*d^4 + 69*a*c*d^2*e^2 + 45*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)])/(693*Sqrt[
c]*e^6*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.481906, antiderivative size = 494, 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 = {735, 815, 844, 719, 424, 419} \[ \frac{8 \sqrt{a+c x^2} \sqrt{d+e x} \left (45 a^2 e^4-24 c d e x \left (2 a e^2+c d^2\right )+69 a c d^2 e^2+32 c^2 d^4\right )}{693 e^5}-\frac{16 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (45 a^2 e^4+69 a c d^2 e^2+32 c^2 d^4\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{693 \sqrt{c} e^6 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{16 \sqrt{-a} \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (93 a^2 e^4+93 a c d^2 e^2+32 c^2 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{693 e^6 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{20 \left (a+c x^2\right )^{3/2} \sqrt{d+e x} \left (9 a e^2+8 c d^2-7 c d e x\right )}{693 e^3}+\frac{2 \left (a+c x^2\right )^{5/2} \sqrt{d+e x}}{11 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*Sqrt[d + e*x]*(32*c^2*d^4 + 69*a*c*d^2*e^2 + 45*a^2*e^4 - 24*c*d*e*(c*d^2 + 2*a*e^2)*x)*Sqrt[a + c*x^2])/(6
93*e^5) + (20*Sqrt[d + e*x]*(8*c*d^2 + 9*a*e^2 - 7*c*d*e*x)*(a + c*x^2)^(3/2))/(693*e^3) + (2*Sqrt[d + e*x]*(a
+ c*x^2)^(5/2))/(11*e) + (16*Sqrt[-a]*Sqrt[c]*d*(32*c^2*d^4 + 93*a*c*d^2*e^2 + 93*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)]
)/(693*e^6*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (16*Sqrt[-a]*(c*d^2 + a*e^2)*
(32*c^2*d^4 + 69*a*c*d^2*e^2 + 45*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)])/(693*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 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 \frac{\left (a+c x^2\right )^{5/2}}{\sqrt{d+e x}} \, dx &=\frac{2 \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{11 e}+\frac{10 \int \frac{(a e-c d x) \left (a+c x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx}{11 e}\\ &=\frac{20 \sqrt{d+e x} \left (8 c d^2+9 a e^2-7 c d e x\right ) \left (a+c x^2\right )^{3/2}}{693 e^3}+\frac{2 \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{11 e}+\frac{40 \int \frac{\left (\frac{1}{2} a c e \left (c d^2+9 a e^2\right )-4 c^2 d \left (c d^2+2 a e^2\right ) x\right ) \sqrt{a+c x^2}}{\sqrt{d+e x}} \, dx}{231 c e^3}\\ &=\frac{8 \sqrt{d+e x} \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4-24 c d e \left (c d^2+2 a e^2\right ) x\right ) \sqrt{a+c x^2}}{693 e^5}+\frac{20 \sqrt{d+e x} \left (8 c d^2+9 a e^2-7 c d e x\right ) \left (a+c x^2\right )^{3/2}}{693 e^3}+\frac{2 \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{11 e}+\frac{32 \int \frac{\frac{1}{4} a c^2 e \left (8 c^2 d^4+21 a c d^2 e^2+45 a^2 e^4\right )-\frac{1}{4} c^3 d \left (32 c^2 d^4+93 a c d^2 e^2+93 a^2 e^4\right ) x}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{693 c^2 e^5}\\ &=\frac{8 \sqrt{d+e x} \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4-24 c d e \left (c d^2+2 a e^2\right ) x\right ) \sqrt{a+c x^2}}{693 e^5}+\frac{20 \sqrt{d+e x} \left (8 c d^2+9 a e^2-7 c d e x\right ) \left (a+c x^2\right )^{3/2}}{693 e^3}+\frac{2 \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{11 e}+\frac{\left (8 \left (c d^2+a e^2\right ) \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{693 e^6}-\frac{\left (8 c d \left (32 c^2 d^4+93 a c d^2 e^2+93 a^2 e^4\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+c x^2}} \, dx}{693 e^6}\\ &=\frac{8 \sqrt{d+e x} \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4-24 c d e \left (c d^2+2 a e^2\right ) x\right ) \sqrt{a+c x^2}}{693 e^5}+\frac{20 \sqrt{d+e x} \left (8 c d^2+9 a e^2-7 c d e x\right ) \left (a+c x^2\right )^{3/2}}{693 e^3}+\frac{2 \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{11 e}-\frac{\left (16 a \sqrt{c} d \left (32 c^2 d^4+93 a c d^2 e^2+93 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 )}{693 \sqrt{-a} 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 \left (c d^2+a e^2\right ) \left (32 c^2 d^4+69 a c d^2 e^2+45 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 )}{693 \sqrt{-a} \sqrt{c} e^6 \sqrt{d+e x} \sqrt{a+c x^2}}\\ &=\frac{8 \sqrt{d+e x} \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4-24 c d e \left (c d^2+2 a e^2\right ) x\right ) \sqrt{a+c x^2}}{693 e^5}+\frac{20 \sqrt{d+e x} \left (8 c d^2+9 a e^2-7 c d e x\right ) \left (a+c x^2\right )^{3/2}}{693 e^3}+\frac{2 \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{11 e}+\frac{16 \sqrt{-a} \sqrt{c} d \left (32 c^2 d^4+93 a c d^2 e^2+93 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 )}{693 e^6 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{a+c x^2}}-\frac{16 \sqrt{-a} \left (c d^2+a e^2\right ) \left (32 c^2 d^4+69 a c d^2 e^2+45 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 )}{693 \sqrt{c} e^6 \sqrt{d+e x} \sqrt{a+c x^2}}\\ \end{align*}

Mathematica [C]  time = 3.67884, size = 634, normalized size = 1.28 \[ \frac{2 \sqrt{d+e x} \left (\frac{8 \sqrt{a} e \sqrt{d+e x} \left (21 i a^{3/2} c d^2 e^3+93 a^2 \sqrt{c} d e^4+45 i a^{5/2} e^5+93 a c^{3/2} d^3 e^2+8 i \sqrt{a} c^2 d^4 e+32 c^{5/2} d^5\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right ),\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}-\frac{8 d e^2 \left (a+c x^2\right ) \left (93 a^2 e^4+93 a c d^2 e^2+32 c^2 d^4\right )}{d+e x}+e^2 \left (a+c x^2\right ) \left (333 a^2 e^4+2 a c e^2 \left (178 d^2-131 d e x+108 e^2 x^2\right )+c^2 \left (80 d^2 e^2 x^2-96 d^3 e x+128 d^4-70 d e^3 x^3+63 e^4 x^4\right )\right )-8 i c d \sqrt{d+e x} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (93 a^2 e^4+93 a c d^2 e^2+32 c^2 d^4\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{693 e^7 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*((-8*d*e^2*(32*c^2*d^4 + 93*a*c*d^2*e^2 + 93*a^2*e^4)*(a + c*x^2))/(d + e*x) + e^2*(a + c*x^2
)*(333*a^2*e^4 + 2*a*c*e^2*(178*d^2 - 131*d*e*x + 108*e^2*x^2) + c^2*(128*d^4 - 96*d^3*e*x + 80*d^2*e^2*x^2 -
70*d*e^3*x^3 + 63*e^4*x^4)) - (8*I)*c*d*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(32*c^2*d^4 + 93*a*c*d^2*e^2 + 93*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))]*Sqrt[d +
e*x]*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)] + (8*Sqrt[a]*e*(32*c^(5/2)*d^5 + (8*I)*Sqrt[a]*c^2*d^4*e + 93*a*c^(3/2)*d^3*e^2 + (21*I)*a^(3
/2)*c*d^2*e^3 + 93*a^2*Sqrt[c]*d*e^4 + (45*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))]*Sqrt[d + e*x]*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]]))/(693
*e^7*Sqrt[a + c*x^2])

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Maple [B]  time = 0.258, 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((c*x^2+a)^(5/2)/(e*x+d)^(1/2),x)

[Out]

2/693*(c*x^2+a)^(1/2)*(e*x+d)^(1/2)*(128*a*c^3*d^5*e^2-912*(-(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^5-8
08*(-(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+104*x^3*a*c^3*d^2*e^5+287*x^2*a^2*c^2*d*e^6+340*x^2*a*c^3
*d^3*e^4+94*x*a^2*c^2*d^2*e^5+32*x*a*c^3*d^4*e^3-53*x^4*a*c^3*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)*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*c)^(1/2)*a^
3*e^7-7*x^6*c^4*d*e^6+279*x^5*a*c^3*e^7+10*x^5*c^4*d^2*e^5-16*x^4*c^4*d^3*e^4+549*x^3*a^2*c^2*e^7+32*x^3*c^4*d
^4*e^3+128*x^2*c^4*d^5*e^2+333*x*a^3*c*e^7+63*x^7*c^4*e^7+1000*(-(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+744*(-(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+1488*(-(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^4-256*(-(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)*El
lipticF((-(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-192*(-(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^2-384*(-(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^6-576*(-(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+256*(-(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+333*a^3*c*d*e^6+356*a^2*c^2*d^3*e^4)/c/e^7/(c*e*x
^3+c*d*x^2+a*e*x+a*d)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\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 (\frac{{\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 \frac{\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 \frac{{\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)