\(\int \sqrt {d+e x} (a+c x^2)^{5/2} \, dx\) [670]

Optimal result

Integrand size = 21, antiderivative size = 566 \[ \int \sqrt {d+e x} \left (a+c x^2\right )^{5/2} \, dx=\frac {8 \sqrt {d+e x} \left (d \left (32 c^2 d^4+113 a c d^2 e^2+177 a^2 e^4\right )-3 e \left (8 c^2 d^4+27 a c d^2 e^2-77 a^2 e^4\right ) x\right ) \sqrt {a+c x^2}}{9009 e^5}+\frac {20 \sqrt {d+e x} \left (4 d \left (2 c d^2+5 a e^2\right )-7 e \left (c d^2-11 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{9009 e^3}-\frac {20 d \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{143 e}+\frac {2 (d+e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 e}+\frac {16 \sqrt {-a} \left (32 c^3 d^6+137 a c^2 d^4 e^2+258 a^2 c d^2 e^4-231 a^3 e^6\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{9009 \sqrt {c} e^6 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {16 \sqrt {-a} d \left (c d^2+a e^2\right ) \left (32 c^2 d^4+113 a c d^2 e^2+177 a^2 e^4\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{9009 \sqrt {c} e^6 \sqrt {d+e x} \sqrt {a+c x^2}} \]

[Out]

2/13*(e*x+d)^(3/2)*(c*x^2+a)^(5/2)/e+20/9009*(4*d*(5*a*e^2+2*c*d^2)-7*e*(-11*a*e^2+c*d^2)*x)*(c*x^2+a)^(3/2)*(
e*x+d)^(1/2)/e^3-20/143*d*(c*x^2+a)^(5/2)*(e*x+d)^(1/2)/e+8/9009*(d*(177*a^2*e^4+113*a*c*d^2*e^2+32*c^2*d^4)-3
*e*(-77*a^2*e^4+27*a*c*d^2*e^2+8*c^2*d^4)*x)*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)/e^5+16/9009*(-231*a^3*e^6+258*a^2*c
*d^2*e^4+137*a*c^2*d^4*e^2+32*c^3*d^6)*EllipticE(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*e/(-a*e+d*(-
a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*(e*x+d)^(1/2)*(1+c*x^2/a)^(1/2)/e^6/c^(1/2)/(c*x^2+a)^(1/2)/((e*x+d)*c^(1
/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)-16/9009*d*(a*e^2+c*d^2)*(177*a^2*e^4+113*a*c*d^2*e^2+32*c^2*d^4)*EllipticF
(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*e/(-a*e+d*(-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*(1+c*x^2/a)
^(1/2)*((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)/e^6/c^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {749, 847, 829, 858, 733, 435, 430} \[ \int \sqrt {d+e x} \left (a+c x^2\right )^{5/2} \, dx=-\frac {16 \sqrt {-a} d \sqrt {\frac {c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (177 a^2 e^4+113 a c d^2 e^2+32 c^2 d^4\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{9009 \sqrt {c} e^6 \sqrt {a+c x^2} \sqrt {d+e x}}+\frac {8 \sqrt {a+c x^2} \sqrt {d+e x} \left (d \left (177 a^2 e^4+113 a c d^2 e^2+32 c^2 d^4\right )-3 e x \left (-77 a^2 e^4+27 a c d^2 e^2+8 c^2 d^4\right )\right )}{9009 e^5}+\frac {16 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (-231 a^3 e^6+258 a^2 c d^2 e^4+137 a c^2 d^4 e^2+32 c^3 d^6\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{9009 \sqrt {c} e^6 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {20 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (4 d \left (5 a e^2+2 c d^2\right )-7 e x \left (c d^2-11 a e^2\right )\right )}{9009 e^3}+\frac {2 \left (a+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac {20 d \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{143 e} \]

[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 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 733

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 749

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 829

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 847

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 858

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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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] (verified)

Result contains complex when optimal does not.

Time = 26.38 (sec) , antiderivative size = 790, normalized size of antiderivative = 1.40 \[ \int \sqrt {d+e x} \left (a+c x^2\right )^{5/2} \, dx=\frac {2 \sqrt {d+e x} \left (e^2 \left (a+c x^2\right ) \left (a^2 e^4 (971 d+2387 e x)+2 a c e^2 \left (266 d^3-197 d^2 e x+163 d e^2 x^2+1078 e^3 x^3\right )+c^2 \left (128 d^5-96 d^4 e x+80 d^3 e^2 x^2-70 d^2 e^3 x^3+63 d e^4 x^4+693 e^5 x^5\right )\right )-\frac {8 \left (e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (32 c^3 d^6+137 a c^2 d^4 e^2+258 a^2 c d^2 e^4-231 a^3 e^6\right ) \left (a+c x^2\right )+\sqrt {c} \left (-32 i c^{7/2} d^7+32 \sqrt {a} c^3 d^6 e-137 i a c^{5/2} d^5 e^2+137 a^{3/2} c^2 d^4 e^3-258 i a^2 c^{3/2} d^3 e^4+258 a^{5/2} c d^2 e^5+231 i a^3 \sqrt {c} d e^6-231 a^{7/2} e^7\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )-\sqrt {a} \sqrt {c} e \left (32 c^3 d^6+8 i \sqrt {a} c^{5/2} d^5 e+137 a c^2 d^4 e^2+32 i a^{3/2} c^{3/2} d^3 e^3+258 a^2 c d^2 e^4+408 i a^{5/2} \sqrt {c} d e^5-231 a^3 e^6\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{c \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} (d+e x)}\right )}{9009 e^7 \sqrt {a+c x^2}} \]

[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])/S
qrt[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[S
qrt[-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*Sqrt
[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(9009*e^7*Sqrt[a + c*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1195\) vs. \(2(482)=964\).

Time = 2.82 (sec) , antiderivative size = 1196, normalized size of antiderivative = 2.11

method	result	size
risch	\(\text {Expression too large to display}\)	\(1196\)
elliptic	\(\text {Expression too large to display}\)	\(1630\)
default	\(\text {Expression too large to display}\)	\(2332\)

[In]

int((c*x^2+a)^(5/2)*(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/9009*(693*c^2*e^5*x^5+63*c^2*d*e^4*x^4+2156*a*c*e^5*x^3-70*c^2*d^2*e^3*x^3+326*a*c*d*e^4*x^2+80*c^2*d^3*e^2*
x^2+2387*a^2*e^5*x-394*a*c*d^2*e^3*x-96*c^2*d^4*e*x+971*a^2*d*e^4+532*a*c*d^3*e^2+128*c^2*d^5)*(e*x+d)^(1/2)*(
c*x^2+a)^(1/2)/e^5+8/9009/e^5*(816*a^3*d*e^5*(d/e-(-a*c)^(1/2)/c)*((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2)*((x-(-a
*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(-d/e+(-a*c)^(1/2)/c))^(1/2)/(c*e*x^3+c*d*x^2+a*
e*x+a*d)^(1/2)*EllipticF(((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2),((-d/e+(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1
/2))+16*d^5*e*c^2*a*(d/e-(-a*c)^(1/2)/c)*((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-d/e-(-a*c)
^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(-d/e+(-a*c)^(1/2)/c))^(1/2)/(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)*EllipticF(
((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2),((-d/e+(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2))+64*d^3*c*e^3*a^2*(d/
e-(-a*c)^(1/2)/c)*((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2)*((x+(-
a*c)^(1/2)/c)/(-d/e+(-a*c)^(1/2)/c))^(1/2)/(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)*EllipticF(((x+d/e)/(d/e-(-a*c)^(1
/2)/c))^(1/2),((-d/e+(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2))+2*(231*a^3*e^6-258*a^2*c*d^2*e^4-137*a*c^2*
d^4*e^2-32*c^3*d^6)*(d/e-(-a*c)^(1/2)/c)*((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-d/e-(-a*c)
^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(-d/e+(-a*c)^(1/2)/c))^(1/2)/(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)*((-d/e-(-a
*c)^(1/2)/c)*EllipticE(((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2),((-d/e+(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2
))+(-a*c)^(1/2)/c*EllipticF(((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2),((-d/e+(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))
^(1/2))))*((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 419, normalized size of antiderivative = 0.74 \[ \int \sqrt {d+e x} \left (a+c x^2\right )^{5/2} \, dx=\frac {2 \, {\left (8 \, {\left (32 \, c^{3} d^{7} + 161 \, a c^{2} d^{5} e^{2} + 354 \, a^{2} c d^{3} e^{4} + 993 \, a^{3} d e^{6}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 24 \, {\left (32 \, c^{3} d^{6} e + 137 \, a c^{2} d^{4} e^{3} + 258 \, a^{2} c d^{2} e^{5} - 231 \, a^{3} e^{7}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (693 \, c^{3} e^{7} x^{5} + 63 \, c^{3} d e^{6} x^{4} + 128 \, c^{3} d^{5} e^{2} + 532 \, a c^{2} d^{3} e^{4} + 971 \, a^{2} c d e^{6} - 14 \, {\left (5 \, c^{3} d^{2} e^{5} - 154 \, a c^{2} e^{7}\right )} x^{3} + 2 \, {\left (40 \, c^{3} d^{3} e^{4} + 163 \, a c^{2} d e^{6}\right )} x^{2} - {\left (96 \, c^{3} d^{4} e^{3} + 394 \, a c^{2} d^{2} e^{5} - 2387 \, a^{2} c e^{7}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{27027 \, c e^{7}} \]

[In]

integrate((c*x^2+a)^(5/2)*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/27027*(8*(32*c^3*d^7 + 161*a*c^2*d^5*e^2 + 354*a^2*c*d^3*e^4 + 993*a^3*d*e^6)*sqrt(c*e)*weierstrassPInverse(
4/3*(c*d^2 - 3*a*e^2)/(c*e^2), -8/27*(c*d^3 + 9*a*d*e^2)/(c*e^3), 1/3*(3*e*x + d)/e) + 24*(32*c^3*d^6*e + 137*
a*c^2*d^4*e^3 + 258*a^2*c*d^2*e^5 - 231*a^3*e^7)*sqrt(c*e)*weierstrassZeta(4/3*(c*d^2 - 3*a*e^2)/(c*e^2), -8/2
7*(c*d^3 + 9*a*d*e^2)/(c*e^3), weierstrassPInverse(4/3*(c*d^2 - 3*a*e^2)/(c*e^2), -8/27*(c*d^3 + 9*a*d*e^2)/(c
*e^3), 1/3*(3*e*x + d)/e)) + 3*(693*c^3*e^7*x^5 + 63*c^3*d*e^6*x^4 + 128*c^3*d^5*e^2 + 532*a*c^2*d^3*e^4 + 971
*a^2*c*d*e^6 - 14*(5*c^3*d^2*e^5 - 154*a*c^2*e^7)*x^3 + 2*(40*c^3*d^3*e^4 + 163*a*c^2*d*e^6)*x^2 - (96*c^3*d^4
*e^3 + 394*a*c^2*d^2*e^5 - 2387*a^2*c*e^7)*x)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(c*e^7)

Sympy [F]

\[ \int \sqrt {d+e x} \left (a+c x^2\right )^{5/2} \, dx=\int \left (a + c x^{2}\right )^{\frac {5}{2}} \sqrt {d + e x}\, dx \]

[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)

Maxima [F]

\[ \int \sqrt {d+e x} \left (a+c x^2\right )^{5/2} \, dx=\int { {\left (c x^{2} + a\right )}^{\frac {5}{2}} \sqrt {e x + d} \,d x } \]

[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)

Giac [F]

\[ \int \sqrt {d+e x} \left (a+c x^2\right )^{5/2} \, dx=\int { {\left (c x^{2} + a\right )}^{\frac {5}{2}} \sqrt {e x + d} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {d+e x} \left (a+c x^2\right )^{5/2} \, dx=\int {\left (c\,x^2+a\right )}^{5/2}\,\sqrt {d+e\,x} \,d x \]

[In]

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

[Out]

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