3.1.82 \(\int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \, dx\) [82]

Optimal. Leaf size=550 \[ -\frac {2 (2 a d-b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}}{15 a e}+\frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{3/2}}{5 e}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (a^2 d^2+b^2 e^2-a e (b d+3 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 a^2 e^2 \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \left (c+b x+a x^2\right )}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 a d-b e) \left (a d^2-e (b d-c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 a^2 e^2 \sqrt {d+e x} \left (c+b x+a x^2\right )} \]

[Out]

2/5*x*(e*x+d)^(3/2)*(a+c/x^2+b/x)^(1/2)/e-2/15*(2*a*d-b*e)*x*(a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2)/a/e-2/15*(a^2*d
^2+b^2*e^2-a*e*(b*d+3*c*e))*x*EllipticE(1/2*((b+2*a*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-
2*e*(-4*a*c+b^2)^(1/2)/(2*a*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2)*(a+c/x^2+b/x)^(1/2)
*(e*x+d)^(1/2)*(-a*(a*x^2+b*x+c)/(-4*a*c+b^2))^(1/2)/a^2/e^2/(a*x^2+b*x+c)/(a*(e*x+d)/(2*a*d-e*(b+(-4*a*c+b^2)
^(1/2))))^(1/2)+2/15*(2*a*d-b*e)*(a*d^2-e*(b*d-c*e))*x*EllipticF(1/2*((b+2*a*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2
)^(1/2))^(1/2)*2^(1/2),(-2*e*(-4*a*c+b^2)^(1/2)/(2*a*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^
(1/2)*(a+c/x^2+b/x)^(1/2)*(-a*(a*x^2+b*x+c)/(-4*a*c+b^2))^(1/2)*(a*(e*x+d)/(2*a*d-e*(b+(-4*a*c+b^2)^(1/2))))^(
1/2)/a^2/e^2/(a*x^2+b*x+c)/(e*x+d)^(1/2)

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Rubi [A]
time = 0.47, antiderivative size = 550, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1587, 748, 846, 857, 732, 435, 430} \begin {gather*} \frac {2 \sqrt {2} x \sqrt {b^2-4 a c} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} (2 a d-b e) \left (a d^2-e (b d-c e)\right ) \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 a^2 e^2 \sqrt {d+e x} \left (a x^2+b x+c\right )}-\frac {2 \sqrt {2} x \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (a^2 d^2-a e (b d+3 c e)+b^2 e^2\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 a^2 e^2 \left (a x^2+b x+c\right ) \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 x (d+e x)^{3/2} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}{5 e}-\frac {2 x \sqrt {d+e x} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} (2 a d-b e)}{15 a e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + c/x^2 + b/x]*x*Sqrt[d + e*x],x]

[Out]

(-2*(2*a*d - b*e)*Sqrt[a + c/x^2 + b/x]*x*Sqrt[d + e*x])/(15*a*e) + (2*Sqrt[a + c/x^2 + b/x]*x*(d + e*x)^(3/2)
)/(5*e) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(a^2*d^2 + b^2*e^2 - a*e*(b*d + 3*c*e))*Sqrt[a + c/x^2 + b/x]*x*Sqrt[d
+ e*x]*Sqrt[-((a*(c + b*x + a*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x)/Sqrt
[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(15*a^2*e^2*Sqrt[(a*(d
+ e*x))/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)]*(c + b*x + a*x^2)) + (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*a*d - b*e)*
(a*d^2 - e*(b*d - c*e))*Sqrt[a + c/x^2 + b/x]*x*Sqrt[(a*(d + e*x))/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-
((a*(c + b*x + a*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x)/Sqrt[b^2 - 4*a*c]
]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(15*a^2*e^2*Sqrt[d + e*x]*(c + b*x
+ a*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 732

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2*Rt[b^2 - 4*a*c, 2]*
(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*
e - e*Rt[b^2 - 4*a*c, 2])))^m)), Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2*c*d - b*e - e*Rt[b^2 - 4*a*c, 2
])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b
, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

Rule 748

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x] - Dist[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*
d - b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || Lt
Q[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 846

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[g*(d + e*x)^m*((a + b*x + 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 + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + 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 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 1587

Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^(mn_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbol
] :> Dist[x^(2*n*FracPart[p])*((a + b/x^n + c/x^(2*n))^FracPart[p]/(c + b*x^n + a*x^(2*n))^FracPart[p]), Int[x
^(m - 2*n*p)*(d + e*x^n)^q*(c + b*x^n + a*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[m
n, -n] && EqQ[mn2, 2*mn] &&  !IntegerQ[p] &&  !IntegerQ[q] && PosQ[n]

Rubi steps

\begin {align*} \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \, dx &=\frac {\left (\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \sqrt {d+e x} \sqrt {c+b x+a x^2} \, dx}{\sqrt {c+b x+a x^2}}\\ &=\frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{3/2}}{5 e}-\frac {\left (\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {\sqrt {d+e x} (b d-2 c e+(2 a d-b e) x)}{\sqrt {c+b x+a x^2}} \, dx}{5 e \sqrt {c+b x+a x^2}}\\ &=-\frac {2 (2 a d-b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}}{15 a e}+\frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{3/2}}{5 e}-\frac {\left (2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {\frac {1}{2} (a d (b d-8 c e)+b e (b d+c e))+\left (a^2 d^2+b^2 e^2-a e (b d+3 c e)\right ) x}{\sqrt {d+e x} \sqrt {c+b x+a x^2}} \, dx}{15 a e \sqrt {c+b x+a x^2}}\\ &=-\frac {2 (2 a d-b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}}{15 a e}+\frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{3/2}}{5 e}+\frac {\left ((2 a d-b e) \left (a d^2-e (b d-c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c+b x+a x^2}} \, dx}{15 a e^2 \sqrt {c+b x+a x^2}}-\frac {\left (2 \left (a^2 d^2+b^2 e^2-a e (b d+3 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c+b x+a x^2}} \, dx}{15 a e^2 \sqrt {c+b x+a x^2}}\\ &=-\frac {2 (2 a d-b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}}{15 a e}+\frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{3/2}}{5 e}-\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} \left (a^2 d^2+b^2 e^2-a e (b d+3 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 a d-b e-\sqrt {b^2-4 a c} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{15 a^2 e^2 \sqrt {\frac {a (d+e x)}{2 a d-b e-\sqrt {b^2-4 a c} e}} \left (c+b x+a x^2\right )}+\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} (2 a d-b e) \left (a d^2-e (b d-c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-b e-\sqrt {b^2-4 a c} e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 a d-b e-\sqrt {b^2-4 a c} e}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{15 a^2 e^2 \sqrt {d+e x} \left (c+b x+a x^2\right )}\\ &=-\frac {2 (2 a d-b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}}{15 a e}+\frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{3/2}}{5 e}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (a^2 d^2+b^2 e^2-a e (b d+3 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 a^2 e^2 \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \left (c+b x+a x^2\right )}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 a d-b e) \left (a d^2-e (b d-c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 a^2 e^2 \sqrt {d+e x} \left (c+b x+a x^2\right )}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 28.29, size = 1046, normalized size = 1.90 \begin {gather*} \frac {1}{15} x \sqrt {d+e x} \sqrt {a+\frac {c+b x}{x^2}} \left (\frac {2 b}{a}+\frac {2 d}{e}+6 x-\frac {4 e^2 \sqrt {\frac {a d^2+e (-b d+c e)}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \left (a^2 d^2+b^2 e^2-a e (b d+3 c e)\right ) (c+x (b+a x))-i \sqrt {2} \left (2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (a^2 d^2+b^2 e^2-a e (b d+3 c e)\right ) (d+e x)^{3/2} \sqrt {\frac {-2 c e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}+2 a d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (d-e x)}{\left (2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 c e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}-2 a d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (-d+e x)}{\left (-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {a d^2-b d e+c e^2}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )+i \sqrt {2} \left (b^2 e^2 \left (-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+a^2 d \left (-8 c e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+a e \left (2 b^2 d e+4 b c e^2-b d \sqrt {\left (b^2-4 a c\right ) e^2}-3 c e \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right ) (d+e x)^{3/2} \sqrt {\frac {-2 c e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}+2 a d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (d-e x)}{\left (2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 c e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}-2 a d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (-d+e x)}{\left (-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {a d^2-b d e+c e^2}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )}{a^2 e^3 \sqrt {\frac {a d^2+e (-b d+c e)}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} (d+e x) (c+x (b+a x))}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + c/x^2 + b/x]*x*Sqrt[d + e*x],x]

[Out]

(x*Sqrt[d + e*x]*Sqrt[a + (c + b*x)/x^2]*((2*b)/a + (2*d)/e + 6*x - (4*e^2*Sqrt[(a*d^2 + e*(-(b*d) + c*e))/(-2
*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(a^2*d^2 + b^2*e^2 - a*e*(b*d + 3*c*e))*(c + x*(b + a*x)) - I*Sqrt[2]*(
2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(a^2*d^2 + b^2*e^2 - a*e*(b*d + 3*c*e))*(d + e*x)^(3/2)*Sqrt[(-2*c*e^2
+ d*Sqrt[(b^2 - 4*a*c)*e^2] + 2*a*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(d - e*x))/((2*a*d - b*e + Sqrt[(b
^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*c*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] - 2*a*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2
]*x + b*e*(-d + e*x))/((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[
(a*d^2 - b*d*e + c*e^2)/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*a*d + b*e + Sqrt[(b^2
- 4*a*c)*e^2])/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] + I*Sqrt[2]*(b^2*e^2*(-(b*e) + Sqrt[(b^2 - 4*a*c)*e^
2]) + a^2*d*(-8*c*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2]) + a*e*(2*b^2*d*e + 4*b*c*e^2 - b*d*Sqrt[(b^2 - 4*a*c)*e^2]
- 3*c*e*Sqrt[(b^2 - 4*a*c)*e^2]))*(d + e*x)^(3/2)*Sqrt[(-2*c*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] + 2*a*d*e*x + e*S
qrt[(b^2 - 4*a*c)*e^2]*x + b*e*(d - e*x))/((2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*c*e^2 +
d*Sqrt[(b^2 - 4*a*c)*e^2] - 2*a*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(-d + e*x))/((-2*a*d + b*e + Sqrt[(
b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(a*d^2 - b*d*e + c*e^2)/(-2*a*d + b*e + Sqrt[
(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*a*d - b*e + Sqrt[(b^2 - 4*
a*c)*e^2]))])/(a^2*e^3*Sqrt[(a*d^2 + e*(-(b*d) + c*e))/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(d + e*x)*(c
+ x*(b + a*x)))))/15

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(4360\) vs. \(2(486)=972\).
time = 0.23, size = 4361, normalized size = 7.93

method	result	size
risch	\(\text {Expression too large to display}\)	\(1711\)
default	\(\text {Expression too large to display}\)	\(4361\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*((a*x^2+b*x+c)/x^2)^(1/2)*x*(e*x+d)^(1/2)*(-10*a^2*b*d*e^3*x^2-3*2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/
2)-2*a*d+e*b))^(1/2)*((-2*a*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*a*x+(-4*a*
c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*
a*d+e*b))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b)/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2))*(-4*a*c+b^2)^(1/2)
*a*b*d^2*e^2+2*2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*((-2*a*x+(-4*a*c+b^2)^(1/2)-b)*e/(2
*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*a*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*
EllipticF(2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b)/(2*a*
d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2))*(-4*a*c+b^2)^(1/2)*a*c*d*e^3+12*2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-
2*a*d+e*b))^(1/2)*((-2*a*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*a*x+(-4*a*c+b
^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d
+e*b))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b)/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2))*a*b*c*d*e^3-8*2^(1/2)
*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*((-2*a*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*a*d-e*b+e*(-4*a*c+b^2
)^(1/2)))^(1/2)*((b+2*a*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*EllipticE(2^(1/2)*(-(e
*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b)/(2*a*d-e*b+e*(-4*a*c+b^2)^(
1/2)))^(1/2))*a*b*c*d*e^3-6*a^2*c*e^4*x^2-2*a*b^2*e^4*x^2-8*a^3*d*e^3*x^3-8*a^2*b*e^4*x^3-2*a^3*d^2*e^2*x^2-2*
a^2*b*d^2*e^2*x-8*a^2*c*d*e^3*x-2*a*b^2*d*e^3*x-2*a*b*c*e^4*x-6*a^3*e^4*x^4+3*2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b
^2)^(1/2)-2*a*d+e*b))^(1/2)*((-2*a*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*a*x
+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(
1/2)-2*a*d+e*b))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b)/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2))*a*b^2*d^2*e
^2+8*2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*((-2*a*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*a*d-e*b+e
*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*a*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*EllipticE(
2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b)/(2*a*d-e*b+e*(-
4*a*c+b^2)^(1/2)))^(1/2))*a^2*b*d^3*e+2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*((-2*a*x+(-4
*a*c+b^2)^(1/2)-b)*e/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*a*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(
1/2)-2*a*d+e*b))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2),(-(e*(-4*a*c+b^2)
^(1/2)-2*a*d+e*b)/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2))*(-4*a*c+b^2)^(1/2)*b^2*d*e^3+2*2^(1/2)*(-(e*x+d)*a/
(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*((-2*a*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/
2)*((b+2*a*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*a/(e*(-
4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b)/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2))
*(-4*a*c+b^2)^(1/2)*a^2*d^3*e-3*2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*((-2*a*x+(-4*a*c+b
^2)^(1/2)-b)*e/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*a*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)-2
*a*d+e*b))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)
-2*a*d+e*b)/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2))*b^3*d*e^3+3*2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a
*d+e*b))^(1/2)*((-2*a*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*a*x+(-4*a*c+b^2)
^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*
b))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b)/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2))*b^2*c*e^4+4*2^(1/2)*(-(e
*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*((-2*a*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/
2)))^(1/2)*((b+2*a*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*EllipticE(2^(1/2)*(-(e*x+d)
*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b)/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2))
)^(1/2))*b^3*d*e^3-4*2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*((-2*a*x+(-4*a*c+b^2)^(1/2)-b
)*e/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*a*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^
(1/2)*EllipticE(2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b)
/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2))*b^2*c*e^4-2*a^2*c*d^2*e^2-2*a*b*c*d*e^3-12*2^(1/2)*(-(e*x+d)*a/(e*(-
4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*((-2*a*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((
b+2*a*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c
+b^2)^(1/2)-2*a*d+e*b))^(1/2),(-(e*(-4*a*c+b^2)...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.11, size = 475, normalized size = 0.86 \begin {gather*} \frac {2 \, {\left ({\left (2 \, a^{3} d^{3} - 3 \, a^{2} b d^{2} e - 3 \, {\left (a b^{2} - 6 \, a^{2} c\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} \sqrt {a} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a^{2} d^{2} - a b d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, a^{2}}, -\frac {4 \, {\left (2 \, a^{3} d^{3} - 3 \, a^{2} b d^{2} e - 3 \, {\left (a b^{2} - 6 \, a^{2} c\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, a^{3}}, \frac {{\left (a d + {\left (3 \, a x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, a}\right ) + 6 \, {\left (a^{3} d^{2} e - a^{2} b d e^{2} + {\left (a b^{2} - 3 \, a^{2} c\right )} e^{3}\right )} \sqrt {a} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (a^{2} d^{2} - a b d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, a^{2}}, -\frac {4 \, {\left (2 \, a^{3} d^{3} - 3 \, a^{2} b d^{2} e - 3 \, {\left (a b^{2} - 6 \, a^{2} c\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a^{2} d^{2} - a b d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, a^{2}}, -\frac {4 \, {\left (2 \, a^{3} d^{3} - 3 \, a^{2} b d^{2} e - 3 \, {\left (a b^{2} - 6 \, a^{2} c\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, a^{3}}, \frac {{\left (a d + {\left (3 \, a x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, a}\right )\right ) + 3 \, {\left (a^{3} d x e^{2} + {\left (3 \, a^{3} x^{2} + a^{2} b x\right )} e^{3}\right )} \sqrt {x e + d} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}\right )} e^{\left (-3\right )}}{45 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45*((2*a^3*d^3 - 3*a^2*b*d^2*e - 3*(a*b^2 - 6*a^2*c)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)*sqrt(a)*e^(1/2)*weierstr
assPInverse(4/3*(a^2*d^2 - a*b*d*e + (b^2 - 3*a*c)*e^2)*e^(-2)/a^2, -4/27*(2*a^3*d^3 - 3*a^2*b*d^2*e - 3*(a*b^
2 - 6*a^2*c)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)*e^(-3)/a^3, 1/3*(a*d + (3*a*x + b)*e)*e^(-1)/a) + 6*(a^3*d^2*e - a
^2*b*d*e^2 + (a*b^2 - 3*a^2*c)*e^3)*sqrt(a)*e^(1/2)*weierstrassZeta(4/3*(a^2*d^2 - a*b*d*e + (b^2 - 3*a*c)*e^2
)*e^(-2)/a^2, -4/27*(2*a^3*d^3 - 3*a^2*b*d^2*e - 3*(a*b^2 - 6*a^2*c)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)*e^(-3)/a^3
, weierstrassPInverse(4/3*(a^2*d^2 - a*b*d*e + (b^2 - 3*a*c)*e^2)*e^(-2)/a^2, -4/27*(2*a^3*d^3 - 3*a^2*b*d^2*e
- 3*(a*b^2 - 6*a^2*c)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)*e^(-3)/a^3, 1/3*(a*d + (3*a*x + b)*e)*e^(-1)/a)) + 3*(a^
3*d*x*e^2 + (3*a^3*x^2 + a^2*b*x)*e^3)*sqrt(x*e + d)*sqrt((a*x^2 + b*x + c)/x^2))*e^(-3)/a^3

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \sqrt {d + e x} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x\,\sqrt {d+e\,x}\,\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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