\(\int \frac {\sqrt {d+e x}}{(a+b x+c x^2)^{5/2}} \, dx\) [2479]
Optimal result
Integrand size = 24, antiderivative size = 605 \[
\int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (9 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (2 c d^2+a e^2\right )-c \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {16 \sqrt {2} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \sqrt {d+e x} \sqrt {a+b x+c x^2}}
\]
[Out]
-2/3*(2*c*x+b)*(e*x+d)^(1/2)/(-4*a*c+b^2)/(c*x^2+b*x+a)^(3/2)-2/3*(9*b^2*c*d*e-4*a*c^2*d*e-b^3*e^2-4*b*c*(a*e^
2+2*c*d^2)-c*(16*c^2*d^2+b^2*e^2-4*c*e*(-3*a*e+4*b*d))*x)*(e*x+d)^(1/2)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)/(c*
x^2+b*x+a)^(1/2)-1/3*(16*c^2*d^2+b^2*e^2-4*c*e*(-3*a*e+4*b*d))*EllipticE(1/2*((b+2*c*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*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(e*x+
d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/(-4*a*c+b^2)^(3/2)/(a*e^2-b*d*e+c*d^2)/(c*x^2+b*x+a)^(1/2)/(c*(
e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)+16/3*(-b*e+2*c*d)*EllipticF(1/2*((b+2*c*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*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-c
*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)/(-4*a*c+b^2)^(3/2)/(e*x+
d)^(1/2)/(c*x^2+b*x+a)^(1/2)
Rubi [A] (verified)
Time = 0.38 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.00, number of
steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {750, 836, 857, 732, 435, 430}
\[
\int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {\sqrt {2} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {16 \sqrt {2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (-c x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-4 b c \left (a e^2+2 c d^2\right )-4 a c^2 d e+b^3 \left (-e^2\right )+9 b^2 c d e\right )}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}
\]
[In]
Int[Sqrt[d + e*x]/(a + b*x + c*x^2)^(5/2),x]
[Out]
(-2*(b + 2*c*x)*Sqrt[d + e*x])/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) - (2*Sqrt[d + e*x]*(9*b^2*c*d*e - 4*a
*c^2*d*e - b^3*e^2 - 4*b*c*(2*c*d^2 + a*e^2) - c*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*a*e))*x))/(3*(b^2 -
4*a*c)^2*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x + c*x^2]) - (Sqrt[2]*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*a*
e))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] +
2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(3*(b^2 - 4
*a*c)^(3/2)*(c*d^2 - b*d*e + a*e^2)*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x
^2]) + (16*Sqrt[2]*(2*c*d - b*e)*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x +
c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2
*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(3*(b^2 - 4*a*c)^(3/2)*Sqrt[d + e*x]*Sqrt[a + b*x
+ 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 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 750
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(b + 2*c
*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m
- 1)*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*x)*(a + b*x + c*x^2)^(p + 1), x], 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] && LtQ[p, -1] && GtQ[m
, 0] && (LtQ[m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && IntQuadraticQ[a, b, c, d, e, m, p, x]
Rule 836
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])
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]
Rubi steps \begin{align*}
\text {integral}& = -\frac {2 (b+2 c x) \sqrt {d+e x}}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \int \frac {-4 c d+\frac {b e}{2}-3 c e x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )} \\ & = -\frac {2 (b+2 c x) \sqrt {d+e x}}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (9 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (2 c d^2+a e^2\right )-c \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}-\frac {4 \int \frac {-\frac {1}{4} c e \left (7 b^2 d e+4 a c d e-8 b \left (c d^2+a e^2\right )\right )+\frac {1}{4} c e \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {2 (b+2 c x) \sqrt {d+e x}}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (9 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (2 c d^2+a e^2\right )-c \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}+\frac {(8 c (2 c d-b e)) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2}-\frac {\left (c \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {2 (b+2 c x) \sqrt {d+e x}}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (9 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (2 c d^2+a e^2\right )-c \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}-\frac {\left (\sqrt {2} \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c 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 c 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 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {a+b x+c x^2}}+\frac {\left (16 \sqrt {2} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {-\frac {c \left (a+b x+c 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 c d-b e-\sqrt {b^2-4 a c} e}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{3 \left (b^2-4 a c\right )^{3/2} \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ & = -\frac {2 (b+2 c x) \sqrt {d+e x}}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (9 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (2 c d^2+a e^2\right )-c \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {16 \sqrt {2} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\
\end{align*}
Mathematica [C] (verified)
Result contains complex when optimal does not.
Time = 20.99 (sec) , antiderivative size = 1323, normalized size of antiderivative = 2.19
\[
\int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {\sqrt {d+e x} \left (a+b x+c x^2\right )^3 \left (-\frac {2 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {2 \left (8 b c^2 d^2-9 b^2 c d e+4 a c^2 d e+b^3 e^2+4 a b c e^2+16 c^3 d^2 x-16 b c^2 d e x+b^2 c e^2 x+12 a c^2 e^2 x\right )}{3 \left (-b^2+4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}\right )}{(a+x (b+c x))^{5/2}}+\frac {(d+e x)^{3/2} \left (a+b x+c x^2\right )^{5/2} \left (-4 \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \left (16 c^2 d^2+b^2 e^2+4 c e (-4 b d+3 a e)\right ) \left (c \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {a e}{d+e x}\right )}{d+e x}\right )+\frac {i \sqrt {2} \left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (16 c^2 d^2+b^2 e^2+4 c e (-4 b d+3 a e)\right ) \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}-\frac {2 a e^2}{d+e x}-2 c d \left (-1+\frac {d}{d+e x}\right )+b e \left (-1+\frac {2 d}{d+e x}\right )}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}+\frac {2 a e^2}{d+e x}+2 c d \left (-1+\frac {d}{d+e x}\right )+b \left (e-\frac {2 d e}{d+e x}\right )}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )}{\sqrt {d+e x}}-\frac {i \sqrt {2} \left (-b^3 e^3+b^2 e^2 \left (2 c d+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+4 b \left (a c e^3-4 c d e \sqrt {\left (b^2-4 a c\right ) e^2}\right )+4 c \left (4 c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}+a e^2 \left (-2 c d+3 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )\right ) \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}-\frac {2 a e^2}{d+e x}-2 c d \left (-1+\frac {d}{d+e x}\right )+b e \left (-1+\frac {2 d}{d+e x}\right )}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}+\frac {2 a e^2}{d+e x}+2 c d \left (-1+\frac {d}{d+e x}\right )+b \left (e-\frac {2 d e}{d+e x}\right )}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )}{\sqrt {d+e x}}\right )}{6 \left (-b^2+4 a c\right )^2 e \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} (a+x (b+c x))^{5/2} \sqrt {\frac {(d+e x)^2 \left (c \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {a e}{d+e x}\right )}{d+e x}\right )}{e^2}}}
\]
[In]
Integrate[Sqrt[d + e*x]/(a + b*x + c*x^2)^(5/2),x]
[Out]
(Sqrt[d + e*x]*(a + b*x + c*x^2)^3*((-2*(b + 2*c*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (2*(8*b*c^2*d^2 -
9*b^2*c*d*e + 4*a*c^2*d*e + b^3*e^2 + 4*a*b*c*e^2 + 16*c^3*d^2*x - 16*b*c^2*d*e*x + b^2*c*e^2*x + 12*a*c^2*e^
2*x))/(3*(-b^2 + 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2))))/(a + x*(b + c*x))^(5/2) + ((d + e*x)^(3
/2)*(a + b*x + c*x^2)^(5/2)*(-4*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(16*
c^2*d^2 + b^2*e^2 + 4*c*e*(-4*b*d + 3*a*e))*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x
)))/(d + e*x)) + (I*Sqrt[2]*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(16*c^2*d^2 + b^2*e^2 + 4*c*e*(-4*b*d + 3*
a*e))*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*
x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/(d + e*x) + 2*c*d*(-1
+ d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticE[I*ArcSinh[(Sqr
t[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e +
Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x] - (I*Sqrt[2]*(-(b^3*e^3) +
b^2*e^2*(2*c*d + Sqrt[(b^2 - 4*a*c)*e^2]) + 4*b*(a*c*e^3 - 4*c*d*e*Sqrt[(b^2 - 4*a*c)*e^2]) + 4*c*(4*c*d^2*Sqr
t[(b^2 - 4*a*c)*e^2] + a*e^2*(-2*c*d + 3*Sqrt[(b^2 - 4*a*c)*e^2])))*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/
(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sq
rt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/(d + e*x) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*
c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e
+ Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b
^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x]))/(6*(-b^2 + 4*a*c)^2*e*(c*d^2 - b*d*e + a*e^2)*Sqrt[(c*d^2 + e*(-(b*d) + a
*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(a + x*(b + c*x))^(5/2)*Sqrt[((d + e*x)^2*(c*(-1 + d/(d + e*x))
^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)))/e^2])
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(1485\) vs. \(2(541)=1082\).
Time = 1.28 (sec) , antiderivative size = 1486, normalized size of antiderivative =
2.46
| | |
method | result | size |
| | |
elliptic |
\(\text {Expression too large to display}\) |
\(1486\) |
default |
\(\text {Expression too large to display}\) |
\(13071\) |
| | |
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[In]
int((e*x+d)^(1/2)/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
[Out]
((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*((4/3/c/(4*a*c-b^2)*x+2/3/c^2*b/(4*a*c-b^2))*(
c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)/(a/c+b/c*x+x^2)^2-2*(c*e*x+c*d)*(-1/3*(12*a*c*e^2+b^2*e^2-16*b*
c*d*e+16*c^2*d^2)/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)/(4*a*c-b^2)*x-1/3*(4*a*b*c
*e^2+4*a*c^2*d*e+b^3*e^2-9*b^2*c*d*e+8*b*c^2*d^2)/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c
*d^2)/(4*a*c-b^2)/c)/((a/c+b/c*x+x^2)*(c*e*x+c*d))^(1/2)+2*(-1/3*(4*a*b*c*e^3-32*a*c^2*d*e^2-b^3*e^3+32*b*c^2*
d^2*e-32*c^3*d^3)/(4*a*c-b^2)/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)-1/3*e*(4*a*b*c
*e^2+4*a*c^2*d*e+b^3*e^2-9*b^2*c*d*e+8*b*c^2*d^2)/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c
*d^2)/(4*a*c-b^2)-2/3*c*d*(12*a*c*e^2+b^2*e^2-16*b*c*d*e+16*c^2*d^2)/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^
2*d^2+b^3*d*e-b^2*c*d^2)/(4*a*c-b^2))*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/
2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*
c+b^2)^(1/2))/c)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)*El
lipticF(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-
b+(-4*a*c+b^2)^(1/2))))^(1/2))-2/3*c*e*(12*a*c*e^2+b^2*e^2-16*b*c*d*e+16*c^2*d^2)/(4*a*c-b^2)/(4*a^2*c*e^2-a*b
^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*
a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1
/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a
*d)^(1/2)*((-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*EllipticE(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),(
(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+1/2/c*(-b+(-4*a*c+b^2)^(1/2))
*EllipticF(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c
*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))))
Fricas [C] (verification not implemented)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 1722, normalized size of antiderivative = 2.85
\[
\int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display}
\]
[In]
integrate((e*x+d)^(1/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
[Out]
2/9*((16*a^2*c^3*d^3 - 24*a^2*b*c^2*d^2*e + (16*c^5*d^3 - 24*b*c^4*d^2*e + 6*(b^2*c^3 + 4*a*c^4)*d*e^2 + (b^3*
c^2 - 12*a*b*c^3)*e^3)*x^4 + 6*(a^2*b^2*c + 4*a^3*c^2)*d*e^2 + (a^2*b^3 - 12*a^3*b*c)*e^3 + 2*(16*b*c^4*d^3 -
24*b^2*c^3*d^2*e + 6*(b^3*c^2 + 4*a*b*c^3)*d*e^2 + (b^4*c - 12*a*b^2*c^2)*e^3)*x^3 + (16*(b^2*c^3 + 2*a*c^4)*d
^3 - 24*(b^3*c^2 + 2*a*b*c^3)*d^2*e + 6*(b^4*c + 6*a*b^2*c^2 + 8*a^2*c^3)*d*e^2 + (b^5 - 10*a*b^3*c - 24*a^2*b
*c^2)*e^3)*x^2 + 2*(16*a*b*c^3*d^3 - 24*a*b^2*c^2*d^2*e + 6*(a*b^3*c + 4*a^2*b*c^2)*d*e^2 + (a*b^4 - 12*a^2*b^
2*c)*e^3)*x)*sqrt(c*e)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/27*(2*c^3
*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)
/(c*e)) + 3*(16*a^2*c^3*d^2*e - 16*a^2*b*c^2*d*e^2 + (16*c^5*d^2*e - 16*b*c^4*d*e^2 + (b^2*c^3 + 12*a*c^4)*e^3
)*x^4 + (a^2*b^2*c + 12*a^3*c^2)*e^3 + 2*(16*b*c^4*d^2*e - 16*b^2*c^3*d*e^2 + (b^3*c^2 + 12*a*b*c^3)*e^3)*x^3
+ (16*(b^2*c^3 + 2*a*c^4)*d^2*e - 16*(b^3*c^2 + 2*a*b*c^3)*d*e^2 + (b^4*c + 14*a*b^2*c^2 + 24*a^2*c^3)*e^3)*x^
2 + 2*(16*a*b*c^3*d^2*e - 16*a*b^2*c^2*d*e^2 + (a*b^3*c + 12*a^2*b*c^2)*e^3)*x)*sqrt(c*e)*weierstrassZeta(4/3*
(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^
2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2
), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), 1/3*(3*c*e
*x + c*d + b*e)/(c*e))) + 3*(8*a^2*b*c^2*e^3 - (b^3*c^2 - 12*a*b*c^3)*d^2*e + (b^4*c - 13*a*b^2*c^2 + 4*a^2*c^
3)*d*e^2 + (16*c^5*d^2*e - 16*b*c^4*d*e^2 + (b^2*c^3 + 12*a*c^4)*e^3)*x^3 + (24*b*c^4*d^2*e - (25*b^2*c^3 - 4*
a*c^4)*d*e^2 + 2*(b^3*c^2 + 8*a*b*c^3)*e^3)*x^2 + (6*(b^2*c^3 + 4*a*c^4)*d^2*e - (7*b^3*c^2 + 20*a*b*c^3)*d*e^
2 + (b^4*c + 3*a*b^2*c^2 + 20*a^2*c^3)*e^3)*x)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d))/(((b^4*c^4 - 8*a*b^2*c^5 +
16*a^2*c^6)*d^2*e - (b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d*e^2 + (a*b^4*c^3 - 8*a^2*b^2*c^4 + 16*a^3*c^5)*e
^3)*x^4 + (a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d^2*e - (a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d*e^2
+ (a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)*e^3 + 2*((b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d^2*e - (b^6*c^2 -
8*a*b^4*c^3 + 16*a^2*b^2*c^4)*d*e^2 + (a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*e^3)*x^3 + ((b^6*c^2 - 6*a*b^
4*c^3 + 32*a^3*c^5)*d^2*e - (b^7*c - 6*a*b^5*c^2 + 32*a^3*b*c^4)*d*e^2 + (a*b^6*c - 6*a^2*b^4*c^2 + 32*a^4*c^4
)*e^3)*x^2 + 2*((a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*d^2*e - (a*b^6*c - 8*a^2*b^4*c^2 + 16*a^3*b^2*c^3)*
d*e^2 + (a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*e^3)*x)
Sympy [F]
\[
\int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {d + e x}}{\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}\, dx
\]
[In]
integrate((e*x+d)**(1/2)/(c*x**2+b*x+a)**(5/2),x)
[Out]
Integral(sqrt(d + e*x)/(a + b*x + c*x**2)**(5/2), x)
Maxima [F]
\[
\int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {e x + d}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x }
\]
[In]
integrate((e*x+d)^(1/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")
[Out]
integrate(sqrt(e*x + d)/(c*x^2 + b*x + a)^(5/2), x)
Giac [F]
\[
\int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {e x + d}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x }
\]
[In]
integrate((e*x+d)^(1/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
[Out]
integrate(sqrt(e*x + d)/(c*x^2 + b*x + a)^(5/2), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {d+e\,x}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x
\]
[In]
int((d + e*x)^(1/2)/(a + b*x + c*x^2)^(5/2),x)
[Out]
int((d + e*x)^(1/2)/(a + b*x + c*x^2)^(5/2), x)