\(\int \frac {b+2 c x}{\sqrt {d+e x} (a+b x+c x^2)^{5/2}} \, dx\) [1653]

Optimal result

Integrand size = 30, antiderivative size = 665 \[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 e \sqrt {d+e x} \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} e (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]

[Out]

-2/3*((-4*a*c+b^2)*(-b*e+c*d)-c*(-4*a*c+b^2)*e*x)*(e*x+d)^(1/2)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(c*x^2+b*x+a)
^(3/2)-2/3*e*(3*b^2*c*d*e-8*a*c^2*d*e-2*b^3*e^2-b*c*(-7*a*e^2+c*d^2)-2*c*(c^2*d^2+b^2*e^2-c*e*(3*a*e+b*d))*x)*
(e*x+d)^(1/2)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^2/(c*x^2+b*x+a)^(1/2)-2/3*e*(c^2*d^2+b^2*e^2-c*e*(3*a*e+b*d))*E
llipticE(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)/(a*e^2-b*d*e+c*d^
2)^2/(-4*a*c+b^2)^(1/2)/(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)+2/3*e*(-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)/(a*e^2-b*d*e+c*d^2)/(-4*a*c+b^2)^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 665, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {836, 857, 732, 435, 430} \[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \sqrt {2} e \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \sqrt {b^2-4 a c} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 \sqrt {2} e \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {2 e \sqrt {d+e x} \left (-2 c x \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )-b c \left (c d^2-7 a e^2\right )-8 a c^2 d e-2 b^3 e^2+3 b^2 c d e\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2} \]

[In]

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

[Out]

(-2*Sqrt[d + e*x]*((b^2 - 4*a*c)*(c*d - b*e) - c*(b^2 - 4*a*c)*e*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*
(a + b*x + c*x^2)^(3/2)) - (2*e*Sqrt[d + e*x]*(3*b^2*c*d*e - 8*a*c^2*d*e - 2*b^3*e^2 - b*c*(c*d^2 - 7*a*e^2) -
2*c*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x + c*x
^2]) - (2*Sqrt[2]*e*(c^2*d^2 + b^2*e^2 - c*e*(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*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[(c*(d + e*x))
/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (2*Sqrt[2]*e*(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 + Sq
rt[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*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^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 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 \sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (b^2-4 a c\right ) e (c d-2 b e)-\frac {3}{2} c \left (b^2-4 a c\right ) e^2 x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {2 \sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 e \sqrt {d+e x} \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x+c x^2}}+\frac {4 \int \frac {-\frac {1}{4} c \left (b^2-4 a c\right ) e^2 \left (b c d^2+b^2 d e-8 a c d e+a b e^2\right )-\frac {1}{2} c \left (b^2-4 a c\right ) e^2 \left (c^2 d^2+b^2 e^2-c e (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 )^2} \\ & = -\frac {2 \sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 e \sqrt {d+e x} \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x+c x^2}}+\frac {(c e (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 ) \left (c d^2-b d e+a e^2\right )}-\frac {\left (2 c e \left (c^2 d^2+b^2 e^2-c e (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 ) \left (c d^2-b d e+a e^2\right )^2} \\ & = -\frac {2 \sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 e \sqrt {d+e x} \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x+c x^2}}-\frac {\left (2 \sqrt {2} e \left (c^2 d^2+b^2 e^2-c e (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 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 \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 (2 \sqrt {2} e (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 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ & = -\frac {2 \sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 e \sqrt {d+e x} \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} e \left (c^2 d^2+b^2 e^2-c e (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 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 \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 {2 \sqrt {2} e (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 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 21.52 (sec) , antiderivative size = 1334, normalized size of antiderivative = 2.01 \[ \int \frac {b+2 c x}{\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 (-c d+b e+c e x)}{3 \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}+\frac {2 \left (b c^2 d^2 e-3 b^2 c d e^2+8 a c^2 d e^2+2 b^3 e^3-7 a b c e^3+2 c^3 d^2 e x-2 b c^2 d e^2 x+2 b^2 c e^3 x-6 a c^2 e^3 x\right )}{3 \left (b^2-4 a c\right ) \left (-c d^2+b d e-a e^2\right )^2 \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 (c^2 d^2+b^2 e^2-c e (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 (c^2 d^2+b^2 e^2-c e (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 )+b c e \left (4 a e^2-d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+c \left (c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}-a e^2 \left (8 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 )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \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[(b + 2*c*x)/(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*(-(c*d) + b*e + c*e*x))/(3*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2)
+ (2*(b*c^2*d^2*e - 3*b^2*c*d*e^2 + 8*a*c^2*d*e^2 + 2*b^3*e^3 - 7*a*b*c*e^3 + 2*c^3*d^2*e*x - 2*b*c^2*d*e^2*x
+ 2*b^2*c*e^3*x - 6*a*c^2*e^3*x))/(3*(b^2 - 4*a*c)*(-(c*d^2) + b*d*e - a*e^2)^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])]*(c^2*d^2 + b^2*e^2 - c*e*(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])*(c^2*d^2 + b^2*e^2 -
c*e*(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[(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] + (I*Sqrt[2]*
(-(b^3*e^3) + b^2*e^2*(2*c*d + Sqrt[(b^2 - 4*a*c)*e^2]) + b*c*e*(4*a*e^2 - d*Sqrt[(b^2 - 4*a*c)*e^2]) + c*(c*d
^2*Sqrt[(b^2 - 4*a*c)*e^2] - a*e^2*(8*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]
)]*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])]*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 + Sq
rt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x]))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^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. \(1551\) vs. \(2(601)=1202\).

Time = 3.85 (sec) , antiderivative size = 1552, normalized size of antiderivative = 2.33

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1552\)
default	\(\text {Expression too large to display}\)	\(13049\)

[In]

int((2*c*x+b)/(c*x^2+b*x+a)^(5/2)/(e*x+d)^(1/2),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)*((2/3/c*e/(a*e^2-b*d*e+c*d^2)*x+2/3*(b*e-c*d)/
(a*e^2-b*d*e+c*d^2)/c^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)*(-2/3
*e*(3*a*c*e^2-b^2*e^2+b*c*d*e-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)/(a*e^
2-b*d*e+c*d^2)*x-1/3*e*(7*a*b*c*e^2-8*a*c^2*d*e-2*b^3*e^2+3*b^2*c*d*e-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)/c/(a*e^2-b*d*e+c*d^2))/((a/c+b/c*x+x^2)*(c*e*x+c*d))^(1/2)+2*(2/3*e*(4*a*
b*c*e^3-2*a*c^2*d*e^2-b^3*e^3+2*b*c^2*d^2*e-2*c^3*d^3)/(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)/(a*e^2-b*d*e+c*d^2)-1/3*e^2*(7*a*b*c*e^2-8*a*c^2*d*e-2*b^3*e^2+3*b^2*c*d*e-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)/(a*e^2-b*d*e+c*d^2)-4/3*c*d*e*(3*a*c*e^2-b^2*e^2+b*c*d*e-
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)/(a*e^2-b*d*e+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)*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))-4/3*c*e^2*(3*a*c*e^2-
b^2*e^2+b*c*d*e-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)/(a*e^2-b*d*e+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)))*E
llipticE(((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 = 0.18 (sec) , antiderivative size = 1918, normalized size of antiderivative = 2.88 \[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

2/9*((2*a^2*c^3*d^3 - 3*a^2*b*c^2*d^2*e + (2*c^5*d^3 - 3*b*c^4*d^2*e - 3*(b^2*c^3 - 6*a*c^4)*d*e^2 + (2*b^3*c^
2 - 9*a*b*c^3)*e^3)*x^4 - 3*(a^2*b^2*c - 6*a^3*c^2)*d*e^2 + (2*a^2*b^3 - 9*a^3*b*c)*e^3 + 2*(2*b*c^4*d^3 - 3*b
^2*c^3*d^2*e - 3*(b^3*c^2 - 6*a*b*c^3)*d*e^2 + (2*b^4*c - 9*a*b^2*c^2)*e^3)*x^3 + (2*(b^2*c^3 + 2*a*c^4)*d^3 -
3*(b^3*c^2 + 2*a*b*c^3)*d^2*e - 3*(b^4*c - 4*a*b^2*c^2 - 12*a^2*c^3)*d*e^2 + (2*b^5 - 5*a*b^3*c - 18*a^2*b*c^
2)*e^3)*x^2 + 2*(2*a*b*c^3*d^3 - 3*a*b^2*c^2*d^2*e - 3*(a*b^3*c - 6*a^2*b*c^2)*d*e^2 + (2*a*b^4 - 9*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)) + 6*(a^2*c^3*d^2*e - a^2*b*c^2*d*e^2 + (c^5*d^2*e - b*c^4*d*e^2 + (b^2*c^3 - 3*a*c^4)*e^3)*x^4 + (a^2*b^2*
c - 3*a^3*c^2)*e^3 + 2*(b*c^4*d^2*e - b^2*c^3*d*e^2 + (b^3*c^2 - 3*a*b*c^3)*e^3)*x^3 + ((b^2*c^3 + 2*a*c^4)*d^
2*e - (b^3*c^2 + 2*a*b*c^3)*d*e^2 + (b^4*c - a*b^2*c^2 - 6*a^2*c^3)*e^3)*x^2 + 2*(a*b*c^3*d^2*e - a*b^2*c^2*d*
e^2 + (a*b^3*c - 3*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), weie
rstrassPInverse(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*((b^2*c^3 - 4*
a*c^4)*d^3 - (2*b^3*c^2 - 7*a*b*c^3)*d^2*e + (b^4*c - 12*a^2*c^3)*d*e^2 - (3*a*b^3*c - 11*a^2*b*c^2)*e^3 - 2*(
c^5*d^2*e - b*c^4*d*e^2 + (b^2*c^3 - 3*a*c^4)*e^3)*x^3 - (3*b*c^4*d^2*e - (5*b^2*c^3 - 8*a*c^4)*d*e^2 + (4*b^3
*c^2 - 13*a*b*c^3)*e^3)*x^2 - 2*((b^2*c^3 - a*c^4)*d^2*e - (2*b^3*c^2 - 5*a*b*c^3)*d*e^2 + (b^4*c - 2*a*b^2*c^
2 - 5*a^2*c^3)*e^3)*x)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d))/((a^2*b^2*c^3 - 4*a^3*c^4)*d^4 - 2*(a^2*b^3*c^2 -
4*a^3*b*c^3)*d^3*e + (a^2*b^4*c - 2*a^3*b^2*c^2 - 8*a^4*c^3)*d^2*e^2 - 2*(a^3*b^3*c - 4*a^4*b*c^2)*d*e^3 + (a^
4*b^2*c - 4*a^5*c^2)*e^4 + ((b^2*c^5 - 4*a*c^6)*d^4 - 2*(b^3*c^4 - 4*a*b*c^5)*d^3*e + (b^4*c^3 - 2*a*b^2*c^4 -
8*a^2*c^5)*d^2*e^2 - 2*(a*b^3*c^3 - 4*a^2*b*c^4)*d*e^3 + (a^2*b^2*c^3 - 4*a^3*c^4)*e^4)*x^4 + 2*((b^3*c^4 - 4
*a*b*c^5)*d^4 - 2*(b^4*c^3 - 4*a*b^2*c^4)*d^3*e + (b^5*c^2 - 2*a*b^3*c^3 - 8*a^2*b*c^4)*d^2*e^2 - 2*(a*b^4*c^2
- 4*a^2*b^2*c^3)*d*e^3 + (a^2*b^3*c^2 - 4*a^3*b*c^3)*e^4)*x^3 + ((b^4*c^3 - 2*a*b^2*c^4 - 8*a^2*c^5)*d^4 - 2*
(b^5*c^2 - 2*a*b^3*c^3 - 8*a^2*b*c^4)*d^3*e + (b^6*c - 12*a^2*b^2*c^3 - 16*a^3*c^4)*d^2*e^2 - 2*(a*b^5*c - 2*a
^2*b^3*c^2 - 8*a^3*b*c^3)*d*e^3 + (a^2*b^4*c - 2*a^3*b^2*c^2 - 8*a^4*c^3)*e^4)*x^2 + 2*((a*b^3*c^3 - 4*a^2*b*c
^4)*d^4 - 2*(a*b^4*c^2 - 4*a^2*b^2*c^3)*d^3*e + (a*b^5*c - 2*a^2*b^3*c^2 - 8*a^3*b*c^3)*d^2*e^2 - 2*(a^2*b^4*c
- 4*a^3*b^2*c^2)*d*e^3 + (a^3*b^3*c - 4*a^4*b*c^2)*e^4)*x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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