\(\int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx\) [2446]

Optimal result

Integrand size = 24, antiderivative size = 617 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx=-\frac {2 \sqrt {a+b x+c x^2}}{5 e (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac {4 \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 e^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 e^2 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 617, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {746, 848, 857, 732, 435, 430} \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx=-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 e^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 e^2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac {4 \sqrt {a+b x+c x^2} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{15 e \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )^2}+\frac {2 \sqrt {a+b x+c x^2} (2 c d-b e)}{15 e (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2}}{5 e (d+e x)^{5/2}} \]

[In]

Int[Sqrt[a + b*x + c*x^2]/(d + e*x)^(7/2),x]

[Out]

(-2*Sqrt[a + b*x + c*x^2])/(5*e*(d + e*x)^(5/2)) + (2*(2*c*d - b*e)*Sqrt[a + b*x + c*x^2])/(15*e*(c*d^2 - b*d*
e + a*e^2)*(d + e*x)^(3/2)) + (4*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*Sqrt[a + b*x + c*x^2])/(15*e*(c*d^2 -
b*d*e + a*e^2)^2*Sqrt[d + e*x]) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(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)/Sqr
t[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)])/(15*e^2*(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]*Sqrt[
b^2 - 4*a*c]*(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)])/(15*e^2*(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 746

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 + 1))), x] - Dist[p/(e*(m + 1)), Int[(d + e*x)^(m + 1)*(b + 2*c*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] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQua
draticQ[a, b, c, d, e, m, p, x]

Rule 848

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*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] && LtQ[m, -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 {a+b x+c x^2}}{5 e (d+e x)^{5/2}}+\frac {\int \frac {b+2 c x}{(d+e x)^{5/2} \sqrt {a+b x+c x^2}} \, dx}{5 e} \\ & = -\frac {2 \sqrt {a+b x+c x^2}}{5 e (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (-b c d+2 b^2 e-6 a c e\right )-\frac {1}{2} c (2 c d-b e) x}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx}{15 e \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {2 \sqrt {a+b x+c x^2}}{5 e (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac {4 \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}+\frac {4 \int \frac {-\frac {1}{4} c \left (b c d^2+b^2 d e-8 a c d e+a b e^2\right )-\frac {1}{2} c \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}{15 e \left (c d^2-b d e+a e^2\right )^2} \\ & = -\frac {2 \sqrt {a+b x+c x^2}}{5 e (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac {4 \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}+\frac {(c (2 c d-b e)) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 e^2 \left (c d^2-b d e+a e^2\right )}-\frac {\left (2 c \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}{15 e^2 \left (c d^2-b d e+a e^2\right )^2} \\ & = -\frac {2 \sqrt {a+b x+c x^2}}{5 e (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac {4 \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} \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 )}{15 e^2 \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} \sqrt {b^2-4 a c} (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 )}{15 e^2 \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 {a+b x+c x^2}}{5 e (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac {4 \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \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 )}{15 e^2 \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} \sqrt {b^2-4 a c} (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 )}{15 e^2 \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 = 32.54 (sec) , antiderivative size = 1252, normalized size of antiderivative = 2.03 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx=\sqrt {d+e x} \sqrt {a+x (b+c x)} \left (-\frac {2}{5 e (d+e x)^3}-\frac {2 (-2 c d+b e)}{15 e \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {4 \left (-c^2 d^2+b c d e-b^2 e^2+3 a c e^2\right )}{15 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)}\right )-\frac {(d+e x)^{3/2} \sqrt {a+x (b+c x)} \left (4 \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}}} \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 )}{15 e^3 \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}}} \sqrt {a+b x+c x^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[a + b*x + c*x^2]/(d + e*x)^(7/2),x]

[Out]

Sqrt[d + e*x]*Sqrt[a + x*(b + c*x)]*(-2/(5*e*(d + e*x)^3) - (2*(-2*c*d + b*e))/(15*e*(c*d^2 - b*d*e + a*e^2)*(
d + e*x)^2) - (4*(-(c^2*d^2) + b*c*d*e - b^2*e^2 + 3*a*c*e^2))/(15*e*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x))) - (
(d + e*x)^(3/2)*Sqrt[a + x*(b + c*x)]*(4*Sqrt[(c*d^2 - b*d*e + a*e^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))*(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 + Sqrt[(b^2 - 4*a*c)*e^2]))]
)/Sqrt[d + e*x]))/(15*e^3*(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])]*Sqrt[a + b*x + c*x^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. \(1148\) vs. \(2(547)=1094\).

Time = 0.90 (sec) , antiderivative size = 1149, normalized size of antiderivative = 1.86

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1149\)
default	\(\text {Expression too large to display}\)	\(12980\)

[In]

int((c*x^2+b*x+a)^(1/2)/(e*x+d)^(7/2),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)*(-2/5/e^4*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x
+a*d)^(1/2)/(x+d/e)^3-2/15*(b*e-2*c*d)/e^3/(a*e^2-b*d*e+c*d^2)*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)
/(x+d/e)^2-4/15*(c*e*x^2+b*e*x+a*e)/(a*e^2-b*d*e+c*d^2)^2/e^2*(3*a*c*e^2-b^2*e^2+b*c*d*e-c^2*d^2)/((x+d/e)*(c*
e*x^2+b*e*x+a*e))^(1/2)+2*(-1/15*c*(b*e-2*c*d)/e^2/(a*e^2-b*d*e+c*d^2)-2/15/e^2*(b*e-c*d)*(3*a*c*e^2-b^2*e^2+b
*c*d*e-c^2*d^2)/(a*e^2-b*d*e+c*d^2)^2+2/15*b/e/(a*e^2-b*d*e+c*d^2)^2*(3*a*c*e^2-b^2*e^2+b*c*d*e-c^2*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))+4/15*c/e*(3
*a*c*e^2-b^2*e^2+b*c*d*e-c^2*d^2)/(a*e^2-b*d*e+c*d^2)^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 = 0.15 (sec) , antiderivative size = 1188, normalized size of antiderivative = 1.93 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

2/45*((2*c^3*d^6 - 3*b*c^2*d^5*e - 3*(b^2*c - 6*a*c^2)*d^4*e^2 + (2*b^3 - 9*a*b*c)*d^3*e^3 + (2*c^3*d^3*e^3 -
3*b*c^2*d^2*e^4 - 3*(b^2*c - 6*a*c^2)*d*e^5 + (2*b^3 - 9*a*b*c)*e^6)*x^3 + 3*(2*c^3*d^4*e^2 - 3*b*c^2*d^3*e^3
- 3*(b^2*c - 6*a*c^2)*d^2*e^4 + (2*b^3 - 9*a*b*c)*d*e^5)*x^2 + 3*(2*c^3*d^5*e - 3*b*c^2*d^4*e^2 - 3*(b^2*c - 6
*a*c^2)*d^3*e^3 + (2*b^3 - 9*a*b*c)*d^2*e^4)*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*(c^3*d^5*e - b*c^2*d^4*e^2 + (b^2*c - 3*a*c^2)*d^3*e^3 + (c^3*d
^2*e^4 - b*c^2*d*e^5 + (b^2*c - 3*a*c^2)*e^6)*x^3 + 3*(c^3*d^3*e^3 - b*c^2*d^2*e^4 + (b^2*c - 3*a*c^2)*d*e^5)*
x^2 + 3*(c^3*d^4*e^2 - b*c^2*d^3*e^3 + (b^2*c - 3*a*c^2)*d^2*e^4)*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*(c^3*d^4*e^2 + b*c^2*d^3*e^3 - 10*a*c^2*d^2*e^4 + 5*a*b*c*d*e^5 - 3*a^2*c*e^6 + 2*(c^3*d^2*e^
4 - b*c^2*d*e^5 + (b^2*c - 3*a*c^2)*e^6)*x^2 + (6*c^3*d^3*e^3 - 7*b*c^2*d^2*e^4 - a*b*c*e^6 + 5*(b^2*c - 2*a*c
^2)*d*e^5)*x)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d))/(c^3*d^7*e^3 - 2*b*c^2*d^6*e^4 - 2*a*b*c*d^4*e^6 + a^2*c*d^
3*e^7 + (b^2*c + 2*a*c^2)*d^5*e^5 + (c^3*d^4*e^6 - 2*b*c^2*d^3*e^7 - 2*a*b*c*d*e^9 + a^2*c*e^10 + (b^2*c + 2*a
*c^2)*d^2*e^8)*x^3 + 3*(c^3*d^5*e^5 - 2*b*c^2*d^4*e^6 - 2*a*b*c*d^2*e^8 + a^2*c*d*e^9 + (b^2*c + 2*a*c^2)*d^3*
e^7)*x^2 + 3*(c^3*d^6*e^4 - 2*b*c^2*d^5*e^5 - 2*a*b*c*d^3*e^7 + a^2*c*d^2*e^8 + (b^2*c + 2*a*c^2)*d^4*e^6)*x)

Sympy [F]

\[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx=\int \frac {\sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \]

[In]

integrate((c*x**2+b*x+a)**(1/2)/(e*x+d)**(7/2),x)

[Out]

Integral(sqrt(a + b*x + c*x**2)/(d + e*x)**(7/2), x)

Maxima [F]

\[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + b*x + a)/(e*x + d)^(7/2), x)

Giac [F]

\[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

integrate(sqrt(c*x^2 + b*x + a)/(e*x + d)^(7/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx=\int \frac {\sqrt {c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^{7/2}} \,d x \]

[In]

int((a + b*x + c*x^2)^(1/2)/(d + e*x)^(7/2),x)

[Out]

int((a + b*x + c*x^2)^(1/2)/(d + e*x)^(7/2), x)