\(\int \frac {(d+e x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx\) [2461]

Optimal result

Integrand size = 24, antiderivative size = 600 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 e \left (71 c^2 d^2+24 b^2 e^2-c e (71 b d+25 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{105 c^3}+\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}+\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^4 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (71 c^2 d^2+24 b^2 e^2-c e (71 b d+25 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]

[Out]

12/35*e*(-b*e+2*c*d)*(e*x+d)^(3/2)*(c*x^2+b*x+a)^(1/2)/c^2+2/7*e*(e*x+d)^(5/2)*(c*x^2+b*x+a)^(1/2)/c+2/105*e*(
71*c^2*d^2+24*b^2*e^2-c*e*(25*a*e+71*b*d))*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^3+8/105*(-b*e+2*c*d)*(11*c^2*d^
2+6*b^2*e^2-c*e*(13*a*e+11*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)/c^4/(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)
-2/105*(a*e^2-b*d*e+c*d^2)*(71*c^2*d^2+24*b^2*e^2-c*e*(25*a*e+71*b*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)
/c^4/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 600, 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 = {756, 846, 857, 732, 435, 430} \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx=-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-c e (25 a e+71 b d)+24 b^2 e^2+71 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {8 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (13 a e+11 b d)+6 b^2 e^2+11 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^4 \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 e \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (25 a e+71 b d)+24 b^2 e^2+71 c^2 d^2\right )}{105 c^3}+\frac {12 e (d+e x)^{3/2} \sqrt {a+b x+c x^2} (2 c d-b e)}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c} \]

[In]

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

[Out]

(2*e*(71*c^2*d^2 + 24*b^2*e^2 - c*e*(71*b*d + 25*a*e))*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])/(105*c^3) + (12*e*
(2*c*d - b*e)*(d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(35*c^2) + (2*e*(d + e*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(7
*c) + (8*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(11*c^2*d^2 + 6*b^2*e^2 - c*e*(11*b*d + 13*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)])/(105*c^4*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]*(c*d^2 - b*d*e + a
*e^2)*(71*c^2*d^2 + 24*b^2*e^2 - c*e*(71*b*d + 25*a*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)])/(105*c^4*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 756

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}+\frac {2 \int \frac {(d+e x)^{3/2} \left (\frac {1}{2} \left (7 c d^2-e (b d+5 a e)\right )+3 e (2 c d-b e) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{7 c} \\ & = \frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}+\frac {4 \int \frac {\sqrt {d+e x} \left (\frac {1}{4} \left (35 c^2 d^3+6 b e^2 (b d+3 a e)-c d e (17 b d+61 a e)\right )+\frac {1}{4} e \left (71 c^2 d^2+24 b^2 e^2-c e (71 b d+25 a e)\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{35 c^2} \\ & = \frac {2 e \left (71 c^2 d^2+24 b^2 e^2-c e (71 b d+25 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{105 c^3}+\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}+\frac {8 \int \frac {\frac {1}{8} \left (105 c^3 d^4-24 b^2 e^3 (b d+a e)-2 c^2 d^2 e (61 b d+127 a e)+c e^2 \left (89 b^2 d^2+150 a b d e+25 a^2 e^2\right )\right )+e (2 c d-b e) \left (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{105 c^3} \\ & = \frac {2 e \left (71 c^2 d^2+24 b^2 e^2-c e (71 b d+25 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{105 c^3}+\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}+\frac {\left (8 (2 c d-b e) \left (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{105 c^3}+\frac {\left (8 \left (-d e (2 c d-b e) \left (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 a e)\right )+\frac {1}{8} e \left (105 c^3 d^4-24 b^2 e^3 (b d+a e)-2 c^2 d^2 e (61 b d+127 a e)+c e^2 \left (89 b^2 d^2+150 a b d e+25 a^2 e^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{105 c^3 e} \\ & = \frac {2 e \left (71 c^2 d^2+24 b^2 e^2-c e (71 b d+25 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{105 c^3}+\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}+\frac {\left (8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 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 )}{105 c^4 \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} \sqrt {b^2-4 a c} \left (-d e (2 c d-b e) \left (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 a e)\right )+\frac {1}{8} e \left (105 c^3 d^4-24 b^2 e^3 (b d+a e)-2 c^2 d^2 e (61 b d+127 a e)+c e^2 \left (89 b^2 d^2+150 a b d e+25 a^2 e^2\right )\right )\right ) \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 )}{105 c^4 e \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ & = \frac {2 e \left (71 c^2 d^2+24 b^2 e^2-c e (71 b d+25 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{105 c^3}+\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}+\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 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 )}{105 c^4 \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} \left (c d^2-b d e+a e^2\right ) \left (71 c^2 d^2-71 b c d e+24 b^2 e^2-25 a c e^2\right ) \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 )}{105 c^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 34.18 (sec) , antiderivative size = 1318, normalized size of antiderivative = 2.20 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {d+e x} \left (a+b x+c x^2\right ) \left (-\frac {2 e \left (-122 c^2 d^2+89 b c d e-24 b^2 e^2+25 a c e^2\right )}{105 c^3}-\frac {4 e^2 (-11 c d+3 b e) x}{35 c^2}+\frac {2 e^3 x^2}{7 c}\right )}{\sqrt {a+x (b+c x)}}+\frac {(d+e x)^{3/2} \sqrt {a+b x+c x^2} \left (-16 (-2 c d+b e) \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 (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 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 {4 i \sqrt {2} (-2 c d+b e) \left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 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 (-105 c^4 d^4+24 b^3 e^3 \left (-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right )-2 c^3 d^2 \left (-105 b d e-127 a e^2+44 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+4 b c e^2 \left (29 b^2 d e+19 a b e^2-23 b d \sqrt {\left (b^2-4 a c\right ) e^2}-13 a e \sqrt {\left (b^2-4 a c\right ) e^2}\right )+c^2 e \left (-221 b^2 d^2 e+2 b d \left (-127 a e^2+66 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+a e \left (-25 a e^2+104 d \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 )}{105 c^4 e \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+x (b+c x)} \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[(d + e*x)^(7/2)/Sqrt[a + b*x + c*x^2],x]

[Out]

(Sqrt[d + e*x]*(a + b*x + c*x^2)*((-2*e*(-122*c^2*d^2 + 89*b*c*d*e - 24*b^2*e^2 + 25*a*c*e^2))/(105*c^3) - (4*
e^2*(-11*c*d + 3*b*e)*x)/(35*c^2) + (2*e^3*x^2)/(7*c)))/Sqrt[a + x*(b + c*x)] + ((d + e*x)^(3/2)*Sqrt[a + b*x
+ c*x^2]*(-16*(-2*c*d + b*e)*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(11*c^2
*d^2 + 6*b^2*e^2 - c*e*(11*b*d + 13*a*e))*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x))
)/(d + e*x)) + ((4*I)*Sqrt[2]*(-2*c*d + b*e)*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(11*c^2*d^2 + 6*b^2*e^2 -
c*e*(11*b*d + 13*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])]*Ellipt
icE[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]*(-105*c^4*d^4 + 24*b^3*e^3*(-(b*e) + Sqrt[(b^2 - 4*a*c)*e^2]) - 2*c^3*d^2*(-105*b*d*e - 127*a*e^2 + 44*d*S
qrt[(b^2 - 4*a*c)*e^2]) + 4*b*c*e^2*(29*b^2*d*e + 19*a*b*e^2 - 23*b*d*Sqrt[(b^2 - 4*a*c)*e^2] - 13*a*e*Sqrt[(b
^2 - 4*a*c)*e^2]) + c^2*e*(-221*b^2*d^2*e + 2*b*d*(-127*a*e^2 + 66*d*Sqrt[(b^2 - 4*a*c)*e^2]) + a*e*(-25*a*e^2
+ 104*d*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])])/Sqr
t[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]))/(105*c^4*e*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[a + x*(b + c*x)
]*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. \(1220\) vs. \(2(530)=1060\).

Time = 1.84 (sec) , antiderivative size = 1221, normalized size of antiderivative = 2.04

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1221\)
risch	\(\text {Expression too large to display}\)	\(2932\)
default	\(\text {Expression too large to display}\)	\(6947\)

[In]

int((e*x+d)^(7/2)/(c*x^2+b*x+a)^(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)*(2/7*e^3/c*x^2*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+
b*d*x+a*d)^(1/2)+2/5*(4*d*e^3-2/7*e^3/c*(3*b*e+3*c*d))/c/e*x*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2
/3*(6*d^2*e^2-2/7*e^3/c*(5/2*a*e+5/2*b*d)-2/5*(4*d*e^3-2/7*e^3/c*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))/c/e*(c*e*x^
3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2*(d^4-2/5*(4*d*e^3-2/7*e^3/c*(3*b*e+3*c*d))/c/e*a*d-2/3*(6*d^2*e^2-2
/7*e^3/c*(5/2*a*e+5/2*b*d)-2/5*(4*d*e^3-2/7*e^3/c*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))/c/e*(1/2*a*e+1/2*b*d))*(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))+2*(4*d^3*e
-4/7*e^3/c*a*d-2/5*(4*d*e^3-2/7*e^3/c*(3*b*e+3*c*d))/c/e*(3/2*a*e+3/2*b*d)-2/3*(6*d^2*e^2-2/7*e^3/c*(5/2*a*e+5
/2*b*d)-2/5*(4*d*e^3-2/7*e^3/c*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))/c/e*(b*e+c*d))*(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 = 0.11 (sec) , antiderivative size = 589, normalized size of antiderivative = 0.98 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 \, {\left ({\left (139 \, c^{4} d^{4} - 278 \, b c^{3} d^{3} e + {\left (347 \, b^{2} c^{2} - 554 \, a c^{3}\right )} d^{2} e^{2} - 2 \, {\left (104 \, b^{3} c - 277 \, a b c^{2}\right )} d e^{3} + {\left (48 \, b^{4} - 176 \, a b^{2} c + 75 \, a^{2} c^{2}\right )} e^{4}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) - 24 \, {\left (22 \, c^{4} d^{3} e - 33 \, b c^{3} d^{2} e^{2} + {\left (23 \, b^{2} c^{2} - 26 \, a c^{3}\right )} d e^{3} - {\left (6 \, b^{3} c - 13 \, a b c^{2}\right )} e^{4}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (15 \, c^{4} e^{4} x^{2} + 122 \, c^{4} d^{2} e^{2} - 89 \, b c^{3} d e^{3} + {\left (24 \, b^{2} c^{2} - 25 \, a c^{3}\right )} e^{4} + 6 \, {\left (11 \, c^{4} d e^{3} - 3 \, b c^{3} e^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}\right )}}{315 \, c^{5} e} \]

[In]

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

[Out]

2/315*((139*c^4*d^4 - 278*b*c^3*d^3*e + (347*b^2*c^2 - 554*a*c^3)*d^2*e^2 - 2*(104*b^3*c - 277*a*b*c^2)*d*e^3
+ (48*b^4 - 176*a*b^2*c + 75*a^2*c^2)*e^4)*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)) - 24*(22*c^4*d^3*e - 33*b*c^3*d^2*e^2 + (23*b^2*c^2 - 26*a*c^3)*d*e^3 -
(6*b^3*c - 13*a*b*c^2)*e^4)*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), weierstrassPIn
verse(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*(15*c^4*e^4*x^2 + 122*c^
4*d^2*e^2 - 89*b*c^3*d*e^3 + (24*b^2*c^2 - 25*a*c^3)*e^4 + 6*(11*c^4*d*e^3 - 3*b*c^3*e^4)*x)*sqrt(c*x^2 + b*x
+ a)*sqrt(e*x + d))/(c^5*e)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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