\(\int \frac {(d+e x)^{7/2}}{(a+b x+c x^2)^{3/2}} \, dx\) [2469]

Optimal result

Integrand size = 24, antiderivative size = 641 \[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {4 e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac {2 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac {\sqrt {2} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt {b^2-4 a c} \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {4 \sqrt {2} \left (c d^2-b d e+a e^2\right ) \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]

[Out]

-2*(e*x+d)^(5/2)*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^(1/2)+2*e*(-b*e+2*c*d)*(e*x+d)^(3/2)*(c
*x^2+b*x+a)^(1/2)/c/(-4*a*c+b^2)+4/3*e*(3*c^2*d^2+2*b^2*e^2-c*e*(5*a*e+3*b*d))*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/
2)/c^2/(-4*a*c+b^2)+1/3*(-b*e+2*c*d)*(3*c^2*d^2+8*b^2*e^2-c*e*(29*a*e+3*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)/c^3/(-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)-4/3*(a*e^2-b*d*e+c*d^2)*(3*c^2*d^2+2*b^2*e^2-c*e*(5*a*e+3*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)*(-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^3/(-4*a*c+b^2)^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 641, 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 = {752, 846, 857, 732, 435, 430} \[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {4 \sqrt {2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {\sqrt {2} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt {b^2-4 a c} \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {4 e \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right )}{3 c^2 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 e (d+e x)^{3/2} \sqrt {a+b x+c x^2} (2 c d-b e)}{c \left (b^2-4 a c\right )} \]

[In]

Int[(d + e*x)^(7/2)/(a + b*x + c*x^2)^(3/2),x]

[Out]

(-2*(d + e*x)^(5/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) + (4*e*(3*c^2*d^2 +
2*b^2*e^2 - c*e*(3*b*d + 5*a*e))*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])/(3*c^2*(b^2 - 4*a*c)) + (2*e*(2*c*d - b
*e)*(d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(c*(b^2 - 4*a*c)) + (Sqrt[2]*(2*c*d - b*e)*(3*c^2*d^2 + 8*b^2*e^2 -
c*e*(3*b*d + 29*a*e))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + S
qrt[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*c^3*Sqrt[b^2 - 4*a*c]*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]
) - (4*Sqrt[2]*(c*d^2 - b*d*e + a*e^2)*(3*c^2*d^2 + 2*b^2*e^2 - c*e*(3*b*d + 5*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)]
)/(3*c^3*Sqrt[b^2 - 4*a*c]*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 752

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(d
*b - 2*a*e + (2*c*d - b*e)*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 - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, 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, 1] && 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 (d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 \int \frac {(d+e x)^{3/2} \left (-\frac {5}{2} e (b d-2 a e)-\frac {5}{2} e (2 c d-b e) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{b^2-4 a c} \\ & = -\frac {2 (d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}-\frac {4 \int \frac {\sqrt {d+e x} \left (-\frac {5}{4} e \left (b^2 d e-16 a c d e+3 b \left (c d^2+a e^2\right )\right )-\frac {5}{2} e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{5 c \left (b^2-4 a c\right )} \\ & = -\frac {2 (d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {4 e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac {2 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}-\frac {8 \int \frac {\frac {5}{8} e \left (4 b^3 d e^2+2 a c e \left (27 c d^2-5 a e^2\right )-b c d \left (3 c d^2+25 a e^2\right )-b^2 \left (9 c d^2 e-4 a e^3\right )\right )-\frac {5}{8} e (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 c^2 \left (b^2-4 a c\right )} \\ & = -\frac {2 (d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {4 e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac {2 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac {\left ((2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{3 c^2 \left (b^2-4 a c\right )}-\frac {\left (8 \left (\frac {5}{8} d e (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )+\frac {5}{8} e^2 \left (4 b^3 d e^2+2 a c e \left (27 c d^2-5 a e^2\right )-b c d \left (3 c d^2+25 a e^2\right )-b^2 \left (9 c d^2 e-4 a e^3\right )\right )\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 c^2 \left (b^2-4 a c\right ) e} \\ & = -\frac {2 (d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {4 e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac {2 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac {\left (\sqrt {2} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 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 c^3 \sqrt {b^2-4 a c} \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} \left (\frac {5}{8} d e (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )+\frac {5}{8} e^2 \left (4 b^3 d e^2+2 a c e \left (27 c d^2-5 a e^2\right )-b c d \left (3 c d^2+25 a e^2\right )-b^2 \left (9 c d^2 e-4 a e^3\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 )}{15 c^3 \sqrt {b^2-4 a c} e \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ & = -\frac {2 (d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {4 e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac {2 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac {\sqrt {2} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 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 c^3 \sqrt {b^2-4 a c} \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 {4 \sqrt {2} \left (c d^2-b d e+a e^2\right ) \left (3 c^2 d^2-3 b c d e+2 b^2 e^2-5 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 )}{3 c^3 \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 23.08 (sec) , antiderivative size = 1394, normalized size of antiderivative = 2.17 \[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {\sqrt {d+e x} \left (a+b x+c x^2\right )^2 \left (\frac {2 e^3}{3 c^2}+\frac {2 \left (b c^2 d^3-6 a c^2 d^2 e+3 a b c d e^2-a b^2 e^3+2 a^2 c e^3+2 c^3 d^3 x-3 b c^2 d^2 e x+3 b^2 c d e^2 x-6 a c^2 d e^2 x-b^3 e^3 x+3 a b c e^3 x\right )}{c^2 \left (-b^2+4 a c\right ) \left (a+b x+c x^2\right )}\right )}{(a+x (b+c x))^{3/2}}+\frac {(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \left (4 (-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 (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 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} (-2 c d+b e) \left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 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 (-8 b^4 e^4+b^3 e^3 \left (27 c d+8 \sqrt {\left (b^2-4 a c\right ) e^2}\right )-b^2 c e^2 \left (27 c d^2-37 a e^2+19 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+2 c^2 \left (-10 a^2 e^4-3 c d^3 \sqrt {\left (b^2-4 a c\right ) e^2}+a d e^2 \left (54 c d+29 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )+b c e \left (9 c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}-a e^2 \left (108 c d+29 \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 c^3 \left (-b^2+4 a c\right ) 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}}} (a+x (b+c x))^{3/2} \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[(d + e*x)^(7/2)/(a + b*x + c*x^2)^(3/2),x]

[Out]

(Sqrt[d + e*x]*(a + b*x + c*x^2)^2*((2*e^3)/(3*c^2) + (2*(b*c^2*d^3 - 6*a*c^2*d^2*e + 3*a*b*c*d*e^2 - a*b^2*e^
3 + 2*a^2*c*e^3 + 2*c^3*d^3*x - 3*b*c^2*d^2*e*x + 3*b^2*c*d*e^2*x - 6*a*c^2*d*e^2*x - b^3*e^3*x + 3*a*b*c*e^3*
x))/(c^2*(-b^2 + 4*a*c)*(a + b*x + c*x^2))))/(a + x*(b + c*x))^(3/2) + ((d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2
)*(4*(-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])]*(3*c^2*d^2 + 8*b
^2*e^2 - c*e*(3*b*d + 29*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)*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(3*c^2*d^2 + 8*b^2*e^2 - c*e*(3*b*d + 29
*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[(Sq
rt[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]*(-8*b^4*e^4 +
b^3*e^3*(27*c*d + 8*Sqrt[(b^2 - 4*a*c)*e^2]) - b^2*c*e^2*(27*c*d^2 - 37*a*e^2 + 19*d*Sqrt[(b^2 - 4*a*c)*e^2])
+ 2*c^2*(-10*a^2*e^4 - 3*c*d^3*Sqrt[(b^2 - 4*a*c)*e^2] + a*d*e^2*(54*c*d + 29*Sqrt[(b^2 - 4*a*c)*e^2])) + b*c
*e*(9*c*d^2*Sqrt[(b^2 - 4*a*c)*e^2] - a*e^2*(108*c*d + 29*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]))/(6*c^3*(-b^2 + 4*a*c)*e*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))^(3/2)*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. \(1363\) vs. \(2(575)=1150\).

Time = 6.82 (sec) , antiderivative size = 1364, normalized size of antiderivative = 2.13

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1364\)
risch	\(\text {Expression too large to display}\)	\(2855\)
default	\(\text {Expression too large to display}\)	\(6487\)

[In]

int((e*x+d)^(7/2)/(c*x^2+b*x+a)^(3/2),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*(c*e*x+c*d)*(-(3*a*b*c*e^3-6*a*c^2*d*e^2-b
^3*e^3+3*b^2*c*d*e^2-3*b*c^2*d^2*e+2*c^3*d^3)/c^3/(4*a*c-b^2)*x-(2*a^2*c*e^3-a*b^2*e^3+3*a*b*c*d*e^2-6*a*c^2*d
^2*e+b*c^2*d^3)/c^3/(4*a*c-b^2))/((a/c+b/c*x+x^2)*(c*e*x+c*d))^(1/2)+2/3/c^2*e^3*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*
x+b*d*x+a*d)^(1/2)+2*(-e^2*(a*c*e^2-b^2*e^2+4*b*c*d*e-6*c^2*d^2)/c^3+(4*a^2*c^2*e^4-5*a*b^2*c*e^4+18*a*b*c^2*d
*e^3-24*a*c^3*d^2*e^2+b^4*e^4-5*b^3*c*d*e^3+9*b^2*c^2*d^2*e^2-6*b*c^3*d^3*e+4*c^4*d^4)/(4*a*c-b^2)/c^3-1/c^2*e
*(2*a^2*c*e^3-a*b^2*e^3+3*a*b*c*d*e^2-6*a*c^2*d^2*e+b*c^2*d^3)/(4*a*c-b^2)-2/c^2*d*(3*a*b*c*e^3-6*a*c^2*d*e^2-
b^3*e^3+3*b^2*c*d*e^2-3*b*c^2*d^2*e+2*c^3*d^3)/(4*a*c-b^2)-2/3/c^2*e^3*(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*(-1/c^2*e^3*(b*e-4*c*d)-(
3*a*b*c*e^3-6*a*c^2*d*e^2-b^3*e^3+3*b^2*c*d*e^2-3*b*c^2*d^2*e+2*c^3*d^3)/c^2*e/(4*a*c-b^2)-2/3/c^2*e^3*(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 = 0.13 (sec) , antiderivative size = 1060, normalized size of antiderivative = 1.65 \[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

-2/9*((6*a*c^4*d^4 - 12*a*b*c^3*d^3*e - (17*a*b^2*c^2 - 104*a^2*c^3)*d^2*e^2 + (23*a*b^3*c - 104*a^2*b*c^2)*d*
e^3 - (8*a*b^4 - 41*a^2*b^2*c + 30*a^3*c^2)*e^4 + (6*c^5*d^4 - 12*b*c^4*d^3*e - (17*b^2*c^3 - 104*a*c^4)*d^2*e
^2 + (23*b^3*c^2 - 104*a*b*c^3)*d*e^3 - (8*b^4*c - 41*a*b^2*c^2 + 30*a^2*c^3)*e^4)*x^2 + (6*b*c^4*d^4 - 12*b^2
*c^3*d^3*e - (17*b^3*c^2 - 104*a*b*c^3)*d^2*e^2 + (23*b^4*c - 104*a*b^2*c^2)*d*e^3 - (8*b^5 - 41*a*b^3*c + 30*
a^2*b*c^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)) + 3*(6*a*c^4*d^3*e - 9*a*b*c^3*d^2*e^2 + (19*a*b^2*c^2 - 58*a^2*c^3)*d*e^3 - (8*a*b^3*c - 29*a^2
*b*c^2)*e^4 + (6*c^5*d^3*e - 9*b*c^4*d^2*e^2 + (19*b^2*c^3 - 58*a*c^4)*d*e^3 - (8*b^3*c^2 - 29*a*b*c^3)*e^4)*x
^2 + (6*b*c^4*d^3*e - 9*b^2*c^3*d^2*e^2 + (19*b^3*c^2 - 58*a*b*c^3)*d*e^3 - (8*b^4*c - 29*a*b^2*c^2)*e^4)*x)*s
qrt(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*(3*b*c^4*d^3*e - 18*a*c^4*d^2*e^2 + 9*a*b*c^3*d*e^3 -
(b^2*c^3 - 4*a*c^4)*e^4*x^2 - 2*(2*a*b^2*c^2 - 5*a^2*c^3)*e^4 + (6*c^5*d^3*e - 9*b*c^4*d^2*e^2 + 9*(b^2*c^3 -
2*a*c^4)*d*e^3 - (4*b^3*c^2 - 13*a*b*c^3)*e^4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d))/((b^2*c^5 - 4*a*c^6)*e
*x^2 + (b^3*c^4 - 4*a*b*c^5)*e*x + (a*b^2*c^4 - 4*a^2*c^5)*e)

Sympy [F]

\[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {7}{2}}}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((e*x+d)**(7/2)/(c*x**2+b*x+a)**(3/2),x)

[Out]

Integral((d + e*x)**(7/2)/(a + b*x + c*x**2)**(3/2), x)

Maxima [F]

\[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(7/2)/(c*x^2 + b*x + a)^(3/2), x)

Giac [F]

\[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

integrate((e*x + d)^(7/2)/(c*x^2 + b*x + a)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{7/2}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]

[In]

int((d + e*x)^(7/2)/(a + b*x + c*x^2)^(3/2),x)

[Out]

int((d + e*x)^(7/2)/(a + b*x + c*x^2)^(3/2), x)