3.643 \(\int \frac {(f+g x)^{5/2}}{(d+e x) \sqrt {a+c x^2}} \, dx\)

Optimal. Leaf size=600 \[ \frac {2 \sqrt {-a} g \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \left (a e^2 g^2+c \left (-3 d^2 g^2+6 d e f g-2 e^2 f^2\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {a \sqrt {c} x}{(-a)^{3/2}}+1}}{\sqrt {2}}\right )|\frac {2 a g}{a g-\sqrt {-a} \sqrt {c} f}\right )}{3 c^{3/2} e^3 \sqrt {a+c x^2} \sqrt {f+g x}}-\frac {2 (e f-d g)^2 \sqrt {\frac {g \left (\sqrt {-a}-\sqrt {c} x\right )}{\sqrt {-a} g+\sqrt {c} f}} \sqrt {-\frac {g \left (\sqrt {-a}+\sqrt {c} x\right )}{\sqrt {c} f-\sqrt {-a} g}} \Pi \left (\frac {e \left (f+\frac {\sqrt {-a} g}{\sqrt {c}}\right )}{e f-d g};\sin ^{-1}\left (\sqrt {\frac {c}{c f+\sqrt {-a} \sqrt {c} g}} \sqrt {f+g x}\right )|\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{e^3 \sqrt {a+c x^2} \sqrt {\frac {c}{\sqrt {-a} \sqrt {c} g+c f}}}-\frac {2 \sqrt {-a} g \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} (7 e f-3 d g) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {a \sqrt {c} x}{(-a)^{3/2}}+1}}{\sqrt {2}}\right )|\frac {2 a g}{a g-\sqrt {-a} \sqrt {c} f}\right )}{3 \sqrt {c} e^2 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}}+\frac {2 g^2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 c e} \]

[Out]

2/3*g^2*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/c/e-2/3*g*(-3*d*g+7*e*f)*EllipticE(1/2*(1+a*x*c^(1/2)/(-a)^(3/2))^(1/2)*
2^(1/2),2^(1/2)*(a*g/(a*g-f*(-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*(g*x+f)^(1/2)*(c*x^2/a+1)^(1/2)/e^2/c^(1/2)
/(c*x^2+a)^(1/2)/((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)+2/3*g*(a*e^2*g^2+c*(-3*d^2*g^2+6*d*e*f*g-2*e
^2*f^2))*EllipticF(1/2*(1+a*x*c^(1/2)/(-a)^(3/2))^(1/2)*2^(1/2),2^(1/2)*(a*g/(a*g-f*(-a)^(1/2)*c^(1/2)))^(1/2)
)*(-a)^(1/2)*(c*x^2/a+1)^(1/2)*((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)/c^(3/2)/e^3/(g*x+f)^(1/2)/(c*x
^2+a)^(1/2)-2*(-d*g+e*f)^2*EllipticPi((g*x+f)^(1/2)*(c/(c*f+g*(-a)^(1/2)*c^(1/2)))^(1/2),e*(f+g*(-a)^(1/2)/c^(
1/2))/(-d*g+e*f),((g*(-a)^(1/2)+f*c^(1/2))/(-g*(-a)^(1/2)+f*c^(1/2)))^(1/2))*(g*((-a)^(1/2)-x*c^(1/2))/(g*(-a)
^(1/2)+f*c^(1/2)))^(1/2)*(-g*((-a)^(1/2)+x*c^(1/2))/(-g*(-a)^(1/2)+f*c^(1/2)))^(1/2)/e^3/(c*x^2+a)^(1/2)/(c/(c
*f+g*(-a)^(1/2)*c^(1/2)))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.95, antiderivative size = 808, normalized size of antiderivative = 1.35, number of steps used = 16, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {958, 719, 419, 933, 168, 538, 537, 424, 743, 844} \[ -\frac {2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} \Pi \left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e};\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right ) (e f-d g)^3}{e^3 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \sqrt {f+g x} \sqrt {c x^2+a}}-\frac {2 \sqrt {-a} g \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right ) (e f-d g)^2}{\sqrt {c} e^3 \sqrt {f+g x} \sqrt {c x^2+a}}-\frac {2 \sqrt {-a} g \sqrt {f+g x} \sqrt {\frac {c x^2}{a}+1} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right ) (e f-d g)}{\sqrt {c} e^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {c x^2+a}}-\frac {8 \sqrt {-a} f g \sqrt {f+g x} \sqrt {\frac {c x^2}{a}+1} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 \sqrt {c} e \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {c x^2+a}}+\frac {2 \sqrt {-a} g \left (c f^2+a g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 c^{3/2} e \sqrt {f+g x} \sqrt {c x^2+a}}+\frac {2 g^2 \sqrt {f+g x} \sqrt {c x^2+a}}{3 c e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*g^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3*c*e) - (8*Sqrt[-a]*f*g*Sqrt[f + g*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[Ar
cSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(3*Sqrt[c]*e*Sqrt[(Sqrt[c]
*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[a + c*x^2]) - (2*Sqrt[-a]*g*(e*f - d*g)*Sqrt[f + g*x]*Sqrt[1 + (c*x
^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(Sqrt[c
]*e^2*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[a + c*x^2]) - (2*Sqrt[-a]*g*(e*f - d*g)^2*Sqrt[(
Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a
]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(Sqrt[c]*e^3*Sqrt[f + g*x]*Sqrt[a + c*x^2]) + (2*Sqrt[-a]*g
*(c*f^2 + a*g^2)*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[
1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(3*c^(3/2)*e*Sqrt[f + g*x]*Sqrt[a +
c*x^2]) - (2*(e*f - d*g)^3*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*EllipticPi[(
2*e)/((Sqrt[c]*d)/Sqrt[-a] + e), ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (2*Sqrt[-a]*g)/(Sqrt[c]*f + S
qrt[-a]*g)])/(e^3*((Sqrt[c]*d)/Sqrt[-a] + e)*Sqrt[f + g*x]*Sqrt[a + c*x^2])

Rule 168

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c
*f)/d, 0]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rule 538

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d*x^2)/c]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rule 719

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(2*a*Rt[-(c/a), 2]*(d + e*x)^m*Sqrt[
1 + (c*x^2)/a])/(c*Sqrt[a + c*x^2]*((c*(d + e*x))/(c*d - a*e*Rt[-(c/a), 2]))^m), Subst[Int[(1 + (2*a*e*Rt[-(c/
a), 2]*x^2)/(c*d - a*e*Rt[-(c/a), 2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(1 - Rt[-(c/a), 2]*x)/2]], x] /; FreeQ[{a,
c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m^2, 1/4]

Rule 743

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + 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) - a*e^
2*(m - 1) + 2*c*d*e*(m + p)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2,
0] && If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m
, p, x]

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 933

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-(c
/a), 2]}, Dist[Sqrt[1 + (c*x^2)/a]/Sqrt[a + c*x^2], Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]
), x], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] &&  !GtQ[a, 0]

Rule 958

Int[((f_.) + (g_.)*(x_))^(n_)/(((d_.) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Int[ExpandIntegra
nd[1/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), (f + g*x)^(n + 1/2)/(d + e*x), x], x] /; FreeQ[{a, c, d, e, f, g}, x] &&
NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[n + 1/2]

Rubi steps

\begin {align*} \int \frac {(f+g x)^{5/2}}{(d+e x) \sqrt {a+c x^2}} \, dx &=\int \left (\frac {g (e f-d g)^2}{e^3 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {(e f-d g)^3}{e^3 (d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {g (e f-d g) \sqrt {f+g x}}{e^2 \sqrt {a+c x^2}}+\frac {g (f+g x)^{3/2}}{e \sqrt {a+c x^2}}\right ) \, dx\\ &=\frac {g \int \frac {(f+g x)^{3/2}}{\sqrt {a+c x^2}} \, dx}{e}+\frac {(g (e f-d g)) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{e^2}+\frac {\left (g (e f-d g)^2\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{e^3}+\frac {(e f-d g)^3 \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{e^3}\\ &=\frac {2 g^2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 c e}+\frac {(2 g) \int \frac {\frac {1}{2} \left (3 c f^2-a g^2\right )+2 c f g x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{3 c e}+\frac {\left ((e f-d g)^3 \sqrt {1+\frac {c x^2}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}} \sqrt {1+\frac {\sqrt {c} x}{\sqrt {-a}}} (d+e x) \sqrt {f+g x}} \, dx}{e^3 \sqrt {a+c x^2}}+\frac {\left (2 a g (e f-d g) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{\sqrt {-a} \sqrt {c} e^2 \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (2 a g (e f-d g)^2 \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{\sqrt {-a} \sqrt {c} e^3 \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=\frac {2 g^2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 c e}-\frac {2 \sqrt {-a} g (e f-d g) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} g (e f-d g)^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^3 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {(4 f g) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{3 e}-\frac {\left (g \left (c f^2+a g^2\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{3 c e}-\frac {\left (2 (e f-d g)^3 \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e-e x^2\right ) \sqrt {f+\frac {\sqrt {-a} g}{\sqrt {c}}-\frac {\sqrt {-a} g x^2}{\sqrt {c}}}} \, dx,x,\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}\right )}{e^3 \sqrt {a+c x^2}}\\ &=\frac {2 g^2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 c e}-\frac {2 \sqrt {-a} g (e f-d g) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} g (e f-d g)^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^3 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\left (2 (e f-d g)^3 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e-e x^2\right ) \sqrt {1-\frac {\sqrt {-a} g x^2}{\sqrt {c} \left (f+\frac {\sqrt {-a} g}{\sqrt {c}}\right )}}} \, dx,x,\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}\right )}{e^3 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\left (8 a f g \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{3 \sqrt {-a} \sqrt {c} e \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}-\frac {\left (2 a g \left (c f^2+a g^2\right ) \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{3 \sqrt {-a} c^{3/2} e \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=\frac {2 g^2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 c e}-\frac {8 \sqrt {-a} f g \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 \sqrt {c} e \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} g (e f-d g) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} g (e f-d g)^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^3 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {2 \sqrt {-a} g \left (c f^2+a g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 c^{3/2} e \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {2 (e f-d g)^3 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \Pi \left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e};\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{e^3 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \sqrt {f+g x} \sqrt {a+c x^2}}\\ \end {align*}

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Mathematica [C]  time = 9.58, size = 1440, normalized size = 2.40 \[ \frac {2 \sqrt {f+g x} \sqrt {c x^2+a} g^2}{3 c e}+\frac {2 (f+g x)^{3/2} \left (\frac {7 c e^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} f^3}{(f+g x)^2}+\frac {3 i c e^2 \sqrt {-\frac {f}{f+g x}-\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}+1} \sqrt {-\frac {f}{f+g x}+\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}+1} \Pi \left (\frac {\sqrt {c} (e f-d g)}{e \left (\sqrt {c} f+i \sqrt {a} g\right )};i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right ) f^2}{\sqrt {f+g x}}-\frac {14 c e^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} f^2}{f+g x}-\frac {3 c d e g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} f^2}{(f+g x)^2}-\frac {6 i c d e g \sqrt {-\frac {f}{f+g x}-\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}+1} \sqrt {-\frac {f}{f+g x}+\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}+1} \Pi \left (\frac {\sqrt {c} (e f-d g)}{e \left (\sqrt {c} f+i \sqrt {a} g\right )};i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right ) f}{\sqrt {f+g x}}+7 c e^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} f+\frac {6 c d e g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} f}{f+g x}+\frac {7 a e^2 g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} f}{(f+g x)^2}+\frac {\sqrt {c} e \left (\sqrt {a} g-i \sqrt {c} f\right ) (7 e f-3 d g) \sqrt {-\frac {f}{f+g x}-\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}+1} \sqrt {-\frac {f}{f+g x}+\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}+1} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}+\frac {i e \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (i \sqrt {a} e g+\sqrt {c} (6 e f-3 d g)\right ) \sqrt {-\frac {f}{f+g x}-\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}+1} \sqrt {-\frac {f}{f+g x}+\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}+1} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}+\frac {3 i c d^2 g^2 \sqrt {-\frac {f}{f+g x}-\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}+1} \sqrt {-\frac {f}{f+g x}+\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}+1} \Pi \left (\frac {\sqrt {c} (e f-d g)}{e \left (\sqrt {c} f+i \sqrt {a} g\right )};i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}-3 c d e g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}-\frac {3 a d e g^3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{(f+g x)^2}\right )}{3 c e^3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \sqrt {\frac {c (f+g x)^2 \left (\frac {f}{f+g x}-1\right )^2}{g^2}+a}} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)^(5/2)/((d + e*x)*Sqrt[a + c*x^2]),x]

[Out]

(2*g^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3*c*e) + (2*(f + g*x)^(3/2)*(7*c*e^2*f*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]
- 3*c*d*e*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + (7*c*e^2*f^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (
3*c*d*e*f^2*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 + (7*a*e^2*f*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])
/(f + g*x)^2 - (3*a*d*e*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (14*c*e^2*f^2*Sqrt[-f - (I*Sqrt[a]
*g)/Sqrt[c]])/(f + g*x) + (6*c*d*e*f*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x) + (Sqrt[c]*e*((-I)*Sqrt[c]*
f + Sqrt[a]*g)*(7*e*f - 3*d*g)*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x)
+ (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqr
t[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + (I*e*(Sqrt[c]*f + I*Sqrt[a]*g)*(I*Sqrt[a]*e*
g + Sqrt[c]*(6*e*f - 3*d*g))*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) +
(I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[
c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + ((3*I)*c*e^2*f^2*Sqrt[1 - f/(f + g*x) - (I*Sqr
t[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticPi[(Sqrt[c]*(e*
f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f
- I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] - ((6*I)*c*d*e*f*g*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]
*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticPi[(Sqrt[c]*(e*f -
d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I
*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + ((3*I)*c*d^2*g^2*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/
(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticPi[(Sqrt[c]*(e*f - d*g)
)/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqr
t[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x]))/(3*c*e^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*Sqrt[a + (c*(f +
g*x)^2*(-1 + f/(f + g*x))^2)/g^2])

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x + f\right )}^{\frac {5}{2}}}{\sqrt {c x^{2} + a} {\left (e x + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.05, size = 3164, normalized size = 5.27 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)*(3*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c
)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticF((-(g*x+f)/(-c*f+(-a*c)^(1/2)*
g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*a*c*d*e*g^3-6*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)
*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2
)*EllipticF((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*a*c*
e^2*f*g^2-(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-
a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticF((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(
1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*(-a*c)^(1/2)*a*e^2*g^3-3*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+
(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticF((-(g*
x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*c^2*d^2*f*g^2+9*(-(g*
x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c
*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticF((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-
a*c)^(1/2)*g))^(1/2))*c^2*d*e*f^2*g-9*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c
)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticF((-(g*x+f)/(-c*f+(-a*c)^(1/2)*
g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*c^2*e^2*f^3+3*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)
*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2
)*EllipticF((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*(-a*
c)^(1/2)*c*d^2*g^3-6*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/
2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticF((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c
*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*(-a*c)^(1/2)*c*d*e*f*g^2+2*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^
(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*El
lipticF((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*(-a*c)^(
1/2)*c*e^2*f^2*g-3*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)
*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticE((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f
+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*a*c*d*e*g^3+7*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-
a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticE((-(g*x+
f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*a*c*e^2*f*g^2-3*(-(g*x+
f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f
+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticE((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*
c)^(1/2)*g))^(1/2))*c^2*d*e*f^2*g+7*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^
(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticE((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)
*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*c^2*e^2*f^3+3*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c
)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*
EllipticPi((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-c*f+(-a*c)^(1/2)*g)/(d*g-e*f)/c*e,(-(-c*f+(-a*c)^(1/2)*g
)/(c*f+(-a*c)^(1/2)*g))^(1/2))*c^2*d^2*f*g^2-6*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(
c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticPi((-(g*x+f)/(-c*f+(-a
*c)^(1/2)*g)*c)^(1/2),(-c*f+(-a*c)^(1/2)*g)/(d*g-e*f)/c*e,(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))
*c^2*d*e*f^2*g+3*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*(
(c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticPi((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-c*f+(-
a*c)^(1/2)*g)/(d*g-e*f)/c*e,(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*c^2*e^2*f^3-3*(-(g*x+f)/(-c*f
+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^
(1/2)*g)*g)^(1/2)*EllipticPi((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-c*f+(-a*c)^(1/2)*g)/(d*g-e*f)/c*e,(-(-
c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*(-a*c)^(1/2)*c*d^2*g^3+6*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(
1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*Ell
ipticPi((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-c*f+(-a*c)^(1/2)*g)/(d*g-e*f)/c*e,(-(-c*f+(-a*c)^(1/2)*g)/(
c*f+(-a*c)^(1/2)*g))^(1/2))*(-a*c)^(1/2)*c*d*e*f*g^2-3*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^
(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticPi((-(g*x+f)/(
-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-c*f+(-a*c)^(1/2)*g)/(d*g-e*f)/c*e,(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g)
)^(1/2))*(-a*c)^(1/2)*c*e^2*f^2*g-c^2*e^2*g^3*x^3-c^2*e^2*f*g^2*x^2-a*c*e^2*g^3*x-a*c*e^2*f*g^2)/e^3/c^2/(c*g*
x^3+c*f*x^2+a*g*x+a*f)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x + f\right )}^{\frac {5}{2}}}{\sqrt {c x^{2} + a} {\left (e x + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (f+g\,x\right )}^{5/2}}{\sqrt {c\,x^2+a}\,\left (d+e\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (f + g x\right )^{\frac {5}{2}}}{\sqrt {a + c x^{2}} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((f + g*x)**(5/2)/(sqrt(a + c*x**2)*(d + e*x)), x)

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