\(\int \frac {(f+g x)^{3/2}}{(d+e x) \sqrt {a+c x^2}} \, dx\) [644]

Optimal result

Integrand size = 28, antiderivative size = 469 \[ \int \frac {(f+g x)^{3/2}}{(d+e x) \sqrt {a+c x^2}} \, dx=-\frac {2 \sqrt {-a} g \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} g (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^2 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {2 (e f-d g)^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticPi}\left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{e^2 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \sqrt {f+g x} \sqrt {a+c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {972, 733, 430, 947, 174, 552, 551, 435} \[ \int \frac {(f+g x)^{3/2}}{(d+e x) \sqrt {a+c x^2}} \, dx=-\frac {2 \sqrt {-a} g \sqrt {\frac {c x^2}{a}+1} (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^2 \sqrt {a+c x^2} \sqrt {f+g x}}-\frac {2 \sqrt {\frac {c x^2}{a}+1} (e f-d g)^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \operatorname {EllipticPi}\left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{e^2 \sqrt {a+c x^2} \sqrt {f+g x} \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right )}-\frac {2 \sqrt {-a} g \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}} \]

[In]

Int[(f + g*x)^(3/2)/((d + e*x)*Sqrt[a + c*x^2]),x]

[Out]

(-2*Sqrt[-a]*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*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[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*Ellipti
cF[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[f +
g*x]*Sqrt[a + c*x^2]) - (2*(e*f - d*g)^2*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 + Sqrt[-a]*g)])/(e^2*((Sqrt[c]*d)/Sqrt[-a] + e)*Sqrt[f + g*x]*Sqrt[a + c*x^2])

Rule 174

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 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 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], 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 552

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

Rule 733

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 947

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 972

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*} \text {integral}& = \int \left (\frac {g (e f-d g)}{e^2 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {(e f-d g)^2}{e^2 (d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {g \sqrt {f+g x}}{e \sqrt {a+c x^2}}\right ) \, dx \\ & = \frac {g \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{e}+\frac {(g (e f-d g)) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{e^2}+\frac {(e f-d g)^2 \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{e^2} \\ & = \frac {\left ((e f-d g)^2 \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^2 \sqrt {a+c x^2}}+\frac {\left (2 a g \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}}\right ) \text {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 \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) \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \text {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^2 \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = -\frac {2 \sqrt {-a} 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 \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 {\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^2 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\left (2 (e f-d g)^2 \sqrt {1+\frac {c x^2}{a}}\right ) \text {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^2 \sqrt {a+c x^2}} \\ & = -\frac {2 \sqrt {-a} 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 \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 {\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^2 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\left (2 (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}}\right ) \text {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^2 \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = -\frac {2 \sqrt {-a} 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 \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 {\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^2 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {2 (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}} \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^2 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \sqrt {f+g x} \sqrt {a+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.87 (sec) , antiderivative size = 927, normalized size of antiderivative = 1.98 \[ \int \frac {(f+g x)^{3/2}}{(d+e x) \sqrt {a+c x^2}} \, dx=\frac {2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-i \sqrt {a} g}} \left (\frac {2 i \sqrt {a} f g \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} g}{i \sqrt {c} f+\sqrt {a} g}\right )}{\sqrt {c} e}-\frac {i \sqrt {a} d g^2 \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} g}{i \sqrt {c} f+\sqrt {a} g}\right )}{\sqrt {c} e^2}+\frac {g \sqrt {\frac {g \left (\sqrt {a}+i \sqrt {c} x\right )}{-i \sqrt {c} f+\sqrt {a} g}} \left (i \sqrt {a}+\sqrt {c} x\right ) \left (\left (\sqrt {c} f+i \sqrt {a} g\right ) E\left (\arcsin \left (\sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-i \sqrt {a} g}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )-i \sqrt {a} g \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-i \sqrt {a} g}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{c e \sqrt {\frac {g \left (\sqrt {a}-i \sqrt {c} x\right )}{i \sqrt {c} f+\sqrt {a} g}}}-\frac {\sqrt {a} f^2 \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {a} e}{i \sqrt {c} d+\sqrt {a} e},\arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} g}{i \sqrt {c} f+\sqrt {a} g}\right )}{i \sqrt {c} d+\sqrt {a} e}+\frac {2 \sqrt {a} d f g \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {a} e}{i \sqrt {c} d+\sqrt {a} e},\arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} g}{i \sqrt {c} f+\sqrt {a} g}\right )}{i \sqrt {c} d e+\sqrt {a} e^2}-\frac {\sqrt {a} d^2 g^2 \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {a} e}{i \sqrt {c} d+\sqrt {a} e},\arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} g}{i \sqrt {c} f+\sqrt {a} g}\right )}{e^2 \left (i \sqrt {c} d+\sqrt {a} e\right )}\right )}{\sqrt {f+g x} \sqrt {a+c x^2}} \]

[In]

Integrate[(f + g*x)^(3/2)/((d + e*x)*Sqrt[a + c*x^2]),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(851\) vs. \(2(382)=764\).

Time = 1.37 (sec) , antiderivative size = 852, normalized size of antiderivative = 1.82

method	result	size
elliptic	\(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 g \left (d g -2 e f \right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{e^{2} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}+\frac {2 g^{2} \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{e \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}+\frac {2 \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \Pi \left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {d}{e}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{e^{3} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}\, \left (-\frac {f}{g}+\frac {d}{e}\right )}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\)	\(852\)
default	\(\frac {2 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, \sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) g}{g \sqrt {-a c}+c f}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) g}{g \sqrt {-a c}-c f}}\, \left (\sqrt {-a c}\, F\left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) d \,g^{2}-\sqrt {-a c}\, F\left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) e f g -\sqrt {-a c}\, \Pi \left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \frac {\left (g \sqrt {-a c}-c f \right ) e}{c \left (d g -e f \right )}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) d \,g^{2}+\sqrt {-a c}\, \Pi \left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \frac {\left (g \sqrt {-a c}-c f \right ) e}{c \left (d g -e f \right )}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) e f g +F\left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) a e \,g^{2}-F\left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) c d f g +2 F\left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) c e \,f^{2}-E\left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) a e \,g^{2}-E\left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) c e \,f^{2}+\Pi \left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \frac {\left (g \sqrt {-a c}-c f \right ) e}{c \left (d g -e f \right )}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) c d f g -\Pi \left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \frac {\left (g \sqrt {-a c}-c f \right ) e}{c \left (d g -e f \right )}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) c e \,f^{2}\right )}{c \,e^{2} \left (c g \,x^{3}+c f \,x^{2}+a g x +f a \right )}\)	\(959\)

[In]

int((g*x+f)^(3/2)/(e*x+d)/(c*x^2+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {(f+g x)^{3/2}}{(d+e x) \sqrt {a+c x^2}} \, dx=\text {Timed out} \]

[In]

integrate((g*x+f)^(3/2)/(e*x+d)/(c*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

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

[In]

integrate((g*x+f)**(3/2)/(e*x+d)/(c*x**2+a)**(1/2),x)

[Out]

Integral((f + g*x)**(3/2)/(sqrt(a + c*x**2)*(d + e*x)), x)

Maxima [F]

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

[In]

integrate((g*x+f)^(3/2)/(e*x+d)/(c*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

integrate((g*x + f)^(3/2)/(sqrt(c*x^2 + a)*(e*x + d)), x)

Giac [F]

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

[In]

integrate((g*x+f)^(3/2)/(e*x+d)/(c*x^2+a)^(1/2),x, algorithm="giac")

[Out]

integrate((g*x + f)^(3/2)/(sqrt(c*x^2 + a)*(e*x + d)), x)

Mupad [F(-1)]

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

[In]

int((f + g*x)^(3/2)/((a + c*x^2)^(1/2)*(d + e*x)),x)

[Out]

int((f + g*x)^(3/2)/((a + c*x^2)^(1/2)*(d + e*x)), x)