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

Optimal. Leaf size=469 \[ -\frac{2 \sqrt{-a} g \sqrt{\frac{c x^2}{a}+1} (e f-d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{c} e^2 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{2 \sqrt{\frac{c x^2}{a}+1} (e f-d g)^2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right )}{e^2 \sqrt{a+c x^2} \sqrt{f+g x} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right )}-\frac{2 \sqrt{-a} g \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{c} e \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}} \]

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.637311, antiderivative size = 469, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {958, 719, 419, 933, 168, 538, 537, 424} \[ -\frac{2 \sqrt{-a} g \sqrt{\frac{c x^2}{a}+1} (e f-d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{c} e^2 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{2 \sqrt{\frac{c x^2}{a}+1} (e f-d g)^2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right )}{e^2 \sqrt{a+c x^2} \sqrt{f+g x} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right )}-\frac{2 \sqrt{-a} g \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{c} e \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}} \]

Antiderivative was successfully verified.

[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 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]

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 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 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 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 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 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 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]

Rubi steps

\begin{align*} \int \frac{(f+g x)^{3/2}}{(d+e x) \sqrt{a+c x^2}} \, dx &=\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 ) \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 \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 ) \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^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 ) \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^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 ) \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^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]  time = 1.0958, size = 927, normalized size = 1.98 \[ \frac{2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f-i \sqrt{a} g}} \left (-\frac{\sqrt{a} \sqrt{\frac{c x^2}{a}+1} \Pi \left (\frac{2 \sqrt{a} e}{i \sqrt{c} d+\sqrt{a} e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{i \sqrt{c} x}{\sqrt{a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{a} g}{i \sqrt{c} f+\sqrt{a} g}\right ) f^2}{i \sqrt{c} d+\sqrt{a} e}+\frac{2 i \sqrt{a} g \sqrt{\frac{c x^2}{a}+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{i \sqrt{c} x}{\sqrt{a}}}}{\sqrt{2}}\right ),\frac{2 \sqrt{a} g}{i \sqrt{c} f+\sqrt{a} g}\right ) f}{\sqrt{c} e}+\frac{2 \sqrt{a} d g \sqrt{\frac{c x^2}{a}+1} \Pi \left (\frac{2 \sqrt{a} e}{i \sqrt{c} d+\sqrt{a} e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{i \sqrt{c} x}{\sqrt{a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{a} g}{i \sqrt{c} f+\sqrt{a} g}\right ) f}{\sqrt{a} e^2+i \sqrt{c} d e}-\frac{i \sqrt{a} d g^2 \sqrt{\frac{c x^2}{a}+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{i \sqrt{c} x}{\sqrt{a}}}}{\sqrt{2}}\right ),\frac{2 \sqrt{a} g}{i \sqrt{c} f+\sqrt{a} g}\right )}{\sqrt{c} e^2}+\frac{g \sqrt{\frac{g \left (i \sqrt{c} x+\sqrt{a}\right )}{\sqrt{a} g-i \sqrt{c} f}} \left (\sqrt{c} x+i \sqrt{a}\right ) \left (\left (\sqrt{c} f+i \sqrt{a} g\right ) E\left (\sin ^{-1}\left (\sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f-i \sqrt{a} g}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )-i \sqrt{a} g \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f-i \sqrt{a} g}}\right ),\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )\right )}{c e \sqrt{\frac{g \left (\sqrt{a}-i \sqrt{c} x\right )}{i \sqrt{c} f+\sqrt{a} g}}}-\frac{\sqrt{a} d^2 g^2 \sqrt{\frac{c x^2}{a}+1} \Pi \left (\frac{2 \sqrt{a} e}{i \sqrt{c} d+\sqrt{a} e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{i \sqrt{c} x}{\sqrt{a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{a} g}{i \sqrt{c} f+\sqrt{a} g}\right )}{e^2 \left (i \sqrt{c} d+\sqrt{a} e\right )}\right )}{\sqrt{f+g x} \sqrt{c x^2+a}} \]

Antiderivative was successfully verified.

[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])

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Maple [B]  time = 0.257, size = 959, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (g x + f\right )}^{\frac{3}{2}}}{\sqrt{c x^{2} + a}{\left (e x + d\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (f + g x\right )^{\frac{3}{2}}}{\sqrt{a + c x^{2}} \left (d + e x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out