3.2858 \(\int \frac{1}{(a+b x)^{5/2} \sqrt{c+d x} \sqrt{e+f x}} \, dx\)

Optimal. Leaf size=437 \[ \frac{2 \sqrt{d} \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} (-3 a d f+b c f+2 b d e) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right ),\frac{f (b c-a d)}{d (b e-a f)}\right )}{3 b \sqrt{c+d x} \sqrt{e+f x} (a d-b c)^{3/2} (b e-a f)}+\frac{4 b \sqrt{c+d x} \sqrt{e+f x} (-2 a d f+b c f+b d e)}{3 \sqrt{a+b x} (b c-a d)^2 (b e-a f)^2}-\frac{2 b \sqrt{c+d x} \sqrt{e+f x}}{3 (a+b x)^{3/2} (b c-a d) (b e-a f)}-\frac{4 \sqrt{d} \sqrt{e+f x} \sqrt{\frac{b (c+d x)}{b c-a d}} (-2 a d f+b c f+b d e) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 \sqrt{c+d x} (a d-b c)^{3/2} (b e-a f)^2 \sqrt{\frac{b (e+f x)}{b e-a f}}} \]

[Out]

(-2*b*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*(b*c - a*d)*(b*e - a*f)*(a + b*x)^(3/2)) + (4*b*(b*d*e + b*c*f - 2*a*d*f
)*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*(b*c - a*d)^2*(b*e - a*f)^2*Sqrt[a + b*x]) - (4*Sqrt[d]*(b*d*e + b*c*f - 2*a
*d*f)*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d
]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(3*(-(b*c) + a*d)^(3/2)*(b*e - a*f)^2*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(
b*e - a*f)]) + (2*Sqrt[d]*(2*b*d*e + b*c*f - 3*a*d*f)*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e
- a*f)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(3*b*(
-(b*c) + a*d)^(3/2)*(b*e - a*f)*Sqrt[c + d*x]*Sqrt[e + f*x])

________________________________________________________________________________________

Rubi [A]  time = 0.496029, antiderivative size = 437, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {104, 152, 158, 114, 113, 121, 120} \[ \frac{4 b \sqrt{c+d x} \sqrt{e+f x} (-2 a d f+b c f+b d e)}{3 \sqrt{a+b x} (b c-a d)^2 (b e-a f)^2}-\frac{2 b \sqrt{c+d x} \sqrt{e+f x}}{3 (a+b x)^{3/2} (b c-a d) (b e-a f)}+\frac{2 \sqrt{d} \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} (-3 a d f+b c f+2 b d e) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b \sqrt{c+d x} \sqrt{e+f x} (a d-b c)^{3/2} (b e-a f)}-\frac{4 \sqrt{d} \sqrt{e+f x} \sqrt{\frac{b (c+d x)}{b c-a d}} (-2 a d f+b c f+b d e) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 \sqrt{c+d x} (a d-b c)^{3/2} (b e-a f)^2 \sqrt{\frac{b (e+f x)}{b e-a f}}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(5/2)*Sqrt[c + d*x]*Sqrt[e + f*x]),x]

[Out]

(-2*b*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*(b*c - a*d)*(b*e - a*f)*(a + b*x)^(3/2)) + (4*b*(b*d*e + b*c*f - 2*a*d*f
)*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*(b*c - a*d)^2*(b*e - a*f)^2*Sqrt[a + b*x]) - (4*Sqrt[d]*(b*d*e + b*c*f - 2*a
*d*f)*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d
]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(3*(-(b*c) + a*d)^(3/2)*(b*e - a*f)^2*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(
b*e - a*f)]) + (2*Sqrt[d]*(2*b*d*e + b*c*f - 3*a*d*f)*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e
- a*f)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(3*b*(
-(b*c) + a*d)^(3/2)*(b*e - a*f)*Sqrt[c + d*x]*Sqrt[e + f*x])

Rule 104

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegersQ[2*m, 2*n, 2*p]

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 158

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol]
:> Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(f*g - e*h)/f, Int[1/(Sqrt[a + b*
x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && SimplerQ[a + b*x, e + f*x] &&
SimplerQ[c + d*x, e + f*x]

Rule 114

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

Rule 113

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

Rule 121

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

Rule 120

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

Rubi steps

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

Mathematica [C]  time = 3.66099, size = 449, normalized size = 1.03 \[ -\frac{2 \left (b^2 (c+d x) (e+f x) \sqrt{\frac{b c}{d}-a} ((b c-a d) (b e-a f)-2 (a+b x) (-2 a d f+b c f+b d e))+(a+b x) \left (-i f (a+b x)^{3/2} (b c-a d) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} (-3 a d f+2 b c f+b d e) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right ),\frac{b d e-a d f}{b c f-a d f}\right )+2 b^2 (c+d x) (e+f x) \sqrt{\frac{b c}{d}-a} (-2 a d f+b c f+b d e)+2 i f (a+b x)^{3/2} (b c-a d) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} (-2 a d f+b c f+b d e) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right )|\frac{b d e-a d f}{b c f-a d f}\right )\right )\right )}{3 b (a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{\frac{b c}{d}-a} (b c-a d)^2 (b e-a f)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(5/2)*Sqrt[c + d*x]*Sqrt[e + f*x]),x]

[Out]

(-2*(b^2*Sqrt[-a + (b*c)/d]*(c + d*x)*(e + f*x)*((b*c - a*d)*(b*e - a*f) - 2*(b*d*e + b*c*f - 2*a*d*f)*(a + b*
x)) + (a + b*x)*(2*b^2*Sqrt[-a + (b*c)/d]*(b*d*e + b*c*f - 2*a*d*f)*(c + d*x)*(e + f*x) + (2*I)*(b*c - a*d)*f*
(b*d*e + b*c*f - 2*a*d*f)*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*
EllipticE[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)] - I*(b*c - a*d)*f*(b*d
*e + 2*b*c*f - 3*a*d*f)*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*El
lipticF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)])))/(3*b*Sqrt[-a + (b*c)/
d]*(b*c - a*d)^2*(b*e - a*f)^2*(a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x])

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Maple [B]  time = 0.076, size = 4067, normalized size = 9.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(-x^2*a*b^3*c*d*f^2+EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-
b*e))^(1/2))*a^2*b^2*c^2*f^2*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(
1/2)+3*x^2*b^4*c*d*e*f-5*x*a^2*b^2*c*d*f^2-5*x*a^2*b^2*d^2*e*f+3*a*b^3*c^2*e*f+3*a*b^3*c*d*e^2+x*b^4*c^2*e*f+3
*EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^4*d^2*f^2*(d*(b*x+a)/(a*d-b*c))^(1/2
)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-4*EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)
*f/d/(a*f-b*e))^(1/2))*a^4*d^2*f^2*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b
*c))^(1/2)+2*x*a*b^3*c*d*e*f+2*EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^2*b^2*
d^2*e^2*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-2*EllipticE((d*(
b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^2*b^2*c^2*f^2*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)
*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-2*EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b
*e))^(1/2))*a^2*b^2*d^2*e^2*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1
/2)-2*EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*x*b^4*c*d*e^2*(d*(b*x+a)/(a*d-b*c
))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-4*EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a
*d-b*c)*f/d/(a*f-b*e))^(1/2))*x*a^3*b*d^2*f^2*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+
c)*b/(a*d-b*c))^(1/2)-2*EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*x*a*b^3*c^2*f^2
*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-2*EllipticE((d*(b*x+a)/
(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*x*a*b^3*d^2*e^2*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f
-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+2*EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1
/2))*x*b^4*c^2*e*f*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+2*Ell
ipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*x*b^4*c*d*e^2*(d*(b*x+a)/(a*d-b*c))^(1/2)*
(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-4*EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f
/d/(a*f-b*e))^(1/2))*a^3*b*c*d*f^2*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b
*c))^(1/2)-5*EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^3*b*d^2*e*f*(d*(b*x+a)/(
a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2
),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^3*c^2*e*f*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d
*x+c)*b/(a*d-b*c))^(1/2)-2*EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^3*c*d*e^
2*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+6*EllipticE((d*(b*x+a)
/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^3*b*c*d*f^2*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-
b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+6*EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/
2))*a^3*b*d^2*e*f*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+2*Elli
pticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^3*c^2*e*f*(d*(b*x+a)/(a*d-b*c))^(1/2)*(
-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+2*EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/
d/(a*f-b*e))^(1/2))*a*b^3*c*d*e^2*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*
c))^(1/2)-b^4*c^2*e^2+x*b^4*c*d*e^2-4*x^3*a*b^3*d^2*f^2+2*x^3*b^4*c*d*f^2+2*x^3*b^4*d^2*e*f-5*x^2*a^2*b^2*d^2*
f^2+3*x*a*b^3*c^2*f^2+3*x*a*b^3*d^2*e^2-x^2*a*b^3*d^2*e*f+3*EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f
/d/(a*f-b*e))^(1/2))*x*a^3*b*d^2*f^2*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d
-b*c))^(1/2)+EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*x*a*b^3*c^2*f^2*(d*(b*x+a)
/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+2*EllipticF((d*(b*x+a)/(a*d-b*c))^
(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*x*a*b^3*d^2*e^2*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2
)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*x*b^4*c^
2*e*f*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+2*x^2*b^4*c^2*f^2+
2*x^2*b^4*d^2*e^2+6*EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*x*a*b^3*c*d*e*f*(d*
(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-8*EllipticE((d*(b*x+a)/(a*d
-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*x*a*b^3*c*d*e*f*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e
))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-4*EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))
*x*a^2*b^2*c*d*f^2*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-5*Ell
ipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*x*a^2*b^2*d^2*e*f*(d*(b*x+a)/(a*d-b*c))^(1
/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+6*EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*
c)*f/d/(a*f-b*e))^(1/2))*x*a^2*b^2*c*d*f^2*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*
b/(a*d-b*c))^(1/2)+6*EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*x*a^2*b^2*d^2*e*f*
(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+6*EllipticF((d*(b*x+a)/(
a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^2*b^2*c*d*e*f*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-
b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-8*EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/
2))*a^2*b^2*c*d*e*f*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-5*a^
2*b^2*c*d*e*f)/(d*f*x^2+c*f*x+d*e*x+c*e)/(a*d-b*c)/(a*f-b*e)/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/(b*x+a)^(3/2)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{2}} \sqrt{d x + c} \sqrt{f x + e}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x + a} \sqrt{d x + c} \sqrt{f x + e}}{b^{3} d f x^{5} + a^{3} c e +{\left (b^{3} d e +{\left (b^{3} c + 3 \, a b^{2} d\right )} f\right )} x^{4} +{\left ({\left (b^{3} c + 3 \, a b^{2} d\right )} e + 3 \,{\left (a b^{2} c + a^{2} b d\right )} f\right )} x^{3} +{\left (3 \,{\left (a b^{2} c + a^{2} b d\right )} e +{\left (3 \, a^{2} b c + a^{3} d\right )} f\right )} x^{2} +{\left (a^{3} c f +{\left (3 \, a^{2} b c + a^{3} d\right )} e\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)/(b^3*d*f*x^5 + a^3*c*e + (b^3*d*e + (b^3*c + 3*a*b^2*d)*f)*
x^4 + ((b^3*c + 3*a*b^2*d)*e + 3*(a*b^2*c + a^2*b*d)*f)*x^3 + (3*(a*b^2*c + a^2*b*d)*e + (3*a^2*b*c + a^3*d)*f
)*x^2 + (a^3*c*f + (3*a^2*b*c + a^3*d)*e)*x), x)

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Sympy [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(1/(b*x+a)**(5/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{2}} \sqrt{d x + c} \sqrt{f x + e}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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