3.29.58 \(\int \frac {1}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx\) [2858]

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

[Out]

-2/3*b*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)^(3/2)+4/3*b*(-2*a*d*f+b*c*f+b*d*e)*(d*x+c)^(1
/2)*(f*x+e)^(1/2)/(-a*d+b*c)^2/(-a*f+b*e)^2/(b*x+a)^(1/2)-4/3*(-2*a*d*f+b*c*f+b*d*e)*EllipticE(d^(1/2)*(b*x+a)
^(1/2)/(a*d-b*c)^(1/2),((-a*d+b*c)*f/d/(-a*f+b*e))^(1/2))*d^(1/2)*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(f*x+e)^(1/2)/(
a*d-b*c)^(3/2)/(-a*f+b*e)^2/(d*x+c)^(1/2)/(b*(f*x+e)/(-a*f+b*e))^(1/2)+2/3*(-3*a*d*f+b*c*f+2*b*d*e)*EllipticF(
d^(1/2)*(b*x+a)^(1/2)/(a*d-b*c)^(1/2),((-a*d+b*c)*f/d/(-a*f+b*e))^(1/2))*d^(1/2)*(b*(d*x+c)/(-a*d+b*c))^(1/2)*
(b*(f*x+e)/(-a*f+b*e))^(1/2)/b/(a*d-b*c)^(3/2)/(-a*f+b*e)/(d*x+c)^(1/2)/(f*x+e)^(1/2)

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Rubi [A]
time = 0.33, antiderivative size = 437, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {106, 157, 164, 115, 114, 122, 121} \begin {gather*} \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 (\text {ArcSin}\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 (\text {ArcSin}\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}}}+\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)} \end {gather*}

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 106

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 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*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)))], 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 115

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 121

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

Rule 122

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 157

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 164

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]

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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 22.55, size = 449, normalized size = 1.03 \begin {gather*} -\frac {2 \left (b^2 \sqrt {-a+\frac {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) \left (2 b^2 \sqrt {-a+\frac {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 {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )-i (b c-a d) f (b d e+2 b c f-3 a d f) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )\right )\right )}{3 b \sqrt {-a+\frac {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}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(3385\) vs. \(2(383)=766\).
time = 0.10, size = 3386, normalized size = 7.75

method	result	size
elliptic	\(\frac {\sqrt {\left (b x +a \right ) \left (d x +c \right ) \left (f x +e \right )}\, \left (-\frac {2 \sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}{3 \left (a^{2} d f -a b c f -a b d e +b^{2} c e \right ) b \left (x +\frac {a}{b}\right )^{2}}-\frac {4 \left (b d f \,x^{2}+b c f x +b d e x +b c e \right ) \left (2 a d f -b c f -b d e \right )}{3 \left (a^{2} d f -a b c f -a b d e +b^{2} c e \right )^{2} \sqrt {\left (x +\frac {a}{b}\right ) \left (b d f \,x^{2}+b c f x +b d e x +b c e \right )}}+\frac {2 \left (-\frac {d f}{3 \left (a^{2} d f -a b c f -a b d e +b^{2} c e \right )}+\frac {2 \left (a d f -b c f -b d e \right ) \left (2 a d f -b c f -b d e \right )}{3 \left (a^{2} d f -a b c f -a b d e +b^{2} c e \right )^{2}}+\frac {2 \left (b c f +b d e \right ) \left (2 a d f -b c f -b d e \right )}{3 \left (a^{2} d f -a b c f -a b d e +b^{2} c e \right )^{2}}\right ) \left (-\frac {c}{d}+\frac {e}{f}\right ) \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {e}{f}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )}{\sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}+\frac {4 b d f \left (2 a d f -b c f -b d e \right ) \left (-\frac {c}{d}+\frac {e}{f}\right ) \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {e}{f}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \left (\left (-\frac {e}{f}+\frac {a}{b}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )-\frac {a \EllipticF \left (\sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )}{b}\right )}{3 \left (a^{2} d f -a b c f -a b d e +b^{2} c e \right )^{2} \sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}\right )}{\sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {f x +e}}\)	\(878\)
default	\(\text {Expression too large to display}\)	\(3386\)

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,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.52, size = 1406, normalized size = 3.22 \begin {gather*} \frac {2 \, {\left (3 \, {\left (2 \, {\left (b^{4} c d - 2 \, a b^{3} d^{2}\right )} f^{2} x + {\left (3 \, a b^{3} c d - 5 \, a^{2} b^{2} d^{2}\right )} f^{2} + {\left (2 \, b^{4} d^{2} f x - {\left (b^{4} c d - 3 \, a b^{3} d^{2}\right )} f\right )} e\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e} + {\left ({\left (2 \, b^{4} c^{2} - 5 \, a b^{3} c d + 5 \, a^{2} b^{2} d^{2}\right )} f^{2} x^{2} + 2 \, {\left (2 \, a b^{3} c^{2} - 5 \, a^{2} b^{2} c d + 5 \, a^{3} b d^{2}\right )} f^{2} x + {\left (2 \, a^{2} b^{2} c^{2} - 5 \, a^{3} b c d + 5 \, a^{4} d^{2}\right )} f^{2} + 2 \, {\left (b^{4} d^{2} x^{2} + 2 \, a b^{3} d^{2} x + a^{2} b^{2} d^{2}\right )} e^{2} + {\left ({\left (b^{4} c d - 5 \, a b^{3} d^{2}\right )} f x^{2} + 2 \, {\left (a b^{3} c d - 5 \, a^{2} b^{2} d^{2}\right )} f x + {\left (a^{2} b^{2} c d - 5 \, a^{3} b d^{2}\right )} f\right )} e\right )} \sqrt {b d f} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} d^{2} e^{2} + {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )} f^{2} - {\left (b^{2} c d + a b d^{2}\right )} f e\right )}}{3 \, b^{2} d^{2} f^{2}}, -\frac {4 \, {\left (2 \, b^{3} d^{3} e^{3} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} f^{3} - 3 \, {\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} f^{2} e - 3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} f e^{2}\right )}}{27 \, b^{3} d^{3} f^{3}}, \frac {3 \, b d f x + b d e + {\left (b c + a d\right )} f}{3 \, b d f}\right ) + 6 \, {\left ({\left (b^{4} c d - 2 \, a b^{3} d^{2}\right )} f^{2} x^{2} + 2 \, {\left (a b^{3} c d - 2 \, a^{2} b^{2} d^{2}\right )} f^{2} x + {\left (a^{2} b^{2} c d - 2 \, a^{3} b d^{2}\right )} f^{2} + {\left (b^{4} d^{2} f x^{2} + 2 \, a b^{3} d^{2} f x + a^{2} b^{2} d^{2} f\right )} e\right )} \sqrt {b d f} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} d^{2} e^{2} + {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )} f^{2} - {\left (b^{2} c d + a b d^{2}\right )} f e\right )}}{3 \, b^{2} d^{2} f^{2}}, -\frac {4 \, {\left (2 \, b^{3} d^{3} e^{3} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} f^{3} - 3 \, {\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} f^{2} e - 3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} f e^{2}\right )}}{27 \, b^{3} d^{3} f^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} d^{2} e^{2} + {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )} f^{2} - {\left (b^{2} c d + a b d^{2}\right )} f e\right )}}{3 \, b^{2} d^{2} f^{2}}, -\frac {4 \, {\left (2 \, b^{3} d^{3} e^{3} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} f^{3} - 3 \, {\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} f^{2} e - 3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} f e^{2}\right )}}{27 \, b^{3} d^{3} f^{3}}, \frac {3 \, b d f x + b d e + {\left (b c + a d\right )} f}{3 \, b d f}\right )\right )\right )}}{9 \, {\left ({\left (a^{2} b^{5} c^{2} d - 2 \, a^{3} b^{4} c d^{2} + a^{4} b^{3} d^{3}\right )} f^{3} x^{2} + 2 \, {\left (a^{3} b^{4} c^{2} d - 2 \, a^{4} b^{3} c d^{2} + a^{5} b^{2} d^{3}\right )} f^{3} x + {\left (a^{4} b^{3} c^{2} d - 2 \, a^{5} b^{2} c d^{2} + a^{6} b d^{3}\right )} f^{3} + {\left ({\left (b^{7} c^{2} d - 2 \, a b^{6} c d^{2} + a^{2} b^{5} d^{3}\right )} f x^{2} + 2 \, {\left (a b^{6} c^{2} d - 2 \, a^{2} b^{5} c d^{2} + a^{3} b^{4} d^{3}\right )} f x + {\left (a^{2} b^{5} c^{2} d - 2 \, a^{3} b^{4} c d^{2} + a^{4} b^{3} d^{3}\right )} f\right )} e^{2} - 2 \, {\left ({\left (a b^{6} c^{2} d - 2 \, a^{2} b^{5} c d^{2} + a^{3} b^{4} d^{3}\right )} f^{2} x^{2} + 2 \, {\left (a^{2} b^{5} c^{2} d - 2 \, a^{3} b^{4} c d^{2} + a^{4} b^{3} d^{3}\right )} f^{2} x + {\left (a^{3} b^{4} c^{2} d - 2 \, a^{4} b^{3} c d^{2} + a^{5} b^{2} d^{3}\right )} f^{2}\right )} e\right )}} \end {gather*}

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]

2/9*(3*(2*(b^4*c*d - 2*a*b^3*d^2)*f^2*x + (3*a*b^3*c*d - 5*a^2*b^2*d^2)*f^2 + (2*b^4*d^2*f*x - (b^4*c*d - 3*a*
b^3*d^2)*f)*e)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e) + ((2*b^4*c^2 - 5*a*b^3*c*d + 5*a^2*b^2*d^2)*f^2*x^2
+ 2*(2*a*b^3*c^2 - 5*a^2*b^2*c*d + 5*a^3*b*d^2)*f^2*x + (2*a^2*b^2*c^2 - 5*a^3*b*c*d + 5*a^4*d^2)*f^2 + 2*(b^4
*d^2*x^2 + 2*a*b^3*d^2*x + a^2*b^2*d^2)*e^2 + ((b^4*c*d - 5*a*b^3*d^2)*f*x^2 + 2*(a*b^3*c*d - 5*a^2*b^2*d^2)*f
*x + (a^2*b^2*c*d - 5*a^3*b*d^2)*f)*e)*sqrt(b*d*f)*weierstrassPInverse(4/3*(b^2*d^2*e^2 + (b^2*c^2 - a*b*c*d +
a^2*d^2)*f^2 - (b^2*c*d + a*b*d^2)*f*e)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*
a^2*b*c*d^2 + 2*a^3*d^3)*f^3 - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*f^2*e - 3*(b^3*c*d^2 + a*b^2*d^3)*f*e
^2)/(b^3*d^3*f^3), 1/3*(3*b*d*f*x + b*d*e + (b*c + a*d)*f)/(b*d*f)) + 6*((b^4*c*d - 2*a*b^3*d^2)*f^2*x^2 + 2*(
a*b^3*c*d - 2*a^2*b^2*d^2)*f^2*x + (a^2*b^2*c*d - 2*a^3*b*d^2)*f^2 + (b^4*d^2*f*x^2 + 2*a*b^3*d^2*f*x + a^2*b^
2*d^2*f)*e)*sqrt(b*d*f)*weierstrassZeta(4/3*(b^2*d^2*e^2 + (b^2*c^2 - a*b*c*d + a^2*d^2)*f^2 - (b^2*c*d + a*b*
d^2)*f*e)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3 -
3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*f^2*e - 3*(b^3*c*d^2 + a*b^2*d^3)*f*e^2)/(b^3*d^3*f^3), weierstrassP
Inverse(4/3*(b^2*d^2*e^2 + (b^2*c^2 - a*b*c*d + a^2*d^2)*f^2 - (b^2*c*d + a*b*d^2)*f*e)/(b^2*d^2*f^2), -4/27*(
2*b^3*d^3*e^3 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3 - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a
^2*b*d^3)*f^2*e - 3*(b^3*c*d^2 + a*b^2*d^3)*f*e^2)/(b^3*d^3*f^3), 1/3*(3*b*d*f*x + b*d*e + (b*c + a*d)*f)/(b*d
*f))))/((a^2*b^5*c^2*d - 2*a^3*b^4*c*d^2 + a^4*b^3*d^3)*f^3*x^2 + 2*(a^3*b^4*c^2*d - 2*a^4*b^3*c*d^2 + a^5*b^2
*d^3)*f^3*x + (a^4*b^3*c^2*d - 2*a^5*b^2*c*d^2 + a^6*b*d^3)*f^3 + ((b^7*c^2*d - 2*a*b^6*c*d^2 + a^2*b^5*d^3)*f
*x^2 + 2*(a*b^6*c^2*d - 2*a^2*b^5*c*d^2 + a^3*b^4*d^3)*f*x + (a^2*b^5*c^2*d - 2*a^3*b^4*c*d^2 + a^4*b^3*d^3)*f
)*e^2 - 2*((a*b^6*c^2*d - 2*a^2*b^5*c*d^2 + a^3*b^4*d^3)*f^2*x^2 + 2*(a^2*b^5*c^2*d - 2*a^3*b^4*c*d^2 + a^4*b^
3*d^3)*f^2*x + (a^3*b^4*c^2*d - 2*a^4*b^3*c*d^2 + a^5*b^2*d^3)*f^2)*e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x\right )^{\frac {5}{2}} \sqrt {c + d x} \sqrt {e + f x}}\, dx \end {gather*}

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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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