3.71 \(\int \frac {1}{(a+b x) (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx\)
Optimal. Leaf size=393 \[ \frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x}}{\sqrt {c+d x} (b c-a d) (d e-c f) (d g-c h)}-\frac {2 d \sqrt {h} \sqrt {c+d x} \sqrt {e h-f g} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {e h-f g}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{\sqrt {g+h x} (b c-a d) (d e-c f) (d g-c h) \sqrt {-\frac {f (c+d x)}{d e-c f}}}-\frac {2 b \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {f} \sqrt {e+f x} \sqrt {g+h x} (b c-a d)^2} \]
[Out]
2*d^2*(f*x+e)^(1/2)*(h*x+g)^(1/2)/(-a*d+b*c)/(-c*f+d*e)/(-c*h+d*g)/(d*x+c)^(1/2)-2*b*EllipticPi(f^(1/2)*(d*x+c
)^(1/2)/(c*f-d*e)^(1/2),-b*(-c*f+d*e)/(-a*d+b*c)/f,((-c*f+d*e)*h/f/(-c*h+d*g))^(1/2))*(c*f-d*e)^(1/2)*(d*(f*x+
e)/(-c*f+d*e))^(1/2)*(d*(h*x+g)/(-c*h+d*g))^(1/2)/(-a*d+b*c)^2/f^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)-2*d*Ellipti
cE(h^(1/2)*(f*x+e)^(1/2)/(e*h-f*g)^(1/2),(-d*(-e*h+f*g)/(-c*f+d*e)/h)^(1/2))*h^(1/2)*(e*h-f*g)^(1/2)*(d*x+c)^(
1/2)*(f*(h*x+g)/(-e*h+f*g))^(1/2)/(-a*d+b*c)/(-c*f+d*e)/(-c*h+d*g)/(-f*(d*x+c)/(-c*f+d*e))^(1/2)/(h*x+g)^(1/2)
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Rubi [A] time = 0.62, antiderivative size = 393, normalized size of antiderivative = 1.00,
number of steps used = 10, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used
= {179, 104, 21, 114, 113, 169, 538, 537} \[ \frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x}}{\sqrt {c+d x} (b c-a d) (d e-c f) (d g-c h)}-\frac {2 d \sqrt {h} \sqrt {c+d x} \sqrt {e h-f g} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {e h-f g}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{\sqrt {g+h x} (b c-a d) (d e-c f) (d g-c h) \sqrt {-\frac {f (c+d x)}{d e-c f}}}-\frac {2 b \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {f} \sqrt {e+f x} \sqrt {g+h x} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In]
Int[1/((a + b*x)*(c + d*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
[Out]
(2*d^2*Sqrt[e + f*x]*Sqrt[g + h*x])/((b*c - a*d)*(d*e - c*f)*(d*g - c*h)*Sqrt[c + d*x]) - (2*d*Sqrt[h]*Sqrt[-(
f*g) + e*h]*Sqrt[c + d*x]*Sqrt[(f*(g + h*x))/(f*g - e*h)]*EllipticE[ArcSin[(Sqrt[h]*Sqrt[e + f*x])/Sqrt[-(f*g)
+ e*h]], -((d*(f*g - e*h))/((d*e - c*f)*h))])/((b*c - a*d)*(d*e - c*f)*(d*g - c*h)*Sqrt[-((f*(c + d*x))/(d*e
- c*f))]*Sqrt[g + h*x]) - (2*b*Sqrt[-(d*e) + c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*
h)]*EllipticPi[-((b*(d*e - c*f))/((b*c - a*d)*f)), ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e -
c*f)*h)/(f*(d*g - c*h))])/((b*c - a*d)^2*Sqrt[f]*Sqrt[e + f*x]*Sqrt[g + h*x])
Rule 21
Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*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 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 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 169
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] && !SimplerQ[e
+ f*x, c + d*x] && !SimplerQ[g + h*x, c + d*x]
Rule 179
Int[(((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_))/(Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)])
, x_Symbol] :> Int[ExpandIntegrand[1/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), (a + b*x)^m*(c + d*x)^(n + 1
/2), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IntegerQ[m] && IntegerQ[n + 1/2]
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]
Rubi steps
\begin {align*} \int \frac {1}{(a+b x) (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx &=\int \left (-\frac {d}{(b c-a d) (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}}+\frac {b}{(b c-a d) (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}\right ) \, dx\\ &=\frac {b \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{b c-a d}-\frac {d \int \frac {1}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{b c-a d}\\ &=\frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (d e-c f) (d g-c h) \sqrt {c+d x}}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {1}{\left (b c-a d-b x^2\right ) \sqrt {e-\frac {c f}{d}+\frac {f x^2}{d}} \sqrt {g-\frac {c h}{d}+\frac {h x^2}{d}}} \, dx,x,\sqrt {c+d x}\right )}{b c-a d}+\frac {(2 d) \int \frac {-\frac {1}{2} c f h-\frac {1}{2} d f h x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{(b c-a d) (d e-c f) (d g-c h)}\\ &=\frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (d e-c f) (d g-c h) \sqrt {c+d x}}-\frac {(d f h) \int \frac {\sqrt {c+d x}}{\sqrt {e+f x} \sqrt {g+h x}} \, dx}{(b c-a d) (d e-c f) (d g-c h)}-\frac {\left (2 b \sqrt {\frac {d (e+f x)}{d e-c f}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b c-a d-b x^2\right ) \sqrt {1+\frac {f x^2}{d \left (e-\frac {c f}{d}\right )}} \sqrt {g-\frac {c h}{d}+\frac {h x^2}{d}}} \, dx,x,\sqrt {c+d x}\right )}{(b c-a d) \sqrt {e+f x}}\\ &=\frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (d e-c f) (d g-c h) \sqrt {c+d x}}-\frac {\left (2 b \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b c-a d-b x^2\right ) \sqrt {1+\frac {f x^2}{d \left (e-\frac {c f}{d}\right )}} \sqrt {1+\frac {h x^2}{d \left (g-\frac {c h}{d}\right )}}} \, dx,x,\sqrt {c+d x}\right )}{(b c-a d) \sqrt {e+f x} \sqrt {g+h x}}-\frac {\left (d f h \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}}\right ) \int \frac {\sqrt {\frac {c f}{-d e+c f}+\frac {d f x}{-d e+c f}}}{\sqrt {e+f x} \sqrt {\frac {f g}{f g-e h}+\frac {f h x}{f g-e h}}} \, dx}{(b c-a d) (d e-c f) (d g-c h) \sqrt {\frac {f (c+d x)}{-d e+c f}} \sqrt {g+h x}}\\ &=\frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (d e-c f) (d g-c h) \sqrt {c+d x}}-\frac {2 d \sqrt {h} \sqrt {-f g+e h} \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {-f g+e h}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{(b c-a d) (d e-c f) (d g-c h) \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}}-\frac {2 b \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d)^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}\\ \end {align*}
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Mathematica [C] time = 4.05, size = 322, normalized size = 0.82 \[ \frac {2 i (c+d x) \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} \left ((a d f-2 b c f+b d e) \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {d g}{h}-c}}{\sqrt {c+d x}}\right ),\frac {d e h-c f h}{d f g-c f h}\right )+f (b c-a d) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {d g}{h}-c}}{\sqrt {c+d x}}\right )|\frac {d e h-c f h}{d f g-c f h}\right )+b (c f-d e) \Pi \left (-\frac {b c h-a d h}{b d g-b c h};i \sinh ^{-1}\left (\frac {\sqrt {\frac {d g}{h}-c}}{\sqrt {c+d x}}\right )|\frac {d e h-c f h}{d f g-c f h}\right )\right )}{\sqrt {e+f x} \sqrt {g+h x} (b c-a d)^2 (c f-d e) \sqrt {\frac {d g}{h}-c}} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((a + b*x)*(c + d*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
[Out]
((2*I)*(c + d*x)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*((b*c - a*d)*f*EllipticE[
I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - c*f*h)] + (b*d*e - 2*b*c*f + a*d*f)*Elli
pticF[I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - c*f*h)] + b*(-(d*e) + c*f)*Ellipti
cPi[-((b*c*h - a*d*h)/(b*d*g - b*c*h)), I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g -
c*f*h)]))/((b*c - a*d)^2*(-(d*e) + c*f)*Sqrt[-c + (d*g)/h]*Sqrt[e + f*x]*Sqrt[g + h*x])
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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(1/(b*x+a)/(d*x+c)^(3/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="fricas")
[Out]
Timed out
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giac [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(1/(b*x+a)/(d*x+c)^(3/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="giac")
[Out]
Timed out
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maple [B] time = 0.08, size = 2842, normalized size = 7.23 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(b*x+a)/(d*x+c)^(3/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x)
[Out]
-2*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)*(-EllipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)
^(1/2))*b*c^2*d*f^2*g*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)/(c*h-d*g)*d)^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)+El
lipticE(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*a*c^2*d*f^2*h*((d*x+c)/(c*f-d*e)*f)^(1/2)
*(-(h*x+g)/(c*h-d*g)*d)^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)-EllipticE(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c
*h-d*g)/f*h)^(1/2))*a*c*d^2*f^2*g*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)/(c*h-d*g)*d)^(1/2)*(-(f*x+e)/(c*f-d*e)
*d)^(1/2)+EllipticE(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*a*d^3*e*f*g*((d*x+c)/(c*f-d*e
)*f)^(1/2)*(-(h*x+g)/(c*h-d*g)*d)^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)+EllipticE(((d*x+c)/(c*f-d*e)*f)^(1/2),((c
*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*c^2*d*f^2*g*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)/(c*h-d*g)*d)^(1/2)*(-(f*x+e)
/(c*f-d*e)*d)^(1/2)-EllipticPi(((d*x+c)/(c*f-d*e)*f)^(1/2),-(c*f-d*e)/(a*d-b*c)*b/f,((c*f-d*e)/(c*h-d*g)/f*h)^
(1/2))*b*c^2*d*f^2*g*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)/(c*h-d*g)*d)^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)+Ell
ipticPi(((d*x+c)/(c*f-d*e)*f)^(1/2),-(c*f-d*e)/(a*d-b*c)*b/f,((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*c*d^2*e^2*h*((
d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)/(c*h-d*g)*d)^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)-EllipticF(((d*x+c)/(c*f-d*
e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*a*c^2*d*f^2*h*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)/(c*h-d*g)*d)^
(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)+EllipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*a*c*
d^2*f^2*g*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)/(c*h-d*g)*d)^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)-EllipticF(((d*
x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*a*d^3*e*f*g*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)/(c*
h-d*g)*d)^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)+EllipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(
1/2))*a*c*d^2*e*f*h*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)/(c*h-d*g)*d)^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)-Elli
pticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*c^2*d*e*f*h*((d*x+c)/(c*f-d*e)*f)^(1/2)*(
-(h*x+g)/(c*h-d*g)*d)^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)+EllipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h
-d*g)/f*h)^(1/2))*b*c*d^2*e*f*g*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)/(c*h-d*g)*d)^(1/2)*(-(f*x+e)/(c*f-d*e)*d
)^(1/2)-EllipticE(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*a*c*d^2*e*f*h*((d*x+c)/(c*f-d*e
)*f)^(1/2)*(-(h*x+g)/(c*h-d*g)*d)^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)+EllipticE(((d*x+c)/(c*f-d*e)*f)^(1/2),((c
*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*c^2*d*e*f*h*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)/(c*h-d*g)*d)^(1/2)*(-(f*x+e)
/(c*f-d*e)*d)^(1/2)-EllipticE(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*c*d^2*e*f*g*((d*x
+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)/(c*h-d*g)*d)^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)-2*EllipticPi(((d*x+c)/(c*f-d*
e)*f)^(1/2),-(c*f-d*e)/(a*d-b*c)*b/f,((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*c^2*d*e*f*h*((d*x+c)/(c*f-d*e)*f)^(1/2
)*(-(h*x+g)/(c*h-d*g)*d)^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)+2*EllipticPi(((d*x+c)/(c*f-d*e)*f)^(1/2),-(c*f-d*e
)/(a*d-b*c)*b/f,((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*c*d^2*e*f*g*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)/(c*h-d*g)
*d)^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)+x*a*d^3*e*f*h-x*b*c*d^2*f^2*g-x^2*b*c*d^2*f^2*h+x^2*a*d^3*f^2*h-b*c*d^2
*e*f*g+x*a*d^3*f^2*g+a*d^3*e*f*g+EllipticPi(((d*x+c)/(c*f-d*e)*f)^(1/2),-(c*f-d*e)/(a*d-b*c)*b/f,((c*f-d*e)/(c
*h-d*g)/f*h)^(1/2))*b*c^3*f^2*h*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)/(c*h-d*g)*d)^(1/2)*(-(f*x+e)/(c*f-d*e)*d
)^(1/2)-EllipticPi(((d*x+c)/(c*f-d*e)*f)^(1/2),-(c*f-d*e)/(a*d-b*c)*b/f,((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*d^3
*e^2*g*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)/(c*h-d*g)*d)^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)+EllipticF(((d*x+c
)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*c^3*f^2*h*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)/(c*h-d
*g)*d)^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)-EllipticE(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2
))*b*c^3*f^2*h*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)/(c*h-d*g)*d)^(1/2)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)-x*b*c*d^2
*e*f*h)/f/(a*d-b*c)^2/(c*h-d*g)/(c*f-d*e)/(d*f*h*x^3+c*f*h*x^2+d*e*h*x^2+d*f*g*x^2+c*e*h*x+c*f*g*x+d*e*g*x+c*e
*g)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x + a\right )} {\left (d x + c\right )}^{\frac {3}{2}} \sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x+a)/(d*x+c)^(3/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="maxima")
[Out]
integrate(1/((b*x + a)*(d*x + c)^(3/2)*sqrt(f*x + e)*sqrt(h*x + g)), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(a + b*x)*(c + d*x)^(3/2)),x)
[Out]
int(1/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(a + b*x)*(c + d*x)^(3/2)), x)
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sympy [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(1/(b*x+a)/(d*x+c)**(3/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)
[Out]
Timed out
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