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

Optimal. Leaf size=625 \[ -\frac {2 d \sqrt {g+h x} (d e-c f) \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {e+f x} \sqrt {b g-a h}}{\sqrt {a+b x} \sqrt {f g-e h}}\right ),-\frac {(b c-a d) (f g-e h)}{(b g-a h) (d e-c f)}\right )}{\sqrt {c+d x} (b c-a d) \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (-2 a d f+b c f+b d e)}{\sqrt {a+b x} (b c-a d) (b e-a f) (b g-a h)}+\frac {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} (-2 a d f+b c f+b d e)}{\sqrt {c+d x} (b c-a d) (b e-a f) (b g-a h)}-\frac {2 \sqrt {a+b x} \sqrt {d g-c h} \sqrt {f g-e h} (-2 a d f+b c f+b d e) \sqrt {-\frac {(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} E\left (\sin ^{-1}\left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{\sqrt {g+h x} (b c-a d) (b e-a f) (b g-a h) \sqrt {\frac {(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}} \]

[Out]

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

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Rubi [A]  time = 1.06, antiderivative size = 625, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1599, 1602, 12, 170, 419, 176, 424} \[ -\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (-2 a d f+b c f+b d e)}{\sqrt {a+b x} (b c-a d) (b e-a f) (b g-a h)}+\frac {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} (-2 a d f+b c f+b d e)}{\sqrt {c+d x} (b c-a d) (b e-a f) (b g-a h)}-\frac {2 d \sqrt {g+h x} (d e-c f) \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} F\left (\sin ^{-1}\left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right )|-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt {c+d x} (b c-a d) \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}-\frac {2 \sqrt {a+b x} \sqrt {d g-c h} \sqrt {f g-e h} (-2 a d f+b c f+b d e) \sqrt {-\frac {(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} E\left (\sin ^{-1}\left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{\sqrt {g+h x} (b c-a d) (b e-a f) (b g-a h) \sqrt {\frac {(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d*(b*d*e + b*c*f - 2*a*d*f)*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((b*c - a*d)*(b*e - a*f)*(b*g - a*h)
*Sqrt[c + d*x]) - (2*b*(b*d*e + b*c*f - 2*a*d*f)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((b*c - a*d)*(b*e
- a*f)*(b*g - a*h)*Sqrt[a + b*x]) - (2*(b*d*e + b*c*f - 2*a*d*f)*Sqrt[d*g - c*h]*Sqrt[f*g - e*h]*Sqrt[a + b*x]
*Sqrt[-(((d*e - c*f)*(g + h*x))/((f*g - e*h)*(c + d*x)))]*EllipticE[ArcSin[(Sqrt[d*g - c*h]*Sqrt[e + f*x])/(Sq
rt[f*g - e*h]*Sqrt[c + d*x])], ((b*c - a*d)*(f*g - e*h))/((b*e - a*f)*(d*g - c*h))])/((b*c - a*d)*(b*e - a*f)*
(b*g - a*h)*Sqrt[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))]*Sqrt[g + h*x]) - (2*d*(d*e - c*f)*Sqrt[((b*e
- a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Sqrt[g + h*x]*EllipticF[ArcSin[(Sqrt[b*g - a*h]*Sqrt[e + f*x])/(Sq
rt[f*g - e*h]*Sqrt[a + b*x])], -(((b*c - a*d)*(f*g - e*h))/((d*e - c*f)*(b*g - a*h)))])/((b*c - a*d)*Sqrt[b*g
- a*h]*Sqrt[f*g - e*h]*Sqrt[c + d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 170

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x
_Symbol] :> Dist[(2*Sqrt[g + h*x]*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/((f*g - e*h)*Sqrt[c +
d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))]), Subst[Int[1/(Sqrt[1 + ((b*c - a*d)*x^2)/(d*e
- c*f)]*Sqrt[1 - ((b*g - a*h)*x^2)/(f*g - e*h)]), x], x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d
, e, f, g, h}, x]

Rule 176

Int[Sqrt[(c_.) + (d_.)*(x_)]/(((a_.) + (b_.)*(x_))^(3/2)*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x
_Symbol] :> Dist[(-2*Sqrt[c + d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))])/((b*e - a*f)*Sqrt
[g + h*x]*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]), Subst[Int[Sqrt[1 + ((b*c - a*d)*x^2)/(d*e -
c*f)]/Sqrt[1 - ((b*g - a*h)*x^2)/(f*g - e*h)], x], x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e
, f, g, h}, x]

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 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]

Rule 1599

Int[(((a_.) + (b_.)*(x_))^(m_)*((A_.) + (B_.)*(x_)))/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(
g_.) + (h_.)*(x_)]), x_Symbol] :> Simp[((A*b^2 - a*b*B)*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g +
h*x])/((m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)), x] - Dist[1/(2*(m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*
h)), Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[A*(2*a^2*d*f*h*(m + 1) - 2*a*b*(
m + 1)*(d*f*g + d*e*h + c*f*h) + b^2*(2*m + 3)*(d*e*g + c*f*g + c*e*h)) - b*B*(a*(d*e*g + c*f*g + c*e*h) + 2*b
*c*e*g*(m + 1)) - 2*((A*b - a*B)*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h + c*f*h)))*x + d*f*h*(2*m + 5)*(A
*b^2 - a*b*B)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B}, x] && IntegerQ[2*m] && LtQ[m, -1]

Rule 1602

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*
(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp[(C*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(b*f*h*Sqrt[c
+ d*x]), x] + (Dist[1/(2*b*d*f*h), Int[(1*Simp[2*A*b*d*f*h - C*(b*d*e*g + a*c*f*h) + (2*b*B*d*f*h - C*(a*d*f*
h + b*(d*f*g + d*e*h + c*f*h)))*x, x])/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] + Dis
t[(C*(d*e - c*f)*(d*g - c*h))/(2*b*d*f*h), Int[Sqrt[a + b*x]/((c + d*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]), x]
, x]) /; FreeQ[{a, b, c, d, e, f, g, h, A, B, C}, x]

Rubi steps

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

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Mathematica [A]  time = 9.18, size = 341, normalized size = 0.55 \[ \frac {2 (e+f x)^{3/2} (g+h x)^{3/2} (b e-a f) \sqrt {\frac {(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \left ((d g-c h) (-2 a d f+b c f+b d e) E\left (\sin ^{-1}\left (\sqrt {\frac {(a f-b e) (g+h x)}{(f g-e h) (a+b x)}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )-d (b g-a h) (d e-c f) \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {(g+h x) (a f-b e)}{(a+b x) (f g-e h)}}\right ),\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )\right )}{(a+b x)^{5/2} \sqrt {c+d x} (b c-a d) (f g-e h)^3 \left (-\frac {(e+f x) (g+h x) (b e-a f) (b g-a h)}{(a+b x)^2 (f g-e h)^2}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*e - a*f)*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*(e + f*x)^(3/2)*(g + h*x)^(3/2)*((b*d*e +
b*c*f - 2*a*d*f)*(d*g - c*h)*EllipticE[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((b*
c - a*d)*(f*g - e*h))/((b*e - a*f)*(d*g - c*h))] - d*(d*e - c*f)*(b*g - a*h)*EllipticF[ArcSin[Sqrt[((-(b*e) +
a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((b*c - a*d)*(f*g - e*h))/((b*e - a*f)*(d*g - c*h))]))/((b*c - a*d)
*(f*g - e*h)^3*(a + b*x)^(5/2)*Sqrt[c + d*x]*(-(((b*e - a*f)*(b*g - a*h)*(e + f*x)*(g + h*x))/((f*g - e*h)^2*(
a + b*x)^2)))^(3/2))

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fricas [F]  time = 7.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (2 \, d f x + d e + c f\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}{b^{2} d f h x^{5} + a^{2} c e g + {\left (b^{2} d f g + {\left (b^{2} d e + {\left (b^{2} c + 2 \, a b d\right )} f\right )} h\right )} x^{4} + {\left ({\left (b^{2} d e + {\left (b^{2} c + 2 \, a b d\right )} f\right )} g + {\left ({\left (b^{2} c + 2 \, a b d\right )} e + {\left (2 \, a b c + a^{2} d\right )} f\right )} h\right )} x^{3} + {\left ({\left ({\left (b^{2} c + 2 \, a b d\right )} e + {\left (2 \, a b c + a^{2} d\right )} f\right )} g + {\left (a^{2} c f + {\left (2 \, a b c + a^{2} d\right )} e\right )} h\right )} x^{2} + {\left (a^{2} c e h + {\left (a^{2} c f + {\left (2 \, a b c + a^{2} d\right )} e\right )} g\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {2 \, d f x + d e + c f}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.14, size = 12900, normalized size = 20.64 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {2 \, d f x + d e + c f}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {c\,f+d\,e+2\,d\,f\,x}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,{\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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