3.1.3 \(\int \frac {A+B x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [3]

Optimal. Leaf size=284 \[ \frac {2 B \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 \sqrt {-d e+c f} (B g-A h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {g+h x}} \]

[Out]

2*B*EllipticE(f^(1/2)*(d*x+c)^(1/2)/(c*f-d*e)^(1/2),((-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)*(h*x+g)^(1/2)/d/h/f^(1/2)/(f*x+e)^(1/2)/(d*(h*x+g)/(-c*h+d*g))^(1/2)-2*(-A*h+B*g)*Ellipt
icF(f^(1/2)*(d*x+c)^(1/2)/(c*f-d*e)^(1/2),((-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)/d/h/f^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)

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Rubi [A]
time = 0.11, antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {164, 115, 114, 122, 121} \begin {gather*} \frac {2 B \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\text {ArcSin}\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 )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 (B g-A h) \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}} F\left (\text {ArcSin}\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 )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {g+h x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

(2*B*Sqrt[-(d*e) + c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[g + h*x]*EllipticE[ArcSin[(Sqrt[f]*Sqrt[c + d*x])
/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(d*Sqrt[f]*h*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g - c
*h)]) - (2*Sqrt[-(d*e) + c*f]*(B*g - A*h)*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)]*Elli
pticF[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(d*Sqrt[f]*h*Sqrt[
e + f*x]*Sqrt[g + h*x])

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 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 {A+B x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx &=\frac {B \int \frac {\sqrt {g+h x}}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{h}+\frac {(-B g+A h) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{h}\\ &=\frac {\left ((-B g+A h) \sqrt {\frac {d (e+f x)}{d e-c f}}\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {g+h x}} \, dx}{h \sqrt {e+f x}}+\frac {\left (B \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x}\right ) \int \frac {\sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}} \, dx}{h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}\\ &=\frac {2 B \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\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 )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}+\frac {\left ((-B g+A h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}}\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}} \, dx}{h \sqrt {e+f x} \sqrt {g+h x}}\\ &=\frac {2 B \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\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 )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 \sqrt {-d e+c f} (B g-A h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} F\left (\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 )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {g+h x}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 19.41, size = 319, normalized size = 1.12 \begin {gather*} -\frac {2 \left (-B d^2 \sqrt {-c+\frac {d e}{f}} (e+f x) (g+h x)-i B (d e-c f) h (c+d x)^{3/2} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right )|\frac {d f g-c f h}{d e h-c f h}\right )+i d (B e-A f) h (c+d x)^{3/2} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right )|\frac {d f g-c f h}{d e h-c f h}\right )\right )}{d^2 \sqrt {-c+\frac {d e}{f}} f h \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

(-2*(-(B*d^2*Sqrt[-c + (d*e)/f]*(e + f*x)*(g + h*x)) - I*B*(d*e - c*f)*h*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f
*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticE[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g -
c*f*h)/(d*e*h - c*f*h)] + I*d*(B*e - A*f)*h*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*
x))/(h*(c + d*x))]*EllipticF[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)]))/(
d^2*Sqrt[-c + (d*e)/f]*f*h*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(565\) vs. \(2(250)=500\).
time = 0.12, size = 566, normalized size = 1.99

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (f x +e \right ) \left (h x +g \right )}\, \left (\frac {2 A \left (-\frac {e}{f}+\frac {g}{h}\right ) \sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{\sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}+\frac {2 B \left (-\frac {e}{f}+\frac {g}{h}\right ) \sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, \left (\left (-\frac {g}{h}+\frac {c}{d}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )-\frac {c \EllipticF \left (\sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{d}\right )}{\sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}\right )}{\sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}\)	\(498\)
default	\(-\frac {2 \left (A \EllipticF \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) d e \,h^{2}-A \EllipticF \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) d f g h -B \EllipticF \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) c e \,h^{2}+B \EllipticF \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) c f g h +B \EllipticE \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) c e \,h^{2}-B \EllipticE \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) c f g h -B \EllipticE \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) d e g h +B \EllipticE \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) d f \,g^{2}\right ) \sqrt {\frac {\left (f x +e \right ) h}{e h -f g}}\, \sqrt {\frac {\left (d x +c \right ) h}{c h -d g}}\, \sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}\, \sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}{h^{2} f d \left (d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g \right )}\)	\(566\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2*(A*EllipticF((-(h*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2))*d*e*h^2-A*EllipticF((-(h*x+g)*f/
(e*h-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2))*d*f*g*h-B*EllipticF((-(h*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*g)*
d/f/(c*h-d*g))^(1/2))*c*e*h^2+B*EllipticF((-(h*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2))*c*f*g*
h+B*EllipticE((-(h*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2))*c*e*h^2-B*EllipticE((-(h*x+g)*f/(e
*h-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2))*c*f*g*h-B*EllipticE((-(h*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*g)*d/
f/(c*h-d*g))^(1/2))*d*e*g*h+B*EllipticE((-(h*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2))*d*f*g^2)
*((f*x+e)*h/(e*h-f*g))^(1/2)*((d*x+c)*h/(c*h-d*g))^(1/2)*(-(h*x+g)*f/(e*h-f*g))^(1/2)/h^2/f/d*(d*x+c)^(1/2)*(f
*x+e)^(1/2)*(h*x+g)^(1/2)/(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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((B*x + A)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.36, size = 725, normalized size = 2.55 \begin {gather*} -\frac {2 \, {\left (3 \, \sqrt {d f h} B d f h {\rm weierstrassZeta}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - c d f^{2} g h + c^{2} f^{2} h^{2} + d^{2} h^{2} e^{2} - {\left (d^{2} f g h + c d f h^{2}\right )} e\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, c d^{2} f^{3} g^{2} h - 3 \, c^{2} d f^{3} g h^{2} + 2 \, c^{3} f^{3} h^{3} + 2 \, d^{3} h^{3} e^{3} - 3 \, {\left (d^{3} f g h^{2} + c d^{2} f h^{3}\right )} e^{2} - 3 \, {\left (d^{3} f^{2} g^{2} h - 4 \, c d^{2} f^{2} g h^{2} + c^{2} d f^{2} h^{3}\right )} e\right )}}{27 \, d^{3} f^{3} h^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - c d f^{2} g h + c^{2} f^{2} h^{2} + d^{2} h^{2} e^{2} - {\left (d^{2} f g h + c d f h^{2}\right )} e\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, c d^{2} f^{3} g^{2} h - 3 \, c^{2} d f^{3} g h^{2} + 2 \, c^{3} f^{3} h^{3} + 2 \, d^{3} h^{3} e^{3} - 3 \, {\left (d^{3} f g h^{2} + c d^{2} f h^{3}\right )} e^{2} - 3 \, {\left (d^{3} f^{2} g^{2} h - 4 \, c d^{2} f^{2} g h^{2} + c^{2} d f^{2} h^{3}\right )} e\right )}}{27 \, d^{3} f^{3} h^{3}}, \frac {3 \, d f h x + d f g + c f h + d h e}{3 \, d f h}\right )\right ) + {\left (B d f g + B d h e + {\left (B c - 3 \, A d\right )} f h\right )} \sqrt {d f h} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - c d f^{2} g h + c^{2} f^{2} h^{2} + d^{2} h^{2} e^{2} - {\left (d^{2} f g h + c d f h^{2}\right )} e\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, c d^{2} f^{3} g^{2} h - 3 \, c^{2} d f^{3} g h^{2} + 2 \, c^{3} f^{3} h^{3} + 2 \, d^{3} h^{3} e^{3} - 3 \, {\left (d^{3} f g h^{2} + c d^{2} f h^{3}\right )} e^{2} - 3 \, {\left (d^{3} f^{2} g^{2} h - 4 \, c d^{2} f^{2} g h^{2} + c^{2} d f^{2} h^{3}\right )} e\right )}}{27 \, d^{3} f^{3} h^{3}}, \frac {3 \, d f h x + d f g + c f h + d h e}{3 \, d f h}\right )\right )}}{3 \, d^{2} f^{2} h^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*sqrt(d*f*h)*B*d*f*h*weierstrassZeta(4/3*(d^2*f^2*g^2 - c*d*f^2*g*h + c^2*f^2*h^2 + d^2*h^2*e^2 - (d^2*
f*g*h + c*d*f*h^2)*e)/(d^2*f^2*h^2), -4/27*(2*d^3*f^3*g^3 - 3*c*d^2*f^3*g^2*h - 3*c^2*d*f^3*g*h^2 + 2*c^3*f^3*
h^3 + 2*d^3*h^3*e^3 - 3*(d^3*f*g*h^2 + c*d^2*f*h^3)*e^2 - 3*(d^3*f^2*g^2*h - 4*c*d^2*f^2*g*h^2 + c^2*d*f^2*h^3
)*e)/(d^3*f^3*h^3), weierstrassPInverse(4/3*(d^2*f^2*g^2 - c*d*f^2*g*h + c^2*f^2*h^2 + d^2*h^2*e^2 - (d^2*f*g*
h + c*d*f*h^2)*e)/(d^2*f^2*h^2), -4/27*(2*d^3*f^3*g^3 - 3*c*d^2*f^3*g^2*h - 3*c^2*d*f^3*g*h^2 + 2*c^3*f^3*h^3
+ 2*d^3*h^3*e^3 - 3*(d^3*f*g*h^2 + c*d^2*f*h^3)*e^2 - 3*(d^3*f^2*g^2*h - 4*c*d^2*f^2*g*h^2 + c^2*d*f^2*h^3)*e)
/(d^3*f^3*h^3), 1/3*(3*d*f*h*x + d*f*g + c*f*h + d*h*e)/(d*f*h))) + (B*d*f*g + B*d*h*e + (B*c - 3*A*d)*f*h)*sq
rt(d*f*h)*weierstrassPInverse(4/3*(d^2*f^2*g^2 - c*d*f^2*g*h + c^2*f^2*h^2 + d^2*h^2*e^2 - (d^2*f*g*h + c*d*f*
h^2)*e)/(d^2*f^2*h^2), -4/27*(2*d^3*f^3*g^3 - 3*c*d^2*f^3*g^2*h - 3*c^2*d*f^3*g*h^2 + 2*c^3*f^3*h^3 + 2*d^3*h^
3*e^3 - 3*(d^3*f*g*h^2 + c*d^2*f*h^3)*e^2 - 3*(d^3*f^2*g^2*h - 4*c*d^2*f^2*g*h^2 + c^2*d*f^2*h^3)*e)/(d^3*f^3*
h^3), 1/3*(3*d*f*h*x + d*f*g + c*f*h + d*h*e)/(d*f*h)))/(d^2*f^2*h^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{\sqrt {c + d x} \sqrt {e + f x} \sqrt {g + h x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)

[Out]

Integral((A + B*x)/(sqrt(c + d*x)*sqrt(e + f*x)*sqrt(g + h*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((B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="giac")

[Out]

integrate((B*x + A)/(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 \begin {gather*} \int \frac {A+B\,x}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\sqrt {c+d\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(c + d*x)^(1/2)),x)

[Out]

int((A + B*x)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(c + d*x)^(1/2)), x)

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