3.127 \(\int \frac{A+B x}{(a+b x)^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx\)

Optimal. Leaf size=678 \[ \frac{\sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left (a^3 (-B) d f h+3 a^2 A b d f h+a b^2 (B (c e h+c f g+d e g)-2 A (c f h+d e h+d f g))-b^3 (2 B c e g-A (c e h+c f g+d e g))\right ) \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)^2 (b e-a f) (b g-a h)}-\frac{b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (A b-a B)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}-\frac{\sqrt{f} (A b-a B) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt{e+f x} \sqrt{g+h x} (b c-a d) (b e-a f)}+\frac{\sqrt{f} \sqrt{g+h x} (A b-a B) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{\sqrt{e+f x} (b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{d (g+h x)}{d g-c h}}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 4.59057, antiderivative size = 678, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left (a^3 (-B) d f h+3 a^2 A b d f h+a b^2 (B (c e h+c f g+d e g)-2 A (c f h+d e h+d f g))-b^3 (2 B c e g-A (c e h+c f g+d e g))\right ) \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)^2 (b e-a f) (b g-a h)}-\frac{b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (A b-a B)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}-\frac{\sqrt{f} (A b-a B) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt{e+f x} \sqrt{g+h x} (b c-a d) (b e-a f)}+\frac{\sqrt{f} \sqrt{g+h x} (A b-a B) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{\sqrt{e+f x} (b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{d (g+h x)}{d g-c h}}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(b*x+a)**2/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 19.3543, size = 14516, normalized size = 21.41 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

Result too large to show

_______________________________________________________________________________________

Maple [B]  time = 0.107, size = 13380, normalized size = 19.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b*x+a)^2/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x)

[Out]

result too large to display

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^2*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^2*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(b*x+a)**2/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{{\left (b x + a\right )}^{2} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^2*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="giac")

[Out]

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