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

Optimal. Leaf size=606 \[ -\frac{2 b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (A b-a B)}{\sqrt{a+b x} (b c-a d) (b e-a f) (b g-a h)}+\frac{2 d \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x} (A b-a B)}{\sqrt{c+d x} (b c-a d) (b e-a f) (b g-a h)}+\frac{2 \sqrt{g+h x} (B c-A d) \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} F\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt{c+d x} (b c-a d) \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}-\frac{2 \sqrt{a+b x} (A b-a B) \sqrt{d g-c h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{\sqrt{g+h x} (b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}} \]

[Out]

(2*(A*b - a*B)*d*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*(A*b - a*B)*Sqrt[c + d*x]*Sqrt[e + f*x]*S
qrt[g + h*x])/((b*c - a*d)*(b*e - a*f)*(b*g - a*h)*Sqrt[a + b*x]) - (2*(A*b - a*
B)*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])/(Sqrt
[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*(B*c - A*d)*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
])/(Sqrt[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)))])

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Rubi [A]  time = 2.44749, antiderivative size = 606, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (A b-a B)}{\sqrt{a+b x} (b c-a d) (b e-a f) (b g-a h)}+\frac{2 d \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x} (A b-a B)}{\sqrt{c+d x} (b c-a d) (b e-a f) (b g-a h)}+\frac{2 \sqrt{g+h x} (B c-A d) \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} F\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt{c+d x} (b c-a d) \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}-\frac{2 \sqrt{a+b x} (A b-a B) \sqrt{d g-c h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{\sqrt{g+h x} (b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*(A*b - a*B)*d*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*(A*b - a*B)*Sqrt[c + d*x]*Sqrt[e + f*x]*S
qrt[g + h*x])/((b*c - a*d)*(b*e - a*f)*(b*g - a*h)*Sqrt[a + b*x]) - (2*(A*b - a*
B)*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])/(Sqrt
[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*(B*c - A*d)*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
])/(Sqrt[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)))])

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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)**(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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Mathematica [B]  time = 17.5008, size = 1749, normalized size = 2.89 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-2*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)*Sqrt[a + b*x]) + ((-2*(A*b - a*B)*(a + b*x)^(5/2)*(d + (b*c)/(
a + b*x) - (a*d)/(a + b*x))*(f + (b*e)/(a + b*x) - (a*f)/(a + b*x))*(h + (b*g)/(
a + b*x) - (a*h)/(a + b*x)))/(Sqrt[c + ((a + b*x)*(d - (a*d)/(a + b*x)))/b]*Sqrt
[e + ((a + b*x)*(f - (a*f)/(a + b*x)))/b]*Sqrt[g + ((a + b*x)*(h - (a*h)/(a + b*
x)))/b]) - (2*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)*(a + b*x)^(3/2)*Sqrt[(d + (b*c
)/(a + b*x) - (a*d)/(a + b*x))*(f + (b*e)/(a + b*x) - (a*f)/(a + b*x))*(h + (b*g
)/(a + b*x) - (a*h)/(a + b*x))]*(-((A*b*Sqrt[((b*c - a*d)*(b*g - a*h)*(-(d/(-(b*
c) + a*d)) + (a + b*x)^(-1)))/(b*d*g - b*c*h)]*(-(f/(-(b*e) + a*f)) + (a + b*x)^
(-1))*Sqrt[(-(h/(-(b*g) + a*h)) + (a + b*x)^(-1))/(f/(-(b*e) + a*f) - h/(-(b*g)
+ a*h))]*(-(((b*d*g - b*c*h)*EllipticE[ArcSin[Sqrt[((b*e - a*f)*(h + (b*g)/(a +
b*x) - (a*h)/(a + b*x)))/(b*(-(f*g) + e*h))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/(
(-(b*e) + a*f)*(-(d*g) + c*h))])/((b*c - a*d)*(b*g - a*h))) - (d*EllipticF[ArcSi
n[Sqrt[((b*e - a*f)*(h + (b*g)/(a + b*x) - (a*h)/(a + b*x)))/(b*(-(f*g) + e*h))]
], ((-(b*c) + a*d)*(-(f*g) + e*h))/((-(b*e) + a*f)*(-(d*g) + c*h))])/(-(b*c) + a
*d)))/(Sqrt[(-(f/(-(b*e) + a*f)) + (a + b*x)^(-1))/(-(f/(-(b*e) + a*f)) + h/(-(b
*g) + a*h))]*Sqrt[(d + (b*c - a*d)/(a + b*x))*(f + (b*e - a*f)/(a + b*x))*(h + (
b*g - a*h)/(a + b*x))])) + (a*B*Sqrt[((b*c - a*d)*(b*g - a*h)*(-(d/(-(b*c) + a*d
)) + (a + b*x)^(-1)))/(b*d*g - b*c*h)]*(-(f/(-(b*e) + a*f)) + (a + b*x)^(-1))*Sq
rt[(-(h/(-(b*g) + a*h)) + (a + b*x)^(-1))/(f/(-(b*e) + a*f) - h/(-(b*g) + a*h))]
*(-(((b*d*g - b*c*h)*EllipticE[ArcSin[Sqrt[((b*e - a*f)*(h + (b*g)/(a + b*x) - (
a*h)/(a + b*x)))/(b*(-(f*g) + e*h))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((-(b*e)
+ a*f)*(-(d*g) + c*h))])/((b*c - a*d)*(b*g - a*h))) - (d*EllipticF[ArcSin[Sqrt[(
(b*e - a*f)*(h + (b*g)/(a + b*x) - (a*h)/(a + b*x)))/(b*(-(f*g) + e*h))]], ((-(b
*c) + a*d)*(-(f*g) + e*h))/((-(b*e) + a*f)*(-(d*g) + c*h))])/(-(b*c) + a*d)))/(S
qrt[(-(f/(-(b*e) + a*f)) + (a + b*x)^(-1))/(-(f/(-(b*e) + a*f)) + h/(-(b*g) + a*
h))]*Sqrt[(d + (b*c - a*d)/(a + b*x))*(f + (b*e - a*f)/(a + b*x))*(h + (b*g - a*
h)/(a + b*x))]) - (B*Sqrt[(-(d/(-(b*c) + a*d)) + (a + b*x)^(-1))/(-(d/(-(b*c) +
a*d)) + h/(-(b*g) + a*h))]*Sqrt[(-(f/(-(b*e) + a*f)) + (a + b*x)^(-1))/(-(f/(-(b
*e) + a*f)) + h/(-(b*g) + a*h))]*(-(h/(-(b*g) + a*h)) + (a + b*x)^(-1))*Elliptic
F[ArcSin[Sqrt[((-(b*e) + a*f)*(-h - (b*g)/(a + b*x) + (a*h)/(a + b*x)))/(b*(-(f*
g) + e*h))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((-(b*e) + a*f)*(-(d*g) + c*h))])/
(Sqrt[(-(h/(-(b*g) + a*h)) + (a + b*x)^(-1))/(f/(-(b*e) + a*f) - h/(-(b*g) + a*h
))]*Sqrt[(d + (b*c - a*d)/(a + b*x))*(f + (b*e - a*f)/(a + b*x))*(h + (b*g - a*h
)/(a + b*x))])))/(Sqrt[c + ((a + b*x)*(d - (a*d)/(a + b*x)))/b]*Sqrt[e + ((a + b
*x)*(f - (a*f)/(a + b*x)))/b]*Sqrt[g + ((a + b*x)*(h - (a*h)/(a + b*x)))/b]))/(b
^2*(-(b*c) + a*d)*(-(b*e) + a*f)*(-(b*g) + a*h))

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Maple [B]  time = 0.2, size = 9328, normalized size = 15.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(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., size = 0, normalized size = 0. \[ \int \frac{B x + A}{{\left (b x + a\right )}^{\frac{3}{2}} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(3/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{B x + A}{{\left (b x + a\right )}^{\frac{3}{2}} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(3/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="fricas")

[Out]

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

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Sympy [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)**(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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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{{\left (b x + a\right )}^{\frac{3}{2}} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(3/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="giac")

[Out]

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