3.121 \(\int \frac{a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{5/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx\)

Optimal. Leaf size=616 \[ -\frac{2 b^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (b B-2 a C)}{\sqrt{a+b x} (b c-a d) (b e-a f) (b g-a h)}+\frac{2 b d \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x} (b B-2 a C)}{\sqrt{c+d x} (b c-a d) (b e-a f) (b g-a h)}+\frac{2 \sqrt{g+h x} (a C d-b B d+b c C) \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} F\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt{c+d x} (b c-a d) \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}-\frac{2 b \sqrt{a+b x} (b B-2 a C) \sqrt{d g-c h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{\sqrt{g+h x} (b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}} \]

[Out]

(2*b*(b*B - 2*a*C)*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^2*(b*B - 2*a*C)*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
*(b*B - 2*a*C)*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*C - b*B*d + a*C*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 = 2.744, antiderivative size = 616, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 62, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ -\frac{2 b^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (b B-2 a C)}{\sqrt{a+b x} (b c-a d) (b e-a f) (b g-a h)}+\frac{2 b d \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x} (b B-2 a C)}{\sqrt{c+d x} (b c-a d) (b e-a f) (b g-a h)}+\frac{2 \sqrt{g+h x} (a C d-b B d+b c C) \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} F\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt{c+d x} (b c-a d) \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}-\frac{2 b \sqrt{a+b x} (b B-2 a C) \sqrt{d g-c h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{\sqrt{g+h x} (b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}} \]

Warning: Unable to verify antiderivative.

[In]  Int[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^(5/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

(2*b*(b*B - 2*a*C)*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^2*(b*B - 2*a*C)*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
*(b*B - 2*a*C)*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*C - b*B*d + a*C*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((C*b**2*x**2+B*b**2*x+B*a*b-C*a**2)/(b*x+a)**(5/2)/(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 = 18.1709, size = 1753, normalized size = 2.85 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^(5/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

(-2*b^2*(b*B - 2*a*C)*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*B) + 2*a*C)*(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]) - ((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))]*(-((b*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*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))])/(-(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))])) + (2*a*C*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[ArcS
in[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))]) - (C*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))
*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))])/(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*(-(b*c) + a*d)*(-(b*e) + a*f)*(-(b*g) + a*h))

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Maple [B]  time = 0.21, size = 9443, normalized size = 15.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^(5/2)/(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. \[ \int \frac{C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\left (b x + a\right )}^{\frac{5}{2}} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/((b*x + a)^(5/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="maxima")

[Out]

integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/((b*x + a)^(5/2)*sqrt(d*x + c)*s
qrt(f*x + e)*sqrt(h*x + g)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/((b*x + a)^(5/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="fricas")

[Out]

integral((C*b*x - C*a + B*b)/((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((C*b**2*x**2+B*b**2*x+B*a*b-C*a**2)/(b*x+a)**(5/2)/(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. \[ \int \frac{C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\left (b x + a\right )}^{\frac{5}{2}} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/((b*x + a)^(5/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="giac")

[Out]

integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/((b*x + a)^(5/2)*sqrt(d*x + c)*s
qrt(f*x + e)*sqrt(h*x + g)), x)