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

Optimal. Leaf size=291 \[ \frac{2 b C \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt{f} h \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}-\frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} (a C h-b B h+b C g) F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt{f} h \sqrt{e+f x} \sqrt{g+h x}} \]

[Out]

(2*b*C*Sqrt[-(d*e) + c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[g + h*x]*Elliptic
E[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*C*g - b*B*h + a*C*h)*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))])/(d*Sqrt[f]*h*Sqrt[e + f*x]*Sqrt[g + h*x])

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Rubi [A]  time = 1.14374, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 60, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2 b C \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt{f} h \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}-\frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} (a C h-b B h+b C g) F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt{f} h \sqrt{e+f x} \sqrt{g+h x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*b*C*Sqrt[-(d*e) + c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[g + h*x]*Elliptic
E[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*C*g - b*B*h + a*C*h)*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))])/(d*Sqrt[f]*h*Sqrt[e + f*x]*Sqrt[g + h*x])

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Rubi in Sympy [A]  time = 150.009, size = 248, normalized size = 0.85 \[ \frac{2 C b \sqrt{\frac{f \left (c + d x\right )}{c f - d e}} \sqrt{g + h x} \sqrt{- c f + d e} E\left (\operatorname{asin}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{- c f + d e}} \right )}\middle | \frac{h \left (c f - d e\right )}{d \left (- e h + f g\right )}\right )}{\sqrt{d} f h \sqrt{\frac{f \left (- g - h x\right )}{e h - f g}} \sqrt{c + d x}} + \frac{2 \sqrt{\frac{d \left (- e - f x\right )}{c f - d e}} \sqrt{\frac{d \left (- g - h x\right )}{c h - d g}} \sqrt{c h - d g} \left (B b h - C a h - C b g\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{h} \sqrt{c + d x}}{\sqrt{c h - d g}} \right )}\middle | \frac{f \left (c h - d g\right )}{h \left (c f - d e\right )}\right )}{d h^{\frac{3}{2}} \sqrt{e + f x} \sqrt{g + h x}} \]

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

[Out]

2*C*b*sqrt(f*(c + d*x)/(c*f - d*e))*sqrt(g + h*x)*sqrt(-c*f + d*e)*elliptic_e(as
in(sqrt(d)*sqrt(e + f*x)/sqrt(-c*f + d*e)), h*(c*f - d*e)/(d*(-e*h + f*g)))/(sqr
t(d)*f*h*sqrt(f*(-g - h*x)/(e*h - f*g))*sqrt(c + d*x)) + 2*sqrt(d*(-e - f*x)/(c*
f - d*e))*sqrt(d*(-g - h*x)/(c*h - d*g))*sqrt(c*h - d*g)*(B*b*h - C*a*h - C*b*g)
*elliptic_f(asin(sqrt(h)*sqrt(c + d*x)/sqrt(c*h - d*g)), f*(c*h - d*g)/(h*(c*f -
d*e)))/(d*h**(3/2)*sqrt(e + f*x)*sqrt(g + h*x))

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Mathematica [C]  time = 3.69264, size = 326, normalized size = 1.12 \[ \frac{2 \left (-i d h (c+d x)^{3/2} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} (a C f-b B f+b C e) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right )|\frac{d f g-c f h}{d e h-c f h}\right )+b C d^2 (e+f x) (g+h x) \sqrt{\frac{d e}{f}-c}+i b C h (c+d x)^{3/2} (d e-c f) \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right )|\frac{d f g-c f h}{d e h-c f h}\right )\right )}{d^2 f h \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} \sqrt{\frac{d e}{f}-c}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*C*d^2*Sqrt[-c + (d*e)/f]*(e + f*x)*(g + h*x) + I*b*C*(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))]*Ell
ipticE[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*C*e - b*B*f + a*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))]*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*S
qrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])

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Maple [B]  time = 0.039, size = 673, normalized size = 2.3 \[ 2\,{\frac{\sqrt{hx+g}\sqrt{fx+e}\sqrt{dx+c}}{hf{d}^{2} \left ( dfh{x}^{3}+cfh{x}^{2}+deh{x}^{2}+dfg{x}^{2}+cehx+cfgx+degx+ceg \right ) } \left ( B{\it EllipticF} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) bcdfh-B{\it EllipticF} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) b{d}^{2}eh-C{\it EllipticF} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) acdfh+C{\it EllipticF} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) a{d}^{2}eh-C{\it EllipticF} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) bcdfg+C{\it EllipticF} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) b{d}^{2}eg-C{\it EllipticE} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) b{c}^{2}fh+C{\it EllipticE} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) bcdeh+C{\it EllipticE} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) bcdfg-C{\it EllipticE} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) b{d}^{2}eg \right ) \sqrt{-{\frac{ \left ( fx+e \right ) d}{cf-de}}}\sqrt{-{\frac{ \left ( hx+g \right ) d}{ch-dg}}}\sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}}} \]

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

[Out]

2*(B*EllipticF(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b*c*
d*f*h-B*EllipticF(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b
*d^2*e*h-C*EllipticF(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2)
)*a*c*d*f*h+C*EllipticF(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1
/2))*a*d^2*e*h-C*EllipticF(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))
^(1/2))*b*c*d*f*g+C*EllipticF(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*
g))^(1/2))*b*d^2*e*g-C*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h
-d*g))^(1/2))*b*c^2*f*h+C*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(
c*h-d*g))^(1/2))*b*c*d*e*h+C*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/
f/(c*h-d*g))^(1/2))*b*c*d*f*g-C*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)
*h/f/(c*h-d*g))^(1/2))*b*d^2*e*g)*(-(f*x+e)*d/(c*f-d*e))^(1/2)*(-(h*x+g)*d/(c*h-
d*g))^(1/2)*((d*x+c)*f/(c*f-d*e))^(1/2)/h/f/d^2*(h*x+g)^(1/2)*(f*x+e)^(1/2)*(d*x
+c)^(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., size = 0, normalized size = 0. \[ \int \frac{C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\left (b x + a\right )} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]

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

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)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="fricas")

[Out]

integral((C*b*x - C*a + B*b)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B b - C a + C b x}{\sqrt{c + d x} \sqrt{e + f x} \sqrt{g + h x}}\, dx \]

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

[Out]

Integral((B*b - C*a + C*b*x)/(sqrt(c + d*x)*sqrt(e + f*x)*sqrt(g + h*x)), x)

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

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)*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)*sqrt(d*x + c)*sqrt(f*
x + e)*sqrt(h*x + g)), x)