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

Optimal. Leaf size=313 \[ \frac{2 B \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b d \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}-\frac{2 \left (A-\frac{a B}{b}\right ) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{\sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 2.65882, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.175 \[ \frac{2 B \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b d \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}-\frac{2 \left (A-\frac{a B}{b}\right ) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{\sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

Timed out

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Mathematica [C]  time = 2.26074, size = 244, normalized size = 0.78 \[ \frac{2 i \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \left (d (a B-A b) \Pi \left (\frac{(b c-a d) f}{b (c f-d e)};i \sinh ^{-1}\left (\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right )|\frac{d f g-c f h}{d e h-c f h}\right )+b (A d-B c) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right )|\frac{d f g-c f h}{d e h-c f h}\right )\right )}{b f \sqrt{g+h x} (a d-b c) \sqrt{\frac{d e}{f}-c} \sqrt{\frac{d (e+f x)}{f (c+d x)}}} \]

Antiderivative was successfully verified.

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

[Out]

((2*I)*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*(b*(-(B*c) + A*d)*Ellipti
cF[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)]
+ (-(A*b) + a*B)*d*EllipticPi[((b*c - a*d)*f)/(b*(-(d*e) + c*f)), I*ArcSinh[Sqr
t[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)]))/(b*(-(b*c) +
a*d)*Sqrt[-c + (d*e)/f]*f*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[g + h*x])

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Maple [B]  time = 0.046, size = 665, normalized size = 2.1 \[ 2\,{\frac{\sqrt{dx+c}\sqrt{fx+e}\sqrt{hx+g}}{ \left ( ad-bc \right ) bfd \left ( dfh{x}^{3}+cfh{x}^{2}+deh{x}^{2}+dfg{x}^{2}+cehx+cfgx+degx+ceg \right ) }\sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}}\sqrt{-{\frac{ \left ( hx+g \right ) d}{ch-dg}}}\sqrt{-{\frac{ \left ( fx+e \right ) d}{cf-de}}} \left ( A{\it EllipticPi} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},-{\frac{ \left ( cf-de \right ) b}{f \left ( ad-bc \right ) }},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) bcdf-A{\it EllipticPi} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},-{\frac{ \left ( cf-de \right ) b}{f \left ( ad-bc \right ) }},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) b{d}^{2}e+B{\it EllipticF} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) acdf-B{\it EllipticF} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) a{d}^{2}e-B{\it EllipticF} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) b{c}^{2}f+B{\it EllipticF} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) bcde-B{\it EllipticPi} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},-{\frac{ \left ( cf-de \right ) b}{f \left ( ad-bc \right ) }},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) acdf+B{\it EllipticPi} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},-{\frac{ \left ( cf-de \right ) b}{f \left ( ad-bc \right ) }},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) a{d}^{2}e \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/d/f/b*((d*x+c)*f/(c*f-d*e))^(1/2)*(-
(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)*d/(c*f-d*e))^(1/2)*(A*EllipticPi(((d*x+c)*f
/(c*f-d*e))^(1/2),-(c*f-d*e)*b/f/(a*d-b*c),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b*c*
d*f-A*EllipticPi(((d*x+c)*f/(c*f-d*e))^(1/2),-(c*f-d*e)*b/f/(a*d-b*c),((c*f-d*e)
*h/f/(c*h-d*g))^(1/2))*b*d^2*e+B*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-B*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-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^2*f+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*e-B*EllipticPi(((d*x+c)*f/(c*f-d*e))^(1/2),-(c*
f-d*e)*b/f/(a*d-b*c),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*a*c*d*f+B*EllipticPi(((d*x
+c)*f/(c*f-d*e))^(1/2),-(c*f-d*e)*b/f/(a*d-b*c),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))
*a*d^2*e)/(a*d-b*c)/(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. \[ \int \frac{B x + A}{{\left (b x + a\right )} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)/((a + 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. \[ \int \frac{B x + A}{{\left (b x + a\right )} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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