3.111 \(\int \frac{1}{(a+b x)^{3/2} (c+d x)^{3/2} \sqrt{e+f x} \sqrt{g+h x}} \, dx\)

Optimal. Leaf size=786 \[ \frac{2 b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} \left (a^2 d^2 f h-a b d^2 (e h+f g)+b^2 \left (c^2 f h-c d (e h+f g)+2 d^2 e g\right )\right )}{\sqrt{a+b x} (b c-a d)^2 (b e-a f) (b g-a h) (d e-c f) (d g-c h)}-\frac{2 \sqrt{c+d x} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}} \left (a^2 d^2 f h-a b d^2 (e h+f g)+b^2 \left (c^2 f h-c d (e h+f g)+2 d^2 e g\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt{g+h x} (b c-a d)^2 (b e-a f) \sqrt{b g-a h} (d e-c f) (d g-c h) \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}}}-\frac{2 b^3 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{\sqrt{a+b x} (b c-a d)^2 (b e-a f) (b g-a h)}-\frac{2 d^3 \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{\sqrt{c+d x} (b c-a d)^2 (d e-c f) (d g-c h)}-\frac{4 b d \sqrt{g+h x} \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} F\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt{c+d x} (b c-a d)^2 \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}} \]

[Out]

(-2*d^3*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((b*c - a*d)^2*(d*e - c*f)*(d
*g - c*h)*Sqrt[c + d*x]) - (2*b^3*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((b
*c - a*d)^2*(b*e - a*f)*(b*g - a*h)*Sqrt[a + b*x]) + (2*b*(a^2*d^2*f*h - a*b*d^2
*(f*g + e*h) + b^2*(2*d^2*e*g + c^2*f*h - c*d*(f*g + e*h)))*Sqrt[c + d*x]*Sqrt[e
+ f*x]*Sqrt[g + h*x])/((b*c - a*d)^2*(b*e - a*f)*(d*e - c*f)*(b*g - a*h)*(d*g -
c*h)*Sqrt[a + b*x]) - (2*Sqrt[f*g - e*h]*(a^2*d^2*f*h - a*b*d^2*(f*g + e*h) + b
^2*(2*d^2*e*g + c^2*f*h - c*d*(f*g + e*h)))*Sqrt[c + d*x]*Sqrt[-(((b*e - a*f)*(g
+ h*x))/((f*g - e*h)*(a + b*x)))]*EllipticE[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)^2*(b*e - a*f)*(d*e - c*f)*Sqrt[b*g - a*h]*(d*g - c*h
)*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Sqrt[g + h*x]) - (4*b*d*
Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Sqrt[g + h*x]*EllipticF[Ar
cSin[(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)^2*Sqrt[b*g - a*h]*S
qrt[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 [F]  time = 0.147196, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \text{Int}\left (\frac{1}{(a+b x)^{3/2} (c+d x)^{3/2} \sqrt{e+f x} \sqrt{g+h x}},x\right ) \]

Verification is Not applicable to the result.

[In]  Int[1/((a + b*x)^(3/2)*(c + d*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

Defer[Int][1/((a + b*x)^(3/2)*(c + d*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]), x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \sqrt{e + f x} \sqrt{g + h x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(3/2)/(d*x+c)**(3/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)

[Out]

Integral(1/((a + b*x)**(3/2)*(c + d*x)**(3/2)*sqrt(e + f*x)*sqrt(g + h*x)), x)

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Mathematica [B]  time = 22.0217, size = 7061, normalized size = 8.98 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((a + b*x)^(3/2)*(c + d*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

Result too large to show

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Maple [B]  time = 0.293, size = 21094, normalized size = 26.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(3/2)/(d*x+c)^(3/2)/(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. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{3}{2}} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*(d*x + c)^(3/2)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(3/2)*(d*x + c)^(3/2)*sqrt(f*x + e)*sqrt(h*x + g)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b d x^{2} + a c +{\left (b c + a d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*(d*x + c)^(3/2)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="fricas")

[Out]

integral(1/((b*d*x^2 + a*c + (b*c + a*d)*x)*sqrt(b*x + a)*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(1/(b*x+a)**(3/2)/(d*x+c)**(3/2)/(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. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{3}{2}} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*(d*x + c)^(3/2)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(3/2)*(d*x + c)^(3/2)*sqrt(f*x + e)*sqrt(h*x + g)), x)