3.71 \(\int \frac{1}{(a+b x) (c+d x)^{3/2} \sqrt{e+f x} \sqrt{g+h x}} \, dx\)
Optimal. Leaf size=393 \[ \frac{2 d^2 \sqrt{e+f x} \sqrt{g+h x}}{\sqrt{c+d x} (b c-a d) (d e-c f) (d g-c h)}-\frac{2 d \sqrt{h} \sqrt{c+d x} \sqrt{e h-f g} \sqrt{\frac{f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac{\sqrt{h} \sqrt{e+f x}}{\sqrt{e h-f g}}\right )|-\frac{d (f g-e h)}{(d e-c f) h}\right )}{\sqrt{g+h x} (b c-a d) (d e-c f) (d g-c h) \sqrt{-\frac{f (c+d x)}{d e-c f}}}-\frac{2 b \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{\sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)^2} \]
[Out]
(2*d^2*Sqrt[e + f*x]*Sqrt[g + h*x])/((b*c - a*d)*(d*e - c*f)*(d*g - c*h)*Sqrt[c
+ d*x]) - (2*d*Sqrt[h]*Sqrt[-(f*g) + e*h]*Sqrt[c + d*x]*Sqrt[(f*(g + h*x))/(f*g
- e*h)]*EllipticE[ArcSin[(Sqrt[h]*Sqrt[e + f*x])/Sqrt[-(f*g) + e*h]], -((d*(f*g
- e*h))/((d*e - c*f)*h))])/((b*c - a*d)*(d*e - c*f)*(d*g - c*h)*Sqrt[-((f*(c + d
*x))/(d*e - c*f))]*Sqrt[g + h*x]) - (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)]*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)^2*Sqrt[f]*Sqrt[e + f*x]*Sqrt[g + h*x])
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Rubi [A] time = 2.53704, antiderivative size = 393, normalized size of antiderivative = 1.,
number of steps used = 10, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229
\[ \frac{2 d^2 \sqrt{e+f x} \sqrt{g+h x}}{\sqrt{c+d x} (b c-a d) (d e-c f) (d g-c h)}-\frac{2 d \sqrt{h} \sqrt{c+d x} \sqrt{e h-f g} \sqrt{\frac{f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac{\sqrt{h} \sqrt{e+f x}}{\sqrt{e h-f g}}\right )|-\frac{d (f g-e h)}{(d e-c f) h}\right )}{\sqrt{g+h x} (b c-a d) (d e-c f) (d g-c h) \sqrt{-\frac{f (c+d x)}{d e-c f}}}-\frac{2 b \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{\sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)*(c + d*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
[Out]
(2*d^2*Sqrt[e + f*x]*Sqrt[g + h*x])/((b*c - a*d)*(d*e - c*f)*(d*g - c*h)*Sqrt[c
+ d*x]) - (2*d*Sqrt[h]*Sqrt[-(f*g) + e*h]*Sqrt[c + d*x]*Sqrt[(f*(g + h*x))/(f*g
- e*h)]*EllipticE[ArcSin[(Sqrt[h]*Sqrt[e + f*x])/Sqrt[-(f*g) + e*h]], -((d*(f*g
- e*h))/((d*e - c*f)*h))])/((b*c - a*d)*(d*e - c*f)*(d*g - c*h)*Sqrt[-((f*(c + d
*x))/(d*e - c*f))]*Sqrt[g + h*x]) - (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)]*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)^2*Sqrt[f]*Sqrt[e + f*x]*Sqrt[g + h*x])
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Rubi in Sympy [A] time = 140.58, size = 316, normalized size = 0.8 \[ - \frac{2 b \sqrt{\frac{d \left (- e - f x\right )}{c f - d e}} \sqrt{\frac{d \left (- g - h x\right )}{c h - d g}} \Pi \left (- \frac{b \left (c h - d g\right )}{h \left (a d - b c\right )}; \operatorname{asin}{\left (\sqrt{\frac{h}{c h - d g}} \sqrt{c + d x} \right )}\middle | \frac{f \left (c h - d g\right )}{h \left (c f - d e\right )}\right )}{\sqrt{\frac{h}{c h - d g}} \sqrt{e + f x} \sqrt{g + h x} \left (a d - b c\right )^{2}} - \frac{2 d^{2} \sqrt{e + f x} \sqrt{g + h x}}{\sqrt{c + d x} \left (a d - b c\right ) \left (c f - d e\right ) \left (c h - d g\right )} + \frac{2 d \sqrt{f} \sqrt{\frac{h \left (e + f x\right )}{e h - f g}} \sqrt{c + d x} \sqrt{- e h + f g} E\left (\operatorname{asin}{\left (\frac{\sqrt{f} \sqrt{g + h x}}{\sqrt{- e h + f g}} \right )}\middle | \frac{d \left (e h - f g\right )}{f \left (c h - d g\right )}\right )}{\sqrt{\frac{h \left (c + d x\right )}{c h - d g}} \sqrt{e + f x} \left (a d - b c\right ) \left (c f - d e\right ) \left (c h - d g\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)/(d*x+c)**(3/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)
[Out]
-2*b*sqrt(d*(-e - f*x)/(c*f - d*e))*sqrt(d*(-g - h*x)/(c*h - d*g))*elliptic_pi(-
b*(c*h - d*g)/(h*(a*d - b*c)), asin(sqrt(h/(c*h - d*g))*sqrt(c + d*x)), f*(c*h -
d*g)/(h*(c*f - d*e)))/(sqrt(h/(c*h - d*g))*sqrt(e + f*x)*sqrt(g + h*x)*(a*d - b
*c)**2) - 2*d**2*sqrt(e + f*x)*sqrt(g + h*x)/(sqrt(c + d*x)*(a*d - b*c)*(c*f - d
*e)*(c*h - d*g)) + 2*d*sqrt(f)*sqrt(h*(e + f*x)/(e*h - f*g))*sqrt(c + d*x)*sqrt(
-e*h + f*g)*elliptic_e(asin(sqrt(f)*sqrt(g + h*x)/sqrt(-e*h + f*g)), d*(e*h - f*
g)/(f*(c*h - d*g)))/(sqrt(h*(c + d*x)/(c*h - d*g))*sqrt(e + f*x)*(a*d - b*c)*(c*
f - d*e)*(c*h - d*g))
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Mathematica [C] time = 14.3383, size = 1698, normalized size = 4.32 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)*(c + d*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
[Out]
(2*d^2*Sqrt[e + f*x]*Sqrt[g + h*x])/((b*c - a*d)*(-(d*e) + c*f)*(-(d*g) + c*h)*S
qrt[c + d*x]) - (2*(-(((c + d*x)^(3/2)*(f + (d*e)/(c + d*x) - (c*f)/(c + d*x))*(
h + (d*g)/(c + d*x) - (c*h)/(c + d*x)))/(Sqrt[e + ((c + d*x)*(f - (c*f)/(c + d*x
)))/d]*Sqrt[g + ((c + d*x)*(h - (c*h)/(c + d*x)))/d])) + ((c + d*x)^3*(b - (b*c)
/(c + d*x) + (a*d)/(c + d*x))*Sqrt[f + (d*e)/(c + d*x) - (c*f)/(c + d*x)]*Sqrt[h
+ (d*g)/(c + d*x) - (c*h)/(c + d*x)]*Sqrt[f*h + (d^2*e*g)/(c + d*x)^2 - (c*d*f*
g)/(c + d*x)^2 - (c*d*e*h)/(c + d*x)^2 + (c^2*f*h)/(c + d*x)^2 + (d*f*g)/(c + d*
x) + (d*e*h)/(c + d*x) - (2*c*f*h)/(c + d*x)]*(((-(b*c) + a*d)*(d*e - c*f)*h^2)/
(d*(b*g - a*h)*Sqrt[f + (d*e)/(c + d*x) - (c*f)/(c + d*x)]*Sqrt[h + (d*g)/(c + d
*x) - (c*h)/(c + d*x)]) + ((d*e - c*f)*Sqrt[h + (d*g)/(c + d*x) - (c*h)/(c + d*x
)])/Sqrt[f + (d*e)/(c + d*x) - (c*f)/(c + d*x)] - (b^2*(d*e - c*f)*(d*g - c*h)*S
qrt[h + (d*g)/(c + d*x) - (c*h)/(c + d*x)])/(d*(b*g - a*h)*(b - (b*c)/(c + d*x)
+ (a*d)/(c + d*x))*Sqrt[f + (d*e)/(c + d*x) - (c*f)/(c + d*x)]))*(((-I)*f*Sqrt[1
- (-(d*e) + c*f)/(f*(c + d*x))]*Sqrt[1 - (-(d*g) + c*h)/(h*(c + d*x))]*(Ellipti
cE[I*ArcSinh[Sqrt[-((-(d*g) + c*h)/h)]/Sqrt[c + d*x]], ((-(d*e) + c*f)*h)/(f*(-(
d*g) + c*h))] - EllipticF[I*ArcSinh[Sqrt[-((-(d*g) + c*h)/h)]/Sqrt[c + d*x]], ((
-(d*e) + c*f)*h)/(f*(-(d*g) + c*h))]))/((b*c - a*d)*(-(d*e) + c*f)*Sqrt[-((-(d*g
) + c*h)/h)]*Sqrt[f*h + (d^2*e*g - c*d*f*g - c*d*e*h + c^2*f*h)/(c + d*x)^2 + (d
*f*g + d*e*h - 2*c*f*h)/(c + d*x)]) + (I*b*Sqrt[1 - (-(d*e) + c*f)/(f*(c + d*x))
]*Sqrt[1 - (-(d*g) + c*h)/(h*(c + d*x))]*EllipticF[I*ArcSinh[Sqrt[-((-(d*g) + c*
h)/h)]/Sqrt[c + d*x]], ((-(d*e) + c*f)*h)/(f*(-(d*g) + c*h))])/((b*c - a*d)^2*Sq
rt[-((-(d*g) + c*h)/h)]*Sqrt[f*h + (d^2*e*g - c*d*f*g - c*d*e*h + c^2*f*h)/(c +
d*x)^2 + (d*f*g + d*e*h - 2*c*f*h)/(c + d*x)]) - (b^2*((I*c*Sqrt[1 - (-(d*e) + c
*f)/(f*(c + d*x))]*Sqrt[1 - (-(d*g) + c*h)/(h*(c + d*x))]*EllipticPi[((b*c - a*d
)*h)/(b*(-(d*g) + c*h)), I*ArcSinh[Sqrt[-((-(d*g) + c*h)/h)]/Sqrt[c + d*x]], ((-
(d*e) + c*f)*h)/(f*(-(d*g) + c*h))])/(Sqrt[-((-(d*g) + c*h)/h)]*Sqrt[f*h + (d^2*
e*g)/(c + d*x)^2 - (c*d*f*g)/(c + d*x)^2 - (c*d*e*h)/(c + d*x)^2 + (c^2*f*h)/(c
+ d*x)^2 + (d*f*g)/(c + d*x) + (d*e*h)/(c + d*x) - (2*c*f*h)/(c + d*x)]) - (I*a*
d*Sqrt[1 - (-(d*e) + c*f)/(f*(c + d*x))]*Sqrt[1 - (-(d*g) + c*h)/(h*(c + d*x))]*
EllipticPi[((b*c - a*d)*h)/(b*(-(d*g) + c*h)), I*ArcSinh[Sqrt[-((-(d*g) + c*h)/h
)]/Sqrt[c + d*x]], ((-(d*e) + c*f)*h)/(f*(-(d*g) + c*h))])/(b*Sqrt[-((-(d*g) + c
*h)/h)]*Sqrt[f*h + (d^2*e*g)/(c + d*x)^2 - (c*d*f*g)/(c + d*x)^2 - (c*d*e*h)/(c
+ d*x)^2 + (c^2*f*h)/(c + d*x)^2 + (d*f*g)/(c + d*x) + (d*e*h)/(c + d*x) - (2*c*
f*h)/(c + d*x)])))/(b*c - a*d)^3))/(Sqrt[e + ((c + d*x)*(f - (c*f)/(c + d*x)))/d
]*Sqrt[g + ((c + d*x)*(h - (c*h)/(c + d*x)))/d])))/((-(b*c) + a*d)*(-(d*e) + c*f
)*(-(d*g) + c*h))
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Maple [B] time = 0.136, size = 2842, normalized size = 7.2 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)/(d*x+c)^(3/2)/(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)*(-x^2*b*c*d^2*f^2*h+x^2*a*d^3*f^2*h
-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^3*e^2*g*((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)-EllipticE(((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^2*e*f*h*((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)+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*d*e*f*h*((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)-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^2*e*f*g*((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)-2*EllipticP
i(((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^2*d*e*f*h*((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)+2*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^2*e*f*g*((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)+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^2*e*f*h*((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)-El
lipticF(((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*d*e*f
*h*((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)+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^2*e*f*g*((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)-x*b*c*d^2*e*f*h+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^3*f^2*h*((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)-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^3*f^2*h*((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)+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^3*f^2
*h*((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)+x*a*d^3*e*f*h-x*b*c*d^2*f^2*g-b*c*d^2*e*f*g+x*a*d^3*f^2*g+a*d^3*e*f*g-E
llipticF(((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*d*f^
2*g*((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)+EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))
*a*c^2*d*f^2*h*((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)-EllipticE(((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^2*f^2*g*((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)+EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)
*h/f/(c*h-d*g))^(1/2))*a*d^3*e*f*g*((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)+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*d*f^2*g*((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)-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^2*d*f^2
*g*((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)+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^2*e^2*h*((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)-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^2*d*f^2*h*((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)+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^2*f^2*g*((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)-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^3*e*f*g*((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))/f/
(c*h-d*g)/(c*f-d*e)/(a*d-b*c)^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. \[ \int \frac{1}{{\left (b x + a\right )}{\left (d x + c\right )}^{\frac{3}{2}} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)*(d*x + c)^(3/2)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="maxima")
[Out]
integrate(1/((b*x + a)*(d*x + c)^(3/2)*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(1/((b*x + a)*(d*x + c)^(3/2)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="fricas")
[Out]
Timed out
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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)/(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. \[ \int \frac{1}{{\left (b x + a\right )}{\left (d x + c\right )}^{\frac{3}{2}} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)*(d*x + c)^(3/2)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="giac")
[Out]
integrate(1/((b*x + a)*(d*x + c)^(3/2)*sqrt(f*x + e)*sqrt(h*x + g)), x)