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

Optimal. Leaf size=449 \[ \frac{2 (b c-a d) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}-\frac{2 (b c-a d) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 d \sqrt{c+d x} \sqrt{e h-f g} \sqrt{\frac{f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac{\sqrt{h} \sqrt{e+f x}}{\sqrt{e h-f g}}\right )|-\frac{d (f g-e h)}{(d e-c f) h}\right )}{b f \sqrt{h} \sqrt{g+h x} \sqrt{-\frac{f (c+d x)}{d e-c f}}} \]

[Out]

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

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Rubi [A]  time = 2.93872, antiderivative size = 449, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229 \[ \frac{2 (b c-a d) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}-\frac{2 (b c-a d) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 d \sqrt{c+d x} \sqrt{e h-f g} \sqrt{\frac{f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac{\sqrt{h} \sqrt{e+f x}}{\sqrt{e h-f g}}\right )|-\frac{d (f g-e h)}{(d e-c f) h}\right )}{b f \sqrt{h} \sqrt{g+h x} \sqrt{-\frac{f (c+d x)}{d e-c f}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 171.238, size = 379, normalized size = 0.84 \[ \frac{2 d \sqrt{\frac{h \left (e + f x\right )}{e h - f g}} \sqrt{c + d x} \sqrt{- e h + f g} E\left (\operatorname{asin}{\left (\frac{\sqrt{f} \sqrt{g + h x}}{\sqrt{- e h + f g}} \right )}\middle | \frac{d \left (e h - f g\right )}{f \left (c h - d g\right )}\right )}{b \sqrt{f} h \sqrt{\frac{h \left (c + d x\right )}{c h - d g}} \sqrt{e + f x}} + \frac{2 \sqrt{\frac{f \left (c + d x\right )}{c f - d e}} \sqrt{\frac{f \left (- g - h x\right )}{e h - f g}} \left (a d - b c\right )^{2} \Pi \left (- \frac{b \left (e h - f g\right )}{h \left (a f - b e\right )}; \operatorname{asin}{\left (\sqrt{\frac{h}{e h - f g}} \sqrt{e + f x} \right )}\middle | \frac{d \left (- e h + f g\right )}{h \left (c f - d e\right )}\right )}{b^{2} \sqrt{\frac{h}{e h - f g}} \sqrt{c + d x} \sqrt{g + h x} \left (a f - b e\right )} - \frac{2 \sqrt{\frac{d \left (- e - f x\right )}{c f - d e}} \sqrt{\frac{d \left (- g - h x\right )}{c h - d g}} \left (a d - b c\right ) \sqrt{c f - d e} F\left (\operatorname{asin}{\left (\frac{\sqrt{f} \sqrt{c + d x}}{\sqrt{c f - d e}} \right )}\middle | \frac{h \left (c f - d e\right )}{f \left (c h - d g\right )}\right )}{b^{2} \sqrt{f} \sqrt{e + f x} \sqrt{g + h x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*d*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)))/(b*sqrt
(f)*h*sqrt(h*(c + d*x)/(c*h - d*g))*sqrt(e + f*x)) + 2*sqrt(f*(c + d*x)/(c*f - d
*e))*sqrt(f*(-g - h*x)/(e*h - f*g))*(a*d - b*c)**2*elliptic_pi(-b*(e*h - f*g)/(h
*(a*f - b*e)), asin(sqrt(h/(e*h - f*g))*sqrt(e + f*x)), d*(-e*h + f*g)/(h*(c*f -
d*e)))/(b**2*sqrt(h/(e*h - f*g))*sqrt(c + d*x)*sqrt(g + h*x)*(a*f - b*e)) - 2*s
qrt(d*(-e - f*x)/(c*f - d*e))*sqrt(d*(-g - h*x)/(c*h - d*g))*(a*d - b*c)*sqrt(c*
f - d*e)*elliptic_f(asin(sqrt(f)*sqrt(c + d*x)/sqrt(c*f - d*e)), h*(c*f - d*e)/(
f*(c*h - d*g)))/(b**2*sqrt(f)*sqrt(e + f*x)*sqrt(g + h*x))

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Mathematica [C]  time = 10.6348, size = 381, normalized size = 0.85 \[ -\frac{2 \sqrt{c+d x} \left (-\frac{b d f (g+h x)}{\sqrt{e+f x}}+\frac{i \sqrt{\frac{f (g+h x)}{h (e+f x)}} \left (f \left (f h (b c-a d)^2 \Pi \left (\frac{(b e-a f) h}{b (e h-f g)};i \sinh ^{-1}\left (\frac{\sqrt{\frac{f g}{h}-e}}{\sqrt{e+f x}}\right )|\frac{(d e-c f) h}{d (e h-f g)}\right )-b \left (a d^2 (e h-f g)+b \left (c^2 f h-2 c d e h+d^2 e g\right )\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{f g}{h}-e}}{\sqrt{e+f x}}\right )|\frac{(d e-c f) h}{d (e h-f g)}\right )\right )-b d^2 (b e-a f) (e h-f g) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{f g}{h}-e}}{\sqrt{e+f x}}\right )|\frac{(d e-c f) h}{d (e h-f g)}\right )\right )}{d (a f-b e) \sqrt{\frac{f g}{h}-e} \sqrt{\frac{f (c+d x)}{d (e+f x)}}}\right )}{b^2 f h \sqrt{g+h x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c + d*x]*(-((b*d*f*(g + h*x))/Sqrt[e + f*x]) + (I*Sqrt[(f*(g + h*x))/(h
*(e + f*x))]*(-(b*d^2*(b*e - a*f)*(-(f*g) + e*h)*EllipticE[I*ArcSinh[Sqrt[-e + (
f*g)/h]/Sqrt[e + f*x]], ((d*e - c*f)*h)/(d*(-(f*g) + e*h))]) + f*(-(b*(a*d^2*(-(
f*g) + e*h) + b*(d^2*e*g - 2*c*d*e*h + c^2*f*h))*EllipticF[I*ArcSinh[Sqrt[-e + (
f*g)/h]/Sqrt[e + f*x]], ((d*e - c*f)*h)/(d*(-(f*g) + e*h))]) + (b*c - a*d)^2*f*h
*EllipticPi[((b*e - a*f)*h)/(b*(-(f*g) + e*h)), I*ArcSinh[Sqrt[-e + (f*g)/h]/Sqr
t[e + f*x]], ((d*e - c*f)*h)/(d*(-(f*g) + e*h))])))/(d*(-(b*e) + a*f)*Sqrt[-e +
(f*g)/h]*Sqrt[(f*(c + d*x))/(d*(e + f*x))])))/(b^2*f*h*Sqrt[g + h*x])

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Maple [B]  time = 0.039, size = 968, normalized size = 2.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(3/2)/(b*x+a)/(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)/h/f/b^2*((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*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))*a*d^2*e*h-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^2*f*h+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*e*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*d*f*g-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+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-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-Elliptic
E(((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+Ellipti
cE(((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-Ellipt
icPi(((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*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))*a*d^2*e*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*c^2*f*h-Ellip
ticPi(((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*e*h)/(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. \[ \int \frac{{\left (d x + c\right )}^{\frac{3}{2}}}{{\left (b x + a\right )} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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