3.419 \(\int \frac{1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=370 \[ -\frac{4 e \sqrt{b x+c x^2} \left (b^2 e^2-b c d e+c^2 d^2\right )}{b^2 d^2 \sqrt{d+e x} (c d-b e)^2}+\frac{4 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{(-b)^{3/2} d^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}-\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{(-b)^{3/2} d \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)} \]

[Out]

(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(b^2*d*(c*d - b*e)*Sqrt[d + e*x]*Sqrt[b
*x + c*x^2]) - (4*e*(c^2*d^2 - b*c*d*e + b^2*e^2)*Sqrt[b*x + c*x^2])/(b^2*d^2*(c
*d - b*e)^2*Sqrt[d + e*x]) + (4*Sqrt[c]*(c^2*d^2 - b*c*d*e + b^2*e^2)*Sqrt[x]*Sq
rt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e
)/(c*d)])/((-b)^(3/2)*d^2*(c*d - b*e)^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (
2*Sqrt[c]*(2*c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[Ar
cSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/((-b)^(3/2)*d*(c*d - b*e)*Sqrt[d
+ e*x]*Sqrt[b*x + c*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 1.17057, antiderivative size = 370, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348 \[ -\frac{4 e \sqrt{b x+c x^2} \left (b^2 e^2-b c d e+c^2 d^2\right )}{b^2 d^2 \sqrt{d+e x} (c d-b e)^2}+\frac{4 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{(-b)^{3/2} d^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}-\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{(-b)^{3/2} d \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(3/2)*(b*x + c*x^2)^(3/2)),x]

[Out]

(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(b^2*d*(c*d - b*e)*Sqrt[d + e*x]*Sqrt[b
*x + c*x^2]) - (4*e*(c^2*d^2 - b*c*d*e + b^2*e^2)*Sqrt[b*x + c*x^2])/(b^2*d^2*(c
*d - b*e)^2*Sqrt[d + e*x]) + (4*Sqrt[c]*(c^2*d^2 - b*c*d*e + b^2*e^2)*Sqrt[x]*Sq
rt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e
)/(c*d)])/((-b)^(3/2)*d^2*(c*d - b*e)^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (
2*Sqrt[c]*(2*c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[Ar
cSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/((-b)^(3/2)*d*(c*d - b*e)*Sqrt[d
+ e*x]*Sqrt[b*x + c*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 164.073, size = 337, normalized size = 0.91 \[ \frac{4 \sqrt{c} \sqrt{x} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (b^{2} e^{2} - b c d e + c^{2} d^{2}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{d^{2} \left (- b\right )^{\frac{3}{2}} \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right )^{2} \sqrt{b x + c x^{2}}} + \frac{2 c \sqrt{x} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - 2 c d\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{e} \sqrt{x}}{\sqrt{- d}} \right )}\middle | \frac{c d}{b e}\right )}{b^{2} \sqrt{e} \sqrt{- d} \sqrt{d + e x} \left (b e - c d\right ) \sqrt{b x + c x^{2}}} - \frac{2 \left (b \left (b e - c d\right ) + c x \left (b e - 2 c d\right )\right )}{b^{2} d \sqrt{d + e x} \left (b e - c d\right ) \sqrt{b x + c x^{2}}} - \frac{4 e \sqrt{b x + c x^{2}} \left (b^{2} e^{2} - b c d e + c^{2} d^{2}\right )}{b^{2} d^{2} \sqrt{d + e x} \left (b e - c d\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*sqrt(c)*sqrt(x)*sqrt(1 + c*x/b)*sqrt(d + e*x)*(b**2*e**2 - b*c*d*e + c**2*d**2
)*elliptic_e(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(d**2*(-b)**(3/2)*sqrt(1
+ e*x/d)*(b*e - c*d)**2*sqrt(b*x + c*x**2)) + 2*c*sqrt(x)*sqrt(1 + c*x/b)*sqrt(
1 + e*x/d)*(b*e - 2*c*d)*elliptic_f(asin(sqrt(e)*sqrt(x)/sqrt(-d)), c*d/(b*e))/(
b**2*sqrt(e)*sqrt(-d)*sqrt(d + e*x)*(b*e - c*d)*sqrt(b*x + c*x**2)) - 2*(b*(b*e
- c*d) + c*x*(b*e - 2*c*d))/(b**2*d*sqrt(d + e*x)*(b*e - c*d)*sqrt(b*x + c*x**2)
) - 4*e*sqrt(b*x + c*x**2)*(b**2*e**2 - b*c*d*e + c**2*d**2)/(b**2*d**2*sqrt(d +
e*x)*(b*e - c*d)**2)

_______________________________________________________________________________________

Mathematica [C]  time = 1.85478, size = 266, normalized size = 0.72 \[ \frac{-2 i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (2 b^2 e^2-3 b c d e+c^2 d^2\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+4 i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+2 b d \left (b^2 e^2+b c e^2 x+c^2 d (d+e x)\right )}{b^2 d^2 \sqrt{x (b+c x)} \sqrt{d+e x} (c d-b e)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(3/2)*(b*x + c*x^2)^(3/2)),x]

[Out]

(2*b*d*(b^2*e^2 + b*c*e^2*x + c^2*d*(d + e*x)) + (4*I)*Sqrt[b/c]*c*e*(c^2*d^2 -
b*c*d*e + b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSi
nh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - (2*I)*Sqrt[b/c]*c*e*(c^2*d^2 - 3*b*c*d*e +
2*b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt
[b/c]/Sqrt[x]], (c*d)/(b*e)])/(b^2*d^2*(c*d - b*e)^2*Sqrt[x*(b + c*x)]*Sqrt[d +
e*x])

_______________________________________________________________________________________

Maple [B]  time = 0.052, size = 698, normalized size = 1.9 \[ -2\,{\frac{\sqrt{x \left ( cx+b \right ) }}{x \left ( cx+b \right ) \left ( be-cd \right ) ^{2}c{b}^{2}{d}^{2}\sqrt{ex+d}} \left ( \sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{3}cd{e}^{2}-3\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{2}{c}^{2}{d}^{2}e+2\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) b{c}^{3}{d}^{3}+2\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{4}{e}^{3}-4\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{3}cd{e}^{2}+4\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{2}{c}^{2}{d}^{2}e-2\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) b{c}^{3}{d}^{3}+2\,{x}^{2}{b}^{2}{c}^{2}{e}^{3}-2\,{x}^{2}b{c}^{3}d{e}^{2}+2\,{x}^{2}{c}^{4}{d}^{2}e+2\,x{b}^{3}c{e}^{3}-x{b}^{2}{c}^{2}d{e}^{2}-xb{c}^{3}{d}^{2}e+2\,x{c}^{4}{d}^{3}+{b}^{3}cd{e}^{2}-2\,{b}^{2}{c}^{2}{d}^{2}e+b{c}^{3}{d}^{3} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(3/2)/(c*x^2+b*x)^(3/2),x)

[Out]

-2/x*(((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((
c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*d*e^2-3*((c*x+b)/b)^(1/2)*(-(e*x+d)
*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(
1/2))*b^2*c^2*d^2*e+2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1
/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^3+2*((c*x+b)/b)^(
1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*
e/(b*e-c*d))^(1/2))*b^4*e^3-4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c
*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*d*e^2+4*((c
*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)
^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^2*e-2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e
-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b
*c^3*d^3+2*x^2*b^2*c^2*e^3-2*x^2*b*c^3*d*e^2+2*x^2*c^4*d^2*e+2*x*b^3*c*e^3-x*b^2
*c^2*d*e^2-x*b*c^3*d^2*e+2*x*c^4*d^3+b^3*c*d*e^2-2*b^2*c^2*d^2*e+b*c^3*d^3)*(x*(
c*x+b))^(1/2)/(c*x+b)/(b*e-c*d)^2/c/b^2/d^2/(e*x+d)^(1/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((c*x^2 + b*x)^(3/2)*(e*x + d)^(3/2)), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (c e x^{3} + b d x +{\left (c d + b e\right )} x^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{e x + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((c*e*x^3 + b*d*x + (c*d + b*e)*x^2)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))
, x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(3/2)/(c*x**2+b*x)**(3/2),x)

[Out]

Integral(1/((x*(b + c*x))**(3/2)*(d + e*x)**(3/2)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((c*x^2 + b*x)^(3/2)*(e*x + d)^(3/2)), x)