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

Optimal. Leaf size=274 \[ -\frac{2 \sqrt{d+e x} (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (c d-b e)}-\frac{4 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{(-b)^{3/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{(-b)^{3/2} d \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)} \]

[Out]

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

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Rubi [A]  time = 0.758157, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304 \[ -\frac{2 \sqrt{d+e x} (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (c d-b e)}-\frac{4 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{(-b)^{3/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{(-b)^{3/2} d \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 99.7055, size = 240, normalized size = 0.88 \[ - \frac{4 \sqrt{c} \sqrt{x} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} F\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{\left (- b\right )^{\frac{3}{2}} \sqrt{d + e x} \sqrt{b x + c x^{2}}} + \frac{2 \sqrt{c} \sqrt{x} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (b e - 2 c d\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{d \left (- b\right )^{\frac{3}{2}} \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right ) \sqrt{b x + c x^{2}}} - \frac{2 \sqrt{d + e x} \left (b \left (b e - c d\right ) + c x \left (b e - 2 c d\right )\right )}{b^{2} d \left (b e - c d\right ) \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 1.98091, size = 220, normalized size = 0.8 \[ \frac{-2 i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-c d) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+2 i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-2 c d) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-2 b c d (d+e x)}{b^2 d \sqrt{x (b+c x)} \sqrt{d+e x} (b e-c d)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*b*c*d*(d + e*x) + (2*I)*Sqrt[b/c]*c*e*(-2*c*d + b*e)*Sqrt[1 + b/(c*x)]*Sqrt[
1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - (2*I
)*Sqrt[b/c]*c*e*(-(c*d) + b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*Ellip
ticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])/(b^2*d*(-(c*d) + b*e)*Sqrt[x*(b
+ c*x)]*Sqrt[d + e*x])

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Maple [B]  time = 0.047, size = 480, normalized size = 1.8 \[ -2\,{\frac{\sqrt{x \left ( cx+b \right ) }}{x \left ( cx+b \right ) \left ( be-cd \right ) c{b}^{2}d\sqrt{ex+d}} \left ( 2\,{b}^{2}d\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) ec-2\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) b{c}^{2}{d}^{2}+\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{3}{e}^{2}-3\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{2}cde+2\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) b{c}^{2}{d}^{2}+{x}^{2}b{c}^{2}{e}^{2}-2\,{x}^{2}{c}^{3}de+x{b}^{2}c{e}^{2}-2\,x{c}^{3}{d}^{2}+{b}^{2}cde-b{c}^{2}{d}^{2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/x*(2*b^2*d*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Elli
pticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*e*c-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^2*d^2+((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*El
lipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*e^2-3*((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*d*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^2*d^2+x^2*b*c^2*e^2
-2*x^2*c^3*d*e+x*b^2*c*e^2-2*x*c^3*d^2+b^2*c*d*e-b*c^2*d^2)*(x*(c*x+b))^(1/2)/(c
*x+b)/(b*e-c*d)/c/b^2/d/(e*x+d)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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