3.2455 \(\int \frac{1}{(d+e x)^{3/2} \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=248 \[ \frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 e \sqrt{a+b x+c x^2}}{\sqrt{d+e x} \left (a e^2-b d e+c d^2\right )} \]

[Out]

(-2*e*Sqrt[a + b*x + c*x^2])/((c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]) + (Sqrt[2]*
Sqrt[b^2 - 4*a*c]*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Ell
ipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]],
(-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/((c*d^2 - b*d*e +
a*e^2)*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c
*x^2])

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Rubi [A]  time = 0.458819, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 e \sqrt{a+b x+c x^2}}{\sqrt{d+e x} \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-2*e*Sqrt[a + b*x + c*x^2])/((c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]) + (Sqrt[2]*
Sqrt[b^2 - 4*a*c]*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Ell
ipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]],
(-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/((c*d^2 - b*d*e +
a*e^2)*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c
*x^2])

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Rubi in Sympy [A]  time = 71.5529, size = 231, normalized size = 0.93 \[ - \frac{2 e \sqrt{a + b x + c x^{2}}}{\sqrt{d + e x} \left (a e^{2} - b d e + c d^{2}\right )} + \frac{\sqrt{2} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \sqrt{d + e x} \sqrt{- 4 a c + b^{2}} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{\frac{b + 2 c x + \sqrt{- 4 a c + b^{2}}}{\sqrt{- 4 a c + b^{2}}}}}{2} \right )}\middle | \frac{2 e \sqrt{- 4 a c + b^{2}}}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}\right )}{\sqrt{\frac{c \left (- d - e x\right )}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} \sqrt{a + b x + c x^{2}} \left (a e^{2} - b d e + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*e*sqrt(a + b*x + c*x**2)/(sqrt(d + e*x)*(a*e**2 - b*d*e + c*d**2)) + sqrt(2)*
sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*sqrt(d + e*x)*sqrt(-4*a*c + b**2)*elli
ptic_e(asin(sqrt(2)*sqrt((b + 2*c*x + sqrt(-4*a*c + b**2))/sqrt(-4*a*c + b**2))/
2), 2*e*sqrt(-4*a*c + b**2)/(b*e - 2*c*d + e*sqrt(-4*a*c + b**2)))/(sqrt(c*(-d -
e*x)/(b*e - 2*c*d + e*sqrt(-4*a*c + b**2)))*sqrt(a + b*x + c*x**2)*(a*e**2 - b*
d*e + c*d**2))

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Mathematica [C]  time = 1.37944, size = 408, normalized size = 1.65 \[ \frac{-\frac{4 e^2 (a+x (b+c x))}{\sqrt{d+e x}}+\frac{i \sqrt{2} \left (e \left (\sqrt{b^2-4 a c}-b\right )+2 c d\right ) \sqrt{\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{e \left (\sqrt{b^2-4 a c}+b\right )-2 c d}} \sqrt{1-\frac{2 c (d+e x)}{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}} \left (E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{\left (b+\sqrt{b^2-4 a c}\right ) e-2 c d}} \sqrt{d+e x}\right )|\frac{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e}\right )-F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{\left (b+\sqrt{b^2-4 a c}\right ) e-2 c d}} \sqrt{d+e x}\right )|\frac{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e}\right )\right )}{\sqrt{\frac{c}{e \left (\sqrt{b^2-4 a c}+b\right )-2 c d}}}}{2 e \sqrt{a+x (b+c x)} \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

((-4*e^2*(a + x*(b + c*x)))/Sqrt[d + e*x] + (I*Sqrt[2]*(2*c*d + (-b + Sqrt[b^2 -
4*a*c])*e)*Sqrt[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*d + (b + Sqrt[b^2 - 4
*a*c])*e)]*Sqrt[1 - (2*c*(d + e*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)]*(Ellip
ticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[d + e*x
]], (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)] -
EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[d
+ e*x]], (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e
)]))/Sqrt[c/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)])/(2*e*(c*d^2 + e*(-(b*d) + a*e
))*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.063, size = 1365, normalized size = 5.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [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/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

Timed out