3.1647 \(\int \frac{b+2 c x}{\sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=290 \[ -\frac{2 \sqrt{d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{2} e \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}} \]

[Out]

(-2*Sqrt[d + e*x]*((b^2 - 4*a*c)*(c*d - b*e) - c*(b^2 - 4*a*c)*e*x))/((b^2 - 4*a
*c)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x + c*x^2]) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*
e*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sq
rt[(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 = 0.53501, antiderivative size = 290, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{\sqrt{2} e \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{d+e x} (-b e+c d-c e x)}{\sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[d + e*x]*(c*d - b*e - c*e*x))/((c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x + c
*x^2]) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*e*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))
/(b^2 - 4*a*c))]*EllipticE[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 = 86.6928, size = 243, normalized size = 0.84 \[ - \frac{\sqrt{2} e \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \sqrt{d + e x} \sqrt{- 4 a c + b^{2}} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{\frac{b + 2 c x + \sqrt{- 4 a c + b^{2}}}{\sqrt{- 4 a c + b^{2}}}}}{2} \right )}\middle | \frac{2 e \sqrt{- 4 a c + b^{2}}}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}\right )}{\sqrt{\frac{c \left (- d - e x\right )}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} \sqrt{a + b x + c x^{2}} \left (a e^{2} - b d e + c d^{2}\right )} + \frac{2 \sqrt{d + e x} \left (b e - c d + c e x\right )}{\sqrt{a + b x + c x^{2}} \left (a e^{2} - b d e + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*e*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*sqrt(d + e*x)*sqrt(-4*a*c +
b**2)*elliptic_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)))/(s
qrt(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)) + 2*sqrt(d + e*x)*(b*e - c*d + c*e*x)/(sqrt(a + b*x +
c*x**2)*(a*e**2 - b*d*e + c*d**2))

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Mathematica [C]  time = 1.89099, size = 405, normalized size = 1.4 \[ \frac{4 \sqrt{d+e x} (b e-c d+c e x)-\frac{i \sqrt{2} \left (e \left (\sqrt{b^2-4 a c}-b\right )+2 c d\right ) \sqrt{\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{e \left (\sqrt{b^2-4 a c}+b\right )-2 c d}} \sqrt{1-\frac{2 c (d+e x)}{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}} \left (E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{\left (b+\sqrt{b^2-4 a c}\right ) e-2 c d}} \sqrt{d+e x}\right )|\frac{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e}\right )-F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{\left (b+\sqrt{b^2-4 a c}\right ) e-2 c d}} \sqrt{d+e x}\right )|\frac{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e}\right )\right )}{\sqrt{\frac{c}{e \left (\sqrt{b^2-4 a c}+b\right )-2 c d}}}}{2 \sqrt{a+x (b+c x)} \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[d + e*x]*(-(c*d) + b*e + c*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)]*(Ellipti
cE[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)] - El
lipticF[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*(c*d^2 + e*(-(b*d) + a*e))*S
qrt[a + x*(b + c*x)])

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Maple [B]  time = 0.094, size = 1366, normalized size = 4.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)/(c*x^2+b*x+a)^(3/2)/(e*x+d)^(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)*
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))*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-b*d*e+c*d^2)*(c*x^2+b*x+a)^(1/2)*(e*x+d)^(1/2)/(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. \[ \int \frac{2 \, c x + b}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}} \sqrt{e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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