3.1360 \(\int \frac{1}{(b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=237 \[ \frac{2 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{c d^{3/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{c d^{3/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt{a+b x+c x^2}}{d \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}} \]

[Out]

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

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Rubi [A]  time = 0.723243, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{c d^{3/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{c d^{3/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt{a+b x+c x^2}}{d \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 127.111, size = 224, normalized size = 0.95 \[ \frac{4 \sqrt{a + b x + c x^{2}}}{d \left (- 4 a c + b^{2}\right ) \sqrt{b d + 2 c d x}} - \frac{2 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{c d^{\frac{3}{2}} \sqrt [4]{- 4 a c + b^{2}} \sqrt{a + b x + c x^{2}}} + \frac{2 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{c d^{\frac{3}{2}} \sqrt [4]{- 4 a c + b^{2}} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 2.72057, size = 243, normalized size = 1.03 \[ -\frac{2 i \left (-\left (b^2-4 a c\right ) \sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )+\left (b^2-4 a c\right ) \sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )+2 i c (a+x (b+c x))\right )}{c d \left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.035, size = 336, normalized size = 1.4 \[{\frac{1}{c{d}^{2} \left ( 2\,{x}^{3}{c}^{2}+3\,{x}^{2}bc+2\,acx+{b}^{2}x+ab \right ) \left ( 4\,ac-{b}^{2} \right ) }\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a} \left ( 4\,{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) ac\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}-{\it EllipticE} \left ({\frac{\sqrt{2}}{2}\sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}},\sqrt{2} \right ){b}^{2}\sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{-{(2\,cx+b){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{{1 \left ( -b-2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}-4\,{c}^{2}{x}^{2}-4\,bxc-4\,ac \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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