3.1375 \(\int \frac{\sqrt{b d+2 c d x}}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=231 \[ -\frac{2 (b d+2 c d x)^{3/2}}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{4 \sqrt{d} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt{d} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.706833, antiderivative size = 231, 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 \[ -\frac{2 (b d+2 c d x)^{3/2}}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{4 \sqrt{d} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt{d} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 122.427, size = 221, normalized size = 0.96 \[ \frac{4 \sqrt{d} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{\sqrt [4]{- 4 a c + b^{2}} \sqrt{a + b x + c x^{2}}} - \frac{4 \sqrt{d} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{\sqrt [4]{- 4 a c + b^{2}} \sqrt{a + b x + c x^{2}}} - \frac{2 \left (b d + 2 c d x\right )^{\frac{3}{2}}}{d \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*sqrt(d)*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)/((-4*a*c + b**2)**(1/4)*sqrt(a +
b*x + c*x**2)) - 4*sqrt(d)*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)/((-4*a*c + b**2)
**(1/4)*sqrt(a + b*x + c*x**2)) - 2*(b*d + 2*c*d*x)**(3/2)/(d*(-4*a*c + b**2)*sq
rt(a + b*x + c*x**2))

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Mathematica [C]  time = 1.7542, size = 238, normalized size = 1.03 \[ -\frac{2 i \sqrt{d (b+2 c x)} \left (\frac{i (b+2 c x)^2}{\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}}-2 \left (b^2-4 a c\right ) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )+2 \left (b^2-4 a c\right ) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}} \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

((-2*I)*Sqrt[d*(b + 2*c*x)]*((I*(b + 2*c*x)^2)/Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a
*c])] + 2*(b^2 - 4*a*c)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*EllipticE[I*A
rcSinh[Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1] - 2*(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]))/((b^2 - 4*a*c)^(3/2)*Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]*
Sqrt[a + x*(b + c*x)])

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Maple [A]  time = 0.025, size = 332, normalized size = 1.4 \[ -2\,{\frac{\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a}}{ \left ( 2\,{x}^{3}{c}^{2}+3\,{x}^{2}bc+2\,acx+{b}^{2}x+ab \right ) \left ( 4\,ac-{b}^{2} \right ) } \left ( 4\,{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) ac\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}-{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){b}^{2}\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}-4\,{c}^{2}{x}^{2}-4\,bxc-{b}^{2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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