3.1387 \(\int \frac{\sqrt{b d+2 c d x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=278 \[ \frac{4 c (b d+2 c d x)^{3/2}}{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (b d+2 c d x)^{3/2}}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{8 c \sqrt{d} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\left (b^2-4 a c\right )^{5/4} \sqrt{a+b x+c x^2}}-\frac{8 c \sqrt{d} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\left (b^2-4 a c\right )^{5/4} \sqrt{a+b x+c x^2}} \]

[Out]

(-2*(b*d + 2*c*d*x)^(3/2))/(3*(b^2 - 4*a*c)*d*(a + b*x + c*x^2)^(3/2)) + (4*c*(b
*d + 2*c*d*x)^(3/2))/((b^2 - 4*a*c)^2*d*Sqrt[a + b*x + c*x^2]) - (8*c*Sqrt[d]*Sq
rt[-((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)^(5/4)*Sqrt[a + b*x + c*x^2])
+ (8*c*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)^(5/4)*Sqrt
[a + b*x + c*x^2])

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Rubi [A]  time = 0.814027, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{4 c (b d+2 c d x)^{3/2}}{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (b d+2 c d x)^{3/2}}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{8 c \sqrt{d} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\left (b^2-4 a c\right )^{5/4} \sqrt{a+b x+c x^2}}-\frac{8 c \sqrt{d} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\left (b^2-4 a c\right )^{5/4} \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*d + 2*c*d*x]/(a + b*x + c*x^2)^(5/2),x]

[Out]

(-2*(b*d + 2*c*d*x)^(3/2))/(3*(b^2 - 4*a*c)*d*(a + b*x + c*x^2)^(3/2)) + (4*c*(b
*d + 2*c*d*x)^(3/2))/((b^2 - 4*a*c)^2*d*Sqrt[a + b*x + c*x^2]) - (8*c*Sqrt[d]*Sq
rt[-((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)^(5/4)*Sqrt[a + b*x + c*x^2])
+ (8*c*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)^(5/4)*Sqrt
[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 145.334, size = 269, normalized size = 0.97 \[ - \frac{8 c \sqrt{d} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{\left (- 4 a c + b^{2}\right )^{\frac{5}{4}} \sqrt{a + b x + c x^{2}}} + \frac{8 c \sqrt{d} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{\left (- 4 a c + b^{2}\right )^{\frac{5}{4}} \sqrt{a + b x + c x^{2}}} + \frac{4 c \left (b d + 2 c d x\right )^{\frac{3}{2}}}{d \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}}} - \frac{2 \left (b d + 2 c d x\right )^{\frac{3}{2}}}{3 d \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]

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)**(5/2),x)

[Out]

-8*c*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)**(5/4)*sqrt(a
+ b*x + c*x**2)) + 8*c*sqrt(d)*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*ellipt
ic_f(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/((-4*a*c +
b**2)**(5/4)*sqrt(a + b*x + c*x**2)) + 4*c*(b*d + 2*c*d*x)**(3/2)/(d*(-4*a*c + b
**2)**2*sqrt(a + b*x + c*x**2)) - 2*(b*d + 2*c*d*x)**(3/2)/(3*d*(-4*a*c + b**2)*
(a + b*x + c*x**2)**(3/2))

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Mathematica [C]  time = 1.23203, size = 222, normalized size = 0.8 \[ \frac{\sqrt{d (b+2 c x)} \left (-\frac{2 (b+2 c x) (a+x (b+c x)) \left (-2 c \left (5 a+3 c x^2\right )+b^2-6 b c x\right )}{3 \left (b^2-4 a c\right )^2}+\frac{8 i c (a+x (b+c x))^2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}}\right )}{(a+x (b+c x))^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*d + 2*c*d*x]/(a + b*x + c*x^2)^(5/2),x]

[Out]

(Sqrt[d*(b + 2*c*x)]*((-2*(b + 2*c*x)*(a + x*(b + c*x))*(b^2 - 6*b*c*x - 2*c*(5*
a + 3*c*x^2)))/(3*(b^2 - 4*a*c)^2) + ((8*I)*c*(a + x*(b + c*x))^2*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] - 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])])))/(a + x*(b + c
*x))^(5/2)

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Maple [B]  time = 0.032, size = 866, normalized size = 3.1 \[ \text{result too large to display} \]

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)^(5/2),x)

[Out]

-2/3*(24*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))*x^2*a*c^3*((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)-6*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))*x^2*b^2*c^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)+24*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))*x*a*b*c^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)-6*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))*x*b^3*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)+24*((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))*a^2*c^2-6*((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))*a*b^2*c-24*c^4*x^4-48*b
*c^3*x^3-40*x^2*a*c^3-26*x^2*b^2*c^2-40*x*a*b*c^2-2*b^3*c*x-10*a*c*b^2+b^4)*(d*(
2*c*x+b))^(1/2)/(4*a*c-b^2)^2/(2*c*x+b)/(c*x^2+b*x+a)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{2 \, c d x + b d}}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \]

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)^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(2*c*d*x + b*d)/(c*x^2 + b*x + a)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{2 \, c d x + b d}}{{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{c x^{2} + b x + a}}, x\right ) \]

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)^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(2*c*d*x + b*d)/((c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2
+ a^2)*sqrt(c*x^2 + b*x + a)), 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*d*x+b*d)**(1/2)/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{2 \, c d x + b d}}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \]

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)^(5/2),x, algorithm="giac")

[Out]

integrate(sqrt(2*c*d*x + b*d)/(c*x^2 + b*x + a)^(5/2), x)