3.1383 \(\int \frac{1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=228 \[ \frac{80 c^2 \sqrt{a+b x+c x^2}}{d \left (b^2-4 a c\right )^3 (b d+2 c d x)^{3/2}}+\frac{40 c \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{d^{5/2} \left (b^2-4 a c\right )^{11/4} \sqrt{a+b x+c x^2}}+\frac{12 c}{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}-\frac{2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}} \]

[Out]

-2/(3*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(3/2)) + (12*c)/((
b^2 - 4*a*c)^2*d*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2]) + (80*c^2*Sqrt[a +
b*x + c*x^2])/((b^2 - 4*a*c)^3*d*(b*d + 2*c*d*x)^(3/2)) + (40*c*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])/((b^2 - 4*a*c)^(11/4)*d^(5/2)*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.52227, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{80 c^2 \sqrt{a+b x+c x^2}}{d \left (b^2-4 a c\right )^3 (b d+2 c d x)^{3/2}}+\frac{40 c \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{d^{5/2} \left (b^2-4 a c\right )^{11/4} \sqrt{a+b x+c x^2}}+\frac{12 c}{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}-\frac{2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(5/2)),x]

[Out]

-2/(3*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(3/2)) + (12*c)/((
b^2 - 4*a*c)^2*d*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2]) + (80*c^2*Sqrt[a +
b*x + c*x^2])/((b^2 - 4*a*c)^3*d*(b*d + 2*c*d*x)^(3/2)) + (40*c*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])/((b^2 - 4*a*c)^(11/4)*d^(5/2)*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 114.377, size = 219, normalized size = 0.96 \[ \frac{80 c^{2} \sqrt{a + b x + c x^{2}}}{d \left (- 4 a c + b^{2}\right )^{3} \left (b d + 2 c d x\right )^{\frac{3}{2}}} + \frac{12 c}{d \left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}} + \frac{40 c \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{d^{\frac{5}{2}} \left (- 4 a c + b^{2}\right )^{\frac{11}{4}} \sqrt{a + b x + c x^{2}}} - \frac{2}{3 d \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{3}{2}} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*c*d*x+b*d)**(5/2)/(c*x**2+b*x+a)**(5/2),x)

[Out]

80*c**2*sqrt(a + b*x + c*x**2)/(d*(-4*a*c + b**2)**3*(b*d + 2*c*d*x)**(3/2)) + 1
2*c/(d*(-4*a*c + b**2)**2*(b*d + 2*c*d*x)**(3/2)*sqrt(a + b*x + c*x**2)) + 40*c*
sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*elliptic_f(asin(sqrt(b*d + 2*c*d*x)/(s
qrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(d**(5/2)*(-4*a*c + b**2)**(11/4)*sqrt(a +
b*x + c*x**2)) - 2/(3*d*(-4*a*c + b**2)*(b*d + 2*c*d*x)**(3/2)*(a + b*x + c*x**2
)**(3/2))

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Mathematica [C]  time = 1.6153, size = 200, normalized size = 0.88 \[ \frac{2 \left ((b+2 c x)^3 (a+x (b+c x)) \left (\frac{4 a c-b^2}{(a+x (b+c x))^2}+\frac{22 c}{a+x (b+c x)}+\frac{32 c^2}{(b+2 c x)^2}\right )+\frac{60 i c (b+2 c x)^{7/2} \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{\sqrt{-\sqrt{b^2-4 a c}}}\right )}{3 \left (b^2-4 a c\right )^3 \sqrt{a+x (b+c x)} (d (b+2 c x))^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(2*((b + 2*c*x)^3*(a + x*(b + c*x))*((32*c^2)/(b + 2*c*x)^2 + (-b^2 + 4*a*c)/(a
+ x*(b + c*x))^2 + (22*c)/(a + x*(b + c*x))) + ((60*I)*c*(b + 2*c*x)^(7/2)*Sqrt[
(c*(a + x*(b + c*x)))/(b + 2*c*x)^2]*EllipticF[I*ArcSinh[Sqrt[-Sqrt[b^2 - 4*a*c]
]/Sqrt[b + 2*c*x]], -1])/Sqrt[-Sqrt[b^2 - 4*a*c]]))/(3*(b^2 - 4*a*c)^3*(d*(b + 2
*c*x))^(5/2)*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.055, size = 797, normalized size = 3.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^(5/2),x)

[Out]

-2/3*(60*((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)*
EllipticF(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))*(-4*a*c+b^2)^(1/2)*x^3*c^3+90*((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)*EllipticF(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))*(-4*a*c+b^2)^(1/2)*x^2*b*c^2+60*((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)*EllipticF(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))*(-4*a*c+b^2)^(1
/2)*x*a*c^2+30*((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)*EllipticF(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))*(-4*a*c+b^2)^(1/2)*x*b^2*c+30*((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)*EllipticF(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))*(-4*a*c+b^2)^(1/2)*a*b*c+120*c^4*x^4+24
0*b*c^3*x^3+168*x^2*a*c^3+138*x^2*b^2*c^2+168*x*a*b*c^2+18*b^3*c*x+32*a^2*c^2+26
*a*c*b^2-b^4)*(d*(2*c*x+b))^(1/2)/d^3/(2*c*x+b)^2/(4*a*c-b^2)^3/(c*x^2+b*x+a)^(3
/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^(5/2)),x, algorithm="maxima")

[Out]

integrate(1/((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^(5/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (4 \, c^{4} d^{2} x^{6} + 12 \, b c^{3} d^{2} x^{5} +{\left (13 \, b^{2} c^{2} + 8 \, a c^{3}\right )} d^{2} x^{4} + a^{2} b^{2} d^{2} + 2 \,{\left (3 \, b^{3} c + 8 \, a b c^{2}\right )} d^{2} x^{3} +{\left (b^{4} + 10 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d^{2} x^{2} + 2 \,{\left (a b^{3} + 2 \, a^{2} b c\right )} d^{2} x\right )} \sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^(5/2)),x, algorithm="fricas")

[Out]

integral(1/((4*c^4*d^2*x^6 + 12*b*c^3*d^2*x^5 + (13*b^2*c^2 + 8*a*c^3)*d^2*x^4 +
a^2*b^2*d^2 + 2*(3*b^3*c + 8*a*b*c^2)*d^2*x^3 + (b^4 + 10*a*b^2*c + 4*a^2*c^2)*
d^2*x^2 + 2*(a*b^3 + 2*a^2*b*c)*d^2*x)*sqrt(2*c*d*x + b*d)*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(1/(2*c*d*x+b*d)**(5/2)/(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. \[ \int \frac{1}{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^(5/2)),x, algorithm="giac")

[Out]

integrate(1/((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^(5/2)), x)