3.1337 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{11/2}} \, dx\)

Optimal. Leaf size=320 \[ \frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{30 c^3 d^{11/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}-\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{30 c^3 d^{11/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2}}{15 c^2 d^5 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}}-\frac{\sqrt{a+b x+c x^2}}{30 c^2 d^3 (b d+2 c d x)^{5/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{9 c d (b d+2 c d x)^{9/2}} \]

[Out]

-Sqrt[a + b*x + c*x^2]/(30*c^2*d^3*(b*d + 2*c*d*x)^(5/2)) + Sqrt[a + b*x + c*x^2
]/(15*c^2*(b^2 - 4*a*c)*d^5*Sqrt[b*d + 2*c*d*x]) - (a + b*x + c*x^2)^(3/2)/(9*c*
d*(b*d + 2*c*d*x)^(9/2)) - (Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Ellipti
cE[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(30*c^3*(b^2
- 4*a*c)^(1/4)*d^(11/2)*Sqrt[a + b*x + c*x^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])/(30*c^3*(b^2 - 4*a*c)^(1/4)*d^(11/2)*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.942078, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{30 c^3 d^{11/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}-\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{30 c^3 d^{11/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2}}{15 c^2 d^5 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}}-\frac{\sqrt{a+b x+c x^2}}{30 c^2 d^3 (b d+2 c d x)^{5/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{9 c d (b d+2 c d x)^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^(11/2),x]

[Out]

-Sqrt[a + b*x + c*x^2]/(30*c^2*d^3*(b*d + 2*c*d*x)^(5/2)) + Sqrt[a + b*x + c*x^2
]/(15*c^2*(b^2 - 4*a*c)*d^5*Sqrt[b*d + 2*c*d*x]) - (a + b*x + c*x^2)^(3/2)/(9*c*
d*(b*d + 2*c*d*x)^(9/2)) - (Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Ellipti
cE[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(30*c^3*(b^2
- 4*a*c)^(1/4)*d^(11/2)*Sqrt[a + b*x + c*x^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])/(30*c^3*(b^2 - 4*a*c)^(1/4)*d^(11/2)*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 174.356, size = 301, normalized size = 0.94 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{9 c d \left (b d + 2 c d x\right )^{\frac{9}{2}}} - \frac{\sqrt{a + b x + c x^{2}}}{30 c^{2} d^{3} \left (b d + 2 c d x\right )^{\frac{5}{2}}} + \frac{\sqrt{a + b x + c x^{2}}}{15 c^{2} d^{5} \left (- 4 a c + b^{2}\right ) \sqrt{b d + 2 c d x}} - \frac{\sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{30 c^{3} d^{\frac{11}{2}} \sqrt [4]{- 4 a c + b^{2}} \sqrt{a + b x + c x^{2}}} + \frac{\sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{30 c^{3} d^{\frac{11}{2}} \sqrt [4]{- 4 a c + b^{2}} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(3/2)/(2*c*d*x+b*d)**(11/2),x)

[Out]

-(a + b*x + c*x**2)**(3/2)/(9*c*d*(b*d + 2*c*d*x)**(9/2)) - sqrt(a + b*x + c*x**
2)/(30*c**2*d**3*(b*d + 2*c*d*x)**(5/2)) + sqrt(a + b*x + c*x**2)/(15*c**2*d**5*
(-4*a*c + b**2)*sqrt(b*d + 2*c*d*x)) - 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)/(30*
c**3*d**(11/2)*(-4*a*c + b**2)**(1/4)*sqrt(a + b*x + c*x**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)/(30*c**3*d**(11/2)*(-4*a*c + b**2)**(1/4)*sqrt(a + b*x + c*
x**2))

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Mathematica [C]  time = 1.60349, size = 229, normalized size = 0.72 \[ \frac{\frac{c (b+2 c x) (a+x (b+c x)) \left (-11 \left (b^2-4 a c\right ) (b+2 c x)^2+5 \left (b^2-4 a c\right )^2+12 (b+2 c x)^4\right )}{b^2-4 a c}-6 i (b+2 c x)^5 \sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{180 c^3 \sqrt{a+x (b+c x)} (d (b+2 c x))^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^(11/2),x]

[Out]

((c*(b + 2*c*x)*(a + x*(b + c*x))*(5*(b^2 - 4*a*c)^2 - 11*(b^2 - 4*a*c)*(b + 2*c
*x)^2 + 12*(b + 2*c*x)^4))/(b^2 - 4*a*c) - (6*I)*(b + 2*c*x)^5*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*Ar
cSinh[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]))/(180*c^3*(d*(b + 2*c*x))^(11/2)*Sqrt[a +
x*(b + c*x)])

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Maple [B]  time = 0.079, size = 1501, normalized size = 4.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^(11/2),x)

[Out]

1/180*(c*x^2+b*x+a)^(1/2)*(d*(2*c*x+b))^(1/2)*(288*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*b^2*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)+384*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^3*a*
b*c^4*((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)+96*
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^3*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)-58*x^2*b^4*c^2-6*b^5*c*x+12*((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^4*c-736*x^3*a*b*c^4-424*x^2*a*b^2*c^3-25
6*a^2*b*c^3*x-80*a^3*c^3-3*((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)*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^6+192*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^4*a*c^5*((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)-48*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^4*b^2*c^4*((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)-576*b*c^5*x^5-368*x^4*a*c^
5-628*x^4*b^2*c^4-296*b^3*c^3*x^3-256*x^2*a^2*c^4-56*a*b^3*c^2*x-192*c^6*x^6-6*a
*b^4*c-4*a^2*b^2*c^2-96*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^3*b^3*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)-72*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^4*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*b^5*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))/d^6/(2*c^2*x^3+3*b*c*x^
2+2*a*c*x+b^2*x+a*b)/(2*c*x+b)^4/(4*a*c-b^2)/c^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^(11/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^(11/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{{\left (32 \, c^{5} d^{5} x^{5} + 80 \, b c^{4} d^{5} x^{4} + 80 \, b^{2} c^{3} d^{5} x^{3} + 40 \, b^{3} c^{2} d^{5} x^{2} + 10 \, b^{4} c d^{5} x + b^{5} d^{5}\right )} \sqrt{2 \, c d x + b d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^(11/2),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)^(3/2)/((32*c^5*d^5*x^5 + 80*b*c^4*d^5*x^4 + 80*b^2*c^
3*d^5*x^3 + 40*b^3*c^2*d^5*x^2 + 10*b^4*c*d^5*x + b^5*d^5)*sqrt(2*c*d*x + b*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((c*x**2+b*x+a)**(3/2)/(2*c*d*x+b*d)**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^(11/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^(11/2), x)