3.2514 \(\int \left (a+b x+c x^2\right )^{5/4} \, dx\)

Optimal. Leaf size=236 \[ \frac{5 \left (b^2-4 a c\right )^{9/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{168 \sqrt{2} c^{9/4} (b+2 c x)}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{84 c^2}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{7 c} \]

[Out]

(-5*(b^2 - 4*a*c)*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4))/(84*c^2) + ((b + 2*c*x)*(
a + b*x + c*x^2)^(5/4))/(7*c) + (5*(b^2 - 4*a*c)^(9/4)*Sqrt[(b + 2*c*x)^2/((b^2
- 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])^2)]*(1 + (2*S
qrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*EllipticF[2*ArcTan[(Sqrt[2]*c^(
1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(168*Sqrt[2]*c^(9/4)*(
b + 2*c*x))

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Rubi [A]  time = 0.408662, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{5 \left (b^2-4 a c\right )^{9/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{168 \sqrt{2} c^{9/4} (b+2 c x)}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{84 c^2}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{7 c} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(5/4),x]

[Out]

(-5*(b^2 - 4*a*c)*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4))/(84*c^2) + ((b + 2*c*x)*(
a + b*x + c*x^2)^(5/4))/(7*c) + (5*(b^2 - 4*a*c)^(9/4)*Sqrt[(b + 2*c*x)^2/((b^2
- 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])^2)]*(1 + (2*S
qrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*EllipticF[2*ArcTan[(Sqrt[2]*c^(
1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(168*Sqrt[2]*c^(9/4)*(
b + 2*c*x))

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Rubi in Sympy [A]  time = 30.0462, size = 282, normalized size = 1.19 \[ \frac{\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{4}}}{7 c} - \frac{5 \left (b + 2 c x\right ) \left (- a c + \frac{b^{2}}{4}\right ) \sqrt [4]{a + b x + c x^{2}}}{21 c^{2}} + \frac{5 \sqrt{2} \sqrt{- \frac{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}{\left (4 a c - b^{2}\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{9}{4}} \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right ) \sqrt{\left (b + 2 c x\right )^{2}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a + b x + c x^{2}}}{\sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | \frac{1}{2}\right )}{336 c^{\frac{9}{4}} \left (b + 2 c x\right ) \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(5/4),x)

[Out]

(b + 2*c*x)*(a + b*x + c*x**2)**(5/4)/(7*c) - 5*(b + 2*c*x)*(-a*c + b**2/4)*(a +
b*x + c*x**2)**(1/4)/(21*c**2) + 5*sqrt(2)*sqrt(-(-4*a*c + b**2 + c*(4*a + 4*b*
x + 4*c*x**2))/((4*a*c - b**2)*(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b
**2) + 1)**2))*(-4*a*c + b**2)**(9/4)*(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*
a*c + b**2) + 1)*sqrt((b + 2*c*x)**2)*elliptic_f(2*atan(sqrt(2)*c**(1/4)*(a + b*
x + c*x**2)**(1/4)/(-4*a*c + b**2)**(1/4)), 1/2)/(336*c**(9/4)*(b + 2*c*x)*sqrt(
-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2)))

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Mathematica [C]  time = 0.38413, size = 181, normalized size = 0.77 \[ \frac{5 \sqrt [4]{2} \left (b^2-4 a c\right )^2 \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )+4 c (b+2 c x) (a+x (b+c x)) \left (4 c \left (8 a+3 c x^2\right )-5 b^2+12 b c x\right )}{336 c^3 (a+x (b+c x))^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(5/4),x]

[Out]

(4*c*(b + 2*c*x)*(a + x*(b + c*x))*(-5*b^2 + 12*b*c*x + 4*c*(8*a + 3*c*x^2)) + 5
*2^(1/4)*(b^2 - 4*a*c)^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*((b + Sqrt[b^2 - 4*a*c]
+ 2*c*x)/Sqrt[b^2 - 4*a*c])^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, (-b + Sqrt[b
^2 - 4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a*c])])/(336*c^3*(a + x*(b + c*x))^(3/4))

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Maple [F]  time = 0.199, size = 0, normalized size = 0. \[ \int \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(5/4),x)

[Out]

int((c*x^2+b*x+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{5}{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/4),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(5/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/4),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)^(5/4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (a + b x + c x^{2}\right )^{\frac{5}{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**(5/4),x)

[Out]

Integral((a + b*x + c*x**2)**(5/4), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{5}{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/4),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(5/4), x)