3.2508 \(\int \left (a+b x+c x^2\right )^{3/4} \, dx\)

Optimal. Leaf size=452 \[ -\frac{3 \sqrt{b^2-4 a c} (b+2 c x) \sqrt [4]{a+b x+c x^2}}{10 c^{3/2} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}-\frac{3 \left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{20 \sqrt{2} c^{7/4} (b+2 c x)}+\frac{3 \left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{10 \sqrt{2} c^{7/4} (b+2 c x)}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{5 c} \]

[Out]

((b + 2*c*x)*(a + b*x + c*x^2)^(3/4))/(5*c) - (3*Sqrt[b^2 - 4*a*c]*(b + 2*c*x)*(
a + b*x + c*x^2)^(1/4))/(10*c^(3/2)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[
b^2 - 4*a*c])) + (3*(b^2 - 4*a*c)^(7/4)*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*Sqrt[c]*Sqrt[a +
b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*EllipticE[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x +
c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(10*Sqrt[2]*c^(7/4)*(b + 2*c*x)) - (3
*(b^2 - 4*a*c)^(7/4)*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*Sqrt[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])/(20*Sqrt[2]*c^(7/4)*(b + 2*c*x))

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Rubi [A]  time = 0.795694, antiderivative size = 452, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ -\frac{3 \sqrt{b^2-4 a c} (b+2 c x) \sqrt [4]{a+b x+c x^2}}{10 c^{3/2} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}-\frac{3 \left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{20 \sqrt{2} c^{7/4} (b+2 c x)}+\frac{3 \left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{10 \sqrt{2} c^{7/4} (b+2 c x)}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{5 c} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(3/4),x]

[Out]

((b + 2*c*x)*(a + b*x + c*x^2)^(3/4))/(5*c) - (3*Sqrt[b^2 - 4*a*c]*(b + 2*c*x)*(
a + b*x + c*x^2)^(1/4))/(10*c^(3/2)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[
b^2 - 4*a*c])) + (3*(b^2 - 4*a*c)^(7/4)*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*Sqrt[c]*Sqrt[a +
b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*EllipticE[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x +
c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(10*Sqrt[2]*c^(7/4)*(b + 2*c*x)) - (3
*(b^2 - 4*a*c)^(7/4)*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*Sqrt[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])/(20*Sqrt[2]*c^(7/4)*(b + 2*c*x))

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Rubi in Sympy [A]  time = 70.4682, size = 586, normalized size = 1.3 \[ \frac{\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{4}}}{5 c} - \frac{3 \sqrt{- 4 a c + b^{2}} \sqrt [4]{a + b x + c x^{2}} \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )} \sqrt{\left (b + 2 c x\right )^{2}}}{10 c^{\frac{3}{2}} \left (b + 2 c x\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )} + \frac{3 \sqrt{2} \sqrt{- \frac{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}{\left (4 a c - b^{2}\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right ) \sqrt{\left (b + 2 c x\right )^{2}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a + b x + c x^{2}}}{\sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | \frac{1}{2}\right )}{20 c^{\frac{7}{4}} \left (b + 2 c x\right ) \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}} - \frac{3 \sqrt{2} \sqrt{- \frac{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}{\left (4 a c - b^{2}\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right ) \sqrt{\left (b + 2 c x\right )^{2}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a + b x + c x^{2}}}{\sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | \frac{1}{2}\right )}{40 c^{\frac{7}{4}} \left (b + 2 c x\right ) \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(3/4),x)

[Out]

(b + 2*c*x)*(a + b*x + c*x**2)**(3/4)/(5*c) - 3*sqrt(-4*a*c + b**2)*(a + b*x + c
*x**2)**(1/4)*sqrt(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2))*sqrt((b + 2*c*x)*
*2)/(10*c**(3/2)*(b + 2*c*x)*(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**
2) + 1)) + 3*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)**(7/4)*(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2) + 1)*sqrt((
b + 2*c*x)**2)*elliptic_e(2*atan(sqrt(2)*c**(1/4)*(a + b*x + c*x**2)**(1/4)/(-4*
a*c + b**2)**(1/4)), 1/2)/(20*c**(7/4)*(b + 2*c*x)*sqrt(-4*a*c + b**2 + c*(4*a +
4*b*x + 4*c*x**2))) - 3*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)**(7/4)*(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)/(40*c**(7/4)*(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 = 0.278341, size = 155, normalized size = 0.34 \[ \frac{4 c (b+2 c x) (a+x (b+c x))-2^{3/4} \left (b^2-4 a c\right ) \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \sqrt [4]{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{20 c^2 \sqrt [4]{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(3/4),x]

[Out]

(4*c*(b + 2*c*x)*(a + x*(b + c*x)) - 2^(3/4)*(b^2 - 4*a*c)*(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])^(1/4)*Hypergeomet
ric2F1[1/4, 3/4, 7/4, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a*c])])/(
20*c^2*(a + x*(b + c*x))^(1/4))

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Maple [F]  time = 0.201, size = 0, normalized size = 0. \[ \int \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(3/4),x)

[Out]

int((c*x^2+b*x+a)^(3/4),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/4),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(3/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/4),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)^(3/4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (a + b x + c x^{2}\right )^{\frac{3}{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**(3/4),x)

[Out]

Integral((a + b*x + c*x**2)**(3/4), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{3}{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/4),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(3/4), x)