3.2501 \(\int (d+e x) \sqrt [4]{a+b x+c x^2} \, dx\)

Optimal. Leaf size=241 \[ -\frac{\left (b^2-4 a c\right )^{5/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{12 \sqrt{2} c^{9/4} (b+2 c x)}+\frac{(b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e)}{6 c^2}+\frac{2 e \left (a+b x+c x^2\right )^{5/4}}{5 c} \]

[Out]

((2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4))/(6*c^2) + (2*e*(a + b*x + c*
x^2)^(5/4))/(5*c) - ((b^2 - 4*a*c)^(5/4)*(2*c*d - b*e)*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])/(12*Sqrt[2]*c^(9/4)*(b
+ 2*c*x))

_______________________________________________________________________________________

Rubi [A]  time = 0.447055, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\left (b^2-4 a c\right )^{5/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{12 \sqrt{2} c^{9/4} (b+2 c x)}+\frac{(b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e)}{6 c^2}+\frac{2 e \left (a+b x+c x^2\right )^{5/4}}{5 c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)*(a + b*x + c*x^2)^(1/4),x]

[Out]

((2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4))/(6*c^2) + (2*e*(a + b*x + c*
x^2)^(5/4))/(5*c) - ((b^2 - 4*a*c)^(5/4)*(2*c*d - b*e)*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])/(12*Sqrt[2]*c^(9/4)*(b
+ 2*c*x))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 36.3355, size = 284, normalized size = 1.18 \[ \frac{2 e \left (a + b x + c x^{2}\right )^{\frac{5}{4}}}{5 c} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right ) \sqrt [4]{a + b x + c x^{2}}}{6 c^{2}} + \frac{\sqrt{2} \sqrt{- \frac{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}{\left (4 a c - b^{2}\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{5}{4}} \left (b e - 2 c d\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right ) \sqrt{\left (b + 2 c x\right )^{2}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a + b x + c x^{2}}}{\sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | \frac{1}{2}\right )}{24 c^{\frac{9}{4}} \left (b + 2 c x\right ) \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)*(c*x**2+b*x+a)**(1/4),x)

[Out]

2*e*(a + b*x + c*x**2)**(5/4)/(5*c) - (b + 2*c*x)*(b*e - 2*c*d)*(a + b*x + c*x**
2)**(1/4)/(6*c**2) + 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)**(5/4)*(b*e - 2*c*d)*(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)/(24*c**(9/4)*(b + 2*c*x)*sqrt(-4
*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2)))

_______________________________________________________________________________________

Mathematica [C]  time = 0.587001, size = 194, normalized size = 0.8 \[ \frac{5 \sqrt [4]{2} \left (b^2-4 a c\right ) \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{3/4} (b e-2 c d) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )+4 c (a+x (b+c x)) \left (4 c (3 a e+c x (5 d+3 e x))-5 b^2 e+2 b c (5 d+e x)\right )}{120 c^3 (a+x (b+c x))^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)*(a + b*x + c*x^2)^(1/4),x]

[Out]

(4*c*(a + x*(b + c*x))*(-5*b^2*e + 2*b*c*(5*d + e*x) + 4*c*(3*a*e + c*x*(5*d + 3
*e*x))) + 5*2^(1/4)*(b^2 - 4*a*c)*(-2*c*d + b*e)*(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])])/(120*c^3*(a
+ x*(b + c*x))^(3/4))

_______________________________________________________________________________________

Maple [F]  time = 0.12, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) \sqrt [4]{c{x}^{2}+bx+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(c*x^2+b*x+a)^(1/4),x)

[Out]

int((e*x+d)*(c*x^2+b*x+a)^(1/4),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}{\left (e x + d\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(1/4)*(e*x + d),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(1/4)*(e*x + d), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}{\left (e x + d\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(1/4)*(e*x + d),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)*(c*x**2+b*x+a)**(1/4),x)

[Out]

Integral((d + e*x)*(a + b*x + c*x**2)**(1/4), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}{\left (e x + d\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(1/4)*(e*x + d),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(1/4)*(e*x + d), x)