3.2500 \(\int (d+e x)^2 \sqrt [4]{a+b x+c x^2} \, dx\)

Optimal. Leaf size=319 \[ -\frac{\left (b^2-4 a c\right )^{5/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+7 b d)+9 b^2 e^2+28 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{168 \sqrt{2} c^{13/4} (b+2 c x)}+\frac{(b+2 c x) \sqrt [4]{a+b x+c x^2} \left (-4 c e (2 a e+7 b d)+9 b^2 e^2+28 c^2 d^2\right )}{84 c^3}+\frac{9 e \left (a+b x+c x^2\right )^{5/4} (2 c d-b e)}{35 c^2}+\frac{2 e (d+e x) \left (a+b x+c x^2\right )^{5/4}}{7 c} \]

[Out]

((28*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(7*b*d + 2*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^
(1/4))/(84*c^3) + (9*e*(2*c*d - b*e)*(a + b*x + c*x^2)^(5/4))/(35*c^2) + (2*e*(d
 + e*x)*(a + b*x + c*x^2)^(5/4))/(7*c) - ((b^2 - 4*a*c)^(5/4)*(28*c^2*d^2 + 9*b^
2*e^2 - 4*c*e*(7*b*d + 2*a*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*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])/(168*Sqrt[2]*c^(13/4)*(b + 2*c*x))

_______________________________________________________________________________________

Rubi [A]  time = 0.894754, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{\left (b^2-4 a c\right )^{5/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+7 b d)+9 b^2 e^2+28 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{168 \sqrt{2} c^{13/4} (b+2 c x)}+\frac{(b+2 c x) \sqrt [4]{a+b x+c x^2} \left (-4 c e (2 a e+7 b d)+9 b^2 e^2+28 c^2 d^2\right )}{84 c^3}+\frac{9 e \left (a+b x+c x^2\right )^{5/4} (2 c d-b e)}{35 c^2}+\frac{2 e (d+e x) \left (a+b x+c x^2\right )^{5/4}}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

((28*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(7*b*d + 2*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^
(1/4))/(84*c^3) + (9*e*(2*c*d - b*e)*(a + b*x + c*x^2)^(5/4))/(35*c^2) + (2*e*(d
 + e*x)*(a + b*x + c*x^2)^(5/4))/(7*c) - ((b^2 - 4*a*c)^(5/4)*(28*c^2*d^2 + 9*b^
2*e^2 - 4*c*e*(7*b*d + 2*a*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*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])/(168*Sqrt[2]*c^(13/4)*(b + 2*c*x))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 75.4358, size = 371, normalized size = 1.16 \[ \frac{2 e \left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{4}}}{7 c} - \frac{9 e \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{4}}}{35 c^{2}} + \frac{\left (b + 2 c x\right ) \sqrt [4]{a + b x + c x^{2}} \left (- 8 a c e^{2} + 9 b^{2} e^{2} - 28 b c d e + 28 c^{2} d^{2}\right )}{84 c^{3}} - \frac{\sqrt{2} \sqrt{- \frac{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}{\left (4 a c - b^{2}\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{5}{4}} \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right ) \left (- 8 a c e^{2} + 9 b^{2} e^{2} - 28 b c d e + 28 c^{2} d^{2}\right ) \sqrt{\left (b + 2 c x\right )^{2}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a + b x + c x^{2}}}{\sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | \frac{1}{2}\right )}{336 c^{\frac{13}{4}} \left (b + 2 c x\right ) \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*e*(d + e*x)*(a + b*x + c*x**2)**(5/4)/(7*c) - 9*e*(b*e - 2*c*d)*(a + b*x + c*x
**2)**(5/4)/(35*c**2) + (b + 2*c*x)*(a + b*x + c*x**2)**(1/4)*(-8*a*c*e**2 + 9*b
**2*e**2 - 28*b*c*d*e + 28*c**2*d**2)/(84*c**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)/s
qrt(-4*a*c + b**2) + 1)**2))*(-4*a*c + b**2)**(5/4)*(2*sqrt(c)*sqrt(a + b*x + c*
x**2)/sqrt(-4*a*c + b**2) + 1)*(-8*a*c*e**2 + 9*b**2*e**2 - 28*b*c*d*e + 28*c**2
*d**2)*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**(13/4)*(b + 2*c*x)*sqrt(-4*a*c +
b**2 + c*(4*a + 4*b*x + 4*c*x**2)))

_______________________________________________________________________________________

Mathematica [C]  time = 0.947535, size = 274, normalized size = 0.86 \[ \frac{4 c (a+x (b+c x)) \left (4 b c \left (c \left (35 d^2+14 d e x+3 e^2 x^2\right )-37 a e^2\right )+8 c^2 \left (a e (42 d+5 e x)+c x \left (35 d^2+42 d e x+15 e^2 x^2\right )\right )+45 b^3 e^2-2 b^2 c e (70 d+9 e x)\right )-5 \sqrt [4]{2} \left (b^2-4 a c\right ) \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{3/4} \left (-4 c e (2 a e+7 b d)+9 b^2 e^2+28 c^2 d^2\right ) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{1680 c^4 (a+x (b+c x))^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(4*c*(a + x*(b + c*x))*(45*b^3*e^2 - 2*b^2*c*e*(70*d + 9*e*x) + 4*b*c*(-37*a*e^2
 + c*(35*d^2 + 14*d*e*x + 3*e^2*x^2)) + 8*c^2*(a*e*(42*d + 5*e*x) + c*x*(35*d^2
+ 42*d*e*x + 15*e^2*x^2))) - 5*2^(1/4)*(b^2 - 4*a*c)*(28*c^2*d^2 + 9*b^2*e^2 - 4
*c*e*(7*b*d + 2*a*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])])/(1680*c^4*(a + x*(b + c*x))^(3/4))

_______________________________________________________________________________________

Maple [F]  time = 0.125, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{2}\sqrt [4]{c{x}^{2}+bx+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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