3.2518 \(\int \frac{(d+e x)^2}{\sqrt [4]{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=573 \[ \frac{(b+2 c x) \sqrt [4]{a+b x+c x^2} \left (-4 c e (2 a e+5 b d)+7 b^2 e^2+20 c^2 d^2\right )}{10 c^{5/2} \sqrt{b^2-4 a c} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}+\frac{\left (b^2-4 a c\right )^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+5 b d)+7 b^2 e^2+20 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{20 \sqrt{2} c^{11/4} (b+2 c x)}-\frac{\left (b^2-4 a c\right )^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+5 b d)+7 b^2 e^2+20 c^2 d^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{10 \sqrt{2} c^{11/4} (b+2 c x)}+\frac{7 e \left (a+b x+c x^2\right )^{3/4} (2 c d-b e)}{15 c^2}+\frac{2 e (d+e x) \left (a+b x+c x^2\right )^{3/4}}{5 c} \]

[Out]

(7*e*(2*c*d - b*e)*(a + b*x + c*x^2)^(3/4))/(15*c^2) + (2*e*(d + e*x)*(a + b*x +
c*x^2)^(3/4))/(5*c) + ((20*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(5*b*d + 2*a*e))*(b + 2*
c*x)*(a + b*x + c*x^2)^(1/4))/(10*c^(5/2)*Sqrt[b^2 - 4*a*c]*(1 + (2*Sqrt[c]*Sqrt
[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])) - ((b^2 - 4*a*c)^(3/4)*(20*c^2*d^2 + 7*b^
2*e^2 - 4*c*e*(5*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])*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^(11/4)*(b + 2*c*x)) + ((b^2 - 4*
a*c)^(3/4)*(20*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(5*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])/(20*Sqrt[2]*c^(11
/4)*(b + 2*c*x))

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Rubi [A]  time = 1.48181, antiderivative size = 573, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{(b+2 c x) \sqrt [4]{a+b x+c x^2} \left (-4 c e (2 a e+5 b d)+7 b^2 e^2+20 c^2 d^2\right )}{10 c^{5/2} \sqrt{b^2-4 a c} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}+\frac{\left (b^2-4 a c\right )^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+5 b d)+7 b^2 e^2+20 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{20 \sqrt{2} c^{11/4} (b+2 c x)}-\frac{\left (b^2-4 a c\right )^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+5 b d)+7 b^2 e^2+20 c^2 d^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{10 \sqrt{2} c^{11/4} (b+2 c x)}+\frac{7 e \left (a+b x+c x^2\right )^{3/4} (2 c d-b e)}{15 c^2}+\frac{2 e (d+e x) \left (a+b x+c x^2\right )^{3/4}}{5 c} \]

Warning: Unable to verify antiderivative.

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

[Out]

(7*e*(2*c*d - b*e)*(a + b*x + c*x^2)^(3/4))/(15*c^2) + (2*e*(d + e*x)*(a + b*x +
c*x^2)^(3/4))/(5*c) + ((20*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(5*b*d + 2*a*e))*(b + 2*
c*x)*(a + b*x + c*x^2)^(1/4))/(10*c^(5/2)*Sqrt[b^2 - 4*a*c]*(1 + (2*Sqrt[c]*Sqrt
[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])) - ((b^2 - 4*a*c)^(3/4)*(20*c^2*d^2 + 7*b^
2*e^2 - 4*c*e*(5*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])*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^(11/4)*(b + 2*c*x)) + ((b^2 - 4*
a*c)^(3/4)*(20*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(5*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])/(20*Sqrt[2]*c^(11
/4)*(b + 2*c*x))

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Rubi in Sympy [A]  time = 139.998, size = 716, normalized size = 1.25 \[ \frac{2 e \left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{4}}}{5 c} - \frac{7 e \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{4}}}{15 c^{2}} + \frac{\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 )} \left (- 8 a c e^{2} + 7 b^{2} e^{2} - 20 b c d e + 20 c^{2} d^{2}\right ) \sqrt{\left (b + 2 c x\right )^{2}}}{10 c^{\frac{5}{2}} \left (b + 2 c x\right ) \sqrt{- 4 a c + b^{2}} \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )} - \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{3}{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} + 7 b^{2} e^{2} - 20 b c d e + 20 c^{2} d^{2}\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{11}{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{\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{3}{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} + 7 b^{2} e^{2} - 20 b c d e + 20 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 )}{40 c^{\frac{11}{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)**(3/4)/(5*c) - 7*e*(b*e - 2*c*d)*(a + b*x + c*x
**2)**(3/4)/(15*c**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))*(-8*a*c*e**2 + 7*b**2*e**2 - 20*b*c*d*e + 20*c**2*d**2)*sqrt(
(b + 2*c*x)**2)/(10*c**(5/2)*(b + 2*c*x)*sqrt(-4*a*c + b**2)*(2*sqrt(c)*sqrt(a +
b*x + c*x**2)/sqrt(-4*a*c + b**2) + 1)) - 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)**(3/4)*(2*sqrt(c)*sqrt(a + b*x + c*x**2)/s
qrt(-4*a*c + b**2) + 1)*(-8*a*c*e**2 + 7*b**2*e**2 - 20*b*c*d*e + 20*c**2*d**2)*
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**(11/4)*(b + 2*c*x)*sqrt(-4*a*c + b**2 + c
*(4*a + 4*b*x + 4*c*x**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)**(3/4)*(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c +
b**2) + 1)*(-8*a*c*e**2 + 7*b**2*e**2 - 20*b*c*d*e + 20*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)/(40*c**(11/4)*(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 = 0.540304, size = 185, normalized size = 0.32 \[ \frac{2^{3/4} \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \sqrt [4]{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}} \left (-4 c e (2 a e+5 b d)+7 b^2 e^2+20 c^2 d^2\right ) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )+4 c e (a+x (b+c x)) (-7 b e+20 c d+6 c e x)}{60 c^3 \sqrt [4]{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(4*c*e*(20*c*d - 7*b*e + 6*c*e*x)*(a + x*(b + c*x)) + 2^(3/4)*(20*c^2*d^2 + 7*b^
2*e^2 - 4*c*e*(5*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])^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, (-b +
Sqrt[b^2 - 4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a*c])])/(60*c^3*(a + x*(b + c*x))^(1/
4))

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Maple [F]  time = 0.122, size = 0, normalized size = 0. \[ \int{ \left ( ex+d \right ) ^{2}{\frac{1}{\sqrt [4]{c{x}^{2}+bx+a}}}}\, dx \]

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)

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

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, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)

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

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, algorithm="giac")

[Out]

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