3.2517 \(\int \frac{(d+e x)^3}{\sqrt [4]{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=637 \[ \frac{(b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e) \left (-4 c e (6 a e+5 b d)+11 b^2 e^2+20 c^2 d^2\right )}{20 c^{7/2} \sqrt{b^2-4 a c} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}+\frac{\left (b^2-4 a c\right )^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (6 a e+5 b d)+11 b^2 e^2+20 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{40 \sqrt{2} c^{15/4} (b+2 c x)}-\frac{\left (b^2-4 a c\right )^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (6 a e+5 b d)+11 b^2 e^2+20 c^2 d^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{20 \sqrt{2} c^{15/4} (b+2 c x)}+\frac{e \left (a+b x+c x^2\right )^{3/4} \left (-2 c e (40 a e+147 b d)+77 b^2 e^2+66 c e x (2 c d-b e)+360 c^2 d^2\right )}{210 c^3}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{3/4}}{7 c} \]

[Out]

(2*e*(d + e*x)^2*(a + b*x + c*x^2)^(3/4))/(7*c) + (e*(360*c^2*d^2 + 77*b^2*e^2 -
2*c*e*(147*b*d + 40*a*e) + 66*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/4))/(21
0*c^3) + ((2*c*d - b*e)*(20*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(5*b*d + 6*a*e))*(b + 2
*c*x)*(a + b*x + c*x^2)^(1/4))/(20*c^(7/2)*Sqrt[b^2 - 4*a*c]*(1 + (2*Sqrt[c]*Sqr
t[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])) - ((b^2 - 4*a*c)^(3/4)*(2*c*d - b*e)*(20
*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(5*b*d + 6*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]*S
qrt[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])/(20*Sqrt[2]*c^(15/4)*(b + 2*c*
x)) + ((b^2 - 4*a*c)^(3/4)*(2*c*d - b*e)*(20*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(5*b*d
+ 6*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])/(40*Sqrt[2]*c^(15/4)*(b + 2*c*x))

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Rubi [A]  time = 1.58279, antiderivative size = 637, 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 \[ \frac{(b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e) \left (-4 c e (6 a e+5 b d)+11 b^2 e^2+20 c^2 d^2\right )}{20 c^{7/2} \sqrt{b^2-4 a c} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}+\frac{\left (b^2-4 a c\right )^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (6 a e+5 b d)+11 b^2 e^2+20 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{40 \sqrt{2} c^{15/4} (b+2 c x)}-\frac{\left (b^2-4 a c\right )^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (6 a e+5 b d)+11 b^2 e^2+20 c^2 d^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{20 \sqrt{2} c^{15/4} (b+2 c x)}+\frac{e \left (a+b x+c x^2\right )^{3/4} \left (-2 c e (40 a e+147 b d)+77 b^2 e^2+66 c e x (2 c d-b e)+360 c^2 d^2\right )}{210 c^3}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{3/4}}{7 c} \]

Warning: Unable to verify antiderivative.

[In]  Int[(d + e*x)^3/(a + b*x + c*x^2)^(1/4),x]

[Out]

(2*e*(d + e*x)^2*(a + b*x + c*x^2)^(3/4))/(7*c) + (e*(360*c^2*d^2 + 77*b^2*e^2 -
2*c*e*(147*b*d + 40*a*e) + 66*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/4))/(21
0*c^3) + ((2*c*d - b*e)*(20*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(5*b*d + 6*a*e))*(b + 2
*c*x)*(a + b*x + c*x^2)^(1/4))/(20*c^(7/2)*Sqrt[b^2 - 4*a*c]*(1 + (2*Sqrt[c]*Sqr
t[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])) - ((b^2 - 4*a*c)^(3/4)*(2*c*d - b*e)*(20
*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(5*b*d + 6*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]*S
qrt[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])/(20*Sqrt[2]*c^(15/4)*(b + 2*c*
x)) + ((b^2 - 4*a*c)^(3/4)*(2*c*d - b*e)*(20*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(5*b*d
+ 6*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])/(40*Sqrt[2]*c^(15/4)*(b + 2*c*x))

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Rubi in Sympy [A]  time = 161.579, size = 790, normalized size = 1.24 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3/(c*x**2+b*x+a)**(1/4),x)

[Out]

2*e*(d + e*x)**2*(a + b*x + c*x**2)**(3/4)/(7*c) + 8*e*(a + b*x + c*x**2)**(3/4)
*(-5*a*c*e**2 + 77*b**2*e**2/16 - 147*b*c*d*e/8 + 45*c**2*d**2/2 - 33*c*e*x*(b*e
- 2*c*d)/8)/(105*c**3) - (b*e - 2*c*d)*(a + b*x + c*x**2)**(1/4)*sqrt(-4*a*c +
b**2 + c*(4*a + 4*b*x + 4*c*x**2))*(-24*a*c*e**2 + 11*b**2*e**2 - 20*b*c*d*e + 2
0*c**2*d**2)*sqrt((b + 2*c*x)**2)/(20*c**(7/2)*(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)*(b*e - 2*c*d)*(2*
sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2) + 1)*(-24*a*c*e**2 + 11*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)/(40*c**(15/4)*
(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)*(b*e - 2*c*d)*
(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2) + 1)*(-24*a*c*e**2 + 11*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)/(80*c**(15/
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 = 0.720484, size = 232, normalized size = 0.36 \[ \frac{7\ 2^{3/4} \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \sqrt [4]{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}} (2 c d-b e) \left (-4 c e (6 a e+5 b d)+11 b^2 e^2+20 c^2 d^2\right ) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )+4 c e (a+x (b+c x)) \left (-2 c e (40 a e+147 b d+33 b e x)+77 b^2 e^2+12 c^2 \left (35 d^2+21 d e x+5 e^2 x^2\right )\right )}{840 c^4 \sqrt [4]{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3/(a + b*x + c*x^2)^(1/4),x]

[Out]

(4*c*e*(a + x*(b + c*x))*(77*b^2*e^2 - 2*c*e*(147*b*d + 40*a*e + 33*b*e*x) + 12*
c^2*(35*d^2 + 21*d*e*x + 5*e^2*x^2)) + 7*2^(3/4)*(2*c*d - b*e)*(20*c^2*d^2 + 11*
b^2*e^2 - 4*c*e*(5*b*d + 6*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])])/(840*c^4*(a + x*(b + c*x))^
(1/4))

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Maple [F]  time = 0.128, size = 0, normalized size = 0. \[ \int{ \left ( ex+d \right ) ^{3}{\frac{1}{\sqrt [4]{c{x}^{2}+bx+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3/(c*x^2+b*x+a)^(1/4),x)

[Out]

int((e*x+d)^3/(c*x^2+b*x+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{3}}{{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*x^2 + b*x + a)^(1/4),x, algorithm="maxima")

[Out]

integrate((e*x + d)^3/(c*x^2 + b*x + a)^(1/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}{{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*x^2 + b*x + a)^(1/4),x, algorithm="fricas")

[Out]

integral((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)/(c*x^2 + b*x + a)^(1/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3/(c*x**2+b*x+a)**(1/4),x)

[Out]

Integral((d + e*x)**3/(a + b*x + c*x**2)**(1/4), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{3}}{{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*x^2 + b*x + a)^(1/4),x, algorithm="giac")

[Out]

integrate((e*x + d)^3/(c*x^2 + b*x + a)^(1/4), x)