3.2523 \(\int \frac{1}{(d+e x)^3 \sqrt [4]{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=1465 \[ \text{result too large to display} \]

[Out]

-(e*(a + b*x + c*x^2)^(3/4))/(2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) - (5*e*(2*c
*d - b*e)*(a + b*x + c*x^2)^(3/4))/(8*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)) + (5*
Sqrt[c]*(2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4))/(8*Sqrt[b^2 - 4*a*c]*
(c*d^2 - b*d*e + a*e^2)^2*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*
c])) + ((-b^2 + 4*a*c)^(1/4)*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*(-
((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4)*ArcTan[((-b^2 + 4*a*c)^(1/4)*Sqrt[e
]*(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4))/(Sqrt[2]*c^(1/4)*(c*d^2 - b*d*e + a*e
^2)^(1/4))])/(32*c^(1/4)*Sqrt[e]*(c*d^2 - b*d*e + a*e^2)^(9/4)*(a + b*x + c*x^2)
^(1/4)) - ((-b^2 + 4*a*c)^(1/4)*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))
*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4)*ArcTanh[((-b^2 + 4*a*c)^(1/4)*Sq
rt[e]*(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4))/(Sqrt[2]*c^(1/4)*(c*d^2 - b*d*e +
a*e^2)^(1/4))])/(32*c^(1/4)*Sqrt[e]*(c*d^2 - b*d*e + a*e^2)^(9/4)*(a + b*x + c*
x^2)^(1/4)) - (5*c^(1/4)*(b^2 - 4*a*c)^(3/4)*(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*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])/(8*Sqrt[2]*(c*d^2
- b*d*e + a*e^2)^2*(b + 2*c*x)) + (5*c^(1/4)*(b^2 - 4*a*c)^(3/4)*(2*c*d - b*e)*S
qrt[(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])*Ellipti
cF[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2]
)/(16*Sqrt[2]*(c*d^2 - b*d*e + a*e^2)^2*(b + 2*c*x)) - (Sqrt[-b^2 + 4*a*c]*(2*c*
d - b*e)*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*Sqrt[(b + 2*c*x)^2/(b^
2 - 4*a*c)]*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4)*EllipticPi[-(Sqrt[-b^
2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^2]), ArcSin[(1 - (b + 2*c*x)^2
/(b^2 - 4*a*c))^(1/4)], -1])/(32*Sqrt[2]*Sqrt[c]*e*(c*d^2 - b*d*e + a*e^2)^(5/2)
*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4)) + (Sqrt[-b^2 + 4*a*c]*(2*c*d - b*e)*(12*c^
2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*(-(
(c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4)*EllipticPi[(Sqrt[-b^2 + 4*a*c]*e)/(2
*Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^2]), ArcSin[(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^
(1/4)], -1])/(32*Sqrt[2]*Sqrt[c]*e*(c*d^2 - b*d*e + a*e^2)^(5/2)*(b + 2*c*x)*(a
+ b*x + c*x^2)^(1/4))

_______________________________________________________________________________________

Rubi [A]  time = 8.68296, antiderivative size = 1465, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 19, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.864 \[ -\frac{5 (2 c d-b e) \left (c x^2+b x+a\right )^{3/4} e}{8 \left (c d^2-b e d+a e^2\right )^2 (d+e x)}-\frac{\left (c x^2+b x+a\right )^{3/4} e}{2 \left (c d^2-b e d+a e^2\right ) (d+e x)^2}-\frac{5 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\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 )}{8 \sqrt{2} \left (c d^2-b e d+a e^2\right )^2 (b+2 c x)}+\frac{5 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\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 )}{16 \sqrt{2} \left (c d^2-b e d+a e^2\right )^2 (b+2 c x)}+\frac{5 \sqrt{c} (2 c d-b e) (b+2 c x) \sqrt [4]{c x^2+b x+a}}{8 \sqrt{b^2-4 a c} \left (c d^2-b e d+a e^2\right )^2 \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right )}+\frac{\sqrt [4]{4 a c-b^2} \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt [4]{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \tan ^{-1}\left (\frac{\sqrt [4]{4 a c-b^2} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{32 \sqrt [4]{c} \left (c d^2-b e d+a e^2\right )^{9/4} \sqrt [4]{c x^2+b x+a} \sqrt{e}}-\frac{\sqrt [4]{4 a c-b^2} \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt [4]{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \tanh ^{-1}\left (\frac{\sqrt [4]{4 a c-b^2} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{32 \sqrt [4]{c} \left (c d^2-b e d+a e^2\right )^{9/4} \sqrt [4]{c x^2+b x+a} \sqrt{e}}-\frac{\sqrt{4 a c-b^2} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \Pi \left (-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{32 \sqrt{2} \sqrt{c} \left (c d^2-b e d+a e^2\right )^{5/2} (b+2 c x) \sqrt [4]{c x^2+b x+a} e}+\frac{\sqrt{4 a c-b^2} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \Pi \left (\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{32 \sqrt{2} \sqrt{c} \left (c d^2-b e d+a e^2\right )^{5/2} (b+2 c x) \sqrt [4]{c x^2+b x+a} e} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(e*(a + b*x + c*x^2)^(3/4))/(2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) - (5*e*(2*c
*d - b*e)*(a + b*x + c*x^2)^(3/4))/(8*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)) + (5*
Sqrt[c]*(2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4))/(8*Sqrt[b^2 - 4*a*c]*
(c*d^2 - b*d*e + a*e^2)^2*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*
c])) + ((-b^2 + 4*a*c)^(1/4)*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*(-
((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4)*ArcTan[((-b^2 + 4*a*c)^(1/4)*Sqrt[e
]*(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4))/(Sqrt[2]*c^(1/4)*(c*d^2 - b*d*e + a*e
^2)^(1/4))])/(32*c^(1/4)*Sqrt[e]*(c*d^2 - b*d*e + a*e^2)^(9/4)*(a + b*x + c*x^2)
^(1/4)) - ((-b^2 + 4*a*c)^(1/4)*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))
*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4)*ArcTanh[((-b^2 + 4*a*c)^(1/4)*Sq
rt[e]*(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4))/(Sqrt[2]*c^(1/4)*(c*d^2 - b*d*e +
a*e^2)^(1/4))])/(32*c^(1/4)*Sqrt[e]*(c*d^2 - b*d*e + a*e^2)^(9/4)*(a + b*x + c*
x^2)^(1/4)) - (5*c^(1/4)*(b^2 - 4*a*c)^(3/4)*(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*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])/(8*Sqrt[2]*(c*d^2
- b*d*e + a*e^2)^2*(b + 2*c*x)) + (5*c^(1/4)*(b^2 - 4*a*c)^(3/4)*(2*c*d - b*e)*S
qrt[(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])*Ellipti
cF[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2]
)/(16*Sqrt[2]*(c*d^2 - b*d*e + a*e^2)^2*(b + 2*c*x)) - (Sqrt[-b^2 + 4*a*c]*(2*c*
d - b*e)*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*Sqrt[(b + 2*c*x)^2/(b^
2 - 4*a*c)]*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4)*EllipticPi[-(Sqrt[-b^
2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^2]), ArcSin[(1 - (b + 2*c*x)^2
/(b^2 - 4*a*c))^(1/4)], -1])/(32*Sqrt[2]*Sqrt[c]*e*(c*d^2 - b*d*e + a*e^2)^(5/2)
*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4)) + (Sqrt[-b^2 + 4*a*c]*(2*c*d - b*e)*(12*c^
2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*(-(
(c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4)*EllipticPi[(Sqrt[-b^2 + 4*a*c]*e)/(2
*Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^2]), ArcSin[(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^
(1/4)], -1])/(32*Sqrt[2]*Sqrt[c]*e*(c*d^2 - b*d*e + a*e^2)^(5/2)*(b + 2*c*x)*(a
+ b*x + c*x^2)^(1/4))

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 0.749595, size = 187, normalized size = 0.13 \[ -\frac{\sqrt{2} \sqrt [4]{\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}} \sqrt [4]{\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}} F_1\left (\frac{5}{2};\frac{1}{4},\frac{1}{4};\frac{7}{2};\frac{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right )}{5 e (d+e x)^2 \sqrt [4]{a+x (b+c x)}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(Sqrt[2]*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(1/4)*((e*(b + Sqr
t[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(1/4)*AppellF1[5/2, 1/4, 1/4, 7/2, (2*c*
d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*e + Sqrt[b^2 - 4*a*c]
*e)/(2*c*d + 2*c*e*x)])/(5*e*(d + e*x)^2*(a + x*(b + c*x))^(1/4))

_______________________________________________________________________________________

Maple [F]  time = 0.135, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( ex+d \right ) ^{3}}{\frac{1}{\sqrt [4]{c{x}^{2}+bx+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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