3.2522 \(\int \frac{1}{(d+e x)^2 \sqrt [4]{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=1280 \[ \text{result too large to display} \]

[Out]

-((e*(a + b*x + c*x^2)^(3/4))/((c*d^2 - b*d*e + a*e^2)*(d + e*x))) + (Sqrt[c]*(b
+ 2*c*x)*(a + b*x + c*x^2)^(1/4))/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^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)
*(2*c*d - b*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))])/(4*c^(1/4)*Sqrt[e]*(c*d^2 - b*d*e + a*e^2)^(5/4)*(
a + b*x + c*x^2)^(1/4)) - ((-b^2 + 4*a*c)^(1/4)*(2*c*d - b*e)*(-((c*(a + b*x + c
*x^2))/(b^2 - 4*a*c)))^(1/4)*ArcTanh[((-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))])/(4
*c^(1/4)*Sqrt[e]*(c*d^2 - b*d*e + a*e^2)^(5/4)*(a + b*x + c*x^2)^(1/4)) - (c^(1/
4)*(b^2 - 4*a*c)^(3/4)*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])/Sq
rt[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])/(Sqrt[2]*(c*d^2 - b*d*e + a*e^2)*(b + 2*c*x)) + (c^(1/
4)*(b^2 - 4*a*c)^(3/4)*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])/Sq
rt[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])/(2*Sqrt[2]*(c*d^2 - b*d*e + a*e^2)*(b + 2*c*x)) - (Sqr
t[-b^2 + 4*a*c]*(2*c*d - b*e)^2*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]*S
qrt[c*d^2 - b*d*e + a*e^2]), ArcSin[(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4)], -1
])/(4*Sqrt[2]*Sqrt[c]*e*(c*d^2 - b*d*e + a*e^2)^(3/2)*(b + 2*c*x)*(a + b*x + c*x
^2)^(1/4)) + (Sqrt[-b^2 + 4*a*c]*(2*c*d - b*e)^2*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])/(4*Sqrt[2]*Sqrt[c]*e*(c*d^2 - b*d*e + a*e^2)^(3/2)*(b + 2*c*x
)*(a + b*x + c*x^2)^(1/4))

_______________________________________________________________________________________

Rubi [A]  time = 6.91225, antiderivative size = 1280, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 18, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.818 \[ -\frac{\sqrt{4 a c-b^2} \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 ) (2 c d-b e)^2}{4 \sqrt{2} \sqrt{c} e \left (c d^2-b e d+a e^2\right )^{3/2} (b+2 c x) \sqrt [4]{c x^2+b x+a}}+\frac{\sqrt{4 a c-b^2} \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 ) (2 c d-b e)^2}{4 \sqrt{2} \sqrt{c} e \left (c d^2-b e d+a e^2\right )^{3/2} (b+2 c x) \sqrt [4]{c x^2+b x+a}}+\frac{\sqrt [4]{4 a c-b^2} \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 ) (2 c d-b e)}{4 \sqrt [4]{c} \sqrt{e} \left (c d^2-b e d+a e^2\right )^{5/4} \sqrt [4]{c x^2+b x+a}}-\frac{\sqrt [4]{4 a c-b^2} \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 ) (2 c d-b e)}{4 \sqrt [4]{c} \sqrt{e} \left (c d^2-b e d+a e^2\right )^{5/4} \sqrt [4]{c x^2+b x+a}}-\frac{\sqrt [4]{c} \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{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 )}{\sqrt{2} \left (c d^2-b e d+a e^2\right ) (b+2 c x)}+\frac{\sqrt [4]{c} \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{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 )}{2 \sqrt{2} \left (c d^2-b e d+a e^2\right ) (b+2 c x)}-\frac{e \left (c x^2+b x+a\right )^{3/4}}{\left (c d^2-b e d+a e^2\right ) (d+e x)}+\frac{\sqrt{c} (b+2 c x) \sqrt [4]{c x^2+b x+a}}{\sqrt{b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

-((e*(a + b*x + c*x^2)^(3/4))/((c*d^2 - b*d*e + a*e^2)*(d + e*x))) + (Sqrt[c]*(b
+ 2*c*x)*(a + b*x + c*x^2)^(1/4))/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^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)
*(2*c*d - b*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))])/(4*c^(1/4)*Sqrt[e]*(c*d^2 - b*d*e + a*e^2)^(5/4)*(
a + b*x + c*x^2)^(1/4)) - ((-b^2 + 4*a*c)^(1/4)*(2*c*d - b*e)*(-((c*(a + b*x + c
*x^2))/(b^2 - 4*a*c)))^(1/4)*ArcTanh[((-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))])/(4
*c^(1/4)*Sqrt[e]*(c*d^2 - b*d*e + a*e^2)^(5/4)*(a + b*x + c*x^2)^(1/4)) - (c^(1/
4)*(b^2 - 4*a*c)^(3/4)*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])/Sq
rt[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])/(Sqrt[2]*(c*d^2 - b*d*e + a*e^2)*(b + 2*c*x)) + (c^(1/
4)*(b^2 - 4*a*c)^(3/4)*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])/Sq
rt[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])/(2*Sqrt[2]*(c*d^2 - b*d*e + a*e^2)*(b + 2*c*x)) - (Sqr
t[-b^2 + 4*a*c]*(2*c*d - b*e)^2*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]*S
qrt[c*d^2 - b*d*e + a*e^2]), ArcSin[(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4)], -1
])/(4*Sqrt[2]*Sqrt[c]*e*(c*d^2 - b*d*e + a*e^2)^(3/2)*(b + 2*c*x)*(a + b*x + c*x
^2)^(1/4)) + (Sqrt[-b^2 + 4*a*c]*(2*c*d - b*e)^2*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])/(4*Sqrt[2]*Sqrt[c]*e*(c*d^2 - b*d*e + a*e^2)^(3/2)*(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)**2/(c*x**2+b*x+a)**(1/4),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 0.529802, size = 187, normalized size = 0.15 \[ -\frac{\sqrt{2} \sqrt [4]{\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}} \sqrt [4]{\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}} F_1\left (\frac{3}{2};\frac{1}{4},\frac{1}{4};\frac{5}{2};\frac{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right )}{3 e (d+e x) \sqrt [4]{a+x (b+c x)}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((d + e*x)^2*(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[3/2, 1/4, 1/4, 5/2, (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)])/(3*e*(d + e*x)*(a + x*(b + c*x))^(1/4))

_______________________________________________________________________________________

Maple [F]  time = 0.134, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( ex+d \right ) ^{2}}{\frac{1}{\sqrt [4]{c{x}^{2}+bx+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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