3.2503 \(\int \frac{\sqrt [4]{a+b x+c x^2}}{d+e x} \, dx\)

Optimal. Leaf size=881 \[ -\frac{\left (4 a c-b^2\right )^{3/4} \sqrt [4]{c d^2-b e d+a e^2} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{4 a c-b^2} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{c^{3/4} e^{3/2} \left (c x^2+b x+a\right )^{3/4}}-\frac{\left (4 a c-b^2\right )^{3/4} \sqrt [4]{c d^2-b e d+a e^2} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{4 a c-b^2} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{c^{3/4} e^{3/2} \left (c x^2+b x+a\right )^{3/4}}-\frac{\sqrt [4]{b^2-4 a c} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt [4]{c} e^2 (b+2 c x)}-\frac{\left (b^2-4 a c\right ) (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \Pi \left (-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} c e^2 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}-\frac{\left (b^2-4 a c\right ) (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \Pi \left (\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} c e^2 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}+\frac{2 \sqrt [4]{c x^2+b x+a}}{e} \]

[Out]

(2*(a + b*x + c*x^2)^(1/4))/e - ((-b^2 + 4*a*c)^(3/4)*(c*d^2 - b*d*e + a*e^2)^(1
/4)*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(3/4)*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))])/(c^(3/4)*e^(3/2)*(a + b*x + c*x^2)^(3/4)) - ((-b^2 + 4*a*c)^(
3/4)*(c*d^2 - b*d*e + a*e^2)^(1/4)*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(3/4
)*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))])/(c^(3/4)*e^(3/2)*(a + b*x + c
*x^2)^(3/4)) - ((b^2 - 4*a*c)^(1/4)*(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])*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])/(Sqrt[2]*c^(1/4)*e^2*(b + 2
*c*x)) - ((b^2 - 4*a*c)*(2*c*d - b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*(-((c*(a
+ b*x + c*x^2))/(b^2 - 4*a*c)))^(3/4)*EllipticPi[-(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqr
t[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])/(Sqrt[2]*c*e^2*(b + 2*c*x)*(a + b*x + c*x^2)^(3/4)) - ((b^2 - 4*a*c)*(2
*c*d - b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*(-((c*(a + b*x + c*x^2))/(b^2 - 4*
a*c)))^(3/4)*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])/(Sqrt[2]*c*e^2*(b
+ 2*c*x)*(a + b*x + c*x^2)^(3/4))

_______________________________________________________________________________________

Rubi [A]  time = 5.97458, antiderivative size = 881, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 17, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.773 \[ -\frac{\left (4 a c-b^2\right )^{3/4} \sqrt [4]{c d^2-b e d+a e^2} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{4 a c-b^2} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{c^{3/4} e^{3/2} \left (c x^2+b x+a\right )^{3/4}}-\frac{\left (4 a c-b^2\right )^{3/4} \sqrt [4]{c d^2-b e d+a e^2} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{4 a c-b^2} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{c^{3/4} e^{3/2} \left (c x^2+b x+a\right )^{3/4}}-\frac{\sqrt [4]{b^2-4 a c} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt [4]{c} e^2 (b+2 c x)}-\frac{\left (b^2-4 a c\right ) (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \Pi \left (-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} c e^2 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}-\frac{\left (b^2-4 a c\right ) (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \Pi \left (\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} c e^2 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}+\frac{2 \sqrt [4]{c x^2+b x+a}}{e} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*(a + b*x + c*x^2)^(1/4))/e - ((-b^2 + 4*a*c)^(3/4)*(c*d^2 - b*d*e + a*e^2)^(1
/4)*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(3/4)*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))])/(c^(3/4)*e^(3/2)*(a + b*x + c*x^2)^(3/4)) - ((-b^2 + 4*a*c)^(
3/4)*(c*d^2 - b*d*e + a*e^2)^(1/4)*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(3/4
)*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))])/(c^(3/4)*e^(3/2)*(a + b*x + c
*x^2)^(3/4)) - ((b^2 - 4*a*c)^(1/4)*(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])*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])/(Sqrt[2]*c^(1/4)*e^2*(b + 2
*c*x)) - ((b^2 - 4*a*c)*(2*c*d - b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*(-((c*(a
+ b*x + c*x^2))/(b^2 - 4*a*c)))^(3/4)*EllipticPi[-(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqr
t[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])/(Sqrt[2]*c*e^2*(b + 2*c*x)*(a + b*x + c*x^2)^(3/4)) - ((b^2 - 4*a*c)*(2
*c*d - b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*(-((c*(a + b*x + c*x^2))/(b^2 - 4*
a*c)))^(3/4)*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])/(Sqrt[2]*c*e^2*(b
+ 2*c*x)*(a + b*x + c*x^2)^(3/4))

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 0.481979, size = 178, normalized size = 0.2 \[ \frac{2 \sqrt{2} \sqrt [4]{a+x (b+c x)} F_1\left (-\frac{1}{2};-\frac{1}{4},-\frac{1}{4};\frac{1}{2};\frac{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right )}{e \sqrt [4]{\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}} \sqrt [4]{\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.155, size = 0, normalized size = 0. \[ \int{\frac{1}{ex+d}\sqrt [4]{c{x}^{2}+bx+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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