3.2504 \(\int \frac{\sqrt [4]{a+b x+c x^2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=944 \[ \frac{\left (b^2-4 a c\right ) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \Pi \left (-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right ) (2 c d-b e)^2}{4 \sqrt{2} c e^2 \left (c d^2-b e d+a e^2\right ) (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}+\frac{\left (b^2-4 a c\right ) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \Pi \left (\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right ) (2 c d-b e)^2}{4 \sqrt{2} c e^2 \left (c d^2-b e d+a e^2\right ) (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}+\frac{\left (4 a c-b^2\right )^{3/4} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{4 a c-b^2} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) (2 c d-b e)}{4 c^{3/4} e^{3/2} \left (c d^2-b e d+a e^2\right )^{3/4} \left (c x^2+b x+a\right )^{3/4}}+\frac{\left (4 a c-b^2\right )^{3/4} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{4 a c-b^2} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) (2 c d-b e)}{4 c^{3/4} e^{3/2} \left (c d^2-b e d+a e^2\right )^{3/4} \left (c x^2+b x+a\right )^{3/4}}+\frac{c^{3/4} \sqrt [4]{b^2-4 a c} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} e^2 (b+2 c x)}-\frac{\sqrt [4]{c x^2+b x+a}}{e (d+e x)} \]

[Out]

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

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Rubi [A]  time = 5.30416, antiderivative size = 944, 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 \[ \frac{\left (b^2-4 a c\right ) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \Pi \left (-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right ) (2 c d-b e)^2}{4 \sqrt{2} c e^2 \left (c d^2-b e d+a e^2\right ) (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}+\frac{\left (b^2-4 a c\right ) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \Pi \left (\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right ) (2 c d-b e)^2}{4 \sqrt{2} c e^2 \left (c d^2-b e d+a e^2\right ) (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}+\frac{\left (4 a c-b^2\right )^{3/4} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{4 a c-b^2} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) (2 c d-b e)}{4 c^{3/4} e^{3/2} \left (c d^2-b e d+a e^2\right )^{3/4} \left (c x^2+b x+a\right )^{3/4}}+\frac{\left (4 a c-b^2\right )^{3/4} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{4 a c-b^2} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) (2 c d-b e)}{4 c^{3/4} e^{3/2} \left (c d^2-b e d+a e^2\right )^{3/4} \left (c x^2+b x+a\right )^{3/4}}+\frac{c^{3/4} \sqrt [4]{b^2-4 a c} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} e^2 (b+2 c x)}-\frac{\sqrt [4]{c x^2+b x+a}}{e (d+e x)} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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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)**2,x)

[Out]

Timed out

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Mathematica [C]  time = 0.460428, size = 185, normalized size = 0.2 \[ -\frac{2 \sqrt{2} \sqrt [4]{a+x (b+c x)} F_1\left (\frac{1}{2};-\frac{1}{4},-\frac{1}{4};\frac{3}{2};\frac{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right )}{e (d+e x) \sqrt [4]{\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}} \sqrt [4]{\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-2*Sqrt[2]*(a + x*(b + c*x))^(1/4)*AppellF1[1/2, -1/4, -1/4, 3/2, (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)*(d + e*x))

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Maple [F]  time = 0.136, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( ex+d \right ) ^{2}}\sqrt [4]{c{x}^{2}+bx+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)^2,x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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