3.2510 \(\int \frac{\left (a+b x+c x^2\right )^{3/4}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=1220 \[ \text{result too large to display} \]

[Out]

-((a + b*x + c*x^2)^(3/4)/(e*(d + e*x))) + (3*Sqrt[c]*(b + 2*c*x)*(a + b*x + c*x
^2)^(1/4))/(Sqrt[b^2 - 4*a*c]*e^2*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^
2 - 4*a*c])) - (3*(-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)*e
^(5/2)*(c*d^2 - b*d*e + a*e^2)^(1/4)*(a + b*x + c*x^2)^(1/4)) + (3*(-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)*e^(5/2)*(c*d^2 - b*d*e + a*e^2)
^(1/4)*(a + b*x + c*x^2)^(1/4)) - (3*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])/Sqrt[b^2 - 4*a*c])*EllipticE[2*ArcTan[(S
qrt[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)) + (3*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]*S
qrt[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])/(2*Sqrt[2]*e^2*(b + 2*c*x)) +
(3*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*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])/(4*Sqrt[2]*Sqrt[c]*e^3*Sqrt[c*d^2 - b*d*e + a*e^2]*(b + 2*c*x)*(a + b*x
+ c*x^2)^(1/4)) - (3*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^3*Sqrt[c*d^2 - b*d*e + a*e^2]*(b
+ 2*c*x)*(a + b*x + c*x^2)^(1/4))

_______________________________________________________________________________________

Rubi [A]  time = 5.94191, antiderivative size = 1220, 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{3 \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^3 \sqrt{c d^2-b e d+a e^2} (b+2 c x) \sqrt [4]{c x^2+b x+a}}-\frac{3 \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^3 \sqrt{c d^2-b e d+a e^2} (b+2 c x) \sqrt [4]{c x^2+b x+a}}-\frac{3 \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} e^{5/2} \sqrt [4]{c d^2-b e d+a e^2} \sqrt [4]{c x^2+b x+a}}+\frac{3 \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} e^{5/2} \sqrt [4]{c d^2-b e d+a e^2} \sqrt [4]{c x^2+b x+a}}-\frac{3 \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} e^2 (b+2 c x)}+\frac{3 \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} e^2 (b+2 c x)}-\frac{\left (c x^2+b x+a\right )^{3/4}}{e (d+e x)}+\frac{3 \sqrt{c} (b+2 c x) \sqrt [4]{c x^2+b x+a}}{\sqrt{b^2-4 a c} e^2 \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[(a + b*x + c*x^2)^(3/4)/(d + e*x)^2,x]

[Out]

-((a + b*x + c*x^2)^(3/4)/(e*(d + e*x))) + (3*Sqrt[c]*(b + 2*c*x)*(a + b*x + c*x
^2)^(1/4))/(Sqrt[b^2 - 4*a*c]*e^2*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^
2 - 4*a*c])) - (3*(-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)*e
^(5/2)*(c*d^2 - b*d*e + a*e^2)^(1/4)*(a + b*x + c*x^2)^(1/4)) + (3*(-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)*e^(5/2)*(c*d^2 - b*d*e + a*e^2)
^(1/4)*(a + b*x + c*x^2)^(1/4)) - (3*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])/Sqrt[b^2 - 4*a*c])*EllipticE[2*ArcTan[(S
qrt[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)) + (3*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]*S
qrt[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])/(2*Sqrt[2]*e^2*(b + 2*c*x)) +
(3*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*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])/(4*Sqrt[2]*Sqrt[c]*e^3*Sqrt[c*d^2 - b*d*e + a*e^2]*(b + 2*c*x)*(a + b*x
+ c*x^2)^(1/4)) - (3*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^3*Sqrt[c*d^2 - b*d*e + a*e^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((c*x**2+b*x+a)**(3/4)/(e*x+d)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 0.441605, size = 185, normalized size = 0.15 \[ \frac{4 \sqrt{2} (a+x (b+c x))^{3/4} F_1\left (-\frac{1}{2};-\frac{3}{4},-\frac{3}{4};\frac{1}{2};\frac{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right )}{e (d+e x) \left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{3/4} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{3/4}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError