3.2530 \(\int \frac{1}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=1134 \[ \text{result too large to display} \]

[Out]

-(e*(a + b*x + c*x^2)^(1/4))/(2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) - (7*e*(2*c
*d - b*e)*(a + b*x + c*x^2)^(1/4))/(8*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)) - (3*
(-b^2 + 4*a*c)^(3/4)*Sqrt[e]*(20*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(5*b*d + 2*a*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))])/(32*c^(3/4)*(c*d^2 - b*d*e + a*e^2)^(11/4)*(a + b*x + c*x^2)^(3/4))
- (3*(-b^2 + 4*a*c)^(3/4)*Sqrt[e]*(20*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(5*b*d + 2*a*
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))])/(32*c^(3/4)*(c*d^2 - b*d*e + a*e^2)^(11/4)*(a + b*x + c*x^2)
^(3/4)) - (7*c^(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*S
qrt[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])/(8*Sqrt[2]*(c*d^2 - b*
d*e + a*e^2)^2*(b + 2*c*x)) - (3*(b^2 - 4*a*c)*(2*c*d - b*e)*(20*c^2*d^2 + 7*b^2
*e^2 - 4*c*e*(5*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)))^(3/4)*EllipticPi[-(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqr
t[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]*c*(c*d^2 - b*d*e + a*e^2)^3*(b + 2*c*x)*(a + b*x + c*x^2)^(3/4)) -
(3*(b^2 - 4*a*c)*(2*c*d - b*e)*(20*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(5*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)))^(3/4)
*EllipticPi[(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^2]), ArcS
in[(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4)], -1])/(32*Sqrt[2]*c*(c*d^2 - b*d*e +
a*e^2)^3*(b + 2*c*x)*(a + b*x + c*x^2)^(3/4))

_______________________________________________________________________________________

Rubi [A]  time = 7.13441, antiderivative size = 1134, 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{7 (2 c d-b e) \sqrt [4]{c x^2+b x+a} e}{8 \left (c d^2-b e d+a e^2\right )^2 (d+e x)}-\frac{\sqrt [4]{c x^2+b x+a} e}{2 \left (c d^2-b e d+a e^2\right ) (d+e x)^2}-\frac{3 \left (4 a c-b^2\right )^{3/4} \left (20 c^2 d^2+7 b^2 e^2-4 c e (5 b d+2 a e)\right ) \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 ) \sqrt{e}}{32 c^{3/4} \left (c d^2-b e d+a e^2\right )^{11/4} \left (c x^2+b x+a\right )^{3/4}}-\frac{3 \left (4 a c-b^2\right )^{3/4} \left (20 c^2 d^2+7 b^2 e^2-4 c e (5 b d+2 a e)\right ) \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 ) \sqrt{e}}{32 c^{3/4} \left (c d^2-b e d+a e^2\right )^{11/4} \left (c x^2+b x+a\right )^{3/4}}-\frac{7 c^{3/4} \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 )}{8 \sqrt{2} \left (c d^2-b e d+a e^2\right )^2 (b+2 c x)}-\frac{3 \left (b^2-4 a c\right ) (2 c d-b e) \left (20 c^2 d^2+7 b^2 e^2-4 c e (5 b d+2 a e)\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 )}{32 \sqrt{2} c \left (c d^2-b e d+a e^2\right )^3 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}-\frac{3 \left (b^2-4 a c\right ) (2 c d-b e) \left (20 c^2 d^2+7 b^2 e^2-4 c e (5 b d+2 a e)\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 )}{32 \sqrt{2} c \left (c d^2-b e d+a e^2\right )^3 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(e*(a + b*x + c*x^2)^(1/4))/(2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) - (7*e*(2*c
*d - b*e)*(a + b*x + c*x^2)^(1/4))/(8*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)) - (3*
(-b^2 + 4*a*c)^(3/4)*Sqrt[e]*(20*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(5*b*d + 2*a*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))])/(32*c^(3/4)*(c*d^2 - b*d*e + a*e^2)^(11/4)*(a + b*x + c*x^2)^(3/4))
- (3*(-b^2 + 4*a*c)^(3/4)*Sqrt[e]*(20*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(5*b*d + 2*a*
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))])/(32*c^(3/4)*(c*d^2 - b*d*e + a*e^2)^(11/4)*(a + b*x + c*x^2)
^(3/4)) - (7*c^(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*S
qrt[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])/(8*Sqrt[2]*(c*d^2 - b*
d*e + a*e^2)^2*(b + 2*c*x)) - (3*(b^2 - 4*a*c)*(2*c*d - b*e)*(20*c^2*d^2 + 7*b^2
*e^2 - 4*c*e*(5*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)))^(3/4)*EllipticPi[-(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqr
t[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]*c*(c*d^2 - b*d*e + a*e^2)^3*(b + 2*c*x)*(a + b*x + c*x^2)^(3/4)) -
(3*(b^2 - 4*a*c)*(2*c*d - b*e)*(20*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(5*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)))^(3/4)
*EllipticPi[(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^2]), ArcS
in[(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4)], -1])/(32*Sqrt[2]*c*(c*d^2 - b*d*e +
a*e^2)^3*(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(1/(e*x+d)**3/(c*x**2+b*x+a)**(3/4),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 0.819985, size = 187, normalized size = 0.16 \[ -\frac{\left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{3/4} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{3/4} F_1\left (\frac{7}{2};\frac{3}{4},\frac{3}{4};\frac{9}{2};\frac{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right )}{7 \sqrt{2} e (d+e x)^2 (a+x (b+c x))^{3/4}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(((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)*AppellF1[7/2, 3/4, 3/4, 9/2, (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)])/(7*Sqrt[2]*e*(d + e*x)^2*(a + x*(b + c*x))^(3/4))

_______________________________________________________________________________________

Maple [F]  time = 0.136, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( ex+d \right ) ^{3}} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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