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

Optimal. Leaf size=179 \[ \frac{\left (21 \sqrt{b} c-5 \sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{3/4}}+\frac{\left (5 \sqrt{a} e+21 \sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{3/4}}+\frac{3 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}+\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\frac{x \left (c+d x+e x^2\right )}{8 a \left (a-b x^4\right )^2} \]

[Out]

(x*(c + d*x + e*x^2))/(8*a*(a - b*x^4)^2) + (x*(7*c + 6*d*x + 5*e*x^2))/(32*a^2*
(a - b*x^4)) + ((21*Sqrt[b]*c - 5*Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(64*a^
(11/4)*b^(3/4)) + ((21*Sqrt[b]*c + 5*Sqrt[a]*e)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(6
4*a^(11/4)*b^(3/4)) + (3*d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(16*a^(5/2)*Sqrt[b])

_______________________________________________________________________________________

Rubi [A]  time = 0.353912, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{\left (21 \sqrt{b} c-5 \sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{3/4}}+\frac{\left (5 \sqrt{a} e+21 \sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{3/4}}+\frac{3 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}+\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\frac{x \left (c+d x+e x^2\right )}{8 a \left (a-b x^4\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(c + d*x + e*x^2))/(8*a*(a - b*x^4)^2) + (x*(7*c + 6*d*x + 5*e*x^2))/(32*a^2*
(a - b*x^4)) + ((21*Sqrt[b]*c - 5*Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(64*a^
(11/4)*b^(3/4)) + ((21*Sqrt[b]*c + 5*Sqrt[a]*e)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(6
4*a^(11/4)*b^(3/4)) + (3*d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(16*a^(5/2)*Sqrt[b])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 53.5629, size = 167, normalized size = 0.93 \[ \frac{x \left (c + d x + e x^{2}\right )}{8 a \left (a - b x^{4}\right )^{2}} + \frac{x \left (7 c + 6 d x + 5 e x^{2}\right )}{32 a^{2} \left (a - b x^{4}\right )} + \frac{3 d \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{16 a^{\frac{5}{2}} \sqrt{b}} - \frac{\left (5 \sqrt{a} e - 21 \sqrt{b} c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{64 a^{\frac{11}{4}} b^{\frac{3}{4}}} + \frac{\left (5 \sqrt{a} e + 21 \sqrt{b} c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{64 a^{\frac{11}{4}} b^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(c + d*x + e*x**2)/(8*a*(a - b*x**4)**2) + x*(7*c + 6*d*x + 5*e*x**2)/(32*a**2
*(a - b*x**4)) + 3*d*atanh(sqrt(b)*x**2/sqrt(a))/(16*a**(5/2)*sqrt(b)) - (5*sqrt
(a)*e - 21*sqrt(b)*c)*atan(b**(1/4)*x/a**(1/4))/(64*a**(11/4)*b**(3/4)) + (5*sqr
t(a)*e + 21*sqrt(b)*c)*atanh(b**(1/4)*x/a**(1/4))/(64*a**(11/4)*b**(3/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.661725, size = 244, normalized size = 1.36 \[ \frac{-\frac{\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (5 a^{3/4} e+21 \sqrt [4]{a} \sqrt{b} c+12 \sqrt{a} \sqrt [4]{b} d\right )}{b^{3/4}}+\frac{\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (5 a^{3/4} e+21 \sqrt [4]{a} \sqrt{b} c-12 \sqrt{a} \sqrt [4]{b} d\right )}{b^{3/4}}+\frac{16 a^2 x (c+x (d+e x))}{\left (a-b x^4\right )^2}+\frac{2 \sqrt [4]{a} \left (21 \sqrt{b} c-5 \sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac{4 a x (7 c+x (6 d+5 e x))}{a-b x^4}+\frac{12 \sqrt{a} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt{b}}}{128 a^3} \]

Antiderivative was successfully verified.

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

[Out]

((16*a^2*x*(c + x*(d + e*x)))/(a - b*x^4)^2 + (4*a*x*(7*c + x*(6*d + 5*e*x)))/(a
- b*x^4) + (2*a^(1/4)*(21*Sqrt[b]*c - 5*Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])
/b^(3/4) - ((21*a^(1/4)*Sqrt[b]*c + 12*Sqrt[a]*b^(1/4)*d + 5*a^(3/4)*e)*Log[a^(1
/4) - b^(1/4)*x])/b^(3/4) + ((21*a^(1/4)*Sqrt[b]*c - 12*Sqrt[a]*b^(1/4)*d + 5*a^
(3/4)*e)*Log[a^(1/4) + b^(1/4)*x])/b^(3/4) + (12*Sqrt[a]*d*Log[Sqrt[a] + Sqrt[b]
*x^2])/Sqrt[b])/(128*a^3)

_______________________________________________________________________________________

Maple [B]  time = 0.007, size = 286, normalized size = 1.6 \[{\frac{cx}{8\,a \left ( b{x}^{4}-a \right ) ^{2}}}-{\frac{7\,cx}{32\,{a}^{2} \left ( b{x}^{4}-a \right ) }}+{\frac{21\,c}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{21\,c}{64\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{d{x}^{2}}{8\,a \left ( b{x}^{4}-a \right ) ^{2}}}-{\frac{3\,d{x}^{2}}{16\,{a}^{2} \left ( b{x}^{4}-a \right ) }}-{\frac{3\,d}{32\,{a}^{2}}\ln \left ({1 \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{e{x}^{3}}{8\,a \left ( b{x}^{4}-a \right ) ^{2}}}-{\frac{5\,e{x}^{3}}{32\,{a}^{2} \left ( b{x}^{4}-a \right ) }}-{\frac{5\,e}{64\,{a}^{2}b}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,e}{128\,{a}^{2}b}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*c*x/a/(b*x^4-a)^2-7/32*c/a^2*x/(b*x^4-a)+21/128*c/a^3*(a/b)^(1/4)*ln((x+(a/b
)^(1/4))/(x-(a/b)^(1/4)))+21/64*c/a^3*(a/b)^(1/4)*arctan(x/(a/b)^(1/4))+1/8*d*x^
2/a/(b*x^4-a)^2-3/16*d/a^2*x^2/(b*x^4-a)-3/32*d/a^2/(a*b)^(1/2)*ln((-a+x^2*(a*b)
^(1/2))/(-a-x^2*(a*b)^(1/2)))+1/8*e*x^3/a/(b*x^4-a)^2-5/32*e/a^2*x^3/(b*x^4-a)-5
/64*e/a^2/b/(a/b)^(1/4)*arctan(x/(a/b)^(1/4))+5/128*e/a^2/b/(a/b)^(1/4)*ln((x+(a
/b)^(1/4))/(x-(a/b)^(1/4)))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError

_______________________________________________________________________________________

Sympy [A]  time = 18.7221, size = 563, normalized size = 3.15 \[ - \operatorname{RootSum}{\left (268435456 t^{4} a^{11} b^{3} + t^{2} \left (- 6881280 a^{6} b^{2} c e - 4718592 a^{6} b^{2} d^{2}\right ) + t \left (- 153600 a^{4} b d e^{2} - 2709504 a^{3} b^{2} c^{2} d\right ) - 625 a^{2} e^{4} + 22050 a b c^{2} e^{2} - 60480 a b c d^{2} e + 20736 a b d^{4} - 194481 b^{2} c^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 262144000 t^{3} a^{10} b^{2} e^{3} - 4624220160 t^{3} a^{9} b^{3} c^{2} e + 12683575296 t^{3} a^{9} b^{3} c d^{2} + 309657600 t^{2} a^{7} b^{2} c d e^{2} - 283115520 t^{2} a^{7} b^{2} d^{3} e - 1820786688 t^{2} a^{6} b^{3} c^{3} d + 5040000 t a^{5} b c e^{4} + 6912000 t a^{5} b d^{2} e^{3} + 118540800 t a^{4} b^{2} c^{3} e^{2} - 365783040 t a^{4} b^{2} c^{2} d^{2} e - 111476736 t a^{4} b^{2} c d^{4} + 522764928 t a^{3} b^{3} c^{5} + 112500 a^{3} d e^{5} - 4536000 a^{2} b c d^{3} e^{2} + 2488320 a^{2} b d^{5} e + 58344300 a b^{2} c^{4} d e - 80015040 a b^{2} c^{3} d^{3}}{15625 a^{3} e^{6} + 275625 a^{2} b c^{2} e^{4} - 3024000 a^{2} b c d^{2} e^{3} + 2073600 a^{2} b d^{4} e^{2} - 4862025 a b^{2} c^{4} e^{2} + 53343360 a b^{2} c^{3} d^{2} e - 36578304 a b^{2} c^{2} d^{4} - 85766121 b^{3} c^{6}} \right )} \right )\right )} - \frac{- 11 a c x - 10 a d x^{2} - 9 a e x^{3} + 7 b c x^{5} + 6 b d x^{6} + 5 b e x^{7}}{32 a^{4} - 64 a^{3} b x^{4} + 32 a^{2} b^{2} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-RootSum(268435456*_t**4*a**11*b**3 + _t**2*(-6881280*a**6*b**2*c*e - 4718592*a*
*6*b**2*d**2) + _t*(-153600*a**4*b*d*e**2 - 2709504*a**3*b**2*c**2*d) - 625*a**2
*e**4 + 22050*a*b*c**2*e**2 - 60480*a*b*c*d**2*e + 20736*a*b*d**4 - 194481*b**2*
c**4, Lambda(_t, _t*log(x + (-262144000*_t**3*a**10*b**2*e**3 - 4624220160*_t**3
*a**9*b**3*c**2*e + 12683575296*_t**3*a**9*b**3*c*d**2 + 309657600*_t**2*a**7*b*
*2*c*d*e**2 - 283115520*_t**2*a**7*b**2*d**3*e - 1820786688*_t**2*a**6*b**3*c**3
*d + 5040000*_t*a**5*b*c*e**4 + 6912000*_t*a**5*b*d**2*e**3 + 118540800*_t*a**4*
b**2*c**3*e**2 - 365783040*_t*a**4*b**2*c**2*d**2*e - 111476736*_t*a**4*b**2*c*d
**4 + 522764928*_t*a**3*b**3*c**5 + 112500*a**3*d*e**5 - 4536000*a**2*b*c*d**3*e
**2 + 2488320*a**2*b*d**5*e + 58344300*a*b**2*c**4*d*e - 80015040*a*b**2*c**3*d*
*3)/(15625*a**3*e**6 + 275625*a**2*b*c**2*e**4 - 3024000*a**2*b*c*d**2*e**3 + 20
73600*a**2*b*d**4*e**2 - 4862025*a*b**2*c**4*e**2 + 53343360*a*b**2*c**3*d**2*e
- 36578304*a*b**2*c**2*d**4 - 85766121*b**3*c**6)))) - (-11*a*c*x - 10*a*d*x**2
- 9*a*e*x**3 + 7*b*c*x**5 + 6*b*d*x**6 + 5*b*e*x**7)/(32*a**4 - 64*a**3*b*x**4 +
32*a**2*b**2*x**8)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.224729, size = 481, normalized size = 2.69 \[ -\frac{5 \, b x^{7} e + 6 \, b d x^{6} + 7 \, b c x^{5} - 9 \, a x^{3} e - 10 \, a d x^{2} - 11 \, a c x}{32 \,{\left (b x^{4} - a\right )}^{2} a^{2}} - \frac{\sqrt{2}{\left (12 \, \sqrt{2} \sqrt{-a b} b^{2} d - 21 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 5 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} b^{3}} - \frac{\sqrt{2}{\left (12 \, \sqrt{2} \sqrt{-a b} b^{2} d - 21 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 5 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} b^{3}} + \frac{\sqrt{2}{\left (21 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 5 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{256 \, a^{3} b^{3}} - \frac{\sqrt{2}{\left (21 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 5 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{256 \, a^{3} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/32*(5*b*x^7*e + 6*b*d*x^6 + 7*b*c*x^5 - 9*a*x^3*e - 10*a*d*x^2 - 11*a*c*x)/((
b*x^4 - a)^2*a^2) - 1/128*sqrt(2)*(12*sqrt(2)*sqrt(-a*b)*b^2*d - 21*(-a*b^3)^(1/
4)*b^2*c - 5*(-a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/(
-a/b)^(1/4))/(a^3*b^3) - 1/128*sqrt(2)*(12*sqrt(2)*sqrt(-a*b)*b^2*d - 21*(-a*b^3
)^(1/4)*b^2*c - 5*(-a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/
4))/(-a/b)^(1/4))/(a^3*b^3) + 1/256*sqrt(2)*(21*(-a*b^3)^(1/4)*b^2*c - 5*(-a*b^3
)^(3/4)*e)*ln(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^3*b^3) - 1/256*sqrt(
2)*(21*(-a*b^3)^(1/4)*b^2*c - 5*(-a*b^3)^(3/4)*e)*ln(x^2 - sqrt(2)*x*(-a/b)^(1/4
) + sqrt(-a/b))/(a^3*b^3)