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

Optimal. Leaf size=220 \[ \frac{\left (77 \sqrt{b} c-15 \sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{3/4}}+\frac{\left (15 \sqrt{a} e+77 \sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{3/4}}+\frac{5 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}+\frac{x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a-b x^4\right )}+\frac{x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a-b x^4\right )^2}+\frac{a f+b x \left (c+d x+e x^2\right )}{12 a b \left (a-b x^4\right )^3} \]

[Out]

(x*(11*c + 10*d*x + 9*e*x^2))/(96*a^2*(a - b*x^4)^2) + (x*(77*c + 60*d*x + 45*e*
x^2))/(384*a^3*(a - b*x^4)) + (a*f + b*x*(c + d*x + e*x^2))/(12*a*b*(a - b*x^4)^
3) + ((77*Sqrt[b]*c - 15*Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(256*a^(15/4)*b
^(3/4)) + ((77*Sqrt[b]*c + 15*Sqrt[a]*e)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(256*a^(1
5/4)*b^(3/4)) + (5*d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(32*a^(7/2)*Sqrt[b])

_______________________________________________________________________________________

Rubi [A]  time = 0.436592, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{\left (77 \sqrt{b} c-15 \sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{3/4}}+\frac{\left (15 \sqrt{a} e+77 \sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{3/4}}+\frac{5 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}+\frac{x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a-b x^4\right )}+\frac{x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a-b x^4\right )^2}+\frac{a f+b x \left (c+d x+e x^2\right )}{12 a b \left (a-b x^4\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(x*(11*c + 10*d*x + 9*e*x^2))/(96*a^2*(a - b*x^4)^2) + (x*(77*c + 60*d*x + 45*e*
x^2))/(384*a^3*(a - b*x^4)) + (a*f + b*x*(c + d*x + e*x^2))/(12*a*b*(a - b*x^4)^
3) + ((77*Sqrt[b]*c - 15*Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(256*a^(15/4)*b
^(3/4)) + ((77*Sqrt[b]*c + 15*Sqrt[a]*e)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(256*a^(1
5/4)*b^(3/4)) + (5*d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(32*a^(7/2)*Sqrt[b])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 72.5857, size = 204, normalized size = 0.93 \[ \frac{a f + b x \left (c + d x + e x^{2}\right )}{12 a b \left (a - b x^{4}\right )^{3}} + \frac{x \left (11 c + 10 d x + 9 e x^{2}\right )}{96 a^{2} \left (a - b x^{4}\right )^{2}} + \frac{x \left (77 c + 60 d x + 45 e x^{2}\right )}{384 a^{3} \left (a - b x^{4}\right )} + \frac{5 d \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{32 a^{\frac{7}{2}} \sqrt{b}} - \frac{\left (15 \sqrt{a} e - 77 \sqrt{b} c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{256 a^{\frac{15}{4}} b^{\frac{3}{4}}} + \frac{\left (15 \sqrt{a} e + 77 \sqrt{b} c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{256 a^{\frac{15}{4}} b^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a*f + b*x*(c + d*x + e*x**2))/(12*a*b*(a - b*x**4)**3) + x*(11*c + 10*d*x + 9*e
*x**2)/(96*a**2*(a - b*x**4)**2) + x*(77*c + 60*d*x + 45*e*x**2)/(384*a**3*(a -
b*x**4)) + 5*d*atanh(sqrt(b)*x**2/sqrt(a))/(32*a**(7/2)*sqrt(b)) - (15*sqrt(a)*e
- 77*sqrt(b)*c)*atan(b**(1/4)*x/a**(1/4))/(256*a**(15/4)*b**(3/4)) + (15*sqrt(a
)*e + 77*sqrt(b)*c)*atanh(b**(1/4)*x/a**(1/4))/(256*a**(15/4)*b**(3/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.62815, size = 286, normalized size = 1.3 \[ \frac{-\frac{3 \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (15 a^{3/4} e+77 \sqrt [4]{a} \sqrt{b} c+40 \sqrt{a} \sqrt [4]{b} d\right )}{b^{3/4}}+\frac{3 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (15 a^{3/4} e+77 \sqrt [4]{a} \sqrt{b} c-40 \sqrt{a} \sqrt [4]{b} d\right )}{b^{3/4}}-\frac{128 a^3 (a f+b x (c+x (d+e x)))}{b \left (b x^4-a\right )^3}+\frac{16 a^2 x (11 c+x (10 d+9 e x))}{\left (a-b x^4\right )^2}+\frac{6 \sqrt [4]{a} \left (77 \sqrt{b} c-15 \sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac{4 a x (77 c+15 x (4 d+3 e x))}{a-b x^4}+\frac{120 \sqrt{a} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt{b}}}{1536 a^4} \]

Antiderivative was successfully verified.

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

[Out]

((4*a*x*(77*c + 15*x*(4*d + 3*e*x)))/(a - b*x^4) + (16*a^2*x*(11*c + x*(10*d + 9
*e*x)))/(a - b*x^4)^2 - (128*a^3*(a*f + b*x*(c + x*(d + e*x))))/(b*(-a + b*x^4)^
3) + (6*a^(1/4)*(77*Sqrt[b]*c - 15*Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/b^(3/
4) - (3*(77*a^(1/4)*Sqrt[b]*c + 40*Sqrt[a]*b^(1/4)*d + 15*a^(3/4)*e)*Log[a^(1/4)
- b^(1/4)*x])/b^(3/4) + (3*(77*a^(1/4)*Sqrt[b]*c - 40*Sqrt[a]*b^(1/4)*d + 15*a^
(3/4)*e)*Log[a^(1/4) + b^(1/4)*x])/b^(3/4) + (120*Sqrt[a]*d*Log[Sqrt[a] + Sqrt[b
]*x^2])/Sqrt[b])/(1536*a^4)

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 287, normalized size = 1.3 \[{\frac{1}{ \left ( b{x}^{4}-a \right ) ^{3}} \left ( -{\frac{15\,{b}^{2}e{x}^{11}}{128\,{a}^{3}}}-{\frac{5\,{b}^{2}d{x}^{10}}{32\,{a}^{3}}}-{\frac{77\,{b}^{2}c{x}^{9}}{384\,{a}^{3}}}+{\frac{21\,be{x}^{7}}{64\,{a}^{2}}}+{\frac{5\,bd{x}^{6}}{12\,{a}^{2}}}+{\frac{33\,bc{x}^{5}}{64\,{a}^{2}}}-{\frac{113\,e{x}^{3}}{384\,a}}-{\frac{11\,d{x}^{2}}{32\,a}}-{\frac{51\,cx}{128\,a}}-{\frac{f}{12\,b}} \right ) }+{\frac{77\,c}{512\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{77\,c}{256\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{5\,d}{64}\ln \left ({1 \left ( -{a}^{4}+{x}^{2}\sqrt{b{a}^{7}} \right ) \left ( -{a}^{4}-{x}^{2}\sqrt{b{a}^{7}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{b{a}^{7}}}}}-{\frac{15\,e}{256\,{a}^{3}b}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{15\,e}{512\,{a}^{3}b}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-15/128*e/a^3*b^2*x^11-5/32*d/a^3*b^2*x^10-77/384*c/a^3*b^2*x^9+21/64/a^2*b*e*x
^7+5/12/a^2*d*b*x^6+33/64/a^2*c*b*x^5-113/384/a*e*x^3-11/32*d/a*x^2-51/128/a*c*x
-1/12*f/b)/(b*x^4-a)^3+77/512*c*(a/b)^(1/4)/a^4*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4
)))+77/256*c*(a/b)^(1/4)/a^4*arctan(x/(a/b)^(1/4))-5/64*d/(b*a^7)^(1/2)*ln((-a^4
+x^2*(b*a^7)^(1/2))/(-a^4-x^2*(b*a^7)^(1/2)))-15/256*e/a^3/b/(a/b)^(1/4)*arctan(
x/(a/b)^(1/4))+15/512*e/a^3/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((f*x^3 + e*x^2 + d*x + c)/(b*x^4 - a)^4,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((f*x^3 + e*x^2 + d*x + c)/(b*x^4 - a)^4,x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.222513, size = 555, normalized size = 2.52 \[ \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{-a b} b^{2} d + 77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c + 15 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{-a b} b^{2} d + 77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c + 15 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac{\sqrt{2}{\left (77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 15 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{1024 \, a^{4} b^{3}} - \frac{\sqrt{2}{\left (77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 15 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{1024 \, a^{4} b^{3}} - \frac{45 \, b^{3} x^{11} e + 60 \, b^{3} d x^{10} + 77 \, b^{3} c x^{9} - 126 \, a b^{2} x^{7} e - 160 \, a b^{2} d x^{6} - 198 \, a b^{2} c x^{5} + 113 \, a^{2} b x^{3} e + 132 \, a^{2} b d x^{2} + 153 \, a^{2} b c x + 32 \, a^{3} f}{384 \,{\left (b x^{4} - a\right )}^{3} a^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/512*sqrt(2)*(40*sqrt(2)*sqrt(-a*b)*b^2*d + 77*(-a*b^3)^(1/4)*b^2*c + 15*(-a*b^
3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(a^4*b
^3) + 1/512*sqrt(2)*(40*sqrt(2)*sqrt(-a*b)*b^2*d + 77*(-a*b^3)^(1/4)*b^2*c + 15*
(-a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/
(a^4*b^3) + 1/1024*sqrt(2)*(77*(-a*b^3)^(1/4)*b^2*c - 15*(-a*b^3)^(3/4)*e)*ln(x^
2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^4*b^3) - 1/1024*sqrt(2)*(77*(-a*b^3)
^(1/4)*b^2*c - 15*(-a*b^3)^(3/4)*e)*ln(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b)
)/(a^4*b^3) - 1/384*(45*b^3*x^11*e + 60*b^3*d*x^10 + 77*b^3*c*x^9 - 126*a*b^2*x^
7*e - 160*a*b^2*d*x^6 - 198*a*b^2*c*x^5 + 113*a^2*b*x^3*e + 132*a^2*b*d*x^2 + 15
3*a^2*b*c*x + 32*a^3*f)/((b*x^4 - a)^3*a^3*b)