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

Optimal. Leaf size=382 \[ -\frac{\left (77 \sqrt{b} c-15 \sqrt{a} e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{15/4} b^{3/4}}+\frac{\left (77 \sqrt{b} c-15 \sqrt{a} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{15/4} b^{3/4}}-\frac{\left (15 \sqrt{a} e+77 \sqrt{b} c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt{2} a^{15/4} b^{3/4}}+\frac{\left (15 \sqrt{a} e+77 \sqrt{b} c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{256 \sqrt{2} a^{15/4} b^{3/4}}+\frac{5 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}+\frac{x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a+b x^4\right )}+\frac{x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}-\frac{a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3} \]

[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) + (5*d*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(32*a^(7/2)*Sqrt[b]) - ((77*Sqrt[b]*c +
15*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(256*Sqrt[2]*a^(15/4)*b^
(3/4)) + ((77*Sqrt[b]*c + 15*Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])
/(256*Sqrt[2]*a^(15/4)*b^(3/4)) - ((77*Sqrt[b]*c - 15*Sqrt[a]*e)*Log[Sqrt[a] - S
qrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(15/4)*b^(3/4)) + ((77*S
qrt[b]*c - 15*Sqrt[a]*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])
/(512*Sqrt[2]*a^(15/4)*b^(3/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.822525, antiderivative size = 382, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44 \[ -\frac{\left (77 \sqrt{b} c-15 \sqrt{a} e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{15/4} b^{3/4}}+\frac{\left (77 \sqrt{b} c-15 \sqrt{a} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{15/4} b^{3/4}}-\frac{\left (15 \sqrt{a} e+77 \sqrt{b} c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt{2} a^{15/4} b^{3/4}}+\frac{\left (15 \sqrt{a} e+77 \sqrt{b} c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{256 \sqrt{2} a^{15/4} b^{3/4}}+\frac{5 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}+\frac{x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a+b x^4\right )}+\frac{x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}-\frac{a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3} \]

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) + (5*d*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(32*a^(7/2)*Sqrt[b]) - ((77*Sqrt[b]*c +
15*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(256*Sqrt[2]*a^(15/4)*b^
(3/4)) + ((77*Sqrt[b]*c + 15*Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])
/(256*Sqrt[2]*a^(15/4)*b^(3/4)) - ((77*Sqrt[b]*c - 15*Sqrt[a]*e)*Log[Sqrt[a] - S
qrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(15/4)*b^(3/4)) + ((77*S
qrt[b]*c - 15*Sqrt[a]*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])
/(512*Sqrt[2]*a^(15/4)*b^(3/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 137.466, size = 364, normalized size = 0.95 \[ - \frac{a f - b x \left (c + d x + e x^{2}\right )}{12 a b \left (a + b x^{4}\right )^{3}} + \frac{x \left (11 c + 10 d x + 9 e x^{2}\right )}{96 a^{2} \left (a + b x^{4}\right )^{2}} + \frac{x \left (77 c + 60 d x + 45 e x^{2}\right )}{384 a^{3} \left (a + b x^{4}\right )} + \frac{5 d \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{32 a^{\frac{7}{2}} \sqrt{b}} + \frac{\sqrt{2} \left (15 \sqrt{a} e - 77 \sqrt{b} c\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{1024 a^{\frac{15}{4}} b^{\frac{3}{4}}} - \frac{\sqrt{2} \left (15 \sqrt{a} e - 77 \sqrt{b} c\right ) \log{\left (\sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{1024 a^{\frac{15}{4}} b^{\frac{3}{4}}} - \frac{\sqrt{2} \left (15 \sqrt{a} e + 77 \sqrt{b} c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{512 a^{\frac{15}{4}} b^{\frac{3}{4}}} + \frac{\sqrt{2} \left (15 \sqrt{a} e + 77 \sqrt{b} c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{512 a^{\frac{15}{4}} b^{\frac{3}{4}}} \]

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*atan(sqrt(b)*x**2/sqrt(a))/(32*a**(7/2)*sqrt(b)) + sqrt(2)*(15*s
qrt(a)*e - 77*sqrt(b)*c)*log(-sqrt(2)*a**(1/4)*b**(3/4)*x + sqrt(a)*sqrt(b) + b*
x**2)/(1024*a**(15/4)*b**(3/4)) - sqrt(2)*(15*sqrt(a)*e - 77*sqrt(b)*c)*log(sqrt
(2)*a**(1/4)*b**(3/4)*x + sqrt(a)*sqrt(b) + b*x**2)/(1024*a**(15/4)*b**(3/4)) -
sqrt(2)*(15*sqrt(a)*e + 77*sqrt(b)*c)*atan(1 - sqrt(2)*b**(1/4)*x/a**(1/4))/(512
*a**(15/4)*b**(3/4)) + sqrt(2)*(15*sqrt(a)*e + 77*sqrt(b)*c)*atan(1 + sqrt(2)*b*
*(1/4)*x/a**(1/4))/(512*a**(15/4)*b**(3/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.929587, size = 379, normalized size = 0.99 \[ \frac{\frac{3 \sqrt{2} \left (15 a^{3/4} e-77 \sqrt [4]{a} \sqrt{b} c\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{b^{3/4}}+\frac{3 \sqrt{2} \left (77 \sqrt [4]{a} \sqrt{b} c-15 a^{3/4} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{b^{3/4}}-\frac{256 a^3 (a f-b x (c+x (d+e x)))}{b \left (a+b x^4\right )^3}+\frac{32 a^2 x (11 c+x (10 d+9 e x))}{\left (a+b x^4\right )^2}-\frac{6 \sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (80 \sqrt [4]{a} \sqrt [4]{b} d+15 \sqrt{2} \sqrt{a} e+77 \sqrt{2} \sqrt{b} c\right )}{b^{3/4}}+\frac{6 \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-80 \sqrt [4]{a} \sqrt [4]{b} d+15 \sqrt{2} \sqrt{a} e+77 \sqrt{2} \sqrt{b} c\right )}{b^{3/4}}+\frac{8 a x (77 c+15 x (4 d+3 e x))}{a+b x^4}}{3072 a^4} \]

Antiderivative was successfully verified.

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

[Out]

((8*a*x*(77*c + 15*x*(4*d + 3*e*x)))/(a + b*x^4) + (32*a^2*x*(11*c + x*(10*d + 9
*e*x)))/(a + b*x^4)^2 - (256*a^3*(a*f - b*x*(c + x*(d + e*x))))/(b*(a + b*x^4)^3
) - (6*a^(1/4)*(77*Sqrt[2]*Sqrt[b]*c + 80*a^(1/4)*b^(1/4)*d + 15*Sqrt[2]*Sqrt[a]
*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/b^(3/4) + (6*a^(1/4)*(77*Sqrt[2]*Sq
rt[b]*c - 80*a^(1/4)*b^(1/4)*d + 15*Sqrt[2]*Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*b^(1/
4)*x)/a^(1/4)])/b^(3/4) + (3*Sqrt[2]*(-77*a^(1/4)*Sqrt[b]*c + 15*a^(3/4)*e)*Log[
Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(3/4) + (3*Sqrt[2]*(77*a^(
1/4)*Sqrt[b]*c - 15*a^(3/4)*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]
*x^2])/b^(3/4))/(3072*a^4)

_______________________________________________________________________________________

Maple [A]  time = 0.022, size = 399, normalized size = 1. \[{\frac{1}{ \left ( b{x}^{4}+a \right ) ^{3}} \left ({\frac{15\,{b}^{2}e{x}^{11}}{128\,{a}^{3}}}+{\frac{5\,{b}^{2}d{x}^{10}}{32\,{a}^{3}}}+{\frac{77\,{b}^{2}c{x}^{9}}{384\,{a}^{3}}}+{\frac{21\,be{x}^{7}}{64\,{a}^{2}}}+{\frac{5\,bd{x}^{6}}{12\,{a}^{2}}}+{\frac{33\,bc{x}^{5}}{64\,{a}^{2}}}+{\frac{113\,e{x}^{3}}{384\,a}}+{\frac{11\,d{x}^{2}}{32\,a}}+{\frac{51\,cx}{128\,a}}-{\frac{f}{12\,b}} \right ) }+{\frac{77\,c\sqrt{2}}{1024\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{77\,c\sqrt{2}}{512\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{77\,c\sqrt{2}}{512\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{5\,d}{32}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{b{a}^{7}}}}}+{\frac{15\,e\sqrt{2}}{1024\,{a}^{3}b}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{15\,e\sqrt{2}}{512\,{a}^{3}b}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{15\,e\sqrt{2}}{512\,{a}^{3}b}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

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+77/1024*c*(a/b)^(1/4)/a^4*2^(1/2)*ln((x^2+(a/b)^(1/4)*x*2^
(1/2)+(a/b)^(1/2))/(x^2-(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2)))+77/512*c*(a/b)^(1/4)
/a^4*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+77/512*c*(a/b)^(1/4)/a^4*2^(1/2)*ar
ctan(2^(1/2)/(a/b)^(1/4)*x-1)+5/32*d/(b*a^7)^(1/2)*arctan(x^2*(b/a)^(1/2))+15/10
24*e/a^3/b/(a/b)^(1/4)*2^(1/2)*ln((x^2-(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2))/(x^2+(
a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2)))+15/512*e/a^3/b/(a/b)^(1/4)*2^(1/2)*arctan(2^(
1/2)/(a/b)^(1/4)*x+1)+15/512*e/a^3/b/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1
/4)*x-1)

_______________________________________________________________________________________

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 = 0.237505, size = 528, normalized size = 1.38 \[ \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{a b} b^{2} d + 77 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 15 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{a b} b^{2} d + 77 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 15 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac{\sqrt{2}{\left (77 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c - 15 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{1024 \, a^{4} b^{3}} - \frac{\sqrt{2}{\left (77 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c - 15 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{1024 \, a^{4} b^{3}} + \frac{45 \, b^{3} x^{11} e + 60 \, b^{3} d x^{10} + 77 \, b^{3} c x^{9} + 126 \, a b^{2} x^{7} e + 160 \, a b^{2} d x^{6} + 198 \, a b^{2} c x^{5} + 113 \, a^{2} b x^{3} e + 132 \, a^{2} b d x^{2} + 153 \, a^{2} b c x - 32 \, a^{3} f}{384 \,{\left (b x^{4} + a\right )}^{3} a^{3} b} \]

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) + 1/38
4*(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)/((b*x^4 + a)^3*a^3*b)