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

Optimal. Leaf size=380 \[ -\frac{\left (7 \sqrt{b} d-5 \sqrt{a} f\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{11/4} b^{7/4}}+\frac{\left (7 \sqrt{b} d-5 \sqrt{a} f\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{11/4} b^{7/4}}-\frac{\left (5 \sqrt{a} f+7 \sqrt{b} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt{2} a^{11/4} b^{7/4}}+\frac{\left (5 \sqrt{a} f+7 \sqrt{b} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{256 \sqrt{2} a^{11/4} b^{7/4}}+\frac{e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{5/2} b^{3/2}}+\frac{x \left (7 d+12 e x+15 f x^2\right )}{384 a^2 b \left (a+b x^4\right )}-\frac{c+d x+e x^2+f x^3}{12 b \left (a+b x^4\right )^3}+\frac{x \left (d+2 e x+3 f x^2\right )}{96 a b \left (a+b x^4\right )^2} \]

[Out]

-(c + d*x + e*x^2 + f*x^3)/(12*b*(a + b*x^4)^3) + (x*(d + 2*e*x + 3*f*x^2))/(96*
a*b*(a + b*x^4)^2) + (x*(7*d + 12*e*x + 15*f*x^2))/(384*a^2*b*(a + b*x^4)) + (e*
ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(32*a^(5/2)*b^(3/2)) - ((7*Sqrt[b]*d + 5*Sqrt[a]*
f)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(256*Sqrt[2]*a^(11/4)*b^(7/4)) + ((7
*Sqrt[b]*d + 5*Sqrt[a]*f)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(256*Sqrt[2]*
a^(11/4)*b^(7/4)) - ((7*Sqrt[b]*d - 5*Sqrt[a]*f)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b
^(1/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(11/4)*b^(7/4)) + ((7*Sqrt[b]*d - 5*Sqrt
[a]*f)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(1
1/4)*b^(7/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.845255, antiderivative size = 380, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.393 \[ -\frac{\left (7 \sqrt{b} d-5 \sqrt{a} f\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{11/4} b^{7/4}}+\frac{\left (7 \sqrt{b} d-5 \sqrt{a} f\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{11/4} b^{7/4}}-\frac{\left (5 \sqrt{a} f+7 \sqrt{b} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt{2} a^{11/4} b^{7/4}}+\frac{\left (5 \sqrt{a} f+7 \sqrt{b} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{256 \sqrt{2} a^{11/4} b^{7/4}}+\frac{e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{5/2} b^{3/2}}+\frac{x \left (7 d+12 e x+15 f x^2\right )}{384 a^2 b \left (a+b x^4\right )}-\frac{c+d x+e x^2+f x^3}{12 b \left (a+b x^4\right )^3}+\frac{x \left (d+2 e x+3 f x^2\right )}{96 a b \left (a+b x^4\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(c + d*x + e*x^2 + f*x^3)/(12*b*(a + b*x^4)^3) + (x*(d + 2*e*x + 3*f*x^2))/(96*
a*b*(a + b*x^4)^2) + (x*(7*d + 12*e*x + 15*f*x^2))/(384*a^2*b*(a + b*x^4)) + (e*
ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(32*a^(5/2)*b^(3/2)) - ((7*Sqrt[b]*d + 5*Sqrt[a]*
f)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(256*Sqrt[2]*a^(11/4)*b^(7/4)) + ((7
*Sqrt[b]*d + 5*Sqrt[a]*f)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(256*Sqrt[2]*
a^(11/4)*b^(7/4)) - ((7*Sqrt[b]*d - 5*Sqrt[a]*f)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b
^(1/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(11/4)*b^(7/4)) + ((7*Sqrt[b]*d - 5*Sqrt
[a]*f)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(1
1/4)*b^(7/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 137.143, size = 359, normalized size = 0.94 \[ - \frac{c + d x + e x^{2} + f x^{3}}{12 b \left (a + b x^{4}\right )^{3}} + \frac{x \left (d + 2 e x + 3 f x^{2}\right )}{96 a b \left (a + b x^{4}\right )^{2}} + \frac{x \left (7 d + 12 e x + 15 f x^{2}\right )}{384 a^{2} b \left (a + b x^{4}\right )} + \frac{e \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{32 a^{\frac{5}{2}} b^{\frac{3}{2}}} + \frac{\sqrt{2} \left (5 \sqrt{a} f - 7 \sqrt{b} d\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{1024 a^{\frac{11}{4}} b^{\frac{7}{4}}} - \frac{\sqrt{2} \left (5 \sqrt{a} f - 7 \sqrt{b} d\right ) \log{\left (\sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{1024 a^{\frac{11}{4}} b^{\frac{7}{4}}} - \frac{\sqrt{2} \left (5 \sqrt{a} f + 7 \sqrt{b} d\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{512 a^{\frac{11}{4}} b^{\frac{7}{4}}} + \frac{\sqrt{2} \left (5 \sqrt{a} f + 7 \sqrt{b} d\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{512 a^{\frac{11}{4}} b^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(c + d*x + e*x**2 + f*x**3)/(12*b*(a + b*x**4)**3) + x*(d + 2*e*x + 3*f*x**2)/(
96*a*b*(a + b*x**4)**2) + x*(7*d + 12*e*x + 15*f*x**2)/(384*a**2*b*(a + b*x**4))
+ e*atan(sqrt(b)*x**2/sqrt(a))/(32*a**(5/2)*b**(3/2)) + sqrt(2)*(5*sqrt(a)*f -
7*sqrt(b)*d)*log(-sqrt(2)*a**(1/4)*b**(3/4)*x + sqrt(a)*sqrt(b) + b*x**2)/(1024*
a**(11/4)*b**(7/4)) - sqrt(2)*(5*sqrt(a)*f - 7*sqrt(b)*d)*log(sqrt(2)*a**(1/4)*b
**(3/4)*x + sqrt(a)*sqrt(b) + b*x**2)/(1024*a**(11/4)*b**(7/4)) - sqrt(2)*(5*sqr
t(a)*f + 7*sqrt(b)*d)*atan(1 - sqrt(2)*b**(1/4)*x/a**(1/4))/(512*a**(11/4)*b**(7
/4)) + sqrt(2)*(5*sqrt(a)*f + 7*sqrt(b)*d)*atan(1 + sqrt(2)*b**(1/4)*x/a**(1/4))
/(512*a**(11/4)*b**(7/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.789085, size = 366, normalized size = 0.96 \[ \frac{-\frac{6 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (16 \sqrt [4]{a} \sqrt [4]{b} e+5 \sqrt{2} \sqrt{a} f+7 \sqrt{2} \sqrt{b} d\right )}{a^{11/4}}+\frac{6 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-16 \sqrt [4]{a} \sqrt [4]{b} e+5 \sqrt{2} \sqrt{a} f+7 \sqrt{2} \sqrt{b} d\right )}{a^{11/4}}+\frac{3 \sqrt{2} \left (5 \sqrt{a} f-7 \sqrt{b} d\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{11/4}}+\frac{3 \sqrt{2} \left (7 \sqrt{b} d-5 \sqrt{a} f\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{11/4}}+\frac{8 b^{3/4} x (7 d+3 x (4 e+5 f x))}{a^2 \left (a+b x^4\right )}-\frac{256 b^{3/4} (c+x (d+x (e+f x)))}{\left (a+b x^4\right )^3}+\frac{32 b^{3/4} x (d+x (2 e+3 f x))}{a \left (a+b x^4\right )^2}}{3072 b^{7/4}} \]

Antiderivative was successfully verified.

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

[Out]

((32*b^(3/4)*x*(d + x*(2*e + 3*f*x)))/(a*(a + b*x^4)^2) + (8*b^(3/4)*x*(7*d + 3*
x*(4*e + 5*f*x)))/(a^2*(a + b*x^4)) - (256*b^(3/4)*(c + x*(d + x*(e + f*x))))/(a
+ b*x^4)^3 - (6*(7*Sqrt[2]*Sqrt[b]*d + 16*a^(1/4)*b^(1/4)*e + 5*Sqrt[2]*Sqrt[a]
*f)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(11/4) + (6*(7*Sqrt[2]*Sqrt[b]*d
- 16*a^(1/4)*b^(1/4)*e + 5*Sqrt[2]*Sqrt[a]*f)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(
1/4)])/a^(11/4) + (3*Sqrt[2]*(-7*Sqrt[b]*d + 5*Sqrt[a]*f)*Log[Sqrt[a] - Sqrt[2]*
a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(11/4) + (3*Sqrt[2]*(7*Sqrt[b]*d - 5*Sqrt[a]
*f)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(11/4))/(3072*b^(7
/4))

_______________________________________________________________________________________

Maple [A]  time = 0.022, size = 401, normalized size = 1.1 \[{\frac{1}{ \left ( b{x}^{4}+a \right ) ^{3}} \left ({\frac{5\,bf{x}^{11}}{128\,{a}^{2}}}+{\frac{be{x}^{10}}{32\,{a}^{2}}}+{\frac{7\,bd{x}^{9}}{384\,{a}^{2}}}+{\frac{7\,f{x}^{7}}{64\,a}}+{\frac{e{x}^{6}}{12\,a}}+{\frac{3\,d{x}^{5}}{64\,a}}-{\frac{5\,f{x}^{3}}{384\,b}}-{\frac{e{x}^{2}}{32\,b}}-{\frac{7\,dx}{128\,b}}-{\frac{c}{12\,b}} \right ) }+{\frac{7\,d\sqrt{2}}{1024\,{a}^{3}b}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{7\,d\sqrt{2}}{512\,{a}^{3}b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{7\,d\sqrt{2}}{512\,{a}^{3}b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{e}{32}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{{a}^{5}{b}^{3}}}}}+{\frac{5\,f\sqrt{2}}{1024\,{a}^{2}{b}^{2}}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,f\sqrt{2}}{512\,{a}^{2}{b}^{2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,f\sqrt{2}}{512\,{a}^{2}{b}^{2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(5/128*f/a^2*b*x^11+1/32/a^2*b*e*x^10+7/384/a^2*d*b*x^9+7/64*f/a*x^7+1/12/a*e*x^
6+3/64*d/a*x^5-5/384*f*x^3/b-1/32*e*x^2/b-7/128*d*x/b-1/12*c/b)/(b*x^4+a)^3+7/10
24*d*(a/b)^(1/4)/a^3/b*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)))+7/512*d*(a/b)^(1/4)/a^3/b*2^(1/2)*arctan(2^(1
/2)/(a/b)^(1/4)*x+1)+7/512*d*(a/b)^(1/4)/a^3/b*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4
)*x-1)+1/32*e/(a^5*b^3)^(1/2)*arctan(x^2*(b/a)^(1/2))+5/1024*f/a^2/b^2/(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)))+5/512*f/a^2/b^2/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+
1)+5/512*f/a^2/b^2/(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)*x^3/(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)*x^3/(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(x**3*(f*x**3+e*x**2+d*x+c)/(b*x**4+a)**4,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.232425, size = 513, normalized size = 1.35 \[ \frac{\sqrt{2}{\left (8 \, \sqrt{2} \sqrt{a b} b^{2} e + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} d + 5 \, \left (a b^{3}\right )^{\frac{3}{4}} f\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{512 \, a^{3} b^{4}} + \frac{\sqrt{2}{\left (8 \, \sqrt{2} \sqrt{a b} b^{2} e + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} d + 5 \, \left (a b^{3}\right )^{\frac{3}{4}} f\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{512 \, a^{3} b^{4}} + \frac{\sqrt{2}{\left (7 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} d - 5 \, \left (a b^{3}\right )^{\frac{3}{4}} f\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{1024 \, a^{3} b^{4}} - \frac{\sqrt{2}{\left (7 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} d - 5 \, \left (a b^{3}\right )^{\frac{3}{4}} f\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{1024 \, a^{3} b^{4}} + \frac{15 \, b^{2} f x^{11} + 12 \, b^{2} x^{10} e + 7 \, b^{2} d x^{9} + 42 \, a b f x^{7} + 32 \, a b x^{6} e + 18 \, a b d x^{5} - 5 \, a^{2} f x^{3} - 12 \, a^{2} x^{2} e - 21 \, a^{2} d x - 32 \, a^{2} c}{384 \,{\left (b x^{4} + a\right )}^{3} a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/512*sqrt(2)*(8*sqrt(2)*sqrt(a*b)*b^2*e + 7*(a*b^3)^(1/4)*b^2*d + 5*(a*b^3)^(3/
4)*f)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^3*b^4) + 1/
512*sqrt(2)*(8*sqrt(2)*sqrt(a*b)*b^2*e + 7*(a*b^3)^(1/4)*b^2*d + 5*(a*b^3)^(3/4)
*f)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^3*b^4) + 1/10
24*sqrt(2)*(7*(a*b^3)^(1/4)*b^2*d - 5*(a*b^3)^(3/4)*f)*ln(x^2 + sqrt(2)*x*(a/b)^
(1/4) + sqrt(a/b))/(a^3*b^4) - 1/1024*sqrt(2)*(7*(a*b^3)^(1/4)*b^2*d - 5*(a*b^3)
^(3/4)*f)*ln(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^3*b^4) + 1/384*(15*b^2*
f*x^11 + 12*b^2*x^10*e + 7*b^2*d*x^9 + 42*a*b*f*x^7 + 32*a*b*x^6*e + 18*a*b*d*x^
5 - 5*a^2*f*x^3 - 12*a^2*x^2*e - 21*a^2*d*x - 32*a^2*c)/((b*x^4 + a)^3*a^2*b)