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

Optimal. Leaf size=310 \[ -\frac{\left (\sqrt{b} d-3 \sqrt{a} f\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\left (\sqrt{b} d-3 \sqrt{a} f\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{3/4} b^{7/4}}-\frac{\left (3 \sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\left (3 \sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{3/4} b^{7/4}}+\frac{e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 \sqrt{a} b^{3/2}}-\frac{c+d x+e x^2+f x^3}{4 b \left (a+b x^4\right )} \]

[Out]

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

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Rubi [A]  time = 0.581243, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ -\frac{\left (\sqrt{b} d-3 \sqrt{a} f\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\left (\sqrt{b} d-3 \sqrt{a} f\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{3/4} b^{7/4}}-\frac{\left (3 \sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\left (3 \sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{3/4} b^{7/4}}+\frac{e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 \sqrt{a} b^{3/2}}-\frac{c+d x+e x^2+f x^3}{4 b \left (a+b x^4\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 100.513, size = 291, normalized size = 0.94 \[ - \frac{c + d x + e x^{2} + f x^{3}}{4 b \left (a + b x^{4}\right )} + \frac{e \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 \sqrt{a} b^{\frac{3}{2}}} + \frac{\sqrt{2} \left (3 \sqrt{a} f - \sqrt{b} d\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{32 a^{\frac{3}{4}} b^{\frac{7}{4}}} - \frac{\sqrt{2} \left (3 \sqrt{a} f - \sqrt{b} d\right ) \log{\left (\sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{32 a^{\frac{3}{4}} b^{\frac{7}{4}}} - \frac{\sqrt{2} \left (3 \sqrt{a} f + \sqrt{b} d\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{3}{4}} b^{\frac{7}{4}}} + \frac{\sqrt{2} \left (3 \sqrt{a} f + \sqrt{b} d\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{3}{4}} b^{\frac{7}{4}}} \]

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)**2,x)

[Out]

-(c + d*x + e*x**2 + f*x**3)/(4*b*(a + b*x**4)) + e*atan(sqrt(b)*x**2/sqrt(a))/(
4*sqrt(a)*b**(3/2)) + sqrt(2)*(3*sqrt(a)*f - sqrt(b)*d)*log(-sqrt(2)*a**(1/4)*b*
*(3/4)*x + sqrt(a)*sqrt(b) + b*x**2)/(32*a**(3/4)*b**(7/4)) - sqrt(2)*(3*sqrt(a)
*f - sqrt(b)*d)*log(sqrt(2)*a**(1/4)*b**(3/4)*x + sqrt(a)*sqrt(b) + b*x**2)/(32*
a**(3/4)*b**(7/4)) - sqrt(2)*(3*sqrt(a)*f + sqrt(b)*d)*atan(1 - sqrt(2)*b**(1/4)
*x/a**(1/4))/(16*a**(3/4)*b**(7/4)) + sqrt(2)*(3*sqrt(a)*f + sqrt(b)*d)*atan(1 +
sqrt(2)*b**(1/4)*x/a**(1/4))/(16*a**(3/4)*b**(7/4))

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Mathematica [A]  time = 0.531374, size = 294, normalized size = 0.95 \[ \frac{-\frac{2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (4 \sqrt [4]{a} \sqrt [4]{b} e+3 \sqrt{2} \sqrt{a} f+\sqrt{2} \sqrt{b} d\right )}{a^{3/4}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-4 \sqrt [4]{a} \sqrt [4]{b} e+3 \sqrt{2} \sqrt{a} f+\sqrt{2} \sqrt{b} d\right )}{a^{3/4}}+\frac{\sqrt{2} \left (3 \sqrt{a} f-\sqrt{b} d\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{3/4}}+\frac{\sqrt{2} \left (\sqrt{b} d-3 \sqrt{a} f\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{3/4}}-\frac{8 b^{3/4} (c+x (d+x (e+f x)))}{a+b x^4}}{32 b^{7/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-8*b^(3/4)*(c + x*(d + x*(e + f*x))))/(a + b*x^4) - (2*(Sqrt[2]*Sqrt[b]*d + 4*
a^(1/4)*b^(1/4)*e + 3*Sqrt[2]*Sqrt[a]*f)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)]
)/a^(3/4) + (2*(Sqrt[2]*Sqrt[b]*d - 4*a^(1/4)*b^(1/4)*e + 3*Sqrt[2]*Sqrt[a]*f)*A
rcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(3/4) + (Sqrt[2]*(-(Sqrt[b]*d) + 3*Sqr
t[a]*f)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(3/4) + (Sqrt[
2]*(Sqrt[b]*d - 3*Sqrt[a]*f)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x
^2])/a^(3/4))/(32*b^(7/4))

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Maple [A]  time = 0.021, size = 333, normalized size = 1.1 \[{\frac{1}{b{x}^{4}+a} \left ( -{\frac{f{x}^{3}}{4\,b}}-{\frac{e{x}^{2}}{4\,b}}-{\frac{dx}{4\,b}}-{\frac{c}{4\,b}} \right ) }+{\frac{d\sqrt{2}}{32\,ab}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{d\sqrt{2}}{16\,ab}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{d\sqrt{2}}{16\,ab}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{e}{4}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{a{b}^{3}}}}}+{\frac{3\,f\sqrt{2}}{32\,{b}^{2}}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3\,f\sqrt{2}}{16\,{b}^{2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3\,f\sqrt{2}}{16\,{b}^{2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

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)^2,x)

[Out]

(-1/4*f*x^3/b-1/4*e*x^2/b-1/4*d*x/b-1/4*c/b)/(b*x^4+a)+1/32*d/b*(a/b)^(1/4)/a*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)))+1/16*d/b*(a/b)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+1/16*d/
b*(a/b)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)+1/4*e/(a*b^3)^(1/2)*arct
an(x^2*(b/a)^(1/2))+3/32*f/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)))+3/16*f/b^2/(a/b)^(1/4)*2^
(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+3/16*f/b^2/(a/b)^(1/4)*2^(1/2)*arctan(2^(1
/2)/(a/b)^(1/4)*x-1)

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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)^2,x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 64.074, size = 508, normalized size = 1.64 \[ \operatorname{RootSum}{\left (65536 t^{4} a^{3} b^{7} + t^{2} \left (3072 a^{2} b^{4} d f + 2048 a^{2} b^{4} e^{2}\right ) + t \left (1152 a^{2} b^{2} e f^{2} - 128 a b^{3} d^{2} e\right ) + 81 a^{2} f^{4} + 18 a b d^{2} f^{2} - 48 a b d e^{2} f + 16 a b e^{4} + b^{2} d^{4}, \left ( t \mapsto t \log{\left (x + \frac{110592 t^{3} a^{4} b^{5} f^{3} - 12288 t^{3} a^{3} b^{6} d^{2} f + 32768 t^{3} a^{3} b^{6} d e^{2} + 13824 t^{2} a^{3} b^{4} d e f^{2} - 12288 t^{2} a^{3} b^{4} e^{3} f + 512 t^{2} a^{2} b^{5} d^{3} e + 3888 t a^{3} b^{2} d f^{4} + 5184 t a^{3} b^{2} e^{2} f^{3} - 576 t a^{2} b^{3} d^{3} f^{2} + 1728 t a^{2} b^{3} d^{2} e^{2} f + 512 t a^{2} b^{3} d e^{4} + 16 t a b^{4} d^{5} + 1458 a^{3} e f^{5} + 360 a^{2} b d e^{3} f^{2} - 192 a^{2} b e^{5} f + 30 a b^{2} d^{4} e f - 40 a b^{2} d^{3} e^{3}}{729 a^{3} f^{6} - 81 a^{2} b d^{2} f^{4} + 864 a^{2} b d e^{2} f^{3} - 576 a^{2} b e^{4} f^{2} - 9 a b^{2} d^{4} f^{2} + 96 a b^{2} d^{3} e^{2} f - 64 a b^{2} d^{2} e^{4} + b^{3} d^{6}} \right )} \right )\right )} - \frac{c + d x + e x^{2} + f x^{3}}{4 a b + 4 b^{2} x^{4}} \]

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)**2,x)

[Out]

RootSum(65536*_t**4*a**3*b**7 + _t**2*(3072*a**2*b**4*d*f + 2048*a**2*b**4*e**2)
+ _t*(1152*a**2*b**2*e*f**2 - 128*a*b**3*d**2*e) + 81*a**2*f**4 + 18*a*b*d**2*f
**2 - 48*a*b*d*e**2*f + 16*a*b*e**4 + b**2*d**4, Lambda(_t, _t*log(x + (110592*_
t**3*a**4*b**5*f**3 - 12288*_t**3*a**3*b**6*d**2*f + 32768*_t**3*a**3*b**6*d*e**
2 + 13824*_t**2*a**3*b**4*d*e*f**2 - 12288*_t**2*a**3*b**4*e**3*f + 512*_t**2*a*
*2*b**5*d**3*e + 3888*_t*a**3*b**2*d*f**4 + 5184*_t*a**3*b**2*e**2*f**3 - 576*_t
*a**2*b**3*d**3*f**2 + 1728*_t*a**2*b**3*d**2*e**2*f + 512*_t*a**2*b**3*d*e**4 +
16*_t*a*b**4*d**5 + 1458*a**3*e*f**5 + 360*a**2*b*d*e**3*f**2 - 192*a**2*b*e**5
*f + 30*a*b**2*d**4*e*f - 40*a*b**2*d**3*e**3)/(729*a**3*f**6 - 81*a**2*b*d**2*f
**4 + 864*a**2*b*d*e**2*f**3 - 576*a**2*b*e**4*f**2 - 9*a*b**2*d**4*f**2 + 96*a*
b**2*d**3*e**2*f - 64*a*b**2*d**2*e**4 + b**3*d**6)))) - (c + d*x + e*x**2 + f*x
**3)/(4*a*b + 4*b**2*x**4)

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GIAC/XCAS [A]  time = 0.232053, size = 409, normalized size = 1.32 \[ -\frac{f x^{3} + x^{2} e + d x + c}{4 \,{\left (b x^{4} + a\right )} b} + \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{a b} b^{2} e + \left (a b^{3}\right )^{\frac{1}{4}} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac{3}{4}} f\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a b^{4}} + \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{a b} b^{2} e + \left (a b^{3}\right )^{\frac{1}{4}} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac{3}{4}} f\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a b^{4}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} d - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} f\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{32 \, a b^{4}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} d - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} f\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{32 \, a b^{4}} \]

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)^2,x, algorithm="giac")

[Out]

-1/4*(f*x^3 + x^2*e + d*x + c)/((b*x^4 + a)*b) + 1/16*sqrt(2)*(2*sqrt(2)*sqrt(a*
b)*b^2*e + (a*b^3)^(1/4)*b^2*d + 3*(a*b^3)^(3/4)*f)*arctan(1/2*sqrt(2)*(2*x + sq
rt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a*b^4) + 1/16*sqrt(2)*(2*sqrt(2)*sqrt(a*b)*b^2*
e + (a*b^3)^(1/4)*b^2*d + 3*(a*b^3)^(3/4)*f)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(
a/b)^(1/4))/(a/b)^(1/4))/(a*b^4) + 1/32*sqrt(2)*((a*b^3)^(1/4)*b^2*d - 3*(a*b^3)
^(3/4)*f)*ln(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a*b^4) - 1/32*sqrt(2)*((a
*b^3)^(1/4)*b^2*d - 3*(a*b^3)^(3/4)*f)*ln(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b
))/(a*b^4)