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

Optimal. Leaf size=340 \[ -\frac{3 \left (\sqrt{b} d-\sqrt{a} f\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{7/4} b^{7/4}}+\frac{3 \left (\sqrt{b} d-\sqrt{a} f\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{7/4} b^{7/4}}-\frac{3 \left (\sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{7/4} b^{7/4}}+\frac{3 \left (\sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{7/4} b^{7/4}}+\frac{e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{3/2} b^{3/2}}-\frac{c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac{x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )} \]

[Out]

-(c + d*x + e*x^2 + f*x^3)/(8*b*(a + b*x^4)^2) + (x*(d + 2*e*x + 3*f*x^2))/(32*a
*b*(a + b*x^4)) + (e*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(16*a^(3/2)*b^(3/2)) - (3*(S
qrt[b]*d + Sqrt[a]*f)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(7/
4)*b^(7/4)) + (3*(Sqrt[b]*d + Sqrt[a]*f)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)]
)/(64*Sqrt[2]*a^(7/4)*b^(7/4)) - (3*(Sqrt[b]*d - Sqrt[a]*f)*Log[Sqrt[a] - Sqrt[2
]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(128*Sqrt[2]*a^(7/4)*b^(7/4)) + (3*(Sqrt[b]*
d - Sqrt[a]*f)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(128*Sqrt
[2]*a^(7/4)*b^(7/4))

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Rubi [A]  time = 0.689006, antiderivative size = 340, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.393 \[ -\frac{3 \left (\sqrt{b} d-\sqrt{a} f\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{7/4} b^{7/4}}+\frac{3 \left (\sqrt{b} d-\sqrt{a} f\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{7/4} b^{7/4}}-\frac{3 \left (\sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{7/4} b^{7/4}}+\frac{3 \left (\sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{7/4} b^{7/4}}+\frac{e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{3/2} b^{3/2}}-\frac{c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac{x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(c + d*x + e*x^2 + f*x^3)/(8*b*(a + b*x^4)^2) + (x*(d + 2*e*x + 3*f*x^2))/(32*a
*b*(a + b*x^4)) + (e*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(16*a^(3/2)*b^(3/2)) - (3*(S
qrt[b]*d + Sqrt[a]*f)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(7/
4)*b^(7/4)) + (3*(Sqrt[b]*d + Sqrt[a]*f)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)]
)/(64*Sqrt[2]*a^(7/4)*b^(7/4)) - (3*(Sqrt[b]*d - Sqrt[a]*f)*Log[Sqrt[a] - Sqrt[2
]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(128*Sqrt[2]*a^(7/4)*b^(7/4)) + (3*(Sqrt[b]*
d - Sqrt[a]*f)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(128*Sqrt
[2]*a^(7/4)*b^(7/4))

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Rubi in Sympy [A]  time = 116.304, size = 320, normalized size = 0.94 \[ - \frac{c + d x + e x^{2} + f x^{3}}{8 b \left (a + b x^{4}\right )^{2}} + \frac{x \left (d + 2 e x + 3 f x^{2}\right )}{32 a b \left (a + b x^{4}\right )} + \frac{e \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{16 a^{\frac{3}{2}} b^{\frac{3}{2}}} + \frac{3 \sqrt{2} \left (\sqrt{a} f - \sqrt{b} d\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{256 a^{\frac{7}{4}} b^{\frac{7}{4}}} - \frac{3 \sqrt{2} \left (\sqrt{a} f - \sqrt{b} d\right ) \log{\left (\sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{256 a^{\frac{7}{4}} b^{\frac{7}{4}}} - \frac{3 \sqrt{2} \left (\sqrt{a} f + \sqrt{b} d\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{7}{4}} b^{\frac{7}{4}}} + \frac{3 \sqrt{2} \left (\sqrt{a} f + \sqrt{b} d\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{7}{4}} b^{\frac{7}{4}}} \]

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)**3,x)

[Out]

-(c + d*x + e*x**2 + f*x**3)/(8*b*(a + b*x**4)**2) + x*(d + 2*e*x + 3*f*x**2)/(3
2*a*b*(a + b*x**4)) + e*atan(sqrt(b)*x**2/sqrt(a))/(16*a**(3/2)*b**(3/2)) + 3*sq
rt(2)*(sqrt(a)*f - sqrt(b)*d)*log(-sqrt(2)*a**(1/4)*b**(3/4)*x + sqrt(a)*sqrt(b)
+ b*x**2)/(256*a**(7/4)*b**(7/4)) - 3*sqrt(2)*(sqrt(a)*f - sqrt(b)*d)*log(sqrt(
2)*a**(1/4)*b**(3/4)*x + sqrt(a)*sqrt(b) + b*x**2)/(256*a**(7/4)*b**(7/4)) - 3*s
qrt(2)*(sqrt(a)*f + sqrt(b)*d)*atan(1 - sqrt(2)*b**(1/4)*x/a**(1/4))/(128*a**(7/
4)*b**(7/4)) + 3*sqrt(2)*(sqrt(a)*f + sqrt(b)*d)*atan(1 + sqrt(2)*b**(1/4)*x/a**
(1/4))/(128*a**(7/4)*b**(7/4))

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Mathematica [A]  time = 0.79978, size = 329, normalized size = 0.97 \[ \frac{-\frac{2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (8 \sqrt [4]{a} \sqrt [4]{b} e+3 \sqrt{2} \sqrt{a} f+3 \sqrt{2} \sqrt{b} d\right )}{a^{7/4}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-8 \sqrt [4]{a} \sqrt [4]{b} e+3 \sqrt{2} \sqrt{a} f+3 \sqrt{2} \sqrt{b} d\right )}{a^{7/4}}+\frac{3 \sqrt{2} \left (\sqrt{a} f-\sqrt{b} d\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{7/4}}+\frac{3 \sqrt{2} \left (\sqrt{b} d-\sqrt{a} f\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{7/4}}-\frac{32 b^{3/4} (c+x (d+x (e+f x)))}{\left (a+b x^4\right )^2}+\frac{8 b^{3/4} x (d+x (2 e+3 f x))}{a \left (a+b x^4\right )}}{256 b^{7/4}} \]

Antiderivative was successfully verified.

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

[Out]

((8*b^(3/4)*x*(d + x*(2*e + 3*f*x)))/(a*(a + b*x^4)) - (32*b^(3/4)*(c + x*(d + x
*(e + f*x))))/(a + b*x^4)^2 - (2*(3*Sqrt[2]*Sqrt[b]*d + 8*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^(7/4) + (2*(3*Sqrt
[2]*Sqrt[b]*d - 8*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^(7/4) + (3*Sqrt[2]*(-(Sqrt[b]*d) + Sqrt[a]*f)*Log[Sqrt[a]
- Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4) + (3*Sqrt[2]*(Sqrt[b]*d - Sq
rt[a]*f)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4))/(256*b
^(7/4))

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Maple [A]  time = 0.018, size = 371, normalized size = 1.1 \[{\frac{1}{ \left ( b{x}^{4}+a \right ) ^{2}} \left ({\frac{3\,f{x}^{7}}{32\,a}}+{\frac{e{x}^{6}}{16\,a}}+{\frac{d{x}^{5}}{32\,a}}-{\frac{f{x}^{3}}{32\,b}}-{\frac{e{x}^{2}}{16\,b}}-{\frac{3\,dx}{32\,b}}-{\frac{c}{8\,b}} \right ) }+{\frac{3\,d\sqrt{2}}{256\,{a}^{2}b}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{3\,d\sqrt{2}}{128\,{a}^{2}b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{3\,d\sqrt{2}}{128\,{a}^{2}b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{e}{16}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{{a}^{3}{b}^{3}}}}}+{\frac{3\,f\sqrt{2}}{256\,a{b}^{2}}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3\,f\sqrt{2}}{128\,a{b}^{2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3\,f\sqrt{2}}{128\,a{b}^{2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

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)^3,x)

[Out]

(3/32*f/a*x^7+1/16/a*e*x^6+1/32*d/a*x^5-1/32*f*x^3/b-1/16*e*x^2/b-3/32*d*x/b-1/8
*c/b)/(b*x^4+a)^2+3/256*d/b/a^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)))+3/128*d/b/a^2*(a/b)^(1/4
)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+3/128*d/b/a^2*(a/b)^(1/4)*2^(1/2)*arct
an(2^(1/2)/(a/b)^(1/4)*x-1)+1/16*e/(a^3*b^3)^(1/2)*arctan(x^2*(b/a)^(1/2))+3/256
*f/b^2/a/(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)))+3/128*f/b^2/a/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2
)/(a/b)^(1/4)*x+1)+3/128*f/b^2/a/(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)^3,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)^3,x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.233291, size = 456, normalized size = 1.34 \[ \frac{3 \, b f x^{7} + 2 \, b x^{6} e + b d x^{5} - a f x^{3} - 2 \, a x^{2} e - 3 \, a d x - 4 \, a c}{32 \,{\left (b x^{4} + a\right )}^{2} a b} + \frac{\sqrt{2}{\left (4 \, \sqrt{2} \sqrt{a b} b^{2} e + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac{3}{4}} f\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{2} b^{4}} + \frac{\sqrt{2}{\left (4 \, \sqrt{2} \sqrt{a b} b^{2} e + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac{3}{4}} f\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{2} b^{4}} + \frac{3 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} d - \left (a b^{3}\right )^{\frac{3}{4}} f\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{256 \, a^{2} b^{4}} - \frac{3 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} d - \left (a b^{3}\right )^{\frac{3}{4}} f\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{256 \, a^{2} b^{4}} \]

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)^3,x, algorithm="giac")

[Out]

1/32*(3*b*f*x^7 + 2*b*x^6*e + b*d*x^5 - a*f*x^3 - 2*a*x^2*e - 3*a*d*x - 4*a*c)/(
(b*x^4 + a)^2*a*b) + 1/128*sqrt(2)*(4*sqrt(2)*sqrt(a*b)*b^2*e + 3*(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^2*b^4) + 1/128*sqrt(2)*(4*sqrt(2)*sqrt(a*b)*b^2*e + 3*(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^2*b^4) + 3/256*sqrt(2)*((a*b^3)^(1/4)*b^2*d - (a*b^3)^(3/4)*f)*ln(x^2 +
sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^2*b^4) - 3/256*sqrt(2)*((a*b^3)^(1/4)*b^2*
d - (a*b^3)^(3/4)*f)*ln(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^2*b^4)