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

Optimal. Leaf size=162 \[ \frac{\sqrt [4]{a} \left (\sqrt{b} d-\sqrt{a} f\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 b^{7/4}}+\frac{\sqrt [4]{a} \left (\sqrt{a} f+\sqrt{b} d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 b^{7/4}}+\frac{\sqrt{a} e \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 b^{3/2}}-\frac{c \log \left (a-b x^4\right )}{4 b}-\frac{d x}{b}-\frac{e x^2}{2 b}-\frac{f x^3}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.483711, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31 \[ \frac{\sqrt [4]{a} \left (\sqrt{b} d-\sqrt{a} f\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 b^{7/4}}+\frac{\sqrt [4]{a} \left (\sqrt{a} f+\sqrt{b} d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 b^{7/4}}+\frac{\sqrt{a} e \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 b^{3/2}}-\frac{c \log \left (a-b x^4\right )}{4 b}-\frac{d x}{b}-\frac{e x^2}{2 b}-\frac{f x^3}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\sqrt [4]{a} \left (\sqrt{a} f - \sqrt{b} d\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{2 b^{\frac{7}{4}}} + \frac{\sqrt [4]{a} \left (\sqrt{a} f + \sqrt{b} d\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{2 b^{\frac{7}{4}}} + \frac{\sqrt{a} e \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 b^{\frac{3}{2}}} - \frac{c \log{\left (a - b x^{4} \right )}}{4 b} - \frac{d x}{b} - \frac{f x^{3}}{3 b} - \frac{\int ^{x^{2}} e\, dx}{2 b} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.149653, size = 221, normalized size = 1.36 \[ \frac{-3 \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (a^{3/4} f+\sqrt [4]{a} \sqrt{b} d+\sqrt{a} \sqrt [4]{b} e\right )+3 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (a^{3/4} f+\sqrt [4]{a} \sqrt{b} d-\sqrt{a} \sqrt [4]{b} e\right )+6 \left (\sqrt [4]{a} \sqrt{b} d-a^{3/4} f\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )-3 b^{3/4} c \log \left (a-b x^4\right )+3 \sqrt{a} \sqrt [4]{b} e \log \left (\sqrt{a}+\sqrt{b} x^2\right )-12 b^{3/4} d x-6 b^{3/4} e x^2-4 b^{3/4} f x^3}{12 b^{7/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-12*b^(3/4)*d*x - 6*b^(3/4)*e*x^2 - 4*b^(3/4)*f*x^3 + 6*(a^(1/4)*Sqrt[b]*d - a^
(3/4)*f)*ArcTan[(b^(1/4)*x)/a^(1/4)] - 3*(a^(1/4)*Sqrt[b]*d + Sqrt[a]*b^(1/4)*e
+ a^(3/4)*f)*Log[a^(1/4) - b^(1/4)*x] + 3*(a^(1/4)*Sqrt[b]*d - Sqrt[a]*b^(1/4)*e
+ a^(3/4)*f)*Log[a^(1/4) + b^(1/4)*x] + 3*Sqrt[a]*b^(1/4)*e*Log[Sqrt[a] + Sqrt[
b]*x^2] - 3*b^(3/4)*c*Log[a - b*x^4])/(12*b^(7/4))

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Maple [A]  time = 0.006, size = 208, normalized size = 1.3 \[ -{\frac{f{x}^{3}}{3\,b}}-{\frac{e{x}^{2}}{2\,b}}-{\frac{dx}{b}}+{\frac{d}{2\,b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{d}{4\,b}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{ae}{4\,b}\ln \left ({1 \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{af}{2\,{b}^{2}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{af}{4\,{b}^{2}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{c\ln \left ( b{x}^{4}-a \right ) }{4\,b}} \]

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),x)

[Out]

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

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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),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),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 24.244, size = 887, normalized size = 5.48 \[ - \operatorname{RootSum}{\left (256 t^{4} b^{7} - 256 t^{3} b^{6} c + t^{2} \left (- 64 a b^{4} d f - 32 a b^{4} e^{2} + 96 b^{5} c^{2}\right ) + t \left (- 16 a^{2} b^{2} e f^{2} + 32 a b^{3} c d f + 16 a b^{3} c e^{2} - 16 a b^{3} d^{2} e - 16 b^{4} c^{3}\right ) - a^{3} f^{4} + 4 a^{2} b c e f^{2} + 2 a^{2} b d^{2} f^{2} - 4 a^{2} b d e^{2} f + a^{2} b e^{4} - 4 a b^{2} c^{2} d f - 2 a b^{2} c^{2} e^{2} + 4 a b^{2} c d^{2} e - a b^{2} d^{4} + b^{3} c^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 64 t^{3} a b^{5} f^{3} - 64 t^{3} b^{6} d^{2} f + 128 t^{3} b^{6} d e^{2} + 48 t^{2} a b^{4} c f^{3} + 48 t^{2} a b^{4} d e f^{2} - 32 t^{2} a b^{4} e^{3} f + 48 t^{2} b^{5} c d^{2} f - 96 t^{2} b^{5} c d e^{2} - 16 t^{2} b^{5} d^{3} e + 12 t a^{2} b^{2} d f^{4} + 12 t a^{2} b^{2} e^{2} f^{3} - 12 t a b^{3} c^{2} f^{3} - 24 t a b^{3} c d e f^{2} + 16 t a b^{3} c e^{3} f + 16 t a b^{3} d^{3} f^{2} - 36 t a b^{3} d^{2} e^{2} f - 8 t a b^{3} d e^{4} - 12 t b^{4} c^{2} d^{2} f + 24 t b^{4} c^{2} d e^{2} + 8 t b^{4} c d^{3} e + 4 t b^{4} d^{5} + 3 a^{3} e f^{5} - 3 a^{2} b c d f^{4} - 3 a^{2} b c e^{2} f^{3} - 5 a^{2} b d e^{3} f^{2} + 2 a^{2} b e^{5} f + a b^{2} c^{3} f^{3} + 3 a b^{2} c^{2} d e f^{2} - 2 a b^{2} c^{2} e^{3} f - 4 a b^{2} c d^{3} f^{2} + 9 a b^{2} c d^{2} e^{2} f + 2 a b^{2} c d e^{4} + 5 a b^{2} d^{4} e f - 5 a b^{2} d^{3} e^{3} + b^{3} c^{3} d^{2} f - 2 b^{3} c^{3} d e^{2} - b^{3} c^{2} d^{3} e - b^{3} c d^{5}}{a^{3} f^{6} + a^{2} b d^{2} f^{4} - 8 a^{2} b d e^{2} f^{3} + 4 a^{2} b e^{4} f^{2} - a b^{2} d^{4} f^{2} + 8 a b^{2} d^{3} e^{2} f - 4 a b^{2} d^{2} e^{4} - b^{3} d^{6}} \right )} \right )\right )} - \frac{d x}{b} - \frac{e x^{2}}{2 b} - \frac{f x^{3}}{3 b} \]

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),x)

[Out]

-RootSum(256*_t**4*b**7 - 256*_t**3*b**6*c + _t**2*(-64*a*b**4*d*f - 32*a*b**4*e
**2 + 96*b**5*c**2) + _t*(-16*a**2*b**2*e*f**2 + 32*a*b**3*c*d*f + 16*a*b**3*c*e
**2 - 16*a*b**3*d**2*e - 16*b**4*c**3) - a**3*f**4 + 4*a**2*b*c*e*f**2 + 2*a**2*
b*d**2*f**2 - 4*a**2*b*d*e**2*f + a**2*b*e**4 - 4*a*b**2*c**2*d*f - 2*a*b**2*c**
2*e**2 + 4*a*b**2*c*d**2*e - a*b**2*d**4 + b**3*c**4, Lambda(_t, _t*log(x + (-64
*_t**3*a*b**5*f**3 - 64*_t**3*b**6*d**2*f + 128*_t**3*b**6*d*e**2 + 48*_t**2*a*b
**4*c*f**3 + 48*_t**2*a*b**4*d*e*f**2 - 32*_t**2*a*b**4*e**3*f + 48*_t**2*b**5*c
*d**2*f - 96*_t**2*b**5*c*d*e**2 - 16*_t**2*b**5*d**3*e + 12*_t*a**2*b**2*d*f**4
+ 12*_t*a**2*b**2*e**2*f**3 - 12*_t*a*b**3*c**2*f**3 - 24*_t*a*b**3*c*d*e*f**2
+ 16*_t*a*b**3*c*e**3*f + 16*_t*a*b**3*d**3*f**2 - 36*_t*a*b**3*d**2*e**2*f - 8*
_t*a*b**3*d*e**4 - 12*_t*b**4*c**2*d**2*f + 24*_t*b**4*c**2*d*e**2 + 8*_t*b**4*c
*d**3*e + 4*_t*b**4*d**5 + 3*a**3*e*f**5 - 3*a**2*b*c*d*f**4 - 3*a**2*b*c*e**2*f
**3 - 5*a**2*b*d*e**3*f**2 + 2*a**2*b*e**5*f + a*b**2*c**3*f**3 + 3*a*b**2*c**2*
d*e*f**2 - 2*a*b**2*c**2*e**3*f - 4*a*b**2*c*d**3*f**2 + 9*a*b**2*c*d**2*e**2*f
+ 2*a*b**2*c*d*e**4 + 5*a*b**2*d**4*e*f - 5*a*b**2*d**3*e**3 + b**3*c**3*d**2*f
- 2*b**3*c**3*d*e**2 - b**3*c**2*d**3*e - b**3*c*d**5)/(a**3*f**6 + a**2*b*d**2*
f**4 - 8*a**2*b*d*e**2*f**3 + 4*a**2*b*e**4*f**2 - a*b**2*d**4*f**2 + 8*a*b**2*d
**3*e**2*f - 4*a*b**2*d**2*e**4 - b**3*d**6)))) - d*x/b - e*x**2/(2*b) - f*x**3/
(3*b)

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GIAC/XCAS [A]  time = 0.241904, size = 443, normalized size = 2.73 \[ -\frac{c{\rm ln}\left ({\left | b x^{4} - a \right |}\right )}{4 \, b} - \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-a b} b^{2} e - \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} d - \left (-a b^{3}\right )^{\frac{3}{4}} f\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, b^{4}} - \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-a b} b^{2} e - \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} d - \left (-a b^{3}\right )^{\frac{3}{4}} f\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, b^{4}} + \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} b^{2} d - \left (-a b^{3}\right )^{\frac{3}{4}} f\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{8 \, b^{4}} - \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} b^{2} d - \left (-a b^{3}\right )^{\frac{3}{4}} f\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{8 \, b^{4}} - \frac{2 \, b^{2} f x^{3} + 3 \, b^{2} x^{2} e + 6 \, b^{2} d x}{6 \, b^{3}} \]

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),x, algorithm="giac")

[Out]

-1/4*c*ln(abs(b*x^4 - a))/b - 1/4*sqrt(2)*(sqrt(2)*sqrt(-a*b)*b^2*e - (-a*b^3)^(
1/4)*b^2*d - (-a*b^3)^(3/4)*f)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/(
-a/b)^(1/4))/b^4 - 1/4*sqrt(2)*(sqrt(2)*sqrt(-a*b)*b^2*e - (-a*b^3)^(1/4)*b^2*d
- (-a*b^3)^(3/4)*f)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4)
)/b^4 + 1/8*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))/b^4 - 1/8*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))/b^4 - 1/6*(2*b^2*f*x^3 + 3
*b^2*x^2*e + 6*b^2*d*x)/b^3