3.187 \(\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{a-b x^4} \, dx\)

Optimal. Leaf size=188 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac{\sqrt{b} (a g+b c)}{\sqrt{a}}+a i+b e\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{\sqrt{b} (a g+b c)}{\sqrt{a}}+a i+b e\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac{(a h+b d) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}-\frac{f \log \left (a-b x^4\right )}{4 b}-\frac{g x}{b}-\frac{h x^2}{2 b}-\frac{i x^3}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.64994, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.22 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac{\sqrt{b} (a g+b c)}{\sqrt{a}}+a i+b e\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{\sqrt{b} (a g+b c)}{\sqrt{a}}+a i+b e\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac{(a h+b d) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}-\frac{f \log \left (a-b x^4\right )}{4 b}-\frac{g x}{b}-\frac{h x^2}{2 b}-\frac{i x^3}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(a - b*x^4),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((i*x**6+h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(-b*x**4+a),x)

[Out]

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

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Mathematica [A]  time = 0.800465, size = 301, normalized size = 1.6 \[ \frac{-\frac{3 \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (a^{5/4} \sqrt [4]{b} h+a^{3/2} i+\sqrt [4]{a} b^{5/4} d+\sqrt{a} b e+a \sqrt{b} g+b^{3/2} c\right )}{a^{3/4}}+\frac{3 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (-a^{5/4} \sqrt [4]{b} h+a^{3/2} i-\sqrt [4]{a} b^{5/4} d+\sqrt{a} b e+a \sqrt{b} g+b^{3/2} c\right )}{a^{3/4}}+\frac{6 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (a^{3/2} (-i)-\sqrt{a} b e+a \sqrt{b} g+b^{3/2} c\right )}{a^{3/4}}-3 b^{3/4} f \log \left (a-b x^4\right )+\frac{3 \sqrt [4]{b} (a h+b d) \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt{a}}-12 b^{3/4} g x-6 b^{3/4} h x^2-4 b^{3/4} i x^3}{12 b^{7/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(a - b*x^4),x]

[Out]

(-12*b^(3/4)*g*x - 6*b^(3/4)*h*x^2 - 4*b^(3/4)*i*x^3 + (6*(b^(3/2)*c - Sqrt[a]*b
*e + a*Sqrt[b]*g - a^(3/2)*i)*ArcTan[(b^(1/4)*x)/a^(1/4)])/a^(3/4) - (3*(b^(3/2)
*c + a^(1/4)*b^(5/4)*d + Sqrt[a]*b*e + a*Sqrt[b]*g + a^(5/4)*b^(1/4)*h + a^(3/2)
*i)*Log[a^(1/4) - b^(1/4)*x])/a^(3/4) + (3*(b^(3/2)*c - a^(1/4)*b^(5/4)*d + Sqrt
[a]*b*e + a*Sqrt[b]*g - a^(5/4)*b^(1/4)*h + a^(3/2)*i)*Log[a^(1/4) + b^(1/4)*x])
/a^(3/4) + (3*b^(1/4)*(b*d + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/Sqrt[a] - 3*b^(3/4
)*f*Log[a - b*x^4])/(12*b^(7/4))

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Maple [B]  time = 0.009, size = 367, normalized size = 2. \[ -{\frac{i{x}^{3}}{3\,b}}-{\frac{h{x}^{2}}{2\,b}}-{\frac{gx}{b}}+{\frac{g}{2\,b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{c}{2\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{g}{4\,b}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{c}{4\,a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{ah}{4\,b}\ln \left ({1 \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{d}{4}\ln \left ({1 \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{ai}{2\,{b}^{2}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{e}{2\,b}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{ai}{4\,{b}^{2}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{e}{4\,b}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{f\ln \left ( b{x}^{4}-a \right ) }{4\,b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a),x)

[Out]

-1/3*i*x^3/b-1/2*h*x^2/b-g*x/b+1/2/b*(a/b)^(1/4)*arctan(x/(a/b)^(1/4))*g+1/2*c*(
a/b)^(1/4)/a*arctan(x/(a/b)^(1/4))+1/4/b*(a/b)^(1/4)*ln((x+(a/b)^(1/4))/(x-(a/b)
^(1/4)))*g+1/4*c*(a/b)^(1/4)/a*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))-1/4/b/(a*b)^(
1/2)*ln((-a+x^2*(a*b)^(1/2))/(-a-x^2*(a*b)^(1/2)))*a*h-1/4*d/(a*b)^(1/2)*ln((-a+
x^2*(a*b)^(1/2))/(-a-x^2*(a*b)^(1/2)))-1/2/b^2/(a/b)^(1/4)*arctan(x/(a/b)^(1/4))
*a*i-1/2*e/b/(a/b)^(1/4)*arctan(x/(a/b)^(1/4))+1/4/b^2/(a/b)^(1/4)*ln((x+(a/b)^(
1/4))/(x-(a/b)^(1/4)))*a*i+1/4*e/b/(a/b)^(1/4)*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)
))-1/4/b*f*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(-(i*x^6 + h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 - a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(i*x^6 + h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 - a),x, algorithm="fricas")

[Out]

Timed out

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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((i*x**6+h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(-b*x**4+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.229693, size = 807, normalized size = 4.29 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(i*x^6 + h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 - a),x, algorithm="giac")

[Out]

1/8*i*(2*sqrt(2)*(-a*b^3)^(3/4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/
(-a/b)^(1/4))/b^4 - sqrt(2)*(-a*b^3)^(3/4)*ln(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqr
t(-a/b))/b^4) + 1/8*i*(2*sqrt(2)*(-a*b^3)^(3/4)*arctan(1/2*sqrt(2)*(2*x - sqrt(2
)*(-a/b)^(1/4))/(-a/b)^(1/4))/b^4 + sqrt(2)*(-a*b^3)^(3/4)*ln(x^2 - sqrt(2)*x*(-
a/b)^(1/4) + sqrt(-a/b))/b^4) - 1/4*f*ln(abs(b*x^4 - a))/b + 1/4*sqrt(2)*(sqrt(2
)*sqrt(-a*b)*b^2*d - sqrt(2)*sqrt(-a*b)*a*b*h + (-a*b^3)^(1/4)*b^2*c + (-a*b^3)^
(1/4)*a*b*g + (-a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/
(-a/b)^(1/4))/(a*b^3) + 1/4*sqrt(2)*(sqrt(2)*sqrt(-a*b)*b^2*d - sqrt(2)*sqrt(-a*
b)*a*b*h + (-a*b^3)^(1/4)*b^2*c + (-a*b^3)^(1/4)*a*b*g + (-a*b^3)^(3/4)*e)*arcta
n(1/2*sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(a*b^3) + 1/8*sqrt(2)*(
(-a*b^3)^(1/4)*b^2*c + (-a*b^3)^(1/4)*a*b*g - (-a*b^3)^(3/4)*e)*ln(x^2 + sqrt(2)
*x*(-a/b)^(1/4) + sqrt(-a/b))/(a*b^3) - 1/8*sqrt(2)*((-a*b^3)^(1/4)*b^2*c + (-a*
b^3)^(1/4)*a*b*g - (-a*b^3)^(3/4)*e)*ln(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b
))/(a*b^3) - 1/6*(2*b^2*i*x^3 + 3*b^2*h*x^2 + 6*b^2*g*x)/b^3