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

Optimal. Leaf size=241 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right )}{64 a^{11/4} b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right )}{64 a^{11/4} b^{5/4}}+\frac{(3 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}+\frac{x \left (2 x (3 b d-a h)-a g+7 b c+5 b e x^2\right )+4 a f}{32 a^2 b \left (a-b x^4\right )}+\frac{x \left (x (a h+b d)+a g+b c+b e x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2} \]

[Out]

(x*(b*c + a*g + (b*d + a*h)*x + b*e*x^2 + b*f*x^3))/(8*a*b*(a - b*x^4)^2) + (4*a
*f + x*(7*b*c - a*g + 2*(3*b*d - a*h)*x + 5*b*e*x^2))/(32*a^2*b*(a - b*x^4)) + (
(21*b*c - 5*Sqrt[a]*Sqrt[b]*e - 3*a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(64*a^(11/4)
*b^(5/4)) + ((21*b*c + 5*Sqrt[a]*Sqrt[b]*e - 3*a*g)*ArcTanh[(b^(1/4)*x)/a^(1/4)]
)/(64*a^(11/4)*b^(5/4)) + ((3*b*d - a*h)*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(16*a^(
5/2)*b^(3/2))

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Rubi [A]  time = 0.710137, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right )}{64 a^{11/4} b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right )}{64 a^{11/4} b^{5/4}}+\frac{(3 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}+\frac{x \left (2 x (3 b d-a h)-a g+7 b c+5 b e x^2\right )+4 a f}{32 a^2 b \left (a-b x^4\right )}+\frac{x \left (x (a h+b d)+a g+b c+b e x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(b*c + a*g + (b*d + a*h)*x + b*e*x^2 + b*f*x^3))/(8*a*b*(a - b*x^4)^2) + (4*a
*f + x*(7*b*c - a*g + 2*(3*b*d - a*h)*x + 5*b*e*x^2))/(32*a^2*b*(a - b*x^4)) + (
(21*b*c - 5*Sqrt[a]*Sqrt[b]*e - 3*a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(64*a^(11/4)
*b^(5/4)) + ((21*b*c + 5*Sqrt[a]*Sqrt[b]*e - 3*a*g)*ArcTanh[(b^(1/4)*x)/a^(1/4)]
)/(64*a^(11/4)*b^(5/4)) + ((3*b*d - a*h)*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(16*a^(
5/2)*b^(3/2))

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Rubi in Sympy [A]  time = 117.367, size = 226, normalized size = 0.94 \[ \frac{x \left (a g + b c + b e x^{2} + b f x^{3} + x \left (a h + b d\right )\right )}{8 a b \left (a - b x^{4}\right )^{2}} + \frac{4 a f - x \left (a g - 7 b c - 5 b e x^{2} + 2 x \left (a h - 3 b d\right )\right )}{32 a^{2} b \left (a - b x^{4}\right )} - \frac{\left (a h - 3 b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{16 a^{\frac{5}{2}} b^{\frac{3}{2}}} - \frac{\left (- 5 \sqrt{a} \sqrt{b} e + 3 a g - 21 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{64 a^{\frac{11}{4}} b^{\frac{5}{4}}} - \frac{\left (5 \sqrt{a} \sqrt{b} e + 3 a g - 21 b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{64 a^{\frac{11}{4}} b^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a*g + b*c + b*e*x**2 + b*f*x**3 + x*(a*h + b*d))/(8*a*b*(a - b*x**4)**2) + (4
*a*f - x*(a*g - 7*b*c - 5*b*e*x**2 + 2*x*(a*h - 3*b*d)))/(32*a**2*b*(a - b*x**4)
) - (a*h - 3*b*d)*atanh(sqrt(b)*x**2/sqrt(a))/(16*a**(5/2)*b**(3/2)) - (-5*sqrt(
a)*sqrt(b)*e + 3*a*g - 21*b*c)*atanh(b**(1/4)*x/a**(1/4))/(64*a**(11/4)*b**(5/4)
) - (5*sqrt(a)*sqrt(b)*e + 3*a*g - 21*b*c)*atan(b**(1/4)*x/a**(1/4))/(64*a**(11/
4)*b**(5/4))

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Mathematica [A]  time = 0.554865, size = 309, normalized size = 1.28 \[ \frac{\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (4 a^{5/4} h-5 \sqrt{a} b^{3/4} e-12 \sqrt [4]{a} b d+3 a \sqrt [4]{b} g-21 b^{5/4} c\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (4 a^{5/4} h+5 \sqrt{a} b^{3/4} e-12 \sqrt [4]{a} b d-3 a \sqrt [4]{b} g+21 b^{5/4} c\right )+\frac{16 a^{7/4} \sqrt{b} (a (f+x (g+h x))+b x (c+x (d+e x)))}{\left (a-b x^4\right )^2}+\frac{4 a^{3/4} \sqrt{b} x (-a (g+2 h x)+7 b c+b x (6 d+5 e x))}{a-b x^4}+2 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right )-4 \sqrt [4]{a} (a h-3 b d) \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{128 a^{11/4} b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((4*a^(3/4)*Sqrt[b]*x*(7*b*c + b*x*(6*d + 5*e*x) - a*(g + 2*h*x)))/(a - b*x^4) +
(16*a^(7/4)*Sqrt[b]*(b*x*(c + x*(d + e*x)) + a*(f + x*(g + h*x))))/(a - b*x^4)^
2 + 2*b^(1/4)*(21*b*c - 5*Sqrt[a]*Sqrt[b]*e - 3*a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)]
+ (-21*b^(5/4)*c - 12*a^(1/4)*b*d - 5*Sqrt[a]*b^(3/4)*e + 3*a*b^(1/4)*g + 4*a^(
5/4)*h)*Log[a^(1/4) - b^(1/4)*x] + (21*b^(5/4)*c - 12*a^(1/4)*b*d + 5*Sqrt[a]*b^
(3/4)*e - 3*a*b^(1/4)*g + 4*a^(5/4)*h)*Log[a^(1/4) + b^(1/4)*x] - 4*a^(1/4)*(-3*
b*d + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/(128*a^(11/4)*b^(3/2))

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Maple [B]  time = 0.017, size = 418, normalized size = 1.7 \[ -{\frac{1}{ \left ( b{x}^{4}-a \right ) ^{2}} \left ({\frac{5\,be{x}^{7}}{32\,{a}^{2}}}-{\frac{ \left ( ah-3\,bd \right ){x}^{6}}{16\,{a}^{2}}}-{\frac{ \left ( ag-7\,bc \right ){x}^{5}}{32\,{a}^{2}}}-{\frac{9\,e{x}^{3}}{32\,a}}-{\frac{ \left ( ah+5\,bd \right ){x}^{2}}{16\,ab}}-{\frac{ \left ( 3\,ag+11\,bc \right ) x}{32\,ab}}-{\frac{f}{8\,b}} \right ) }-{\frac{3\,g}{64\,{a}^{2}b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{21\,c}{64\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{3\,g}{128\,{a}^{2}b}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{21\,c}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{ah}{32}\ln \left ({1 \left ( -{a}^{3}b+{x}^{2}\sqrt{{a}^{5}{b}^{3}} \right ) \left ( -{a}^{3}b-{x}^{2}\sqrt{{a}^{5}{b}^{3}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{5}{b}^{3}}}}}-{\frac{3\,bd}{32}\ln \left ({1 \left ( -{a}^{3}b+{x}^{2}\sqrt{{a}^{5}{b}^{3}} \right ) \left ( -{a}^{3}b-{x}^{2}\sqrt{{a}^{5}{b}^{3}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{5}{b}^{3}}}}}-{\frac{5\,e}{64\,{a}^{2}b}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,e}{128\,{a}^{2}b}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(5/32/a^2*b*e*x^7-1/16*(a*h-3*b*d)/a^2*x^6-1/32*(a*g-7*b*c)/a^2*x^5-9/32/a*e*x^
3-1/16*(a*h+5*b*d)/a/b*x^2-1/32*(3*a*g+11*b*c)/a/b*x-1/8*f/b)/(b*x^4-a)^2-3/64*(
a/b)^(1/4)/a^2/b*arctan(x/(a/b)^(1/4))*g+21/64*c/a^3*(a/b)^(1/4)*arctan(x/(a/b)^
(1/4))-3/128*(a/b)^(1/4)/a^2/b*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))*g+21/128*c/a^
3*(a/b)^(1/4)*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))+1/32/(a^5*b^3)^(1/2)*ln((-a^3*
b+x^2*(a^5*b^3)^(1/2))/(-a^3*b-x^2*(a^5*b^3)^(1/2)))*a*h-3/32*b*d/(a^5*b^3)^(1/2
)*ln((-a^3*b+x^2*(a^5*b^3)^(1/2))/(-a^3*b-x^2*(a^5*b^3)^(1/2)))-5/64*e/a^2/b/(a/
b)^(1/4)*arctan(x/(a/b)^(1/4))+5/128*e/a^2/b/(a/b)^(1/4)*ln((x+(a/b)^(1/4))/(x-(
a/b)^(1/4)))

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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(-(h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/(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(-(h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/(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((h*x**5+g*x**4+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 = 0.22894, size = 656, normalized size = 2.72 \[ \frac{\sqrt{2}{\left (12 \, \sqrt{2} \sqrt{-a b} b^{2} d - 4 \, \sqrt{2} \sqrt{-a b} a b h + 21 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 3 \, \left (-a b^{3}\right )^{\frac{1}{4}} a b g + 5 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} b^{3}} + \frac{\sqrt{2}{\left (12 \, \sqrt{2} \sqrt{-a b} b^{2} d - 4 \, \sqrt{2} \sqrt{-a b} a b h + 21 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 3 \, \left (-a b^{3}\right )^{\frac{1}{4}} a b g + 5 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} b^{3}} + \frac{\sqrt{2}{\left (21 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 3 \, \left (-a b^{3}\right )^{\frac{1}{4}} a b g - 5 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{256 \, a^{3} b^{3}} - \frac{\sqrt{2}{\left (21 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 3 \, \left (-a b^{3}\right )^{\frac{1}{4}} a b g - 5 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{256 \, a^{3} b^{3}} - \frac{5 \, b^{2} x^{7} e + 6 \, b^{2} d x^{6} - 2 \, a b h x^{6} + 7 \, b^{2} c x^{5} - a b g x^{5} - 9 \, a b x^{3} e - 10 \, a b d x^{2} - 2 \, a^{2} h x^{2} - 11 \, a b c x - 3 \, a^{2} g x - 4 \, a^{2} f}{32 \,{\left (b x^{4} - a\right )}^{2} a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/128*sqrt(2)*(12*sqrt(2)*sqrt(-a*b)*b^2*d - 4*sqrt(2)*sqrt(-a*b)*a*b*h + 21*(-a
*b^3)^(1/4)*b^2*c - 3*(-a*b^3)^(1/4)*a*b*g + 5*(-a*b^3)^(3/4)*e)*arctan(1/2*sqrt
(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(a^3*b^3) + 1/128*sqrt(2)*(12*sqr
t(2)*sqrt(-a*b)*b^2*d - 4*sqrt(2)*sqrt(-a*b)*a*b*h + 21*(-a*b^3)^(1/4)*b^2*c - 3
*(-a*b^3)^(1/4)*a*b*g + 5*(-a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(-
a/b)^(1/4))/(-a/b)^(1/4))/(a^3*b^3) + 1/256*sqrt(2)*(21*(-a*b^3)^(1/4)*b^2*c - 3
*(-a*b^3)^(1/4)*a*b*g - 5*(-a*b^3)^(3/4)*e)*ln(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sq
rt(-a/b))/(a^3*b^3) - 1/256*sqrt(2)*(21*(-a*b^3)^(1/4)*b^2*c - 3*(-a*b^3)^(1/4)*
a*b*g - 5*(-a*b^3)^(3/4)*e)*ln(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^3*b
^3) - 1/32*(5*b^2*x^7*e + 6*b^2*d*x^6 - 2*a*b*h*x^6 + 7*b^2*c*x^5 - a*b*g*x^5 -
9*a*b*x^3*e - 10*a*b*d*x^2 - 2*a^2*h*x^2 - 11*a*b*c*x - 3*a^2*g*x - 4*a^2*f)/((b
*x^4 - a)^2*a^2*b)