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

Optimal. Leaf size=413 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{128 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{128 \sqrt{2} a^{11/4} b^{5/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{64 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{64 \sqrt{2} a^{11/4} b^{5/4}}+\frac{(a h+3 b d) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}-\frac{4 a f-x \left (2 x (a h+3 b d)+a g+7 b c+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac{x \left (x (b d-a h)-a g+b c+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2} \]

[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)) + (
(3*b*d + a*h)*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(16*a^(5/2)*b^(3/2)) - ((21*b*c + 5
*Sqrt[a]*Sqrt[b]*e + 3*a*g)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]
*a^(11/4)*b^(5/4)) + ((21*b*c + 5*Sqrt[a]*Sqrt[b]*e + 3*a*g)*ArcTan[1 + (Sqrt[2]
*b^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*b^(5/4)) - ((21*b*c - 5*Sqrt[a]*Sqrt[
b]*e + 3*a*g)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(128*Sqrt[
2]*a^(11/4)*b^(5/4)) + ((21*b*c - 5*Sqrt[a]*Sqrt[b]*e + 3*a*g)*Log[Sqrt[a] + Sqr
t[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(128*Sqrt[2]*a^(11/4)*b^(5/4))

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Rubi [A]  time = 1.12168, antiderivative size = 413, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.314 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{128 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{128 \sqrt{2} a^{11/4} b^{5/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{64 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{64 \sqrt{2} a^{11/4} b^{5/4}}+\frac{(a h+3 b d) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}-\frac{4 a f-x \left (2 x (a h+3 b d)+a g+7 b c+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac{x \left (x (b d-a h)-a g+b c+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2} \]

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)) + (
(3*b*d + a*h)*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(16*a^(5/2)*b^(3/2)) - ((21*b*c + 5
*Sqrt[a]*Sqrt[b]*e + 3*a*g)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]
*a^(11/4)*b^(5/4)) + ((21*b*c + 5*Sqrt[a]*Sqrt[b]*e + 3*a*g)*ArcTan[1 + (Sqrt[2]
*b^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*b^(5/4)) - ((21*b*c - 5*Sqrt[a]*Sqrt[
b]*e + 3*a*g)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(128*Sqrt[
2]*a^(11/4)*b^(5/4)) + ((21*b*c - 5*Sqrt[a]*Sqrt[b]*e + 3*a*g)*Log[Sqrt[a] + Sqr
t[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(128*Sqrt[2]*a^(11/4)*b^(5/4))

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Rubi in Sympy [A]  time = 170.297, size = 400, normalized size = 0.97 \[ - \frac{x \left (a g - b c - b e x^{2} - b f x^{3} + x \left (a h - b d\right )\right )}{8 a b \left (a + b x^{4}\right )^{2}} - \frac{4 a f - x \left (a g + 7 b c + 5 b e x^{2} + 2 x \left (a h + 3 b d\right )\right )}{32 a^{2} b \left (a + b x^{4}\right )} + \frac{\left (a h + 3 b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{16 a^{\frac{5}{2}} b^{\frac{3}{2}}} - \frac{\sqrt{2} \left (- 5 \sqrt{a} \sqrt{b} e + 3 a g + 21 b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{256 a^{\frac{11}{4}} b^{\frac{5}{4}}} + \frac{\sqrt{2} \left (- 5 \sqrt{a} \sqrt{b} e + 3 a g + 21 b c\right ) \log{\left (\sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{256 a^{\frac{11}{4}} b^{\frac{5}{4}}} - \frac{\sqrt{2} \left (5 \sqrt{a} \sqrt{b} e + 3 a g + 21 b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{11}{4}} b^{\frac{5}{4}}} + \frac{\sqrt{2} \left (5 \sqrt{a} \sqrt{b} e + 3 a g + 21 b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{11}{4}} b^{\frac{5}{4}}} \]

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)*atan(sqrt(b)*x**2/sqrt(a))/(16*a**(5/2)*b**(3/2)) - sqrt(2)*(
-5*sqrt(a)*sqrt(b)*e + 3*a*g + 21*b*c)*log(-sqrt(2)*a**(1/4)*b**(3/4)*x + sqrt(a
)*sqrt(b) + b*x**2)/(256*a**(11/4)*b**(5/4)) + sqrt(2)*(-5*sqrt(a)*sqrt(b)*e + 3
*a*g + 21*b*c)*log(sqrt(2)*a**(1/4)*b**(3/4)*x + sqrt(a)*sqrt(b) + b*x**2)/(256*
a**(11/4)*b**(5/4)) - sqrt(2)*(5*sqrt(a)*sqrt(b)*e + 3*a*g + 21*b*c)*atan(1 - sq
rt(2)*b**(1/4)*x/a**(1/4))/(128*a**(11/4)*b**(5/4)) + sqrt(2)*(5*sqrt(a)*sqrt(b)
*e + 3*a*g + 21*b*c)*atan(1 + sqrt(2)*b**(1/4)*x/a**(1/4))/(128*a**(11/4)*b**(5/
4))

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Mathematica [A]  time = 0.673992, size = 411, normalized size = 1. \[ \frac{-2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (8 a^{5/4} h+5 \sqrt{2} \sqrt{a} b^{3/4} e+24 \sqrt [4]{a} b d+3 \sqrt{2} a \sqrt [4]{b} g+21 \sqrt{2} b^{5/4} c\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-8 a^{5/4} h+5 \sqrt{2} \sqrt{a} b^{3/4} e-24 \sqrt [4]{a} b d+3 \sqrt{2} a \sqrt [4]{b} g+21 \sqrt{2} b^{5/4} c\right )-\frac{32 a^{7/4} \sqrt{b} (a (f+x (g+h x))-b x (c+x (d+e x)))}{\left (a+b x^4\right )^2}+\frac{8 a^{3/4} \sqrt{b} x (a (g+2 h x)+7 b c+b x (6 d+5 e x))}{a+b x^4}+\sqrt{2} \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (5 \sqrt{a} \sqrt{b} e-3 a g-21 b c\right )+\sqrt{2} \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{256 a^{11/4} b^{3/2}} \]

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]

((8*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) -
(32*a^(7/4)*Sqrt[b]*(-(b*x*(c + x*(d + e*x))) + a*(f + x*(g + h*x))))/(a + b*x^
4)^2 - 2*(21*Sqrt[2]*b^(5/4)*c + 24*a^(1/4)*b*d + 5*Sqrt[2]*Sqrt[a]*b^(3/4)*e +
3*Sqrt[2]*a*b^(1/4)*g + 8*a^(5/4)*h)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2
*(21*Sqrt[2]*b^(5/4)*c - 24*a^(1/4)*b*d + 5*Sqrt[2]*Sqrt[a]*b^(3/4)*e + 3*Sqrt[2
]*a*b^(1/4)*g - 8*a^(5/4)*h)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + Sqrt[2]*b
^(1/4)*(-21*b*c + 5*Sqrt[a]*Sqrt[b]*e - 3*a*g)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(
1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*b^(1/4)*(21*b*c - 5*Sqrt[a]*Sqrt[b]*e + 3*a*g)*L
og[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(256*a^(11/4)*b^(3/2))

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Maple [A]  time = 0.019, size = 562, normalized size = 1.4 \[ \text{result too large to display} \]

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+3/128*(
a/b)^(1/4)/a^2/b*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)*g+21/128*c/a^3*(a/b)^(1
/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+3/128*(a/b)^(1/4)/a^2/b*2^(1/2)*arct
an(2^(1/2)/(a/b)^(1/4)*x-1)*g+21/128*c/a^3*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a
/b)^(1/4)*x-1)+3/256*(a/b)^(1/4)/a^2/b*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)))*g+21/256*c/a^3*(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)))+1/16/(a^5*b^3)^(1/2)*arctan(x^2*(b/a)^(1/2))*a*h+3/16*b*d/(a^5*b^3)^(1
/2)*arctan(x^2*(b/a)^(1/2))+5/256*e/a^2/b/(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)))+5/128*e/a^2/b/
(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+5/128*e/a^2/b/(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((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 = 0.224841, size = 620, normalized size = 1.5 \[ \frac{\sqrt{2}{\left (12 \, \sqrt{2} \sqrt{a b} b^{2} d + 4 \, \sqrt{2} \sqrt{a b} a b h + 21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g + 5 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} b^{3}} + \frac{\sqrt{2}{\left (12 \, \sqrt{2} \sqrt{a b} b^{2} d + 4 \, \sqrt{2} \sqrt{a b} a b h + 21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g + 5 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} b^{3}} + \frac{\sqrt{2}{\left (21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g - 5 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{256 \, a^{3} b^{3}} - \frac{\sqrt{2}{\left (21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g - 5 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{256 \, a^{3} b^{3}} + \frac{5 \, b^{2} x^{7} e + 6 \, b^{2} d x^{6} + 2 \, a b h x^{6} + 7 \, b^{2} c x^{5} + a b g x^{5} + 9 \, a b x^{3} e + 10 \, a b d x^{2} - 2 \, a^{2} h x^{2} + 11 \, a b c x - 3 \, a^{2} g x - 4 \, a^{2} f}{32 \,{\left (b x^{4} + a\right )}^{2} a^{2} b} \]

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*sqrt(2)*sq
rt(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) + sqrt(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)