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

Optimal. Leaf size=268 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac{(3 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}+\frac{x \left (2 x (3 b d-a h)+x^2 (5 b e-3 a i)-a g+7 b c\right )+4 a f}{32 a^2 b \left (a-b x^4\right )}+\frac{x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{8 a b \left (a-b x^4\right )^2} \]

[Out]

(x*(b*c + a*g + (b*d + a*h)*x + (b*e + a*i)*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 - 3*a*i)*x^2))/(32*a^2*
b*(a - b*x^4)) - ((5*b*e - (3*Sqrt[b]*(7*b*c - a*g))/Sqrt[a] - 3*a*i)*ArcTan[(b^
(1/4)*x)/a^(1/4)])/(64*a^(9/4)*b^(7/4)) + ((5*b*e + (3*Sqrt[b]*(7*b*c - a*g))/Sq
rt[a] - 3*a*i)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(64*a^(9/4)*b^(7/4)) + ((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 = 0.959422, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac{(3 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}+\frac{x \left (2 x (3 b d-a h)+x^2 (5 b e-3 a i)-a g+7 b c\right )+4 a f}{32 a^2 b \left (a-b x^4\right )}+\frac{x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{8 a b \left (a-b x^4\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(b*c + a*g + (b*d + a*h)*x + (b*e + a*i)*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 - 3*a*i)*x^2))/(32*a^2*
b*(a - b*x^4)) - ((5*b*e - (3*Sqrt[b]*(7*b*c - a*g))/Sqrt[a] - 3*a*i)*ArcTan[(b^
(1/4)*x)/a^(1/4)])/(64*a^(9/4)*b^(7/4)) + ((5*b*e + (3*Sqrt[b]*(7*b*c - a*g))/Sq
rt[a] - 3*a*i)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(64*a^(9/4)*b^(7/4)) + ((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 = 127.972, size = 252, normalized size = 0.94 \[ \frac{x \left (a g + b c + b f x^{3} + x^{2} \left (a i + b e\right ) + x \left (a h + b d\right )\right )}{8 a b \left (a - b x^{4}\right )^{2}} + \frac{4 a f - x \left (a g - 7 b c + x^{2} \left (3 a i - 5 b e\right ) + 2 x \left (a h - 3 b d\right )\right )}{32 a^{2} b \left (a - b x^{4}\right )} - \frac{\left (a h - 3 b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{16 a^{\frac{5}{2}} b^{\frac{3}{2}}} + \frac{\left (\sqrt{a} \left (6 a - 5 b e\right ) - 3 a \sqrt{b} g + 21 b^{\frac{3}{2}} c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{64 a^{\frac{11}{4}} b^{\frac{7}{4}}} - \frac{\left (\sqrt{a} \left (6 a - 5 b e\right ) + 3 a \sqrt{b} g - 21 b^{\frac{3}{2}} c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{64 a^{\frac{11}{4}} b^{\frac{7}{4}}} \]

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)**3,x)

[Out]

x*(a*g + b*c + b*f*x**3 + x**2*(a*i + b*e) + x*(a*h + b*d))/(8*a*b*(a - b*x**4)*
*2) + (4*a*f - x*(a*g - 7*b*c + x**2*(3*a*i - 5*b*e) + 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)) + (sqrt(a)*(6*a - 5*b*e) - 3*a*sqrt(b)*g + 21*b**(3/2)*c)*atan(b**(1/4)*x
/a**(1/4))/(64*a**(11/4)*b**(7/4)) - (sqrt(a)*(6*a - 5*b*e) + 3*a*sqrt(b)*g - 21
*b**(3/2)*c)*atanh(b**(1/4)*x/a**(1/4))/(64*a**(11/4)*b**(7/4))

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Mathematica [A]  time = 0.673875, size = 359, normalized size = 1.34 \[ \frac{\frac{16 a^{7/4} b^{3/4} (a (f+x (g+x (h+i x)))+b x (c+x (d+e x)))}{\left (a-b x^4\right )^2}-\frac{4 a^{3/4} b^{3/4} x (a (g+x (2 h+3 i x))-b (7 c+x (6 d+5 e x)))}{a-b x^4}+\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (4 a^{5/4} \sqrt [4]{b} h+3 a^{3/2} i-12 \sqrt [4]{a} b^{5/4} d-5 \sqrt{a} b e+3 a \sqrt{b} g-21 b^{3/2} c\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (4 a^{5/4} \sqrt [4]{b} h-3 a^{3/2} i-12 \sqrt [4]{a} b^{5/4} d+5 \sqrt{a} b e-3 a \sqrt{b} g+21 b^{3/2} c\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (3 a^{3/2} i-5 \sqrt{a} b e-3 a \sqrt{b} g+21 b^{3/2} c\right )-4 \sqrt [4]{a} \sqrt [4]{b} (a h-3 b d) \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{128 a^{11/4} b^{7/4}} \]

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)^3,x]

[Out]

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

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Maple [B]  time = 0.017, size = 501, normalized size = 1.9 \[ \text{result too large to display} \]

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)^3,x)

[Out]

-(-1/32*(3*a*i-5*b*e)/a^2*x^7-1/16*(a*h-3*b*d)/a^2*x^6-1/32*(a*g-7*b*c)/a^2*x^5-
1/32*(a*i+9*b*e)/a/b*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)))+3/64/a/b^2/(a/b)^(1/4)*arctan(x/(a/b)^(1/4))*i-5/64*e/a^2/b/(a/b)^(1/4)*a
rctan(x/(a/b)^(1/4))-3/128/a/b^2/(a/b)^(1/4)*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))
*i+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(-(i*x^6 + 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(-(i*x^6 + 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((i*x**6+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 = 0.235573, size = 942, normalized size = 3.51 \[ \text{result too large to display} \]

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)^3,x, algorithm="giac")

[Out]

-3/256*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))/(a^2*b^4) - sqrt(2)*(-a*b^3)^(3/4)*ln(x^2 + sqrt(2)*x*(-a/b)^(1
/4) + sqrt(-a/b))/(a^2*b^4)) - 3/256*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))/(a^2*b^4) + sqrt(2)*(-a*b^3)^(3/4
)*ln(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^2*b^4)) + 1/128*sqrt(2)*(12*s
qrt(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)*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) + 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*(3*a*b*i
*x^7 - 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 + a^2*i
*x^3 + 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)