3.197 \(\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a+b x^4\right )^2} \, dx\)

Optimal. Leaf size=417 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (\sqrt{b} (a g+3 b c)-\sqrt{a} (3 a i+b e)\right )}{16 \sqrt{2} a^{7/4} b^{7/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (\sqrt{b} (a g+3 b c)-\sqrt{a} (3 a i+b e)\right )}{16 \sqrt{2} a^{7/4} b^{7/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{b} (a g+3 b c)+\sqrt{a} (3 a i+b e)\right )}{8 \sqrt{2} a^{7/4} b^{7/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt{b} (a g+3 b c)+\sqrt{a} (3 a i+b e)\right )}{8 \sqrt{2} a^{7/4} b^{7/4}}+\frac{(a h+b d) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}}+\frac{j \log \left (a+b x^4\right )}{4 b^2}+\frac{x \left (x (b d-a h)+x^2 (b e-a i)+x^3 (b f-a j)-a g+b c\right )}{4 a b \left (a+b x^4\right )} \]

[Out]

(x*(b*c - a*g + (b*d - a*h)*x + (b*e - a*i)*x^2 + (b*f - a*j)*x^3))/(4*a*b*(a +
b*x^4)) + ((b*d + a*h)*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(4*a^(3/2)*b^(3/2)) - ((Sq
rt[b]*(3*b*c + a*g) + Sqrt[a]*(b*e + 3*a*i))*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1
/4)])/(8*Sqrt[2]*a^(7/4)*b^(7/4)) + ((Sqrt[b]*(3*b*c + a*g) + Sqrt[a]*(b*e + 3*a
*i))*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(7/4)) - ((Sq
rt[b]*(3*b*c + a*g) - Sqrt[a]*(b*e + 3*a*i))*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/
4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*b^(7/4)) + ((Sqrt[b]*(3*b*c + a*g) - Sq
rt[a]*(b*e + 3*a*i))*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16
*Sqrt[2]*a^(7/4)*b^(7/4)) + (j*Log[a + b*x^4])/(4*b^2)

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Rubi [A]  time = 1.27795, antiderivative size = 417, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.244 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (\sqrt{b} (a g+3 b c)-\sqrt{a} (3 a i+b e)\right )}{16 \sqrt{2} a^{7/4} b^{7/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (\sqrt{b} (a g+3 b c)-\sqrt{a} (3 a i+b e)\right )}{16 \sqrt{2} a^{7/4} b^{7/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{b} (a g+3 b c)+\sqrt{a} (3 a i+b e)\right )}{8 \sqrt{2} a^{7/4} b^{7/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt{b} (a g+3 b c)+\sqrt{a} (3 a i+b e)\right )}{8 \sqrt{2} a^{7/4} b^{7/4}}+\frac{(a h+b d) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}}+\frac{j \log \left (a+b x^4\right )}{4 b^2}+\frac{x \left (x (b d-a h)+x^2 (b e-a i)+x^3 (b f-a j)-a g+b c\right )}{4 a b \left (a+b x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(a + b*x^4)^2,x]

[Out]

(x*(b*c - a*g + (b*d - a*h)*x + (b*e - a*i)*x^2 + (b*f - a*j)*x^3))/(4*a*b*(a +
b*x^4)) + ((b*d + a*h)*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(4*a^(3/2)*b^(3/2)) - ((Sq
rt[b]*(3*b*c + a*g) + Sqrt[a]*(b*e + 3*a*i))*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1
/4)])/(8*Sqrt[2]*a^(7/4)*b^(7/4)) + ((Sqrt[b]*(3*b*c + a*g) + Sqrt[a]*(b*e + 3*a
*i))*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(7/4)) - ((Sq
rt[b]*(3*b*c + a*g) - Sqrt[a]*(b*e + 3*a*i))*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/
4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*b^(7/4)) + ((Sqrt[b]*(3*b*c + a*g) - Sq
rt[a]*(b*e + 3*a*i))*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16
*Sqrt[2]*a^(7/4)*b^(7/4)) + (j*Log[a + b*x^4])/(4*b^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((j*x**7+i*x**6+h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**4+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.708339, size = 460, normalized size = 1.1 \[ \frac{-\frac{2 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (4 a^{5/4} \sqrt [4]{b} h+3 \sqrt{2} a^{3/2} i+4 \sqrt [4]{a} b^{5/4} d+\sqrt{2} \sqrt{a} b e+\sqrt{2} a \sqrt{b} g+3 \sqrt{2} b^{3/2} c\right )}{a^{7/4}}+\frac{2 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-4 a^{5/4} \sqrt [4]{b} h+3 \sqrt{2} a^{3/2} i-4 \sqrt [4]{a} b^{5/4} d+\sqrt{2} \sqrt{a} b e+\sqrt{2} a \sqrt{b} g+3 \sqrt{2} b^{3/2} c\right )}{a^{7/4}}+\frac{\sqrt{2} \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (3 a^{3/2} i+\sqrt{a} b e-a \sqrt{b} g-3 b^{3/2} c\right )}{a^{7/4}}+\frac{\sqrt{2} \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-3 a^{3/2} i-\sqrt{a} b e+a \sqrt{b} g+3 b^{3/2} c\right )}{a^{7/4}}+\frac{8 \left (a^2 j-a b (f+x (g+x (h+i x)))+b^2 x (c+x (d+e x))\right )}{a \left (a+b x^4\right )}+8 j \log \left (a+b x^4\right )}{32 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(a + b*x^4)^2,x]

[Out]

((8*(a^2*j + b^2*x*(c + x*(d + e*x)) - a*b*(f + x*(g + x*(h + i*x)))))/(a*(a + b
*x^4)) - (2*b^(1/4)*(3*Sqrt[2]*b^(3/2)*c + 4*a^(1/4)*b^(5/4)*d + Sqrt[2]*Sqrt[a]
*b*e + Sqrt[2]*a*Sqrt[b]*g + 4*a^(5/4)*b^(1/4)*h + 3*Sqrt[2]*a^(3/2)*i)*ArcTan[1
- (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) + (2*b^(1/4)*(3*Sqrt[2]*b^(3/2)*c - 4*a
^(1/4)*b^(5/4)*d + Sqrt[2]*Sqrt[a]*b*e + Sqrt[2]*a*Sqrt[b]*g - 4*a^(5/4)*b^(1/4)
*h + 3*Sqrt[2]*a^(3/2)*i)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) + (Sq
rt[2]*b^(1/4)*(-3*b^(3/2)*c + Sqrt[a]*b*e - a*Sqrt[b]*g + 3*a^(3/2)*i)*Log[Sqrt[
a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4) + (Sqrt[2]*b^(1/4)*(3*b^(
3/2)*c - Sqrt[a]*b*e + a*Sqrt[b]*g - 3*a^(3/2)*i)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*
b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4) + 8*j*Log[a + b*x^4])/(32*b^2)

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Maple [B]  time = 0.016, size = 682, normalized size = 1.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((j*x^7+i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^4+a)^2,x)

[Out]

(-1/4*(a*i-b*e)/a/b*x^3-1/4*(a*h-b*d)/a/b*x^2-1/4*(a*g-b*c)/a/b*x+1/4*(a*j-b*f)/
b^2)/(b*x^4+a)+1/16/b/a*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)*g+3/
16*c/a^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)+1/32/b/a*(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)))*g+3/32*c/a^2*(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/b/a*(a/b)^(1/4)*2^(1/2)*
arctan(2^(1/2)/(a/b)^(1/4)*x+1)*g+3/16*c/a^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/
(a/b)^(1/4)*x+1)+1/4/(a^3*b^3)^(1/2)*arctan(x^2*(b/a)^(1/2))*a*h+1/4*b*d/(a^3*b^
3)^(1/2)*arctan(x^2*(b/a)^(1/2))+3/32/b^2/(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)))*i+1/32*e/a/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)))+3/16/b^2/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x
-1)*i+1/16*e/a/b/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)+3/16/b^2/(a
/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)*i+1/16*e/a/b/(a/b)^(1/4)*2^(1/
2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+1/4*j/b^2*ln(a*b*(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((j*x^7 + i*x^6 + h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 + a)^2,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((j*x^7 + i*x^6 + h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 + a)^2,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((j*x**7+i*x**6+h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**4+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.228776, size = 833, normalized size = 2. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((j*x^7 + i*x^6 + h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 + a)^2,x, algorithm="giac")

[Out]

3/32*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*b^4) - sqrt(2)*(a*b^3)^(3/4)*ln(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqr
t(a/b))/(a*b^4)) + 3/32*i*(2*sqrt(2)*(a*b^3)^(3/4)*arctan(1/2*sqrt(2)*(2*x - sqr
t(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a*b^4) + sqrt(2)*(a*b^3)^(3/4)*ln(x^2 - sqrt(2)*
x*(a/b)^(1/4) + sqrt(a/b))/(a*b^4)) + 1/4*j*ln(abs(b*x^4 + a))/b^2 - 1/4*((a*i -
b*e)*x^3 - (b*d - a*h)*x^2 - (b*c - a*g)*x + (a*b*f - a^2*j)/b)/((b*x^4 + a)*a*
b) + 1/16*sqrt(2)*(2*sqrt(2)*sqrt(a*b)*b^2*d + 2*sqrt(2)*sqrt(a*b)*a*b*h + 3*(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^2*b^3) + 1/16*sqrt(2)*(2*sqrt(2)*sqrt(a
*b)*b^2*d + 2*sqrt(2)*sqrt(a*b)*a*b*h + 3*(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^2*b^3) + 1/32*sqrt(2)*(3*(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^2*b^3) - 1/32*sqrt(2)
*(3*(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^2*b^3)