3.51 \(\int \frac{a+b x^4}{c+d x^4} \, dx\)

Optimal. Leaf size=223 \[ \frac{(b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{3/4} d^{5/4}}-\frac{(b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{3/4} d^{5/4}}+\frac{(b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{3/4} d^{5/4}}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{2 \sqrt{2} c^{3/4} d^{5/4}}+\frac{b x}{d} \]

[Out]

(b*x)/d + ((b*c - a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(2*Sqrt[2]*c^(3/
4)*d^(5/4)) - ((b*c - a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(2*Sqrt[2]*c
^(3/4)*d^(5/4)) + ((b*c - a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]
*x^2])/(4*Sqrt[2]*c^(3/4)*d^(5/4)) - ((b*c - a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*
d^(1/4)*x + Sqrt[d]*x^2])/(4*Sqrt[2]*c^(3/4)*d^(5/4))

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Rubi [A]  time = 0.328354, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412 \[ \frac{(b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{3/4} d^{5/4}}-\frac{(b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{3/4} d^{5/4}}+\frac{(b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{3/4} d^{5/4}}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{2 \sqrt{2} c^{3/4} d^{5/4}}+\frac{b x}{d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)/(c + d*x^4),x]

[Out]

(b*x)/d + ((b*c - a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(2*Sqrt[2]*c^(3/
4)*d^(5/4)) - ((b*c - a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(2*Sqrt[2]*c
^(3/4)*d^(5/4)) + ((b*c - a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]
*x^2])/(4*Sqrt[2]*c^(3/4)*d^(5/4)) - ((b*c - a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*
d^(1/4)*x + Sqrt[d]*x^2])/(4*Sqrt[2]*c^(3/4)*d^(5/4))

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Rubi in Sympy [A]  time = 57.1622, size = 204, normalized size = 0.91 \[ \frac{b x}{d} - \frac{\sqrt{2} \left (a d - b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x + \sqrt{c} + \sqrt{d} x^{2} \right )}}{8 c^{\frac{3}{4}} d^{\frac{5}{4}}} + \frac{\sqrt{2} \left (a d - b c\right ) \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x + \sqrt{c} + \sqrt{d} x^{2} \right )}}{8 c^{\frac{3}{4}} d^{\frac{5}{4}}} - \frac{\sqrt{2} \left (a d - b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}} \right )}}{4 c^{\frac{3}{4}} d^{\frac{5}{4}}} + \frac{\sqrt{2} \left (a d - b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}} \right )}}{4 c^{\frac{3}{4}} d^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)/(d*x**4+c),x)

[Out]

b*x/d - sqrt(2)*(a*d - b*c)*log(-sqrt(2)*c**(1/4)*d**(1/4)*x + sqrt(c) + sqrt(d)
*x**2)/(8*c**(3/4)*d**(5/4)) + sqrt(2)*(a*d - b*c)*log(sqrt(2)*c**(1/4)*d**(1/4)
*x + sqrt(c) + sqrt(d)*x**2)/(8*c**(3/4)*d**(5/4)) - sqrt(2)*(a*d - b*c)*atan(1
- sqrt(2)*d**(1/4)*x/c**(1/4))/(4*c**(3/4)*d**(5/4)) + sqrt(2)*(a*d - b*c)*atan(
1 + sqrt(2)*d**(1/4)*x/c**(1/4))/(4*c**(3/4)*d**(5/4))

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Mathematica [A]  time = 0.227625, size = 196, normalized size = 0.88 \[ \frac{\sqrt{2} (b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )-\sqrt{2} (b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )+2 \sqrt{2} (b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )-2 \sqrt{2} (b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )+8 b c^{3/4} \sqrt [4]{d} x}{8 c^{3/4} d^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)/(c + d*x^4),x]

[Out]

(8*b*c^(3/4)*d^(1/4)*x + 2*Sqrt[2]*(b*c - a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^
(1/4)] - 2*Sqrt[2]*(b*c - a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)] + Sqrt[2]
*(b*c - a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2] - Sqrt[2]*(b
*c - a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(8*c^(3/4)*d^(
5/4))

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Maple [A]  time = 0.01, size = 266, normalized size = 1.2 \[{\frac{bx}{d}}+{\frac{\sqrt{2}a}{4\,c}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }-{\frac{\sqrt{2}b}{4\,d}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{\sqrt{2}a}{8\,c}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}b}{8\,d}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}a}{4\,c}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }-{\frac{\sqrt{2}b}{4\,d}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)/(d*x^4+c),x)

[Out]

b*x/d+1/4*(c/d)^(1/4)/c*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x-1)*a-1/4/d*(c/d)^(1
/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x-1)*b+1/8*(c/d)^(1/4)/c*2^(1/2)*ln((x^2+
(c/d)^(1/4)*x*2^(1/2)+(c/d)^(1/2))/(x^2-(c/d)^(1/4)*x*2^(1/2)+(c/d)^(1/2)))*a-1/
8/d*(c/d)^(1/4)*2^(1/2)*ln((x^2+(c/d)^(1/4)*x*2^(1/2)+(c/d)^(1/2))/(x^2-(c/d)^(1
/4)*x*2^(1/2)+(c/d)^(1/2)))*b+1/4*(c/d)^(1/4)/c*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/
4)*x+1)*a-1/4/d*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x+1)*b

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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((b*x^4 + a)/(d*x^4 + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239847, size = 743, normalized size = 3.33 \[ -\frac{4 \, d \left (-\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5}}\right )^{\frac{1}{4}} \arctan \left (-\frac{c d \left (-\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5}}\right )^{\frac{1}{4}}}{{\left (b c - a d\right )} x +{\left (b c - a d\right )} \sqrt{\frac{c^{2} d^{2} \sqrt{-\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5}}} +{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}}}\right ) - d \left (-\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5}}\right )^{\frac{1}{4}} \log \left (c d \left (-\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5}}\right )^{\frac{1}{4}} -{\left (b c - a d\right )} x\right ) + d \left (-\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5}}\right )^{\frac{1}{4}} \log \left (-c d \left (-\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5}}\right )^{\frac{1}{4}} -{\left (b c - a d\right )} x\right ) - 4 \, b x}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)/(d*x^4 + c),x, algorithm="fricas")

[Out]

-1/4*(4*d*(-(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d
^4)/(c^3*d^5))^(1/4)*arctan(-c*d*(-(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2
- 4*a^3*b*c*d^3 + a^4*d^4)/(c^3*d^5))^(1/4)/((b*c - a*d)*x + (b*c - a*d)*sqrt((c
^2*d^2*sqrt(-(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*
d^4)/(c^3*d^5)) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^2)/(b^2*c^2 - 2*a*b*c*d + a^
2*d^2)))) - d*(-(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a
^4*d^4)/(c^3*d^5))^(1/4)*log(c*d*(-(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2
- 4*a^3*b*c*d^3 + a^4*d^4)/(c^3*d^5))^(1/4) - (b*c - a*d)*x) + d*(-(b^4*c^4 - 4*
a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(c^3*d^5))^(1/4)*log(
-c*d*(-(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(
c^3*d^5))^(1/4) - (b*c - a*d)*x) - 4*b*x)/d

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Sympy [A]  time = 2.23477, size = 87, normalized size = 0.39 \[ \frac{b x}{d} + \operatorname{RootSum}{\left (256 t^{4} c^{3} d^{5} + a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}, \left ( t \mapsto t \log{\left (\frac{4 t c d}{a d - b c} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)/(d*x**4+c),x)

[Out]

b*x/d + RootSum(256*_t**4*c**3*d**5 + a**4*d**4 - 4*a**3*b*c*d**3 + 6*a**2*b**2*
c**2*d**2 - 4*a*b**3*c**3*d + b**4*c**4, Lambda(_t, _t*log(4*_t*c*d/(a*d - b*c)
+ x)))

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GIAC/XCAS [A]  time = 0.220027, size = 331, normalized size = 1.48 \[ \frac{b x}{d} - \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b c - \left (c d^{3}\right )^{\frac{1}{4}} a d\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{4 \, c d^{2}} - \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b c - \left (c d^{3}\right )^{\frac{1}{4}} a d\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{4 \, c d^{2}} - \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b c - \left (c d^{3}\right )^{\frac{1}{4}} a d\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{c}{d}\right )^{\frac{1}{4}} + \sqrt{\frac{c}{d}}\right )}{8 \, c d^{2}} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b c - \left (c d^{3}\right )^{\frac{1}{4}} a d\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{c}{d}\right )^{\frac{1}{4}} + \sqrt{\frac{c}{d}}\right )}{8 \, c d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)/(d*x^4 + c),x, algorithm="giac")

[Out]

b*x/d - 1/4*sqrt(2)*((c*d^3)^(1/4)*b*c - (c*d^3)^(1/4)*a*d)*arctan(1/2*sqrt(2)*(
2*x + sqrt(2)*(c/d)^(1/4))/(c/d)^(1/4))/(c*d^2) - 1/4*sqrt(2)*((c*d^3)^(1/4)*b*c
- (c*d^3)^(1/4)*a*d)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(c/d)^(1/4))/(c/d)^(1/4)
)/(c*d^2) - 1/8*sqrt(2)*((c*d^3)^(1/4)*b*c - (c*d^3)^(1/4)*a*d)*ln(x^2 + sqrt(2)
*x*(c/d)^(1/4) + sqrt(c/d))/(c*d^2) + 1/8*sqrt(2)*((c*d^3)^(1/4)*b*c - (c*d^3)^(
1/4)*a*d)*ln(x^2 - sqrt(2)*x*(c/d)^(1/4) + sqrt(c/d))/(c*d^2)