3.58 \(\int \frac{\left (a+b x^4\right )^2}{c+d x^4} \, dx\)

Optimal. Leaf size=253 \[ -\frac{(b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{3/4} d^{9/4}}+\frac{(b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{3/4} d^{9/4}}-\frac{(b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{3/4} d^{9/4}}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{2 \sqrt{2} c^{3/4} d^{9/4}}-\frac{b x (b c-2 a d)}{d^2}+\frac{b^2 x^5}{5 d} \]

[Out]

-((b*(b*c - 2*a*d)*x)/d^2) + (b^2*x^5)/(5*d) - ((b*c - a*d)^2*ArcTan[1 - (Sqrt[2
]*d^(1/4)*x)/c^(1/4)])/(2*Sqrt[2]*c^(3/4)*d^(9/4)) + ((b*c - a*d)^2*ArcTan[1 + (
Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(2*Sqrt[2]*c^(3/4)*d^(9/4)) - ((b*c - a*d)^2*Log[Sq
rt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(4*Sqrt[2]*c^(3/4)*d^(9/4)) +
((b*c - a*d)^2*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(4*Sqrt[2
]*c^(3/4)*d^(9/4))

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Rubi [A]  time = 0.414989, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ -\frac{(b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{3/4} d^{9/4}}+\frac{(b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{3/4} d^{9/4}}-\frac{(b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{3/4} d^{9/4}}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{2 \sqrt{2} c^{3/4} d^{9/4}}-\frac{b x (b c-2 a d)}{d^2}+\frac{b^2 x^5}{5 d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^2/(c + d*x^4),x]

[Out]

-((b*(b*c - 2*a*d)*x)/d^2) + (b^2*x^5)/(5*d) - ((b*c - a*d)^2*ArcTan[1 - (Sqrt[2
]*d^(1/4)*x)/c^(1/4)])/(2*Sqrt[2]*c^(3/4)*d^(9/4)) + ((b*c - a*d)^2*ArcTan[1 + (
Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(2*Sqrt[2]*c^(3/4)*d^(9/4)) - ((b*c - a*d)^2*Log[Sq
rt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(4*Sqrt[2]*c^(3/4)*d^(9/4)) +
((b*c - a*d)^2*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(4*Sqrt[2
]*c^(3/4)*d^(9/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b^{2} x^{5}}{5 d} + \frac{\left (2 a d - b c\right ) \int b\, dx}{d^{2}} - \frac{\sqrt{2} \left (a d - b c\right )^{2} \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x + \sqrt{c} + \sqrt{d} x^{2} \right )}}{8 c^{\frac{3}{4}} d^{\frac{9}{4}}} + \frac{\sqrt{2} \left (a d - b c\right )^{2} \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x + \sqrt{c} + \sqrt{d} x^{2} \right )}}{8 c^{\frac{3}{4}} d^{\frac{9}{4}}} - \frac{\sqrt{2} \left (a d - b c\right )^{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}} \right )}}{4 c^{\frac{3}{4}} d^{\frac{9}{4}}} + \frac{\sqrt{2} \left (a d - b c\right )^{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}} \right )}}{4 c^{\frac{3}{4}} d^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**2/(d*x**4+c),x)

[Out]

b**2*x**5/(5*d) + (2*a*d - b*c)*Integral(b, x)/d**2 - sqrt(2)*(a*d - b*c)**2*log
(-sqrt(2)*c**(1/4)*d**(1/4)*x + sqrt(c) + sqrt(d)*x**2)/(8*c**(3/4)*d**(9/4)) +
sqrt(2)*(a*d - b*c)**2*log(sqrt(2)*c**(1/4)*d**(1/4)*x + sqrt(c) + sqrt(d)*x**2)
/(8*c**(3/4)*d**(9/4)) - sqrt(2)*(a*d - b*c)**2*atan(1 - sqrt(2)*d**(1/4)*x/c**(
1/4))/(4*c**(3/4)*d**(9/4)) + sqrt(2)*(a*d - b*c)**2*atan(1 + sqrt(2)*d**(1/4)*x
/c**(1/4))/(4*c**(3/4)*d**(9/4))

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Mathematica [A]  time = 0.192653, size = 231, normalized size = 0.91 \[ \frac{-40 b c^{3/4} \sqrt [4]{d} x (b c-2 a d)-5 \sqrt{2} (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )+5 \sqrt{2} (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )-10 \sqrt{2} (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )+10 \sqrt{2} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )+8 b^2 c^{3/4} d^{5/4} x^5}{40 c^{3/4} d^{9/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^2/(c + d*x^4),x]

[Out]

(-40*b*c^(3/4)*d^(1/4)*(b*c - 2*a*d)*x + 8*b^2*c^(3/4)*d^(5/4)*x^5 - 10*Sqrt[2]*
(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)] + 10*Sqrt[2]*(b*c - a*d)^2
*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)] - 5*Sqrt[2]*(b*c - a*d)^2*Log[Sqrt[c] -
Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2] + 5*Sqrt[2]*(b*c - a*d)^2*Log[Sqrt[c]
+ Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(40*c^(3/4)*d^(9/4))

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Maple [B]  time = 0.002, size = 436, normalized size = 1.7 \[{\frac{{b}^{2}{x}^{5}}{5\,d}}+2\,{\frac{abx}{d}}-{\frac{{b}^{2}xc}{{d}^{2}}}+{\frac{\sqrt{2}{a}^{2}}{4\,c}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }-{\frac{\sqrt{2}ab}{2\,d}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{c\sqrt{2}{b}^{2}}{4\,{d}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{\sqrt{2}{a}^{2}}{4\,c}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }-{\frac{\sqrt{2}ab}{2\,d}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{c\sqrt{2}{b}^{2}}{4\,{d}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{\sqrt{2}{a}^{2}}{8\,c}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}ab}{4\,d}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{c\sqrt{2}{b}^{2}}{8\,{d}^{2}}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^2/(d*x^4+c),x)

[Out]

1/5*b^2*x^5/d+2*b/d*a*x-b^2/d^2*x*c+1/4*(c/d)^(1/4)/c*2^(1/2)*arctan(2^(1/2)/(c/
d)^(1/4)*x+1)*a^2-1/2/d*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x+1)*a*b+
1/4/d^2*(c/d)^(1/4)*c*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x+1)*b^2+1/4*(c/d)^(1/4
)/c*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x-1)*a^2-1/2/d*(c/d)^(1/4)*2^(1/2)*arctan
(2^(1/2)/(c/d)^(1/4)*x-1)*a*b+1/4/d^2*(c/d)^(1/4)*c*2^(1/2)*arctan(2^(1/2)/(c/d)
^(1/4)*x-1)*b^2+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^2-1/4/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)))*
a*b+1/8/d^2*(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)))*b^2

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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)^2/(d*x^4 + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240962, size = 1457, normalized size = 5.76 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^2/(d*x^4 + c),x, algorithm="fricas")

[Out]

1/20*(4*b^2*d*x^5 - 20*d^2*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*
a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 -
8*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^9))^(1/4)*arctan(c*d^2*(-(b^8*c^8 - 8*a*b^7*c^7
*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c
^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^9))^(1/4)/((b^2*c^
2 - 2*a*b*c*d + a^2*d^2)*x + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt((c^2*d^4*sqrt(
-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4
*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(c
^3*d^9)) + (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)*x^2)/(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))
)) + 5*d^2*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3
+ 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 +
a^8*d^8)/(c^3*d^9))^(1/4)*log(c*d^2*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6
*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2
*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^9))^(1/4) + (b^2*c^2 - 2*a*b*c*d + a^
2*d^2)*x) - 5*d^2*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c
^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*
c*d^7 + a^8*d^8)/(c^3*d^9))^(1/4)*log(-c*d^2*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2
*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28
*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^9))^(1/4) + (b^2*c^2 - 2*a*b*
c*d + a^2*d^2)*x) - 20*(b^2*c - 2*a*b*d)*x)/d^2

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Sympy [A]  time = 3.58949, size = 187, normalized size = 0.74 \[ \frac{b^{2} x^{5}}{5 d} + \operatorname{RootSum}{\left (256 t^{4} c^{3} d^{9} + a^{8} d^{8} - 8 a^{7} b c d^{7} + 28 a^{6} b^{2} c^{2} d^{6} - 56 a^{5} b^{3} c^{3} d^{5} + 70 a^{4} b^{4} c^{4} d^{4} - 56 a^{3} b^{5} c^{5} d^{3} + 28 a^{2} b^{6} c^{6} d^{2} - 8 a b^{7} c^{7} d + b^{8} c^{8}, \left ( t \mapsto t \log{\left (\frac{4 t c d^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )} \right )\right )} + \frac{x \left (2 a b d - b^{2} c\right )}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**2/(d*x**4+c),x)

[Out]

b**2*x**5/(5*d) + RootSum(256*_t**4*c**3*d**9 + a**8*d**8 - 8*a**7*b*c*d**7 + 28
*a**6*b**2*c**2*d**6 - 56*a**5*b**3*c**3*d**5 + 70*a**4*b**4*c**4*d**4 - 56*a**3
*b**5*c**5*d**3 + 28*a**2*b**6*c**6*d**2 - 8*a*b**7*c**7*d + b**8*c**8, Lambda(_
t, _t*log(4*_t*c*d**2/(a**2*d**2 - 2*a*b*c*d + b**2*c**2) + x))) + x*(2*a*b*d -
b**2*c)/d**2

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GIAC/XCAS [A]  time = 0.220611, size = 477, normalized size = 1.89 \[ \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{4 \, c d^{3}} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{4 \, c d^{3}} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{c}{d}\right )^{\frac{1}{4}} + \sqrt{\frac{c}{d}}\right )}{8 \, c d^{3}} - \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{c}{d}\right )^{\frac{1}{4}} + \sqrt{\frac{c}{d}}\right )}{8 \, c d^{3}} + \frac{b^{2} d^{4} x^{5} - 5 \, b^{2} c d^{3} x + 10 \, a b d^{4} x}{5 \, d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^2/(d*x^4 + c),x, algorithm="giac")

[Out]

1/4*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d + (c*d^3)^(1/4)*a^2
*d^2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(c/d)^(1/4))/(c/d)^(1/4))/(c*d^3) + 1/4*
sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d + (c*d^3)^(1/4)*a^2*d^2
)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(c/d)^(1/4))/(c/d)^(1/4))/(c*d^3) + 1/8*sqrt
(2)*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d + (c*d^3)^(1/4)*a^2*d^2)*ln
(x^2 + sqrt(2)*x*(c/d)^(1/4) + sqrt(c/d))/(c*d^3) - 1/8*sqrt(2)*((c*d^3)^(1/4)*b
^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d + (c*d^3)^(1/4)*a^2*d^2)*ln(x^2 - sqrt(2)*x*(c/
d)^(1/4) + sqrt(c/d))/(c*d^3) + 1/5*(b^2*d^4*x^5 - 5*b^2*c*d^3*x + 10*a*b*d^4*x)
/d^5