3.60 \(\int \frac{\left (a+b x^4\right )^2}{\left (c+d x^4\right )^3} \, dx\)

Optimal. Leaf size=349 \[ -\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{128 \sqrt{2} c^{11/4} d^{9/4}}+\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{128 \sqrt{2} c^{11/4} d^{9/4}}-\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt{2} c^{11/4} d^{9/4}}+\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{64 \sqrt{2} c^{11/4} d^{9/4}}-\frac{x (b c-a d) (7 a d+5 b c)}{32 c^2 d^2 \left (c+d x^4\right )}-\frac{x \left (a+b x^4\right ) (b c-a d)}{8 c d \left (c+d x^4\right )^2} \]

[Out]

-((b*c - a*d)*x*(a + b*x^4))/(8*c*d*(c + d*x^4)^2) - ((b*c - a*d)*(5*b*c + 7*a*d
)*x)/(32*c^2*d^2*(c + d*x^4)) - ((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*ArcTan[1 -
(Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(64*Sqrt[2]*c^(11/4)*d^(9/4)) + ((5*b^2*c^2 + 6*a
*b*c*d + 21*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(64*Sqrt[2]*c^(11/
4)*d^(9/4)) - ((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4
)*d^(1/4)*x + Sqrt[d]*x^2])/(128*Sqrt[2]*c^(11/4)*d^(9/4)) + ((5*b^2*c^2 + 6*a*b
*c*d + 21*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(128*
Sqrt[2]*c^(11/4)*d^(9/4))

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Rubi [A]  time = 0.549538, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{128 \sqrt{2} c^{11/4} d^{9/4}}+\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{128 \sqrt{2} c^{11/4} d^{9/4}}-\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt{2} c^{11/4} d^{9/4}}+\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{64 \sqrt{2} c^{11/4} d^{9/4}}-\frac{x (b c-a d) (7 a d+5 b c)}{32 c^2 d^2 \left (c+d x^4\right )}-\frac{x \left (a+b x^4\right ) (b c-a d)}{8 c d \left (c+d x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^2/(c + d*x^4)^3,x]

[Out]

-((b*c - a*d)*x*(a + b*x^4))/(8*c*d*(c + d*x^4)^2) - ((b*c - a*d)*(5*b*c + 7*a*d
)*x)/(32*c^2*d^2*(c + d*x^4)) - ((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*ArcTan[1 -
(Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(64*Sqrt[2]*c^(11/4)*d^(9/4)) + ((5*b^2*c^2 + 6*a
*b*c*d + 21*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(64*Sqrt[2]*c^(11/
4)*d^(9/4)) - ((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4
)*d^(1/4)*x + Sqrt[d]*x^2])/(128*Sqrt[2]*c^(11/4)*d^(9/4)) + ((5*b^2*c^2 + 6*a*b
*c*d + 21*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(128*
Sqrt[2]*c^(11/4)*d^(9/4))

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Rubi in Sympy [A]  time = 82.4633, size = 337, normalized size = 0.97 \[ \frac{x \left (a + b x^{4}\right ) \left (a d - b c\right )}{8 c d \left (c + d x^{4}\right )^{2}} + \frac{x \left (a d - b c\right ) \left (7 a d + 5 b c\right )}{32 c^{2} d^{2} \left (c + d x^{4}\right )} - \frac{\sqrt{2} \left (21 a^{2} d^{2} + 6 a b c d + 5 b^{2} c^{2}\right ) \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x + \sqrt{c} + \sqrt{d} x^{2} \right )}}{256 c^{\frac{11}{4}} d^{\frac{9}{4}}} + \frac{\sqrt{2} \left (21 a^{2} d^{2} + 6 a b c d + 5 b^{2} c^{2}\right ) \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x + \sqrt{c} + \sqrt{d} x^{2} \right )}}{256 c^{\frac{11}{4}} d^{\frac{9}{4}}} - \frac{\sqrt{2} \left (21 a^{2} d^{2} + 6 a b c d + 5 b^{2} c^{2}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}} \right )}}{128 c^{\frac{11}{4}} d^{\frac{9}{4}}} + \frac{\sqrt{2} \left (21 a^{2} d^{2} + 6 a b c d + 5 b^{2} c^{2}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}} \right )}}{128 c^{\frac{11}{4}} d^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**2/(d*x**4+c)**3,x)

[Out]

x*(a + b*x**4)*(a*d - b*c)/(8*c*d*(c + d*x**4)**2) + x*(a*d - b*c)*(7*a*d + 5*b*
c)/(32*c**2*d**2*(c + d*x**4)) - sqrt(2)*(21*a**2*d**2 + 6*a*b*c*d + 5*b**2*c**2
)*log(-sqrt(2)*c**(1/4)*d**(1/4)*x + sqrt(c) + sqrt(d)*x**2)/(256*c**(11/4)*d**(
9/4)) + sqrt(2)*(21*a**2*d**2 + 6*a*b*c*d + 5*b**2*c**2)*log(sqrt(2)*c**(1/4)*d*
*(1/4)*x + sqrt(c) + sqrt(d)*x**2)/(256*c**(11/4)*d**(9/4)) - sqrt(2)*(21*a**2*d
**2 + 6*a*b*c*d + 5*b**2*c**2)*atan(1 - sqrt(2)*d**(1/4)*x/c**(1/4))/(128*c**(11
/4)*d**(9/4)) + sqrt(2)*(21*a**2*d**2 + 6*a*b*c*d + 5*b**2*c**2)*atan(1 + sqrt(2
)*d**(1/4)*x/c**(1/4))/(128*c**(11/4)*d**(9/4))

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Mathematica [A]  time = 0.320891, size = 319, normalized size = 0.91 \[ \frac{-\sqrt{2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )+\sqrt{2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )-2 \sqrt{2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )+2 \sqrt{2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )-\frac{8 c^{3/4} \sqrt [4]{d} x \left (-7 a^2 d^2-2 a b c d+9 b^2 c^2\right )}{c+d x^4}+\frac{32 c^{7/4} \sqrt [4]{d} x (b c-a d)^2}{\left (c+d x^4\right )^2}}{256 c^{11/4} d^{9/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^2/(c + d*x^4)^3,x]

[Out]

((32*c^(7/4)*d^(1/4)*(b*c - a*d)^2*x)/(c + d*x^4)^2 - (8*c^(3/4)*d^(1/4)*(9*b^2*
c^2 - 2*a*b*c*d - 7*a^2*d^2)*x)/(c + d*x^4) - 2*Sqrt[2]*(5*b^2*c^2 + 6*a*b*c*d +
21*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)] + 2*Sqrt[2]*(5*b^2*c^2 + 6*
a*b*c*d + 21*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)] - Sqrt[2]*(5*b^2*c
^2 + 6*a*b*c*d + 21*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x
^2] + Sqrt[2]*(5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)
*d^(1/4)*x + Sqrt[d]*x^2])/(256*c^(11/4)*d^(9/4))

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Maple [A]  time = 0.002, size = 499, normalized size = 1.4 \[{\frac{1}{ \left ( d{x}^{4}+c \right ) ^{2}} \left ({\frac{ \left ( 7\,{a}^{2}{d}^{2}+2\,cabd-9\,{b}^{2}{c}^{2} \right ){x}^{5}}{32\,{c}^{2}d}}+{\frac{ \left ( 11\,{a}^{2}{d}^{2}-6\,cabd-5\,{b}^{2}{c}^{2} \right ) x}{32\,{d}^{2}c}} \right ) }+{\frac{21\,\sqrt{2}{a}^{2}}{128\,{c}^{3}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{3\,\sqrt{2}ab}{64\,{c}^{2}d}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{5\,\sqrt{2}{b}^{2}}{128\,{d}^{2}c}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{21\,\sqrt{2}{a}^{2}}{256\,{c}^{3}}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{3\,\sqrt{2}ab}{128\,{c}^{2}d}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{5\,\sqrt{2}{b}^{2}}{256\,{d}^{2}c}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{21\,\sqrt{2}{a}^{2}}{128\,{c}^{3}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}ab}{64\,{c}^{2}d}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{5\,\sqrt{2}{b}^{2}}{128\,{d}^{2}c}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^2/(d*x^4+c)^3,x)

[Out]

(1/32*(7*a^2*d^2+2*a*b*c*d-9*b^2*c^2)/c^2/d*x^5+1/32*(11*a^2*d^2-6*a*b*c*d-5*b^2
*c^2)/d^2/c*x)/(d*x^4+c)^2+21/128/c^3*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(
1/4)*x-1)*a^2+3/64/c^2/d*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x-1)*a*b
+5/128/c/d^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x-1)*b^2+21/256/c^3*
(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^2+3/128/c^2/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+5/256/c/d^2*
(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^2+21/128/c^3*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^
(1/4)*x+1)*a^2+3/64/c^2/d*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x+1)*a*
b+5/128/c/d^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x+1)*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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258428, size = 1692, normalized size = 4.85 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^2/(d*x^4 + c)^3,x, algorithm="fricas")

[Out]

-1/128*(4*(9*b^2*c^2*d - 2*a*b*c*d^2 - 7*a^2*d^3)*x^5 + 4*(c^2*d^4*x^8 + 2*c^3*d
^3*x^4 + c^4*d^2)*(-(625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42
120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 176904*a^5*b^3*c^3*d^5 + 280476*a
^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(c^11*d^9))^(1/4)*arctan(c
^3*d^2*(-(625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5
*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 176904*a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*
d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(c^11*d^9))^(1/4)/((5*b^2*c^2 + 6*a*b
*c*d + 21*a^2*d^2)*x + (5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*sqrt((c^6*d^4*sqrt(-
(625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3
+ 112806*a^4*b^4*c^4*d^4 + 176904*a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222
264*a^7*b*c*d^7 + 194481*a^8*d^8)/(c^11*d^9)) + (25*b^4*c^4 + 60*a*b^3*c^3*d + 2
46*a^2*b^2*c^2*d^2 + 252*a^3*b*c*d^3 + 441*a^4*d^4)*x^2)/(25*b^4*c^4 + 60*a*b^3*
c^3*d + 246*a^2*b^2*c^2*d^2 + 252*a^3*b*c*d^3 + 441*a^4*d^4)))) - (c^2*d^4*x^8 +
2*c^3*d^3*x^4 + c^4*d^2)*(-(625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*
d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 176904*a^5*b^3*c^3*d^5 +
280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(c^11*d^9))^(1/4)*
log(c^3*d^2*(-(625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^
3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 176904*a^5*b^3*c^3*d^5 + 280476*a^6*b^2
*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(c^11*d^9))^(1/4) + (5*b^2*c^2 +
6*a*b*c*d + 21*a^2*d^2)*x) + (c^2*d^4*x^8 + 2*c^3*d^3*x^4 + c^4*d^2)*(-(625*b^8
*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806
*a^4*b^4*c^4*d^4 + 176904*a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*
b*c*d^7 + 194481*a^8*d^8)/(c^11*d^9))^(1/4)*log(-c^3*d^2*(-(625*b^8*c^8 + 3000*a
*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*
d^4 + 176904*a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194
481*a^8*d^8)/(c^11*d^9))^(1/4) + (5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*x) + 4*(5*
b^2*c^3 + 6*a*b*c^2*d - 11*a^2*c*d^2)*x)/(c^2*d^4*x^8 + 2*c^3*d^3*x^4 + c^4*d^2)

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Sympy [A]  time = 20.5634, size = 264, normalized size = 0.76 \[ \frac{x^{5} \left (7 a^{2} d^{3} + 2 a b c d^{2} - 9 b^{2} c^{2} d\right ) + x \left (11 a^{2} c d^{2} - 6 a b c^{2} d - 5 b^{2} c^{3}\right )}{32 c^{4} d^{2} + 64 c^{3} d^{3} x^{4} + 32 c^{2} d^{4} x^{8}} + \operatorname{RootSum}{\left (268435456 t^{4} c^{11} d^{9} + 194481 a^{8} d^{8} + 222264 a^{7} b c d^{7} + 280476 a^{6} b^{2} c^{2} d^{6} + 176904 a^{5} b^{3} c^{3} d^{5} + 112806 a^{4} b^{4} c^{4} d^{4} + 42120 a^{3} b^{5} c^{5} d^{3} + 15900 a^{2} b^{6} c^{6} d^{2} + 3000 a b^{7} c^{7} d + 625 b^{8} c^{8}, \left ( t \mapsto t \log{\left (\frac{128 t c^{3} d^{2}}{21 a^{2} d^{2} + 6 a b c d + 5 b^{2} c^{2}} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**2/(d*x**4+c)**3,x)

[Out]

(x**5*(7*a**2*d**3 + 2*a*b*c*d**2 - 9*b**2*c**2*d) + x*(11*a**2*c*d**2 - 6*a*b*c
**2*d - 5*b**2*c**3))/(32*c**4*d**2 + 64*c**3*d**3*x**4 + 32*c**2*d**4*x**8) + R
ootSum(268435456*_t**4*c**11*d**9 + 194481*a**8*d**8 + 222264*a**7*b*c*d**7 + 28
0476*a**6*b**2*c**2*d**6 + 176904*a**5*b**3*c**3*d**5 + 112806*a**4*b**4*c**4*d*
*4 + 42120*a**3*b**5*c**5*d**3 + 15900*a**2*b**6*c**6*d**2 + 3000*a*b**7*c**7*d
+ 625*b**8*c**8, Lambda(_t, _t*log(128*_t*c**3*d**2/(21*a**2*d**2 + 6*a*b*c*d +
5*b**2*c**2) + x)))

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GIAC/XCAS [A]  time = 0.223269, size = 549, normalized size = 1.57 \[ \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{128 \, c^{3} d^{3}} + \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{128 \, c^{3} d^{3}} + \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{c}{d}\right )^{\frac{1}{4}} + \sqrt{\frac{c}{d}}\right )}{256 \, c^{3} d^{3}} - \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{c}{d}\right )^{\frac{1}{4}} + \sqrt{\frac{c}{d}}\right )}{256 \, c^{3} d^{3}} - \frac{9 \, b^{2} c^{2} d x^{5} - 2 \, a b c d^{2} x^{5} - 7 \, a^{2} d^{3} x^{5} + 5 \, b^{2} c^{3} x + 6 \, a b c^{2} d x - 11 \, a^{2} c d^{2} x}{32 \,{\left (d x^{4} + c\right )}^{2} c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^2/(d*x^4 + c)^3,x, algorithm="giac")

[Out]

1/128*sqrt(2)*(5*(c*d^3)^(1/4)*b^2*c^2 + 6*(c*d^3)^(1/4)*a*b*c*d + 21*(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^3*d^
3) + 1/128*sqrt(2)*(5*(c*d^3)^(1/4)*b^2*c^2 + 6*(c*d^3)^(1/4)*a*b*c*d + 21*(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
^3*d^3) + 1/256*sqrt(2)*(5*(c*d^3)^(1/4)*b^2*c^2 + 6*(c*d^3)^(1/4)*a*b*c*d + 21*
(c*d^3)^(1/4)*a^2*d^2)*ln(x^2 + sqrt(2)*x*(c/d)^(1/4) + sqrt(c/d))/(c^3*d^3) - 1
/256*sqrt(2)*(5*(c*d^3)^(1/4)*b^2*c^2 + 6*(c*d^3)^(1/4)*a*b*c*d + 21*(c*d^3)^(1/
4)*a^2*d^2)*ln(x^2 - sqrt(2)*x*(c/d)^(1/4) + sqrt(c/d))/(c^3*d^3) - 1/32*(9*b^2*
c^2*d*x^5 - 2*a*b*c*d^2*x^5 - 7*a^2*d^3*x^5 + 5*b^2*c^3*x + 6*a*b*c^2*d*x - 11*a
^2*c*d^2*x)/((d*x^4 + c)^2*c^2*d^2)