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

Optimal. Leaf size=341 \[ -\frac{\left (21 \sqrt{b} c-5 \sqrt{a} e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} b^{3/4}}+\frac{\left (21 \sqrt{b} c-5 \sqrt{a} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} b^{3/4}}-\frac{\left (5 \sqrt{a} e+21 \sqrt{b} c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} b^{3/4}}+\frac{\left (5 \sqrt{a} e+21 \sqrt{b} c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{11/4} b^{3/4}}+\frac{3 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}+\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}+\frac{x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2} \]

[Out]

(x*(c + d*x + e*x^2))/(8*a*(a + b*x^4)^2) + (x*(7*c + 6*d*x + 5*e*x^2))/(32*a^2*
(a + b*x^4)) + (3*d*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(16*a^(5/2)*Sqrt[b]) - ((21*S
qrt[b]*c + 5*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(
11/4)*b^(3/4)) + ((21*Sqrt[b]*c + 5*Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^
(1/4)])/(64*Sqrt[2]*a^(11/4)*b^(3/4)) - ((21*Sqrt[b]*c - 5*Sqrt[a]*e)*Log[Sqrt[a
] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(128*Sqrt[2]*a^(11/4)*b^(3/4)) + (
(21*Sqrt[b]*c - 5*Sqrt[a]*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x
^2])/(128*Sqrt[2]*a^(11/4)*b^(3/4))

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Rubi [A]  time = 0.692528, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{\left (21 \sqrt{b} c-5 \sqrt{a} e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} b^{3/4}}+\frac{\left (21 \sqrt{b} c-5 \sqrt{a} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} b^{3/4}}-\frac{\left (5 \sqrt{a} e+21 \sqrt{b} c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} b^{3/4}}+\frac{\left (5 \sqrt{a} e+21 \sqrt{b} c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{11/4} b^{3/4}}+\frac{3 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}+\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}+\frac{x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(c + d*x + e*x^2))/(8*a*(a + b*x^4)^2) + (x*(7*c + 6*d*x + 5*e*x^2))/(32*a^2*
(a + b*x^4)) + (3*d*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(16*a^(5/2)*Sqrt[b]) - ((21*S
qrt[b]*c + 5*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(
11/4)*b^(3/4)) + ((21*Sqrt[b]*c + 5*Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^
(1/4)])/(64*Sqrt[2]*a^(11/4)*b^(3/4)) - ((21*Sqrt[b]*c - 5*Sqrt[a]*e)*Log[Sqrt[a
] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(128*Sqrt[2]*a^(11/4)*b^(3/4)) + (
(21*Sqrt[b]*c - 5*Sqrt[a]*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x
^2])/(128*Sqrt[2]*a^(11/4)*b^(3/4))

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Rubi in Sympy [A]  time = 103.555, size = 326, normalized size = 0.96 \[ \frac{x \left (c + d x + e x^{2}\right )}{8 a \left (a + b x^{4}\right )^{2}} + \frac{x \left (7 c + 6 d x + 5 e x^{2}\right )}{32 a^{2} \left (a + b x^{4}\right )} + \frac{3 d \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{16 a^{\frac{5}{2}} \sqrt{b}} + \frac{\sqrt{2} \left (5 \sqrt{a} e - 21 \sqrt{b} c\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{256 a^{\frac{11}{4}} b^{\frac{3}{4}}} - \frac{\sqrt{2} \left (5 \sqrt{a} e - 21 \sqrt{b} c\right ) \log{\left (\sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{256 a^{\frac{11}{4}} b^{\frac{3}{4}}} - \frac{\sqrt{2} \left (5 \sqrt{a} e + 21 \sqrt{b} c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{11}{4}} b^{\frac{3}{4}}} + \frac{\sqrt{2} \left (5 \sqrt{a} e + 21 \sqrt{b} c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{11}{4}} b^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(c + d*x + e*x**2)/(8*a*(a + b*x**4)**2) + x*(7*c + 6*d*x + 5*e*x**2)/(32*a**2
*(a + b*x**4)) + 3*d*atan(sqrt(b)*x**2/sqrt(a))/(16*a**(5/2)*sqrt(b)) + sqrt(2)*
(5*sqrt(a)*e - 21*sqrt(b)*c)*log(-sqrt(2)*a**(1/4)*b**(3/4)*x + sqrt(a)*sqrt(b)
+ b*x**2)/(256*a**(11/4)*b**(3/4)) - sqrt(2)*(5*sqrt(a)*e - 21*sqrt(b)*c)*log(sq
rt(2)*a**(1/4)*b**(3/4)*x + sqrt(a)*sqrt(b) + b*x**2)/(256*a**(11/4)*b**(3/4)) -
sqrt(2)*(5*sqrt(a)*e + 21*sqrt(b)*c)*atan(1 - sqrt(2)*b**(1/4)*x/a**(1/4))/(128
*a**(11/4)*b**(3/4)) + sqrt(2)*(5*sqrt(a)*e + 21*sqrt(b)*c)*atan(1 + sqrt(2)*b**
(1/4)*x/a**(1/4))/(128*a**(11/4)*b**(3/4))

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Mathematica [A]  time = 0.631277, size = 337, normalized size = 0.99 \[ \frac{\frac{\sqrt{2} \left (5 a^{3/4} e-21 \sqrt [4]{a} \sqrt{b} c\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{b^{3/4}}+\frac{\sqrt{2} \left (21 \sqrt [4]{a} \sqrt{b} c-5 a^{3/4} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{b^{3/4}}+\frac{32 a^2 x (c+x (d+e x))}{\left (a+b x^4\right )^2}-\frac{2 \sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (24 \sqrt [4]{a} \sqrt [4]{b} d+5 \sqrt{2} \sqrt{a} e+21 \sqrt{2} \sqrt{b} c\right )}{b^{3/4}}+\frac{2 \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-24 \sqrt [4]{a} \sqrt [4]{b} d+5 \sqrt{2} \sqrt{a} e+21 \sqrt{2} \sqrt{b} c\right )}{b^{3/4}}+\frac{8 a x (7 c+x (6 d+5 e x))}{a+b x^4}}{256 a^3} \]

Antiderivative was successfully verified.

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

[Out]

((32*a^2*x*(c + x*(d + e*x)))/(a + b*x^4)^2 + (8*a*x*(7*c + x*(6*d + 5*e*x)))/(a
+ b*x^4) - (2*a^(1/4)*(21*Sqrt[2]*Sqrt[b]*c + 24*a^(1/4)*b^(1/4)*d + 5*Sqrt[2]*
Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/b^(3/4) + (2*a^(1/4)*(21*Sqr
t[2]*Sqrt[b]*c - 24*a^(1/4)*b^(1/4)*d + 5*Sqrt[2]*Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]
*b^(1/4)*x)/a^(1/4)])/b^(3/4) + (Sqrt[2]*(-21*a^(1/4)*Sqrt[b]*c + 5*a^(3/4)*e)*L
og[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(3/4) + (Sqrt[2]*(21*a^
(1/4)*Sqrt[b]*c - 5*a^(3/4)*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]
*x^2])/b^(3/4))/(256*a^3)

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Maple [A]  time = 0.007, size = 396, normalized size = 1.2 \[{\frac{cx}{8\,a \left ( b{x}^{4}+a \right ) ^{2}}}+{\frac{7\,cx}{32\,{a}^{2} \left ( b{x}^{4}+a \right ) }}+{\frac{21\,c\sqrt{2}}{256\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{21\,c\sqrt{2}}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{21\,c\sqrt{2}}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{d{x}^{2}}{8\,a \left ( b{x}^{4}+a \right ) ^{2}}}+{\frac{3\,d{x}^{2}}{16\,{a}^{2} \left ( b{x}^{4}+a \right ) }}+{\frac{3\,d}{16\,{a}^{2}}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{e{x}^{3}}{8\,a \left ( b{x}^{4}+a \right ) ^{2}}}+{\frac{5\,e{x}^{3}}{32\,{a}^{2} \left ( b{x}^{4}+a \right ) }}+{\frac{5\,e\sqrt{2}}{256\,{a}^{2}b}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,e\sqrt{2}}{128\,{a}^{2}b}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,e\sqrt{2}}{128\,{a}^{2}b}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*c*x/a/(b*x^4+a)^2+7/32*c/a^2*x/(b*x^4+a)+21/256*c/a^3*(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))
)+21/128*c/a^3*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+21/128*c/a^3*
(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)+1/8*d*x^2/a/(b*x^4+a)^2+3/16
*d/a^2*x^2/(b*x^4+a)+3/16*d/a^2/(a*b)^(1/2)*arctan(x^2*(b/a)^(1/2))+1/8*e*x^3/a/
(b*x^4+a)^2+5/32*e/a^2*x^3/(b*x^4+a)+5/256*e/a^2/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)))+5/128
*e/a^2/b/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+5/128*e/a^2/b/(a/b)
^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)

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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((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((e*x^2 + d*x + c)/(b*x^4 + a)^3,x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 18.332, size = 558, normalized size = 1.64 \[ \operatorname{RootSum}{\left (268435456 t^{4} a^{11} b^{3} + t^{2} \left (6881280 a^{6} b^{2} c e + 4718592 a^{6} b^{2} d^{2}\right ) + t \left (153600 a^{4} b d e^{2} - 2709504 a^{3} b^{2} c^{2} d\right ) + 625 a^{2} e^{4} + 22050 a b c^{2} e^{2} - 60480 a b c d^{2} e + 20736 a b d^{4} + 194481 b^{2} c^{4}, \left ( t \mapsto t \log{\left (x + \frac{262144000 t^{3} a^{10} b^{2} e^{3} - 4624220160 t^{3} a^{9} b^{3} c^{2} e + 12683575296 t^{3} a^{9} b^{3} c d^{2} + 309657600 t^{2} a^{7} b^{2} c d e^{2} - 283115520 t^{2} a^{7} b^{2} d^{3} e + 1820786688 t^{2} a^{6} b^{3} c^{3} d + 5040000 t a^{5} b c e^{4} + 6912000 t a^{5} b d^{2} e^{3} - 118540800 t a^{4} b^{2} c^{3} e^{2} + 365783040 t a^{4} b^{2} c^{2} d^{2} e + 111476736 t a^{4} b^{2} c d^{4} + 522764928 t a^{3} b^{3} c^{5} + 112500 a^{3} d e^{5} + 4536000 a^{2} b c d^{3} e^{2} - 2488320 a^{2} b d^{5} e + 58344300 a b^{2} c^{4} d e - 80015040 a b^{2} c^{3} d^{3}}{15625 a^{3} e^{6} - 275625 a^{2} b c^{2} e^{4} + 3024000 a^{2} b c d^{2} e^{3} - 2073600 a^{2} b d^{4} e^{2} - 4862025 a b^{2} c^{4} e^{2} + 53343360 a b^{2} c^{3} d^{2} e - 36578304 a b^{2} c^{2} d^{4} + 85766121 b^{3} c^{6}} \right )} \right )\right )} + \frac{11 a c x + 10 a d x^{2} + 9 a e x^{3} + 7 b c x^{5} + 6 b d x^{6} + 5 b e x^{7}}{32 a^{4} + 64 a^{3} b x^{4} + 32 a^{2} b^{2} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(268435456*_t**4*a**11*b**3 + _t**2*(6881280*a**6*b**2*c*e + 4718592*a**6
*b**2*d**2) + _t*(153600*a**4*b*d*e**2 - 2709504*a**3*b**2*c**2*d) + 625*a**2*e*
*4 + 22050*a*b*c**2*e**2 - 60480*a*b*c*d**2*e + 20736*a*b*d**4 + 194481*b**2*c**
4, Lambda(_t, _t*log(x + (262144000*_t**3*a**10*b**2*e**3 - 4624220160*_t**3*a**
9*b**3*c**2*e + 12683575296*_t**3*a**9*b**3*c*d**2 + 309657600*_t**2*a**7*b**2*c
*d*e**2 - 283115520*_t**2*a**7*b**2*d**3*e + 1820786688*_t**2*a**6*b**3*c**3*d +
5040000*_t*a**5*b*c*e**4 + 6912000*_t*a**5*b*d**2*e**3 - 118540800*_t*a**4*b**2
*c**3*e**2 + 365783040*_t*a**4*b**2*c**2*d**2*e + 111476736*_t*a**4*b**2*c*d**4
+ 522764928*_t*a**3*b**3*c**5 + 112500*a**3*d*e**5 + 4536000*a**2*b*c*d**3*e**2
- 2488320*a**2*b*d**5*e + 58344300*a*b**2*c**4*d*e - 80015040*a*b**2*c**3*d**3)/
(15625*a**3*e**6 - 275625*a**2*b*c**2*e**4 + 3024000*a**2*b*c*d**2*e**3 - 207360
0*a**2*b*d**4*e**2 - 4862025*a*b**2*c**4*e**2 + 53343360*a*b**2*c**3*d**2*e - 36
578304*a*b**2*c**2*d**4 + 85766121*b**3*c**6)))) + (11*a*c*x + 10*a*d*x**2 + 9*a
*e*x**3 + 7*b*c*x**5 + 6*b*d*x**6 + 5*b*e*x**7)/(32*a**4 + 64*a**3*b*x**4 + 32*a
**2*b**2*x**8)

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GIAC/XCAS [A]  time = 0.224588, size = 454, normalized size = 1.33 \[ \frac{5 \, b x^{7} e + 6 \, b d x^{6} + 7 \, b c x^{5} + 9 \, a x^{3} e + 10 \, a d x^{2} + 11 \, a c x}{32 \,{\left (b x^{4} + a\right )}^{2} a^{2}} + \frac{\sqrt{2}{\left (12 \, \sqrt{2} \sqrt{a b} b^{2} d + 21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 5 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} b^{3}} + \frac{\sqrt{2}{\left (12 \, \sqrt{2} \sqrt{a b} b^{2} d + 21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 5 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} b^{3}} + \frac{\sqrt{2}{\left (21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c - 5 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{256 \, a^{3} b^{3}} - \frac{\sqrt{2}{\left (21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c - 5 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{256 \, a^{3} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32*(5*b*x^7*e + 6*b*d*x^6 + 7*b*c*x^5 + 9*a*x^3*e + 10*a*d*x^2 + 11*a*c*x)/((b
*x^4 + a)^2*a^2) + 1/128*sqrt(2)*(12*sqrt(2)*sqrt(a*b)*b^2*d + 21*(a*b^3)^(1/4)*
b^2*c + 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 + 21*(a*b^3)^(1/4)*
b^2*c + 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 - 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 - 5*(a*b^3)^(3/4)*e)*ln(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^
3*b^3)