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

Optimal. Leaf size=417 \[ \frac{\sqrt{a} d^3 \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{b} \left (a d^4+b c^4\right )}-\frac{c^2 d \log \left (a+b x^4\right )}{4 \left (a d^4+b c^4\right )}+\frac{c^2 d \log (c+d x)}{a d^4+b c^4}+\frac{c \left (\sqrt{a} d^2+\sqrt{b} c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )}-\frac{c \left (\sqrt{a} d^2+\sqrt{b} c^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )}-\frac{c \left (\sqrt{b} c^2-\sqrt{a} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )}+\frac{c \left (\sqrt{b} c^2-\sqrt{a} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )} \]

[Out]

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

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Rubi [A]  time = 1.17695, antiderivative size = 417, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55 \[ \frac{\sqrt{a} d^3 \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{b} \left (a d^4+b c^4\right )}-\frac{c^2 d \log \left (a+b x^4\right )}{4 \left (a d^4+b c^4\right )}+\frac{c^2 d \log (c+d x)}{a d^4+b c^4}+\frac{c \left (\sqrt{a} d^2+\sqrt{b} c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )}-\frac{c \left (\sqrt{a} d^2+\sqrt{b} c^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )}-\frac{c \left (\sqrt{b} c^2-\sqrt{a} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )}+\frac{c \left (\sqrt{b} c^2-\sqrt{a} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 148.362, size = 381, normalized size = 0.91 \[ \frac{\sqrt{a} d^{3} \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{b} \left (a d^{4} + b c^{4}\right )} - \frac{c^{2} d \log{\left (a + b x^{4} \right )}}{4 \left (a d^{4} + b c^{4}\right )} + \frac{c^{2} d \log{\left (c + d x \right )}}{a d^{4} + b c^{4}} + \frac{\sqrt{2} c \left (\sqrt{a} d^{2} - \sqrt{b} c^{2}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{4 \sqrt [4]{a} \sqrt [4]{b} \left (a d^{4} + b c^{4}\right )} - \frac{\sqrt{2} c \left (\sqrt{a} d^{2} - \sqrt{b} c^{2}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{4 \sqrt [4]{a} \sqrt [4]{b} \left (a d^{4} + b c^{4}\right )} + \frac{\sqrt{2} c \left (\sqrt{a} d^{2} + \sqrt{b} c^{2}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{8 \sqrt [4]{a} \sqrt [4]{b} \left (a d^{4} + b c^{4}\right )} - \frac{\sqrt{2} c \left (\sqrt{a} d^{2} + \sqrt{b} c^{2}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{8 \sqrt [4]{a} \sqrt [4]{b} \left (a d^{4} + b c^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a)*d**3*atan(sqrt(b)*x**2/sqrt(a))/(2*sqrt(b)*(a*d**4 + b*c**4)) - c**2*d*l
og(a + b*x**4)/(4*(a*d**4 + b*c**4)) + c**2*d*log(c + d*x)/(a*d**4 + b*c**4) + s
qrt(2)*c*(sqrt(a)*d**2 - sqrt(b)*c**2)*atan(1 - sqrt(2)*b**(1/4)*x/a**(1/4))/(4*
a**(1/4)*b**(1/4)*(a*d**4 + b*c**4)) - sqrt(2)*c*(sqrt(a)*d**2 - sqrt(b)*c**2)*a
tan(1 + sqrt(2)*b**(1/4)*x/a**(1/4))/(4*a**(1/4)*b**(1/4)*(a*d**4 + b*c**4)) + s
qrt(2)*c*(sqrt(a)*d**2 + sqrt(b)*c**2)*log(-sqrt(2)*a**(1/4)*b**(3/4)*x + sqrt(a
)*sqrt(b) + b*x**2)/(8*a**(1/4)*b**(1/4)*(a*d**4 + b*c**4)) - sqrt(2)*c*(sqrt(a)
*d**2 + sqrt(b)*c**2)*log(sqrt(2)*a**(1/4)*b**(3/4)*x + sqrt(a)*sqrt(b) + b*x**2
)/(8*a**(1/4)*b**(1/4)*(a*d**4 + b*c**4))

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Mathematica [A]  time = 0.486246, size = 370, normalized size = 0.89 \[ \frac{-2 \left (2 a^{3/4} d^3-\sqrt{2} \sqrt{a} \sqrt [4]{b} c d^2+\sqrt{2} b^{3/4} c^3\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 \left (-2 a^{3/4} d^3-\sqrt{2} \sqrt{a} \sqrt [4]{b} c d^2+\sqrt{2} b^{3/4} c^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )+\sqrt [4]{b} c \left (\sqrt{2} \left (\sqrt{a} d^2+\sqrt{b} c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )-\sqrt{2} \sqrt{b} c^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )-2 \sqrt [4]{a} \sqrt [4]{b} c d \log \left (a+b x^4\right )+8 \sqrt [4]{a} \sqrt [4]{b} c d \log (c+d x)-\sqrt{2} \sqrt{a} d^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )\right )}{8 \sqrt [4]{a} \sqrt{b} \left (a d^4+b c^4\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.019, size = 422, normalized size = 1. \[ -{\frac{c{d}^{2}\sqrt{2}}{4\,a{d}^{4}+4\,b{c}^{4}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }-{\frac{c{d}^{2}\sqrt{2}}{8\,a{d}^{4}+8\,b{c}^{4}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{c{d}^{2}\sqrt{2}}{4\,a{d}^{4}+4\,b{c}^{4}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{a{d}^{3}}{2\,a{d}^{4}+2\,b{c}^{4}}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{c}^{3}\sqrt{2}}{8\,a{d}^{4}+8\,b{c}^{4}}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{{c}^{3}\sqrt{2}}{4\,a{d}^{4}+4\,b{c}^{4}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{{c}^{3}\sqrt{2}}{4\,a{d}^{4}+4\,b{c}^{4}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{{c}^{2}d\ln \left ( b{x}^{4}+a \right ) }{4\,a{d}^{4}+4\,b{c}^{4}}}+{\frac{{c}^{2}d\ln \left ( dx+c \right ) }{a{d}^{4}+b{c}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(d*x+c)/(b*x^4+a),x)

[Out]

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

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

[Out]

Exception raised: NotImplementedError

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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(x**2/(d*x+c)/(b*x**4+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.294233, size = 575, normalized size = 1.38 \[ \frac{c^{2} d^{2}{\rm ln}\left ({\left | d x + c \right |}\right )}{b c^{4} d + a d^{5}} - \frac{c^{2} d{\rm ln}\left ({\left | b x^{4} + a \right |}\right )}{4 \,{\left (b c^{4} + a d^{4}\right )}} - \frac{{\left (\sqrt{2} a^{2} b^{3} d - \left (a b^{3}\right )^{\frac{3}{4}} a b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \,{\left (\sqrt{2} a^{2} b^{4} c^{2} + \sqrt{2} \sqrt{a b} a^{2} b^{3} d^{2} - 2 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b^{3} c d\right )}} + \frac{{\left (\sqrt{2} a^{2} b^{3} d + \left (a b^{3}\right )^{\frac{3}{4}} a b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \,{\left (\sqrt{2} a^{2} b^{4} c^{2} + \sqrt{2} \sqrt{a b} a^{2} b^{3} d^{2} + 2 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b^{3} c d\right )}} - \frac{{\left (\left (a b^{3}\right )^{\frac{1}{4}} a b c d^{2} + \left (a b^{3}\right )^{\frac{3}{4}} c^{3}\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{4 \,{\left (\sqrt{2} a b^{3} c^{4} + \sqrt{2} a^{2} b^{2} d^{4}\right )}} + \frac{{\left (\left (a b^{3}\right )^{\frac{1}{4}} a b c d^{2} + \left (a b^{3}\right )^{\frac{3}{4}} c^{3}\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{4 \,{\left (\sqrt{2} a b^{3} c^{4} + \sqrt{2} a^{2} b^{2} d^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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