3.113 \(\int \frac{1}{a+b (c+d x)^4} \, dx\)

Optimal. Leaf size=221 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} d}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} d} \]

[Out]

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

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Rubi [A]  time = 0.3728, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} d}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*(c + d*x)^4)^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 52.4997, size = 202, normalized size = 0.91 \[ - \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (- c - d x\right ) + \sqrt{a} + \sqrt{b} \left (c + d x\right )^{2} \right )}}{8 a^{\frac{3}{4}} \sqrt [4]{b} d} + \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (c + d x\right ) + \sqrt{a} + \sqrt{b} \left (c + d x\right )^{2} \right )}}{8 a^{\frac{3}{4}} \sqrt [4]{b} d} - \frac{\sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \left (- c - d x\right )}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} \sqrt [4]{b} d} + \frac{\sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \left (c + d x\right )}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} \sqrt [4]{b} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b*(d*x+c)**4),x)

[Out]

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

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Mathematica [A]  time = 0.135701, size = 161, normalized size = 0.73 \[ \frac{-\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )+\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )-2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*(c + d*x)^4)^(-1),x]

[Out]

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

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Maple [C]  time = 0.004, size = 94, normalized size = 0.4 \[{\frac{1}{4\,bd}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}b{d}^{4}+4\,{{\it \_Z}}^{3}bc{d}^{3}+6\,{{\it \_Z}}^{2}b{c}^{2}{d}^{2}+4\,{\it \_Z}\,b{c}^{3}d+b{c}^{4}+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{d}^{3}{{\it \_R}}^{3}+3\,c{d}^{2}{{\it \_R}}^{2}+3\,{c}^{2}d{\it \_R}+{c}^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*(d*x+c)^4),x)

[Out]

1/4/b/d*sum(1/(_R^3*d^3+3*_R^2*c*d^2+3*_R*c^2*d+c^3)*ln(x-_R),_R=RootOf(_Z^4*b*d
^4+4*_Z^3*b*c*d^3+6*_Z^2*b*c^2*d^2+4*_Z*b*c^3*d+b*c^4+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x + c\right )}^{4} b + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((d*x + c)^4*b + a),x, algorithm="maxima")

[Out]

integrate(1/((d*x + c)^4*b + a), x)

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Fricas [A]  time = 0.264368, size = 215, normalized size = 0.97 \[ -\left (-\frac{1}{a^{3} b d^{4}}\right )^{\frac{1}{4}} \arctan \left (\frac{a d \left (-\frac{1}{a^{3} b d^{4}}\right )^{\frac{1}{4}}}{d x + d \sqrt{\frac{a^{2} d^{2} \sqrt{-\frac{1}{a^{3} b d^{4}}} + d^{2} x^{2} + 2 \, c d x + c^{2}}{d^{2}}} + c}\right ) + \frac{1}{4} \, \left (-\frac{1}{a^{3} b d^{4}}\right )^{\frac{1}{4}} \log \left (a d \left (-\frac{1}{a^{3} b d^{4}}\right )^{\frac{1}{4}} + d x + c\right ) - \frac{1}{4} \, \left (-\frac{1}{a^{3} b d^{4}}\right )^{\frac{1}{4}} \log \left (-a d \left (-\frac{1}{a^{3} b d^{4}}\right )^{\frac{1}{4}} + d x + c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((d*x + c)^4*b + a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.838331, size = 26, normalized size = 0.12 \[ \frac{\operatorname{RootSum}{\left (256 t^{4} a^{3} b + 1, \left ( t \mapsto t \log{\left (x + \frac{4 t a + c}{d} \right )} \right )\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b*(d*x+c)**4),x)

[Out]

RootSum(256*_t**4*a**3*b + 1, Lambda(_t, _t*log(x + (4*_t*a + c)/d)))/d

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x + c\right )}^{4} b + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((d*x + c)^4*b + a),x, algorithm="giac")

[Out]

integrate(1/((d*x + c)^4*b + a), x)