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

Optimal. Leaf size=393 \[ -\frac{\sqrt [4]{b} c \left (\sqrt{a}-\sqrt{b} c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \left (a+b c^4\right )}+\frac{\sqrt [4]{b} c \left (\sqrt{a}-\sqrt{b} c^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \left (a+b c^4\right )}+\frac{\sqrt [4]{b} c \left (\sqrt{a}+\sqrt{b} c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \left (a+b c^4\right )}-\frac{\sqrt [4]{b} c \left (\sqrt{a}+\sqrt{b} c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \left (a+b c^4\right )}-\frac{\log \left (a+b (c+d x)^4\right )}{4 \left (a+b c^4\right )}+\frac{\log (x)}{a+b c^4}-\frac{\sqrt{b} c^2 \tan ^{-1}\left (\frac{\sqrt{b} (c+d x)^2}{\sqrt{a}}\right )}{2 \sqrt{a} \left (a+b c^4\right )} \]

[Out]

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

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Rubi [A]  time = 0.98461, antiderivative size = 393, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 12, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.706 \[ -\frac{\sqrt [4]{b} c \left (\sqrt{a}-\sqrt{b} c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \left (a+b c^4\right )}+\frac{\sqrt [4]{b} c \left (\sqrt{a}-\sqrt{b} c^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \left (a+b c^4\right )}+\frac{\sqrt [4]{b} c \left (\sqrt{a}+\sqrt{b} c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \left (a+b c^4\right )}-\frac{\sqrt [4]{b} c \left (\sqrt{a}+\sqrt{b} c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \left (a+b c^4\right )}-\frac{\log \left (a+b (c+d x)^4\right )}{4 \left (a+b c^4\right )}+\frac{\log (x)}{a+b c^4}-\frac{\sqrt{b} c^2 \tan ^{-1}\left (\frac{\sqrt{b} (c+d x)^2}{\sqrt{a}}\right )}{2 \sqrt{a} \left (a+b c^4\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 115.583, size = 360, normalized size = 0.92 \[ \frac{\log{\left (- d x \right )}}{a + b c^{4}} - \frac{\log{\left (a + b \left (c + d x\right )^{4} \right )}}{4 \left (a + b c^{4}\right )} - \frac{\sqrt{b} c^{2} \operatorname{atan}{\left (\frac{\sqrt{b} \left (c + d x\right )^{2}}{\sqrt{a}} \right )}}{2 \sqrt{a} \left (a + b c^{4}\right )} - \frac{\sqrt{2} \sqrt [4]{b} c \left (\sqrt{a} - \sqrt{b} c^{2}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} \left (c + d x\right ) + \sqrt{a} \sqrt{b} + b \left (c + d x\right )^{2} \right )}}{8 a^{\frac{3}{4}} \left (a + b c^{4}\right )} + \frac{\sqrt{2} \sqrt [4]{b} c \left (\sqrt{a} - \sqrt{b} c^{2}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} \left (c + d x\right ) + \sqrt{a} \sqrt{b} + b \left (c + d x\right )^{2} \right )}}{8 a^{\frac{3}{4}} \left (a + b c^{4}\right )} + \frac{\sqrt{2} \sqrt [4]{b} c \left (\sqrt{a} + \sqrt{b} c^{2}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \left (c + d x\right )}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} \left (a + b c^{4}\right )} - \frac{\sqrt{2} \sqrt [4]{b} c \left (\sqrt{a} + \sqrt{b} c^{2}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \left (c + d x\right )}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} \left (a + b c^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.105736, size = 163, normalized size = 0.41 \[ -\frac{\text{RootSum}\left [\text{$\#$1}^4 b d^4+4 \text{$\#$1}^3 b c d^3+6 \text{$\#$1}^2 b c^2 d^2+4 \text{$\#$1} b c^3 d+a+b c^4\&,\frac{\text{$\#$1}^3 d^3 \log (x-\text{$\#$1})+4 \text{$\#$1}^2 c d^2 \log (x-\text{$\#$1})+4 c^3 \log (x-\text{$\#$1})+6 \text{$\#$1} c^2 d \log (x-\text{$\#$1})}{\text{$\#$1}^3 d^3+3 \text{$\#$1}^2 c d^2+3 \text{$\#$1} c^2 d+c^3}\&\right ]-4 \log (x)}{4 \left (a+b c^4\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(-4*Log[x] + RootSum[a + b*c^4 + 4*b*c^3*d*#1 + 6*b*c^2*d^2*#1^2 + 4*b*c*d^3*#1
^3 + b*d^4*#1^4 & , (4*c^3*Log[x - #1] + 6*c^2*d*Log[x - #1]*#1 + 4*c*d^2*Log[x
- #1]*#1^2 + d^3*Log[x - #1]*#1^3)/(c^3 + 3*c^2*d*#1 + 3*c*d^2*#1^2 + d^3*#1^3)
& ])/(4*(a + b*c^4))

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Maple [C]  time = 0.011, size = 139, normalized size = 0.4 \[{\frac{\ln \left ( x \right ) }{b{c}^{4}+a}}-{\frac{1}{4\,b{c}^{4}+4\,a}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}b{d}^{4}+4\,{{\it \_Z}}^{3}bc{d}^{3}+6\,{{\it \_Z}}^{2}b{c}^{2}{d}^{2}+4\,{\it \_Z}\,b{c}^{3}d+b{c}^{4}+a \right ) }{\frac{ \left ({{\it \_R}}^{3}{d}^{3}+4\,{{\it \_R}}^{2}c{d}^{2}+6\,{\it \_R}\,{c}^{2}d+4\,{c}^{3} \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{3}{d}^{3}+3\,{{\it \_R}}^{2}c{d}^{2}+3\,{\it \_R}\,{c}^{2}d+{c}^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b*(d*x+c)^4),x)

[Out]

ln(x)/(b*c^4+a)-1/4/(b*c^4+a)*sum((_R^3*d^3+4*_R^2*c*d^2+6*_R*c^2*d+4*c^3)/(_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. \[ -\frac{b d \int \frac{d^{3} x^{3} + 4 \, c d^{2} x^{2} + 6 \, c^{2} d x + 4 \, c^{3}}{b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a}\,{d x}}{b c^{4} + a} + \frac{\log \left (x\right )}{b c^{4} + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*d*integrate((d^3*x^3 + 4*c*d^2*x^2 + 6*c^2*d*x + 4*c^3)/(b*d^4*x^4 + 4*b*c*d^
3*x^3 + 6*b*c^2*d^2*x^2 + 4*b*c^3*d*x + b*c^4 + a), x)/(b*c^4 + a) + log(x)/(b*c
^4 + a)

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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(1/(((d*x + c)^4*b + a)*x),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(1/x/(a+b*(d*x+c)**4),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(((d*x + c)^4*b + a)*x), x)