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

Optimal. Leaf size=261 \[ \frac{c \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}-\frac{c \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}+\frac{c \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}-\frac{c \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b} d^2} \]

[Out]

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

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Rubi [A]  time = 0.543396, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ \frac{c \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}-\frac{c \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}+\frac{c \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}-\frac{c \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b} d^2} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*(c + d*x)^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 67.3942, size = 250, normalized size = 0.96 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{b} \left (c + d x\right )^{2}}{\sqrt{a}} \right )}}{2 \sqrt{a} \sqrt{b} d^{2}} + \frac{\sqrt{2} c \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (- c - d x\right ) + \sqrt{a} + \sqrt{b} \left (c + d x\right )^{2} \right )}}{8 a^{\frac{3}{4}} \sqrt [4]{b} d^{2}} - \frac{\sqrt{2} c \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (c + d x\right ) + \sqrt{a} + \sqrt{b} \left (c + d x\right )^{2} \right )}}{8 a^{\frac{3}{4}} \sqrt [4]{b} d^{2}} + \frac{\sqrt{2} c \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \left (- c - d x\right )}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} \sqrt [4]{b} d^{2}} - \frac{\sqrt{2} c \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \left (c + d x\right )}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} \sqrt [4]{b} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0430668, size = 104, normalized size = 0.4 \[ \frac{\text{RootSum}\left [\text{$\#$1}^4 b d^4+4 \text{$\#$1}^3 b c d^3+6 \text{$\#$1}^2 b c^2 d^2+4 \text{$\#$1} b c^3 d+a+b c^4\&,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{\text{$\#$1}^3 d^3+3 \text{$\#$1}^2 c d^2+3 \text{$\#$1} c^2 d+c^3}\&\right ]}{4 b d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.006, size = 95, normalized size = 0.4 \[{\frac{1}{4\,bd}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}b{d}^{4}+4\,{{\it \_Z}}^{3}bc{d}^{3}+6\,{{\it \_Z}}^{2}b{c}^{2}{d}^{2}+4\,{\it \_Z}\,b{c}^{3}d+b{c}^{4}+a \right ) }{\frac{{\it \_R}\,\ln \left ( x-{\it \_R} \right ) }{{d}^{3}{{\it \_R}}^{3}+3\,c{d}^{2}{{\it \_R}}^{2}+3\,{c}^{2}d{\it \_R}+{c}^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b*(d*x+c)^4),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/((d*x + c)^4*b + a), x)

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 2.92449, size = 131, normalized size = 0.5 \[ \operatorname{RootSum}{\left (256 t^{4} a^{3} b^{2} d^{8} + 32 t^{2} a^{2} b d^{4} - 16 t a b c^{2} d^{2} + a + b c^{4}, \left ( t \mapsto t \log{\left (x + \frac{128 t^{3} a^{3} b d^{6} + 16 t^{2} a^{2} b c^{2} d^{4} + 8 t a^{2} d^{2} + 4 t a b c^{4} d^{2} - a c^{2} - b c^{6}}{4 a c d - b c^{5} d} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a+b*(d*x+c)**4),x)

[Out]

RootSum(256*_t**4*a**3*b**2*d**8 + 32*_t**2*a**2*b*d**4 - 16*_t*a*b*c**2*d**2 +
a + b*c**4, Lambda(_t, _t*log(x + (128*_t**3*a**3*b*d**6 + 16*_t**2*a**2*b*c**2*
d**4 + 8*_t*a**2*d**2 + 4*_t*a*b*c**4*d**2 - a*c**2 - b*c**6)/(4*a*c*d - b*c**5*
d))))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/((d*x + c)^4*b + a), x)