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 = 0.185432, 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, Rules used = {247, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} d}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} d} \]

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)

Rule 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{a+b (c+d x)^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{a+b x^4} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,c+d x\right )}{2 \sqrt{a} d}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,c+d x\right )}{2 \sqrt{a} d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,c+d x\right )}{4 \sqrt{a} \sqrt{b} d}+\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,c+d x\right )}{4 \sqrt{a} \sqrt{b} d}-\frac{\operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,c+d x\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d}-\frac{\operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,c+d x\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d}\\ &=-\frac{\log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d}+\frac{\log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,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{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,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 (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 (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{\log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d}+\frac{\log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d}\\ \end{align*}

Mathematica [A]  time = 0.0709959, 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 = 0.003, size = 94, normalized size = 0.4 \begin{align*}{\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\,{\it \_R}\,{c}^{2}d+{c}^{3}}}} \end{align*}

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. \begin{align*} \int \frac{1}{{\left (d x + c\right )}^{4} b + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((d*x + c)^4*b + a), x)

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Fricas [A]  time = 1.55748, size = 443, normalized size = 2. \begin{align*} \left (-\frac{1}{a^{3} b d^{4}}\right )^{\frac{1}{4}} \arctan \left (a^{2} b d^{4} \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}}} \left (-\frac{1}{a^{3} b d^{4}}\right )^{\frac{3}{4}} -{\left (a^{2} b d^{4} x + a^{2} b c d^{3}\right )} \left (-\frac{1}{a^{3} b d^{4}}\right )^{\frac{3}{4}}\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1/(a^3*b*d^4))^(1/4)*arctan(a^2*b*d^4*sqrt((a^2*d^2*sqrt(-1/(a^3*b*d^4)) + d^2*x^2 + 2*c*d*x + c^2)/d^2)*(-1
/(a^3*b*d^4))^(3/4) - (a^2*b*d^4*x + a^2*b*c*d^3)*(-1/(a^3*b*d^4))^(3/4)) + 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 = 0.284529, size = 26, normalized size = 0.12 \begin{align*} \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} \end{align*}

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 [A]  time = 1.13681, size = 196, normalized size = 0.89 \begin{align*} \frac{1}{4} \, i \left (-\frac{1}{a^{3} b d^{4}}\right )^{\frac{1}{4}} \log \left (b d i x + b c i - \left (-a b^{3}\right )^{\frac{1}{4}}\right ) - \frac{1}{4} \, i \left (-\frac{1}{a^{3} b d^{4}}\right )^{\frac{1}{4}} \log \left (-b d i x - b c i - \left (-a b^{3}\right )^{\frac{1}{4}}\right ) + \frac{1}{4} \, \left (-\frac{1}{a^{3} b d^{4}}\right )^{\frac{1}{4}} \log \left ({\left | b d x + b c + \left (-a b^{3}\right )^{\frac{1}{4}} \right |}\right ) - \frac{1}{4} \, \left (-\frac{1}{a^{3} b d^{4}}\right )^{\frac{1}{4}} \log \left ({\left | -b d x - b c + \left (-a b^{3}\right )^{\frac{1}{4}} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*i*(-1/(a^3*b*d^4))^(1/4)*log(b*d*i*x + b*c*i - (-a*b^3)^(1/4)) - 1/4*i*(-1/(a^3*b*d^4))^(1/4)*log(-b*d*i*x
- b*c*i - (-a*b^3)^(1/4)) + 1/4*(-1/(a^3*b*d^4))^(1/4)*log(abs(b*d*x + b*c + (-a*b^3)^(1/4))) - 1/4*(-1/(a^3*
b*d^4))^(1/4)*log(abs(-b*d*x - b*c + (-a*b^3)^(1/4)))