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

Optimal. Leaf size=318 \[ \frac{\left (\sqrt{a}-\sqrt{b} c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4} d^3}-\frac{\left (\sqrt{a}-\sqrt{b} c^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4} d^3}-\frac{\left (\sqrt{a}+\sqrt{b} c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{3/4} d^3}+\frac{\left (\sqrt{a}+\sqrt{b} c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} b^{3/4} d^3}-\frac{c \tan ^{-1}\left (\frac{\sqrt{b} (c+d x)^2}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} d^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.312481, antiderivative size = 318, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.588, Rules used = {371, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ \frac{\left (\sqrt{a}-\sqrt{b} c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4} d^3}-\frac{\left (\sqrt{a}-\sqrt{b} c^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4} d^3}-\frac{\left (\sqrt{a}+\sqrt{b} c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{3/4} d^3}+\frac{\left (\sqrt{a}+\sqrt{b} c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} b^{3/4} d^3}-\frac{c \tan ^{-1}\left (\frac{\sqrt{b} (c+d x)^2}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} d^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*(c + d*x)^4),x]

[Out]

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

Rule 371

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]
] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]

Rule 1876

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[(x^ii*(Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii
]*x^(n/2)))/(a + b*x^n), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ
[n/2, 0] && Expon[Pq, x] < n

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 205

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

Rule 1168

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

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])

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]

Rubi steps

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

Mathematica [C]  time = 0.0325013, size = 106, normalized size = 0.33 \[ \frac{\text{RootSum}\left [6 \text{$\#$1}^2 b c^2 d^2+4 \text{$\#$1}^3 b c d^3+\text{$\#$1}^4 b d^4+4 \text{$\#$1} b c^3 d+a+b c^4\& ,\frac{\text{$\#$1}^2 \log (x-\text{$\#$1})}{3 \text{$\#$1}^2 c d^2+\text{$\#$1}^3 d^3+3 \text{$\#$1} c^2 d+c^3}\& \right ]}{4 b d} \]

Antiderivative was successfully verified.

[In]

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

________________________________________________________________________________________

Maple [C]  time = 0.003, size = 97, normalized size = 0.3 \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{{{\it \_R}}^{2}\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(x^2/(a+b*(d*x+c)^4),x)

[Out]

1/4/b/d*sum(_R^2/(_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))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\left (d x + c\right )}^{4} b + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/((d*x + c)^4*b + a), x)

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [A]  time = 2.54595, size = 274, normalized size = 0.86 \begin{align*} \operatorname{RootSum}{\left (256 t^{4} a^{3} b^{3} d^{12} + 192 t^{2} a^{2} b^{2} c^{2} d^{6} + t \left (- 32 a^{2} b c d^{3} + 32 a b^{2} c^{5} d^{3}\right ) + a^{2} + 2 a b c^{4} + b^{2} c^{8}, \left ( t \mapsto t \log{\left (x + \frac{64 t^{3} a^{4} b^{2} d^{9} + 448 t^{3} a^{3} b^{3} c^{4} d^{9} + 160 t^{2} a^{3} b^{2} c^{3} d^{6} - 32 t^{2} a^{2} b^{3} c^{7} d^{6} + 60 t a^{3} b c^{2} d^{3} + 256 t a^{2} b^{2} c^{6} d^{3} + 4 t a b^{3} c^{10} d^{3} - 5 a^{3} c - 9 a^{2} b c^{5} - 3 a b^{2} c^{9} + b^{3} c^{13}}{a^{3} d - 33 a^{2} b c^{4} d - 33 a b^{2} c^{8} d + b^{3} c^{12} d} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(256*_t**4*a**3*b**3*d**12 + 192*_t**2*a**2*b**2*c**2*d**6 + _t*(-32*a**2*b*c*d**3 + 32*a*b**2*c**5*d**
3) + a**2 + 2*a*b*c**4 + b**2*c**8, Lambda(_t, _t*log(x + (64*_t**3*a**4*b**2*d**9 + 448*_t**3*a**3*b**3*c**4*
d**9 + 160*_t**2*a**3*b**2*c**3*d**6 - 32*_t**2*a**2*b**3*c**7*d**6 + 60*_t*a**3*b*c**2*d**3 + 256*_t*a**2*b**
2*c**6*d**3 + 4*_t*a*b**3*c**10*d**3 - 5*a**3*c - 9*a**2*b*c**5 - 3*a*b**2*c**9 + b**3*c**13)/(a**3*d - 33*a**
2*b*c**4*d - 33*a*b**2*c**8*d + b**3*c**12*d))))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\left (d x + c\right )}^{4} b + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/((d*x + c)^4*b + a), x)