3.110 \(\int \frac{x^3}{a+b (c+d x)^4} \, dx\)

Optimal. Leaf size=356 \[ -\frac{c \left (3 \sqrt{a}-\sqrt{b} c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4} d^4}+\frac{c \left (3 \sqrt{a}-\sqrt{b} c^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4} d^4}+\frac{c \left (3 \sqrt{a}+\sqrt{b} c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{3/4} d^4}-\frac{c \left (3 \sqrt{a}+\sqrt{b} c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} b^{3/4} d^4}+\frac{3 c^2 \tan ^{-1}\left (\frac{\sqrt{b} (c+d x)^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b} d^4}+\frac{\log \left (a+b (c+d x)^4\right )}{4 b d^4} \]

[Out]

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

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Rubi [A]  time = 0.425632, antiderivative size = 356, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.706, Rules used = {371, 1876, 1248, 635, 205, 260, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{c \left (3 \sqrt{a}-\sqrt{b} c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4} d^4}+\frac{c \left (3 \sqrt{a}-\sqrt{b} c^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4} d^4}+\frac{c \left (3 \sqrt{a}+\sqrt{b} c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{3/4} d^4}-\frac{c \left (3 \sqrt{a}+\sqrt{b} c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} b^{3/4} d^4}+\frac{3 c^2 \tan ^{-1}\left (\frac{\sqrt{b} (c+d x)^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b} d^4}+\frac{\log \left (a+b (c+d x)^4\right )}{4 b d^4} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*(c + d*x)^4),x]

[Out]

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

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 1248

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

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 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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

Mathematica [C]  time = 0.0420945, size = 106, normalized size = 0.3 \[ \frac{\text{RootSum}\left [6 \text{$\#$1}^2 b c^2 d^2+4 \text{$\#$1}^3 b c d^3+\text{$\#$1}^4 b d^4+4 \text{$\#$1} b c^3 d+a+b c^4\& ,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})}{3 \text{$\#$1}^2 c d^2+\text{$\#$1}^3 d^3+3 \text{$\#$1} c^2 d+c^3}\& \right ]}{4 b d} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(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]*#1^3)/(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 = 0.012, size = 97, normalized size = 0.3 \begin{align*}{\frac{1}{4\,bd}\sum _{{\it \_R}={\it RootOf} \left ( b{d}^{4}{{\it \_Z}}^{4}+4\,bc{d}^{3}{{\it \_Z}}^{3}+6\,b{c}^{2}{d}^{2}{{\it \_Z}}^{2}+4\,b{c}^{3}d{\it \_Z}+b{c}^{4}+a \right ) }{\frac{{{\it \_R}}^{3}\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^3/(a+b*(d*x+c)^4),x)

[Out]

1/4/b/d*sum(_R^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. \begin{align*} \int \frac{x^{3}}{{\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^3/(a+b*(d*x+c)^4),x, algorithm="maxima")

[Out]

integrate(x^3/((d*x + c)^4*b + a), x)

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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^3/(a+b*(d*x+c)^4),x, algorithm="fricas")

[Out]

Timed out

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Sympy [A]  time = 3.39874, size = 374, normalized size = 1.05 \begin{align*} \operatorname{RootSum}{\left (256 t^{4} a^{3} b^{4} d^{16} - 256 t^{3} a^{3} b^{3} d^{12} + t^{2} \left (96 a^{3} b^{2} d^{8} + 480 a^{2} b^{3} c^{4} d^{8}\right ) + t \left (- 16 a^{3} b d^{4} + 192 a^{2} b^{2} c^{4} d^{4} - 48 a b^{3} c^{8} d^{4}\right ) + a^{3} + 3 a^{2} b c^{4} + 3 a b^{2} c^{8} + b^{3} c^{12}, \left ( t \mapsto t \log{\left (x + \frac{- 1728 t^{3} a^{4} b^{3} d^{12} - 960 t^{3} a^{3} b^{4} c^{4} d^{12} + 1296 t^{2} a^{4} b^{2} d^{8} + 2016 t^{2} a^{3} b^{3} c^{4} d^{8} - 48 t^{2} a^{2} b^{4} c^{8} d^{8} - 324 t a^{4} b d^{4} - 4716 t a^{3} b^{2} c^{4} d^{4} - 1452 t a^{2} b^{3} c^{8} d^{4} - 4 t a b^{4} c^{12} d^{4} + 27 a^{4} - 390 a^{3} b c^{4} - 444 a^{2} b^{2} c^{8} - 26 a b^{3} c^{12} + b^{4} c^{16}}{729 a^{3} b c^{3} d - 1053 a^{2} b^{2} c^{7} d - 117 a b^{3} c^{11} d + b^{4} c^{15} d} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(a+b*(d*x+c)**4),x)

[Out]

RootSum(256*_t**4*a**3*b**4*d**16 - 256*_t**3*a**3*b**3*d**12 + _t**2*(96*a**3*b**2*d**8 + 480*a**2*b**3*c**4*
d**8) + _t*(-16*a**3*b*d**4 + 192*a**2*b**2*c**4*d**4 - 48*a*b**3*c**8*d**4) + a**3 + 3*a**2*b*c**4 + 3*a*b**2
*c**8 + b**3*c**12, Lambda(_t, _t*log(x + (-1728*_t**3*a**4*b**3*d**12 - 960*_t**3*a**3*b**4*c**4*d**12 + 1296
*_t**2*a**4*b**2*d**8 + 2016*_t**2*a**3*b**3*c**4*d**8 - 48*_t**2*a**2*b**4*c**8*d**8 - 324*_t*a**4*b*d**4 - 4
716*_t*a**3*b**2*c**4*d**4 - 1452*_t*a**2*b**3*c**8*d**4 - 4*_t*a*b**4*c**12*d**4 + 27*a**4 - 390*a**3*b*c**4
- 444*a**2*b**2*c**8 - 26*a*b**3*c**12 + b**4*c**16)/(729*a**3*b*c**3*d - 1053*a**2*b**2*c**7*d - 117*a*b**3*c
**11*d + b**4*c**15*d))))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\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^3/(a+b*(d*x+c)^4),x, algorithm="giac")

[Out]

integrate(x^3/((d*x + c)^4*b + a), x)