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

Optimal. Leaf size=234 \[ -\frac{\left (3 \sqrt [3]{a} b^{2/3} c^2+a+b c^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{4/3} d^4}+\frac{\left (3 \sqrt [3]{a} b^{2/3} c^2+a+b c^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} b^{4/3} d^4}+\frac{\left (-3 \sqrt [3]{a} b^{2/3} c^2+a+b c^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{4/3} d^4}-\frac{c \log \left (a+b (c+d x)^3\right )}{b d^4}+\frac{x}{b d^3} \]

[Out]

x/(b*d^3) + ((a - 3*a^(1/3)*b^(2/3)*c^2 + b*c^3)*ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))])/(S
qrt[3]*a^(2/3)*b^(4/3)*d^4) - ((a + 3*a^(1/3)*b^(2/3)*c^2 + b*c^3)*Log[a^(1/3) + b^(1/3)*(c + d*x)])/(3*a^(2/3
)*b^(4/3)*d^4) + ((a + 3*a^(1/3)*b^(2/3)*c^2 + b*c^3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d
*x)^2])/(6*a^(2/3)*b^(4/3)*d^4) - (c*Log[a + b*(c + d*x)^3])/(b*d^4)

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Rubi [A]  time = 0.371634, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.588, Rules used = {371, 1887, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac{\left (3 \sqrt [3]{a} b^{2/3} c^2+a+b c^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{4/3} d^4}+\frac{\left (3 \sqrt [3]{a} b^{2/3} c^2+a+b c^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} b^{4/3} d^4}+\frac{\left (-3 \sqrt [3]{a} b^{2/3} c^2+a+b c^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{4/3} d^4}-\frac{c \log \left (a+b (c+d x)^3\right )}{b d^4}+\frac{x}{b d^3} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*(c + d*x)^3),x]

[Out]

x/(b*d^3) + ((a - 3*a^(1/3)*b^(2/3)*c^2 + b*c^3)*ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))])/(S
qrt[3]*a^(2/3)*b^(4/3)*d^4) - ((a + 3*a^(1/3)*b^(2/3)*c^2 + b*c^3)*Log[a^(1/3) + b^(1/3)*(c + d*x)])/(3*a^(2/3
)*b^(4/3)*d^4) + ((a + 3*a^(1/3)*b^(2/3)*c^2 + b*c^3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d
*x)^2])/(6*a^(2/3)*b^(4/3)*d^4) - (c*Log[a + b*(c + d*x)^3])/(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 1887

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x
] && PolyQ[Pq, x] && IntegerQ[n]

Rule 1871

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 1860

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, -Dist[(r*(B*r - A*s))/(3*a*s), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) + s
*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[a/
b]

Rule 31

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

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

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 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 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]

Rubi steps

\begin{align*} \int \frac{x^3}{a+b (c+d x)^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(-c+x)^3}{a+b x^3} \, dx,x,c+d x\right )}{d^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{b}-\frac{a+b c^3-3 b c^2 x+3 b c x^2}{b \left (a+b x^3\right )}\right ) \, dx,x,c+d x\right )}{d^4}\\ &=\frac{x}{b d^3}-\frac{\operatorname{Subst}\left (\int \frac{a+b c^3-3 b c^2 x+3 b c x^2}{a+b x^3} \, dx,x,c+d x\right )}{b d^4}\\ &=\frac{x}{b d^3}-\frac{\operatorname{Subst}\left (\int \frac{a+b c^3-3 b c^2 x}{a+b x^3} \, dx,x,c+d x\right )}{b d^4}-\frac{(3 c) \operatorname{Subst}\left (\int \frac{x^2}{a+b x^3} \, dx,x,c+d x\right )}{d^4}\\ &=\frac{x}{b d^3}-\frac{c \log \left (a+b (c+d x)^3\right )}{b d^4}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt [3]{a} \left (-3 \sqrt [3]{a} b c^2+2 \sqrt [3]{b} \left (a+b c^3\right )\right )+\sqrt [3]{b} \left (-3 \sqrt [3]{a} b c^2-\sqrt [3]{b} \left (a+b c^3\right )\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{3 a^{2/3} b^{4/3} d^4}-\frac{\left (a+3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{3 a^{2/3} b d^4}\\ &=\frac{x}{b d^3}-\frac{\left (a+3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{4/3} d^4}-\frac{c \log \left (a+b (c+d x)^3\right )}{b d^4}-\frac{\left (a-3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{2 \sqrt [3]{a} b d^4}+\frac{\left (a+3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \operatorname{Subst}\left (\int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{6 a^{2/3} b^{4/3} d^4}\\ &=\frac{x}{b d^3}-\frac{\left (a+3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{4/3} d^4}+\frac{\left (a+3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} b^{4/3} d^4}-\frac{c \log \left (a+b (c+d x)^3\right )}{b d^4}-\frac{\left (a-3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{a^{2/3} b^{4/3} d^4}\\ &=\frac{x}{b d^3}+\frac{\left (a-3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} a^{2/3} b^{4/3} d^4}-\frac{\left (a+3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{4/3} d^4}+\frac{\left (a+3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} b^{4/3} d^4}-\frac{c \log \left (a+b (c+d x)^3\right )}{b d^4}\\ \end{align*}

Mathematica [C]  time = 0.0474439, size = 132, normalized size = 0.56 \[ -\frac{\text{RootSum}\left [3 \text{$\#$1}^2 b c d^2+\text{$\#$1}^3 b d^3+3 \text{$\#$1} b c^2 d+a+b c^3\& ,\frac{3 \text{$\#$1}^2 b c d^2 \log (x-\text{$\#$1})+a \log (x-\text{$\#$1})+3 \text{$\#$1} b c^2 d \log (x-\text{$\#$1})+b c^3 \log (x-\text{$\#$1})}{\text{$\#$1}^2 d^2+2 \text{$\#$1} c d+c^2}\& \right ]-3 b d x}{3 b^2 d^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*(c + d*x)^3),x]

[Out]

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

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Maple [C]  time = 0.006, size = 108, normalized size = 0.5 \begin{align*}{\frac{x}{b{d}^{3}}}+{\frac{1}{3\,{b}^{2}{d}^{4}}\sum _{{\it \_R}={\it RootOf} \left ( b{d}^{3}{{\it \_Z}}^{3}+3\,bc{d}^{2}{{\it \_Z}}^{2}+3\,b{c}^{2}d{\it \_Z}+b{c}^{3}+a \right ) }{\frac{ \left ( -3\,{{\it \_R}}^{2}bc{d}^{2}-3\,{\it \_R}\,b{c}^{2}d-b{c}^{3}-a \right ) \ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a+b*(d*x+c)^3),x)

[Out]

x/b/d^3+1/3/b^2/d^4*sum((-3*_R^2*b*c*d^2-3*_R*b*c^2*d-b*c^3-a)/(_R^2*d^2+2*_R*c*d+c^2)*ln(x-_R),_R=RootOf(_Z^3
*b*d^3+3*_Z^2*b*c*d^2+3*_Z*b*c^2*d+b*c^3+a))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError

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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)^3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [A]  time = 2.52364, size = 238, normalized size = 1.02 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} a^{2} b^{4} d^{12} + 81 t^{2} a^{2} b^{3} c d^{8} + t \left (54 a^{2} b^{2} c^{2} d^{4} - 27 a b^{3} c^{5} d^{4}\right ) + a^{3} + 3 a^{2} b c^{3} + 3 a b^{2} c^{6} + b^{3} c^{9}, \left ( t \mapsto t \log{\left (x + \frac{- 27 t^{2} a^{2} b^{3} c^{2} d^{8} - 3 t a^{3} b d^{4} - 60 t a^{2} b^{2} c^{3} d^{4} - 3 t a b^{3} c^{6} d^{4} - 2 a^{3} c - 12 a^{2} b c^{4} - 9 a b^{2} c^{7} + b^{3} c^{10}}{a^{3} d + 3 a^{2} b c^{3} d - 24 a b^{2} c^{6} d + b^{3} c^{9} d} \right )} \right )\right )} + \frac{x}{b d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(27*_t**3*a**2*b**4*d**12 + 81*_t**2*a**2*b**3*c*d**8 + _t*(54*a**2*b**2*c**2*d**4 - 27*a*b**3*c**5*d**
4) + a**3 + 3*a**2*b*c**3 + 3*a*b**2*c**6 + b**3*c**9, Lambda(_t, _t*log(x + (-27*_t**2*a**2*b**3*c**2*d**8 -
3*_t*a**3*b*d**4 - 60*_t*a**2*b**2*c**3*d**4 - 3*_t*a*b**3*c**6*d**4 - 2*a**3*c - 12*a**2*b*c**4 - 9*a*b**2*c*
*7 + b**3*c**10)/(a**3*d + 3*a**2*b*c**3*d - 24*a*b**2*c**6*d + b**3*c**9*d)))) + x/(b*d**3)

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

[Out]

integrate(x^3/((d*x + c)^3*b + a), x)