\(\int \frac {x^3}{a+b (c+d x)^3} \, dx\) [103]

Optimal result

Integrand size = 17, antiderivative size = 234 \[ \int \frac {x^3}{a+b (c+d x)^3} \, dx=\frac {x}{b d^3}+\frac {\left (a-3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{4/3} d^4}-\frac {\left (a+3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{4/3} d^4}+\frac {\left (a+3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} b^{4/3} d^4}-\frac {c \log \left (a+b (c+d x)^3\right )}{b d^4} \]

[Out]

x/b/d^3-1/3*(a+3*a^(1/3)*b^(2/3)*c^2+b*c^3)*ln(a^(1/3)+b^(1/3)*(d*x+c))/a^(2/3)/b^(4/3)/d^4+1/6*(a+3*a^(1/3)*b
^(2/3)*c^2+b*c^3)*ln(a^(2/3)-a^(1/3)*b^(1/3)*(d*x+c)+b^(2/3)*(d*x+c)^2)/a^(2/3)/b^(4/3)/d^4-c*ln(a+b*(d*x+c)^3
)/b/d^4+1/3*(a-3*a^(1/3)*b^(2/3)*c^2+b*c^3)*arctan(1/3*(a^(1/3)-2*b^(1/3)*(d*x+c))/a^(1/3)*3^(1/2))/a^(2/3)/b^
(4/3)/d^4*3^(1/2)

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {378, 1901, 1885, 1874, 31, 648, 631, 210, 642, 266} \[ \int \frac {x^3}{a+b (c+d x)^3} \, dx=\frac {\left (-3 \sqrt [3]{a} b^{2/3} c^2+a+b c^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{4/3} d^4}-\frac {\left (3 \sqrt [3]{a} b^{2/3} c^2+a+b c^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{4/3} d^4}+\frac {\left (3 \sqrt [3]{a} b^{2/3} c^2+a+b c^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} b^{4/3} d^4}-\frac {c \log \left (a+b (c+d x)^3\right )}{b d^4}+\frac {x}{b d^3} \]

[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 31

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

Rule 210

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

Rule 266

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 378

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 631

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 642

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 648

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 1874

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 1885

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 1901

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(-c+x)^3}{a+b x^3} \, dx,x,c+d x\right )}{d^4} \\ & = \frac {\text {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 {\text {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 {\text {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) \text {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 {\text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.04 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.56 \[ \int \frac {x^3}{a+b (c+d x)^3} \, dx=-\frac {-3 b d x+\text {RootSum}\left [a+b c^3+3 b c^2 d \text {$\#$1}+3 b c d^2 \text {$\#$1}^2+b d^3 \text {$\#$1}^3\&,\frac {a \log (x-\text {$\#$1})+b c^3 \log (x-\text {$\#$1})+3 b c^2 d \log (x-\text {$\#$1}) \text {$\#$1}+3 b c d^2 \log (x-\text {$\#$1}) \text {$\#$1}^2}{c^2+2 c d \text {$\#$1}+d^2 \text {$\#$1}^2}\&\right ]}{3 b^2 d^4} \]

[In]

Integrate[x^3/(a + b*(c + d*x)^3),x]

[Out]

-1/3*(-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) & ])/(b^2*d^4)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.07 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.46

method	result	size
default	\(\frac {x}{b \,d^{3}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +b \,c^{3}+a \right )}{\sum }\frac {\left (-3 \textit {\_R}^{2} b c \,d^{2}-3 \textit {\_R} b \,c^{2} d -b \,c^{3}-a \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}}{3 b^{2} d^{4}}\)	\(108\)
risch	\(\frac {x}{b \,d^{3}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +b \,c^{3}+a \right )}{\sum }\frac {\left (-3 \textit {\_R}^{2} b c \,d^{2}-3 \textit {\_R} b \,c^{2} d -b \,c^{3}-a \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}}{3 b^{2} d^{4}}\)	\(108\)

[In]

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

[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))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 10.97 (sec) , antiderivative size = 6315, normalized size of antiderivative = 26.99 \[ \int \frac {x^3}{a+b (c+d x)^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [A] (verification not implemented)

Time = 1.64 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.02 \[ \int \frac {x^3}{a+b (c+d x)^3} \, dx=\operatorname {RootSum} {\left (27 t^{3} a^{2} b^{4} d^{12} + 81 t^{2} a^{2} b^{3} c d^{8} + t \left (54 a^{2} b^{2} c^{2} d^{4} - 27 a b^{3} c^{5} d^{4}\right ) + a^{3} + 3 a^{2} b c^{3} + 3 a b^{2} c^{6} + b^{3} c^{9}, \left ( t \mapsto t \log {\left (x + \frac {- 27 t^{2} a^{2} b^{3} c^{2} d^{8} - 3 t a^{3} b d^{4} - 60 t a^{2} b^{2} c^{3} d^{4} - 3 t a b^{3} c^{6} d^{4} - 2 a^{3} c - 12 a^{2} b c^{4} - 9 a b^{2} c^{7} + b^{3} c^{10}}{a^{3} d + 3 a^{2} b c^{3} d - 24 a b^{2} c^{6} d + b^{3} c^{9} d} \right )} \right )\right )} + \frac {x}{b d^{3}} \]

[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)

Maxima [F]

\[ \int \frac {x^3}{a+b (c+d x)^3} \, dx=\int { \frac {x^{3}}{{\left (d x + c\right )}^{3} b + a} \,d x } \]

[In]

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

[Out]

x/(b*d^3) - integrate((3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)/(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c
^3 + a), x)/(b*d^3)

Giac [F]

\[ \int \frac {x^3}{a+b (c+d x)^3} \, dx=\int { \frac {x^{3}}{{\left (d x + c\right )}^{3} b + a} \,d x } \]

[In]

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

[Out]

integrate(x^3/((d*x + c)^3*b + a), x)

Mupad [B] (verification not implemented)

Time = 9.62 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.60 \[ \int \frac {x^3}{a+b (c+d x)^3} \, dx=\left (\sum _{k=1}^3\ln \left (\frac {3\,\left (b\,c^5+a\,c^2\right )}{d^2}-\mathrm {root}\left (27\,a^2\,b^4\,d^{12}\,z^3+81\,a^2\,b^3\,c\,d^8\,z^2+54\,a^2\,b^2\,c^2\,d^4\,z-27\,a\,b^3\,c^5\,d^4\,z+3\,a\,b^2\,c^6+3\,a^2\,b\,c^3+b^3\,c^9+a^3,z,k\right )\,\left (\frac {3\,\left (b^2\,c^4\,d^4-5\,a\,b\,c\,d^4\right )}{d^2}+\frac {3\,x\,\left (b^2\,c^3\,d^4+a\,b\,d^4\right )}{d}-\mathrm {root}\left (27\,a^2\,b^4\,d^{12}\,z^3+81\,a^2\,b^3\,c\,d^8\,z^2+54\,a^2\,b^2\,c^2\,d^4\,z-27\,a\,b^3\,c^5\,d^4\,z+3\,a\,b^2\,c^6+3\,a^2\,b\,c^3+b^3\,c^9+a^3,z,k\right )\,a\,b^2\,d^6\,9\right )-\frac {3\,x\,\left (a\,c-2\,b\,c^4\right )}{d}\right )\,\mathrm {root}\left (27\,a^2\,b^4\,d^{12}\,z^3+81\,a^2\,b^3\,c\,d^8\,z^2+54\,a^2\,b^2\,c^2\,d^4\,z-27\,a\,b^3\,c^5\,d^4\,z+3\,a\,b^2\,c^6+3\,a^2\,b\,c^3+b^3\,c^9+a^3,z,k\right )\right )+\frac {x}{b\,d^3} \]

[In]

int(x^3/(a + b*(c + d*x)^3),x)

[Out]

symsum(log((3*(a*c^2 + b*c^5))/d^2 - root(27*a^2*b^4*d^12*z^3 + 81*a^2*b^3*c*d^8*z^2 + 54*a^2*b^2*c^2*d^4*z -
27*a*b^3*c^5*d^4*z + 3*a*b^2*c^6 + 3*a^2*b*c^3 + b^3*c^9 + a^3, z, k)*((3*(b^2*c^4*d^4 - 5*a*b*c*d^4))/d^2 + (
3*x*(b^2*c^3*d^4 + a*b*d^4))/d - 9*root(27*a^2*b^4*d^12*z^3 + 81*a^2*b^3*c*d^8*z^2 + 54*a^2*b^2*c^2*d^4*z - 27
*a*b^3*c^5*d^4*z + 3*a*b^2*c^6 + 3*a^2*b*c^3 + b^3*c^9 + a^3, z, k)*a*b^2*d^6) - (3*x*(a*c - 2*b*c^4))/d)*root
(27*a^2*b^4*d^12*z^3 + 81*a^2*b^3*c*d^8*z^2 + 54*a^2*b^2*c^2*d^4*z - 27*a*b^3*c^5*d^4*z + 3*a*b^2*c^6 + 3*a^2*
b*c^3 + b^3*c^9 + a^3, z, k), k, 1, 3) + x/(b*d^3)