\(\int \frac {c+d x}{(a+b x^3)^4} \, dx\) [10]

Optimal result

Integrand size = 15, antiderivative size = 240 \[ \int \frac {c+d x}{\left (a+b x^3\right )^4} \, dx=\frac {x (c+d x)}{9 a \left (a+b x^3\right )^3}+\frac {x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac {2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}-\frac {2 \left (20 \sqrt [3]{b} c+7 \sqrt [3]{a} d\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{81 \sqrt {3} a^{11/3} b^{2/3}}+\frac {2 \left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac {\left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}} \]

[Out]

1/9*x*(d*x+c)/a/(b*x^3+a)^3+1/54*x*(7*d*x+8*c)/a^2/(b*x^3+a)^2+2/81*x*(7*d*x+10*c)/a^3/(b*x^3+a)+2/243*(20*b^(
1/3)*c-7*a^(1/3)*d)*ln(a^(1/3)+b^(1/3)*x)/a^(11/3)/b^(2/3)-1/243*(20*b^(1/3)*c-7*a^(1/3)*d)*ln(a^(2/3)-a^(1/3)
*b^(1/3)*x+b^(2/3)*x^2)/a^(11/3)/b^(2/3)-2/243*(20*b^(1/3)*c+7*a^(1/3)*d)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(
1/3)*3^(1/2))/a^(11/3)/b^(2/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {1869, 1874, 31, 648, 631, 210, 642} \[ \int \frac {c+d x}{\left (a+b x^3\right )^4} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (7 \sqrt [3]{a} d+20 \sqrt [3]{b} c\right )}{81 \sqrt {3} a^{11/3} b^{2/3}}-\frac {\left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}}+\frac {2 \left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}+\frac {2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}+\frac {x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac {x (c+d x)}{9 a \left (a+b x^3\right )^3} \]

[In]

Int[(c + d*x)/(a + b*x^3)^4,x]

[Out]

(x*(c + d*x))/(9*a*(a + b*x^3)^3) + (x*(8*c + 7*d*x))/(54*a^2*(a + b*x^3)^2) + (2*x*(10*c + 7*d*x))/(81*a^3*(a
+ b*x^3)) - (2*(20*b^(1/3)*c + 7*a^(1/3)*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(81*Sqrt[3]*a^
(11/3)*b^(2/3)) + (2*(20*b^(1/3)*c - 7*a^(1/3)*d)*Log[a^(1/3) + b^(1/3)*x])/(243*a^(11/3)*b^(2/3)) - ((20*b^(1
/3)*c - 7*a^(1/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(243*a^(11/3)*b^(2/3))

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 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 1869

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-x)*Pq*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] +
Dist[1/(a*n*(p + 1)), Int[ExpandToSum[n*(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b
}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 1]

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]

Rubi steps \begin{align*} \text {integral}& = \frac {x (c+d x)}{9 a \left (a+b x^3\right )^3}-\frac {\int \frac {-8 c-7 d x}{\left (a+b x^3\right )^3} \, dx}{9 a} \\ & = \frac {x (c+d x)}{9 a \left (a+b x^3\right )^3}+\frac {x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac {\int \frac {40 c+28 d x}{\left (a+b x^3\right )^2} \, dx}{54 a^2} \\ & = \frac {x (c+d x)}{9 a \left (a+b x^3\right )^3}+\frac {x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac {2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}-\frac {\int \frac {-80 c-28 d x}{a+b x^3} \, dx}{162 a^3} \\ & = \frac {x (c+d x)}{9 a \left (a+b x^3\right )^3}+\frac {x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac {2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}-\frac {\int \frac {\sqrt [3]{a} \left (-160 \sqrt [3]{b} c-28 \sqrt [3]{a} d\right )+\sqrt [3]{b} \left (80 \sqrt [3]{b} c-28 \sqrt [3]{a} d\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{486 a^{11/3} \sqrt [3]{b}}+\frac {\left (2 \left (20 c-\frac {7 \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{243 a^{11/3}} \\ & = \frac {x (c+d x)}{9 a \left (a+b x^3\right )^3}+\frac {x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac {2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}+\frac {2 \left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac {\left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \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}{243 a^{11/3} b^{2/3}}+\frac {\left (20 c+\frac {7 \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{81 a^{10/3}} \\ & = \frac {x (c+d x)}{9 a \left (a+b x^3\right )^3}+\frac {x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac {2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}+\frac {2 \left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac {\left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}}+\frac {\left (2 \left (20 \sqrt [3]{b} c+7 \sqrt [3]{a} d\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{81 a^{11/3} b^{2/3}} \\ & = \frac {x (c+d x)}{9 a \left (a+b x^3\right )^3}+\frac {x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac {2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}-\frac {2 \left (20 \sqrt [3]{b} c+7 \sqrt [3]{a} d\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{81 \sqrt {3} a^{11/3} b^{2/3}}+\frac {2 \left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac {\left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.95 \[ \int \frac {c+d x}{\left (a+b x^3\right )^4} \, dx=\frac {\frac {54 a^3 x (c+d x)}{\left (a+b x^3\right )^3}+\frac {9 a^2 x (8 c+7 d x)}{\left (a+b x^3\right )^2}+\frac {12 a x (10 c+7 d x)}{a+b x^3}-\frac {4 \sqrt {3} \sqrt [3]{a} \left (20 \sqrt [3]{b} c+7 \sqrt [3]{a} d\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}+\frac {4 \left (20 \sqrt [3]{a} \sqrt [3]{b} c-7 a^{2/3} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac {2 \left (-20 \sqrt [3]{a} \sqrt [3]{b} c+7 a^{2/3} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}}{486 a^4} \]

[In]

Integrate[(c + d*x)/(a + b*x^3)^4,x]

[Out]

((54*a^3*x*(c + d*x))/(a + b*x^3)^3 + (9*a^2*x*(8*c + 7*d*x))/(a + b*x^3)^2 + (12*a*x*(10*c + 7*d*x))/(a + b*x
^3) - (4*Sqrt[3]*a^(1/3)*(20*b^(1/3)*c + 7*a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + (
4*(20*a^(1/3)*b^(1/3)*c - 7*a^(2/3)*d)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + (2*(-20*a^(1/3)*b^(1/3)*c + 7*a^(2/
3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3))/(486*a^4)

Maple [C] (verified)

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

Time = 9.84 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.46

method	result	size
risch	\(\frac {\frac {14 d \,b^{2} x^{8}}{81 a^{3}}+\frac {20 c \,b^{2} x^{7}}{81 a^{3}}+\frac {77 b d \,x^{5}}{162 a^{2}}+\frac {52 b c \,x^{4}}{81 a^{2}}+\frac {67 d \,x^{2}}{162 a}+\frac {41 c x}{81 a}}{\left (b \,x^{3}+a \right )^{3}}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (7 \textit {\_R} d +20 c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{243 a^{3} b}\)	\(110\)
default	\(c \left (\frac {x}{9 a \left (b \,x^{3}+a \right )^{3}}+\frac {\frac {4 x}{27 a \left (b \,x^{3}+a \right )^{2}}+\frac {8 \left (\frac {5 x}{18 a \left (b \,x^{3}+a \right )}+\frac {5 \left (\frac {2 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{6 a}\right )}{9 a}}{a}\right )+d \left (\frac {x^{2}}{9 a \left (b \,x^{3}+a \right )^{3}}+\frac {\frac {7 x^{2}}{54 a \left (b \,x^{3}+a \right )^{2}}+\frac {7 \left (\frac {2 x^{2}}{9 a \left (b \,x^{3}+a \right )}+\frac {2 \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 a}\right )}{9 a}}{a}\right )\)	\(318\)

[In]

int((d*x+c)/(b*x^3+a)^4,x,method=_RETURNVERBOSE)

[Out]

(14/81*d/a^3*b^2*x^8+20/81*c/a^3*b^2*x^7+77/162*b*d/a^2*x^5+52/81*b*c/a^2*x^4+67/162*d/a*x^2+41/81*c/a*x)/(b*x
^3+a)^3+2/243/a^3/b*sum((7*_R*d+20*c)/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.94 (sec) , antiderivative size = 2308, normalized size of antiderivative = 9.62 \[ \int \frac {c+d x}{\left (a+b x^3\right )^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/972*(168*b^2*d*x^8 + 240*b^2*c*x^7 + 462*a*b*d*x^5 + 624*a*b*c*x^4 + 402*a^2*d*x^2 + 492*a^2*c*x - 2*(a^3*b^
3*x^9 + 3*a^4*b^2*x^6 + 3*a^5*b*x^3 + a^6)*(4^(1/3)*(I*sqrt(3) + 1)*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (80
00*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^7*b*((8000*b*c^3 + 343*a*d^3)/(a
^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3)))*log(7/4*(4^(1/3)*(I*sqrt(3) + 1)*((8000*b*c^3 + 343*a*
d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^7*b*((8000*
b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3)))^2*a^8*b*d - 400*(4^(1/3)*(I*sqrt(
3) + 1)*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*d*(-
I*sqrt(3) + 1)/(a^7*b*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3)))*a^4*
b*c^2 + 7840*a*c*d^2 + 4*(8000*b*c^3 + 343*a*d^3)*x) + ((a^3*b^3*x^9 + 3*a^4*b^2*x^6 + 3*a^5*b*x^3 + a^6)*(4^(
1/3)*(I*sqrt(3) + 1)*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3) - 140*4
^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^7*b*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))
^(1/3))) + 3*sqrt(1/3)*(a^3*b^3*x^9 + 3*a^4*b^2*x^6 + 3*a^5*b*x^3 + a^6)*sqrt(-((4^(1/3)*(I*sqrt(3) + 1)*((800
0*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*d*(-I*sqrt(3) + 1
)/(a^7*b*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3)))^2*a^7*b + 8960*c*
d)/(a^7*b)))*log(-7/4*(4^(1/3)*(I*sqrt(3) + 1)*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)
/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^7*b*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^
3 - 343*a*d^3)/(a^11*b^2))^(1/3)))^2*a^8*b*d + 400*(4^(1/3)*(I*sqrt(3) + 1)*((8000*b*c^3 + 343*a*d^3)/(a^11*b^
2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^7*b*((8000*b*c^3 + 343*a
*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3)))*a^4*b*c^2 - 7840*a*c*d^2 + 8*(8000*b*c^3 + 343
*a*d^3)*x + 3/4*sqrt(1/3)*(7*(4^(1/3)*(I*sqrt(3) + 1)*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343
*a*d^3)/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^7*b*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (80
00*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3)))*a^8*b*d + 1600*a^4*b*c^2)*sqrt(-((4^(1/3)*(I*sqrt(3) + 1)*((8000*b*c
^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^
7*b*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3)))^2*a^7*b + 8960*c*d)/(a
^7*b))) + ((a^3*b^3*x^9 + 3*a^4*b^2*x^6 + 3*a^5*b*x^3 + a^6)*(4^(1/3)*(I*sqrt(3) + 1)*((8000*b*c^3 + 343*a*d^3
)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^7*b*((8000*b*c
^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3))) - 3*sqrt(1/3)*(a^3*b^3*x^9 + 3*a^4*b
^2*x^6 + 3*a^5*b*x^3 + a^6)*sqrt(-((4^(1/3)*(I*sqrt(3) + 1)*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3
- 343*a*d^3)/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^7*b*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2)
+ (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3)))^2*a^7*b + 8960*c*d)/(a^7*b)))*log(-7/4*(4^(1/3)*(I*sqrt(3) + 1
)*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*d*(-I*sqrt
(3) + 1)/(a^7*b*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3)))^2*a^8*b*d
+ 400*(4^(1/3)*(I*sqrt(3) + 1)*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/
3) - 140*4^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^7*b*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(
a^11*b^2))^(1/3)))*a^4*b*c^2 - 7840*a*c*d^2 + 8*(8000*b*c^3 + 343*a*d^3)*x - 3/4*sqrt(1/3)*(7*(4^(1/3)*(I*sqrt
(3) + 1)*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*d*(
-I*sqrt(3) + 1)/(a^7*b*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3)))*a^8
*b*d + 1600*a^4*b*c^2)*sqrt(-((4^(1/3)*(I*sqrt(3) + 1)*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 34
3*a*d^3)/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^7*b*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8
000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3)))^2*a^7*b + 8960*c*d)/(a^7*b))))/(a^3*b^3*x^9 + 3*a^4*b^2*x^6 + 3*a^5
*b*x^3 + a^6)

Sympy [A] (verification not implemented)

Time = 0.71 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.77 \[ \int \frac {c+d x}{\left (a+b x^3\right )^4} \, dx=\operatorname {RootSum} {\left (14348907 t^{3} a^{11} b^{2} + 408240 t a^{4} b c d + 2744 a d^{3} - 64000 b c^{3}, \left ( t \mapsto t \log {\left (x + \frac {413343 t^{2} a^{8} b d + 194400 t a^{4} b c^{2} + 7840 a c d^{2}}{1372 a d^{3} + 32000 b c^{3}} \right )} \right )\right )} + \frac {82 a^{2} c x + 67 a^{2} d x^{2} + 104 a b c x^{4} + 77 a b d x^{5} + 40 b^{2} c x^{7} + 28 b^{2} d x^{8}}{162 a^{6} + 486 a^{5} b x^{3} + 486 a^{4} b^{2} x^{6} + 162 a^{3} b^{3} x^{9}} \]

[In]

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

[Out]

RootSum(14348907*_t**3*a**11*b**2 + 408240*_t*a**4*b*c*d + 2744*a*d**3 - 64000*b*c**3, Lambda(_t, _t*log(x + (
413343*_t**2*a**8*b*d + 194400*_t*a**4*b*c**2 + 7840*a*c*d**2)/(1372*a*d**3 + 32000*b*c**3)))) + (82*a**2*c*x
+ 67*a**2*d*x**2 + 104*a*b*c*x**4 + 77*a*b*d*x**5 + 40*b**2*c*x**7 + 28*b**2*d*x**8)/(162*a**6 + 486*a**5*b*x*
*3 + 486*a**4*b**2*x**6 + 162*a**3*b**3*x**9)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x}{\left (a+b x^3\right )^4} \, dx=\frac {28 \, b^{2} d x^{8} + 40 \, b^{2} c x^{7} + 77 \, a b d x^{5} + 104 \, a b c x^{4} + 67 \, a^{2} d x^{2} + 82 \, a^{2} c x}{162 \, {\left (a^{3} b^{3} x^{9} + 3 \, a^{4} b^{2} x^{6} + 3 \, a^{5} b x^{3} + a^{6}\right )}} + \frac {2 \, \sqrt {3} {\left (7 \, d \left (\frac {a}{b}\right )^{\frac {1}{3}} + 20 \, c\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{243 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (7 \, d \left (\frac {a}{b}\right )^{\frac {1}{3}} - 20 \, c\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{243 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {2 \, {\left (7 \, d \left (\frac {a}{b}\right )^{\frac {1}{3}} - 20 \, c\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{243 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

[In]

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

[Out]

1/162*(28*b^2*d*x^8 + 40*b^2*c*x^7 + 77*a*b*d*x^5 + 104*a*b*c*x^4 + 67*a^2*d*x^2 + 82*a^2*c*x)/(a^3*b^3*x^9 +
3*a^4*b^2*x^6 + 3*a^5*b*x^3 + a^6) + 2/243*sqrt(3)*(7*d*(a/b)^(1/3) + 20*c)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1
/3))/(a/b)^(1/3))/(a^3*b*(a/b)^(2/3)) + 1/243*(7*d*(a/b)^(1/3) - 20*c)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/
(a^3*b*(a/b)^(2/3)) - 2/243*(7*d*(a/b)^(1/3) - 20*c)*log(x + (a/b)^(1/3))/(a^3*b*(a/b)^(2/3))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.91 \[ \int \frac {c+d x}{\left (a+b x^3\right )^4} \, dx=-\frac {2 \, \sqrt {3} {\left (20 \, b c - 7 \, \left (-a b^{2}\right )^{\frac {1}{3}} d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{243 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3}} - \frac {{\left (20 \, b c + 7 \, \left (-a b^{2}\right )^{\frac {1}{3}} d\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{243 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3}} - \frac {2 \, {\left (7 \, d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 20 \, c\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{243 \, a^{4}} + \frac {28 \, b^{2} d x^{8} + 40 \, b^{2} c x^{7} + 77 \, a b d x^{5} + 104 \, a b c x^{4} + 67 \, a^{2} d x^{2} + 82 \, a^{2} c x}{162 \, {\left (b x^{3} + a\right )}^{3} a^{3}} \]

[In]

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

[Out]

-2/243*sqrt(3)*(20*b*c - 7*(-a*b^2)^(1/3)*d)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(
2/3)*a^3) - 1/243*(20*b*c + 7*(-a*b^2)^(1/3)*d)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(2/3)*a^3)
- 2/243*(7*d*(-a/b)^(1/3) + 20*c)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^4 + 1/162*(28*b^2*d*x^8 + 40*b^2*c
*x^7 + 77*a*b*d*x^5 + 104*a*b*c*x^4 + 67*a^2*d*x^2 + 82*a^2*c*x)/((b*x^3 + a)^3*a^3)

Mupad [B] (verification not implemented)

Time = 9.34 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x}{\left (a+b x^3\right )^4} \, dx=\left (\sum _{k=1}^3\ln \left (\frac {b\,\left (560\,c\,d+196\,d^2\,x+{\mathrm {root}\left (14348907\,a^{11}\,b^2\,z^3+408240\,a^4\,b\,c\,d\,z-64000\,b\,c^3+2744\,a\,d^3,z,k\right )}^2\,a^7\,b\,59049+\mathrm {root}\left (14348907\,a^{11}\,b^2\,z^3+408240\,a^4\,b\,c\,d\,z-64000\,b\,c^3+2744\,a\,d^3,z,k\right )\,a^3\,b\,c\,x\,9720\right )}{a^6\,6561}\right )\,\mathrm {root}\left (14348907\,a^{11}\,b^2\,z^3+408240\,a^4\,b\,c\,d\,z-64000\,b\,c^3+2744\,a\,d^3,z,k\right )\right )+\frac {\frac {67\,d\,x^2}{162\,a}+\frac {41\,c\,x}{81\,a}+\frac {20\,b^2\,c\,x^7}{81\,a^3}+\frac {14\,b^2\,d\,x^8}{81\,a^3}+\frac {52\,b\,c\,x^4}{81\,a^2}+\frac {77\,b\,d\,x^5}{162\,a^2}}{a^3+3\,a^2\,b\,x^3+3\,a\,b^2\,x^6+b^3\,x^9} \]

[In]

int((c + d*x)/(a + b*x^3)^4,x)

[Out]

symsum(log((b*(560*c*d + 196*d^2*x + 59049*root(14348907*a^11*b^2*z^3 + 408240*a^4*b*c*d*z - 64000*b*c^3 + 274
4*a*d^3, z, k)^2*a^7*b + 9720*root(14348907*a^11*b^2*z^3 + 408240*a^4*b*c*d*z - 64000*b*c^3 + 2744*a*d^3, z, k
)*a^3*b*c*x))/(6561*a^6))*root(14348907*a^11*b^2*z^3 + 408240*a^4*b*c*d*z - 64000*b*c^3 + 2744*a*d^3, z, k), k
, 1, 3) + ((67*d*x^2)/(162*a) + (41*c*x)/(81*a) + (20*b^2*c*x^7)/(81*a^3) + (14*b^2*d*x^8)/(81*a^3) + (52*b*c*
x^4)/(81*a^2) + (77*b*d*x^5)/(162*a^2))/(a^3 + b^3*x^9 + 3*a^2*b*x^3 + 3*a*b^2*x^6)