3.1.14 \(\int \frac {(c+d x^3)^4}{a+b x^3} \, dx\) [14]

Optimal. Leaf size=252 \[ \frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^4}{4 b^3}+\frac {d^3 (4 b c-a d) x^7}{7 b^2}+\frac {d^4 x^{10}}{10 b}-\frac {(b c-a d)^4 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{13/3}}+\frac {(b c-a d)^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{13/3}}-\frac {(b c-a d)^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{13/3}} \]

[Out]

d*(-a*d+2*b*c)*(a^2*d^2-2*a*b*c*d+2*b^2*c^2)*x/b^4+1/4*d^2*(a^2*d^2-4*a*b*c*d+6*b^2*c^2)*x^4/b^3+1/7*d^3*(-a*d
+4*b*c)*x^7/b^2+1/10*d^4*x^10/b+1/3*(-a*d+b*c)^4*ln(a^(1/3)+b^(1/3)*x)/a^(2/3)/b^(13/3)-1/6*(-a*d+b*c)^4*ln(a^
(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(2/3)/b^(13/3)-1/3*(-a*d+b*c)^4*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3
)*3^(1/2))/a^(2/3)/b^(13/3)*3^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.13, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {398, 206, 31, 648, 631, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) (b c-a d)^4}{\sqrt {3} a^{2/3} b^{13/3}}-\frac {(b c-a d)^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{13/3}}+\frac {(b c-a d)^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{13/3}}+\frac {d x (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right )}{b^4}+\frac {d^2 x^4 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{4 b^3}+\frac {d^3 x^7 (4 b c-a d)}{7 b^2}+\frac {d^4 x^{10}}{10 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3)^4/(a + b*x^3),x]

[Out]

(d*(2*b*c - a*d)*(2*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x)/b^4 + (d^2*(6*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*x^4)/(4*b^3
) + (d^3*(4*b*c - a*d)*x^7)/(7*b^2) + (d^4*x^10)/(10*b) - ((b*c - a*d)^4*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[
3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*b^(13/3)) + ((b*c - a*d)^4*Log[a^(1/3) + b^(1/3)*x])/(3*a^(2/3)*b^(13/3)) - ((b
*c - a*d)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(2/3)*b^(13/3))

Rule 31

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

Rule 206

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], 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 398

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

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]

Rubi steps

\begin {align*} \int \frac {\left (c+d x^3\right )^4}{a+b x^3} \, dx &=\int \left (\frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right )}{b^4}+\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^3}{b^3}+\frac {d^3 (4 b c-a d) x^6}{b^2}+\frac {d^4 x^9}{b}+\frac {b^4 c^4-4 a b^3 c^3 d+6 a^2 b^2 c^2 d^2-4 a^3 b c d^3+a^4 d^4}{b^4 \left (a+b x^3\right )}\right ) \, dx\\ &=\frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^4}{4 b^3}+\frac {d^3 (4 b c-a d) x^7}{7 b^2}+\frac {d^4 x^{10}}{10 b}+\frac {(b c-a d)^4 \int \frac {1}{a+b x^3} \, dx}{b^4}\\ &=\frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^4}{4 b^3}+\frac {d^3 (4 b c-a d) x^7}{7 b^2}+\frac {d^4 x^{10}}{10 b}+\frac {(b c-a d)^4 \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{2/3} b^4}+\frac {(b c-a d)^4 \int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{2/3} b^4}\\ &=\frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^4}{4 b^3}+\frac {d^3 (4 b c-a d) x^7}{7 b^2}+\frac {d^4 x^{10}}{10 b}+\frac {(b c-a d)^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{13/3}}-\frac {(b c-a d)^4 \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}{6 a^{2/3} b^{13/3}}+\frac {(b c-a d)^4 \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 \sqrt [3]{a} b^4}\\ &=\frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^4}{4 b^3}+\frac {d^3 (4 b c-a d) x^7}{7 b^2}+\frac {d^4 x^{10}}{10 b}+\frac {(b c-a d)^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{13/3}}-\frac {(b c-a d)^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{13/3}}+\frac {(b c-a d)^4 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{2/3} b^{13/3}}\\ &=\frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^4}{4 b^3}+\frac {d^3 (4 b c-a d) x^7}{7 b^2}+\frac {d^4 x^{10}}{10 b}-\frac {(b c-a d)^4 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{13/3}}+\frac {(b c-a d)^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{13/3}}-\frac {(b c-a d)^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{13/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.09, size = 253, normalized size = 1.00 \begin {gather*} \frac {420 \sqrt [3]{b} d \left (4 b^3 c^3-6 a b^2 c^2 d+4 a^2 b c d^2-a^3 d^3\right ) x+105 b^{4/3} d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^4+60 b^{7/3} d^3 (4 b c-a d) x^7+42 b^{10/3} d^4 x^{10}+\frac {140 \sqrt {3} (b c-a d)^4 \tan ^{-1}\left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{2/3}}+\frac {140 (b c-a d)^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}-\frac {70 (b c-a d)^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}}{420 b^{13/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3)^4/(a + b*x^3),x]

[Out]

(420*b^(1/3)*d*(4*b^3*c^3 - 6*a*b^2*c^2*d + 4*a^2*b*c*d^2 - a^3*d^3)*x + 105*b^(4/3)*d^2*(6*b^2*c^2 - 4*a*b*c*
d + a^2*d^2)*x^4 + 60*b^(7/3)*d^3*(4*b*c - a*d)*x^7 + 42*b^(10/3)*d^4*x^10 + (140*Sqrt[3]*(b*c - a*d)^4*ArcTan
[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/a^(2/3) + (140*(b*c - a*d)^4*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) -
(70*(b*c - a*d)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3))/(420*b^(13/3))

________________________________________________________________________________________

Maple [A]
time = 0.33, size = 272, normalized size = 1.08

method	result	size
risch	\(\frac {d^{4} x^{10}}{10 b}-\frac {d^{4} x^{7} a}{7 b^{2}}+\frac {4 d^{3} x^{7} c}{7 b}-\frac {d^{3} x^{4} a c}{b^{2}}+\frac {3 d^{2} x^{4} c^{2}}{2 b}+\frac {d^{4} x^{4} a^{2}}{4 b^{3}}-\frac {d^{4} a^{3} x}{b^{4}}+\frac {4 d^{3} a^{2} c x}{b^{3}}-\frac {6 d^{2} a \,c^{2} x}{b^{2}}+\frac {4 d \,c^{3} x}{b}+\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 b^{5}}\)	\(201\)
default	\(-\frac {d \left (-\frac {d^{3} x^{10} b^{3}}{10}+\frac {\left (\left (a d -2 b c \right ) b^{2} d^{2}-2 b^{3} c \,d^{2}\right ) x^{7}}{7}+\frac {\left (2 \left (a d -2 b c \right ) b^{2} c d -b d \left (a^{2} d^{2}-2 a b c d +2 b^{2} c^{2}\right )\right ) x^{4}}{4}+\left (a d -2 b c \right ) \left (a^{2} d^{2}-2 a b c d +2 b^{2} c^{2}\right ) x \right )}{b^{4}}+\frac {\left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{b^{4}}\)	\(272\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d/b^4*(-1/10*d^3*x^10*b^3+1/7*((a*d-2*b*c)*b^2*d^2-2*b^3*c*d^2)*x^7+1/4*(2*(a*d-2*b*c)*b^2*c*d-b*d*(a^2*d^2-2
*a*b*c*d+2*b^2*c^2))*x^4+(a*d-2*b*c)*(a^2*d^2-2*a*b*c*d+2*b^2*c^2)*x)+(1/3/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/6
/b/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x
-1)))*(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/b^4

________________________________________________________________________________________

Maxima [A]
time = 0.57, size = 364, normalized size = 1.44 \begin {gather*} \frac {14 \, b^{3} d^{4} x^{10} + 20 \, {\left (4 \, b^{3} c d^{3} - a b^{2} d^{4}\right )} x^{7} + 35 \, {\left (6 \, b^{3} c^{2} d^{2} - 4 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{4} + 140 \, {\left (4 \, b^{3} c^{3} d - 6 \, a b^{2} c^{2} d^{2} + 4 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x}{140 \, b^{4}} + \frac {\sqrt {3} {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\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 )}{3 \, b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/140*(14*b^3*d^4*x^10 + 20*(4*b^3*c*d^3 - a*b^2*d^4)*x^7 + 35*(6*b^3*c^2*d^2 - 4*a*b^2*c*d^3 + a^2*b*d^4)*x^4
+ 140*(4*b^3*c^3*d - 6*a*b^2*c^2*d^2 + 4*a^2*b*c*d^3 - a^3*d^4)*x)/b^4 + 1/3*sqrt(3)*(b^4*c^4 - 4*a*b^3*c^3*d
+ 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^5*(a/b)
^(2/3)) - 1/6*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*log(x^2 - x*(a/b)^(1/3)
+ (a/b)^(2/3))/(b^5*(a/b)^(2/3)) + 1/3*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)
*log(x + (a/b)^(1/3))/(b^5*(a/b)^(2/3))

________________________________________________________________________________________

Fricas [A]
time = 2.88, size = 873, normalized size = 3.46 \begin {gather*} \left [\frac {42 \, a^{2} b^{4} d^{4} x^{10} + 60 \, {\left (4 \, a^{2} b^{4} c d^{3} - a^{3} b^{3} d^{4}\right )} x^{7} + 105 \, {\left (6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}\right )} x^{4} + 210 \, \sqrt {\frac {1}{3}} {\left (a b^{5} c^{4} - 4 \, a^{2} b^{4} c^{3} d + 6 \, a^{3} b^{3} c^{2} d^{2} - 4 \, a^{4} b^{2} c d^{3} + a^{5} b d^{4}\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{3} + a}\right ) - 70 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 140 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right ) + 420 \, {\left (4 \, a^{2} b^{4} c^{3} d - 6 \, a^{3} b^{3} c^{2} d^{2} + 4 \, a^{4} b^{2} c d^{3} - a^{5} b d^{4}\right )} x}{420 \, a^{2} b^{5}}, \frac {42 \, a^{2} b^{4} d^{4} x^{10} + 60 \, {\left (4 \, a^{2} b^{4} c d^{3} - a^{3} b^{3} d^{4}\right )} x^{7} + 105 \, {\left (6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}\right )} x^{4} + 420 \, \sqrt {\frac {1}{3}} {\left (a b^{5} c^{4} - 4 \, a^{2} b^{4} c^{3} d + 6 \, a^{3} b^{3} c^{2} d^{2} - 4 \, a^{4} b^{2} c d^{3} + a^{5} b d^{4}\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - 70 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 140 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right ) + 420 \, {\left (4 \, a^{2} b^{4} c^{3} d - 6 \, a^{3} b^{3} c^{2} d^{2} + 4 \, a^{4} b^{2} c d^{3} - a^{5} b d^{4}\right )} x}{420 \, a^{2} b^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/420*(42*a^2*b^4*d^4*x^10 + 60*(4*a^2*b^4*c*d^3 - a^3*b^3*d^4)*x^7 + 105*(6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^
3 + a^4*b^2*d^4)*x^4 + 210*sqrt(1/3)*(a*b^5*c^4 - 4*a^2*b^4*c^3*d + 6*a^3*b^3*c^2*d^2 - 4*a^4*b^2*c*d^3 + a^5*
b*d^4)*sqrt(-(a^2*b)^(1/3)/b)*log((2*a*b*x^3 - 3*(a^2*b)^(1/3)*a*x - a^2 + 3*sqrt(1/3)*(2*a*b*x^2 + (a^2*b)^(2
/3)*x - (a^2*b)^(1/3)*a)*sqrt(-(a^2*b)^(1/3)/b))/(b*x^3 + a)) - 70*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^
2 - 4*a^3*b*c*d^3 + a^4*d^4)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 140*(b^4*c^4 - 4
*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)) + 420*(4*
a^2*b^4*c^3*d - 6*a^3*b^3*c^2*d^2 + 4*a^4*b^2*c*d^3 - a^5*b*d^4)*x)/(a^2*b^5), 1/420*(42*a^2*b^4*d^4*x^10 + 60
*(4*a^2*b^4*c*d^3 - a^3*b^3*d^4)*x^7 + 105*(6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*x^4 + 420*sqrt(
1/3)*(a*b^5*c^4 - 4*a^2*b^4*c^3*d + 6*a^3*b^3*c^2*d^2 - 4*a^4*b^2*c*d^3 + a^5*b*d^4)*sqrt((a^2*b)^(1/3)/b)*arc
tan(sqrt(1/3)*(2*(a^2*b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) - 70*(b^4*c^4 - 4*a*b^3*c^3*d +
6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) +
140*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)
^(2/3)) + 420*(4*a^2*b^4*c^3*d - 6*a^3*b^3*c^2*d^2 + 4*a^4*b^2*c*d^3 - a^5*b*d^4)*x)/(a^2*b^5)]

________________________________________________________________________________________

Sympy [A]
time = 0.97, size = 371, normalized size = 1.47 \begin {gather*} x^{7} \left (- \frac {a d^{4}}{7 b^{2}} + \frac {4 c d^{3}}{7 b}\right ) + x^{4} \left (\frac {a^{2} d^{4}}{4 b^{3}} - \frac {a c d^{3}}{b^{2}} + \frac {3 c^{2} d^{2}}{2 b}\right ) + x \left (- \frac {a^{3} d^{4}}{b^{4}} + \frac {4 a^{2} c d^{3}}{b^{3}} - \frac {6 a c^{2} d^{2}}{b^{2}} + \frac {4 c^{3} d}{b}\right ) + \operatorname {RootSum} {\left (27 t^{3} a^{2} b^{13} - a^{12} d^{12} + 12 a^{11} b c d^{11} - 66 a^{10} b^{2} c^{2} d^{10} + 220 a^{9} b^{3} c^{3} d^{9} - 495 a^{8} b^{4} c^{4} d^{8} + 792 a^{7} b^{5} c^{5} d^{7} - 924 a^{6} b^{6} c^{6} d^{6} + 792 a^{5} b^{7} c^{7} d^{5} - 495 a^{4} b^{8} c^{8} d^{4} + 220 a^{3} b^{9} c^{9} d^{3} - 66 a^{2} b^{10} c^{10} d^{2} + 12 a b^{11} c^{11} d - b^{12} c^{12}, \left ( t \mapsto t \log {\left (\frac {3 t a b^{4}}{a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}} + x \right )} \right )\right )} + \frac {d^{4} x^{10}}{10 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**7*(-a*d**4/(7*b**2) + 4*c*d**3/(7*b)) + x**4*(a**2*d**4/(4*b**3) - a*c*d**3/b**2 + 3*c**2*d**2/(2*b)) + x*(
-a**3*d**4/b**4 + 4*a**2*c*d**3/b**3 - 6*a*c**2*d**2/b**2 + 4*c**3*d/b) + RootSum(27*_t**3*a**2*b**13 - a**12*
d**12 + 12*a**11*b*c*d**11 - 66*a**10*b**2*c**2*d**10 + 220*a**9*b**3*c**3*d**9 - 495*a**8*b**4*c**4*d**8 + 79
2*a**7*b**5*c**5*d**7 - 924*a**6*b**6*c**6*d**6 + 792*a**5*b**7*c**7*d**5 - 495*a**4*b**8*c**8*d**4 + 220*a**3
*b**9*c**9*d**3 - 66*a**2*b**10*c**10*d**2 + 12*a*b**11*c**11*d - b**12*c**12, Lambda(_t, _t*log(3*_t*a*b**4/(
a**4*d**4 - 4*a**3*b*c*d**3 + 6*a**2*b**2*c**2*d**2 - 4*a*b**3*c**3*d + b**4*c**4) + x))) + d**4*x**10/(10*b)

________________________________________________________________________________________

Giac [A]
time = 0.71, size = 391, normalized size = 1.55 \begin {gather*} -\frac {\sqrt {3} {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\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 )}{3 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3}} - \frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3}} - \frac {{\left (b^{10} c^{4} - 4 \, a b^{9} c^{3} d + 6 \, a^{2} b^{8} c^{2} d^{2} - 4 \, a^{3} b^{7} c d^{3} + a^{4} b^{6} d^{4}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{10}} + \frac {14 \, b^{9} d^{4} x^{10} + 80 \, b^{9} c d^{3} x^{7} - 20 \, a b^{8} d^{4} x^{7} + 210 \, b^{9} c^{2} d^{2} x^{4} - 140 \, a b^{8} c d^{3} x^{4} + 35 \, a^{2} b^{7} d^{4} x^{4} + 560 \, b^{9} c^{3} d x - 840 \, a b^{8} c^{2} d^{2} x + 560 \, a^{2} b^{7} c d^{3} x - 140 \, a^{3} b^{6} d^{4} x}{140 \, b^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(3)*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*arctan(1/3*sqrt(3)*(2*x +
(-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(2/3)*b^3) - 1/6*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b
*c*d^3 + a^4*d^4)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(2/3)*b^3) - 1/3*(b^10*c^4 - 4*a*b^9*c^3*
d + 6*a^2*b^8*c^2*d^2 - 4*a^3*b^7*c*d^3 + a^4*b^6*d^4)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^10) + 1/14
0*(14*b^9*d^4*x^10 + 80*b^9*c*d^3*x^7 - 20*a*b^8*d^4*x^7 + 210*b^9*c^2*d^2*x^4 - 140*a*b^8*c*d^3*x^4 + 35*a^2*
b^7*d^4*x^4 + 560*b^9*c^3*d*x - 840*a*b^8*c^2*d^2*x + 560*a^2*b^7*c*d^3*x - 140*a^3*b^6*d^4*x)/b^10

________________________________________________________________________________________

Mupad [B]
time = 1.43, size = 250, normalized size = 0.99 \begin {gather*} x\,\left (\frac {4\,c^3\,d}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^4}{b^2}-\frac {4\,c\,d^3}{b}\right )}{b}+\frac {6\,c^2\,d^2}{b}\right )}{b}\right )-x^7\,\left (\frac {a\,d^4}{7\,b^2}-\frac {4\,c\,d^3}{7\,b}\right )+x^4\,\left (\frac {a\,\left (\frac {a\,d^4}{b^2}-\frac {4\,c\,d^3}{b}\right )}{4\,b}+\frac {3\,c^2\,d^2}{2\,b}\right )+\frac {d^4\,x^{10}}{10\,b}+\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,{\left (a\,d-b\,c\right )}^4}{3\,a^{2/3}\,b^{13/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,{\left (a\,d-b\,c\right )}^4}{a^{2/3}\,b^{13/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^4}{3\,a^{2/3}\,b^{13/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^3)^4/(a + b*x^3),x)

[Out]

x*((4*c^3*d)/b - (a*((a*((a*d^4)/b^2 - (4*c*d^3)/b))/b + (6*c^2*d^2)/b))/b) - x^7*((a*d^4)/(7*b^2) - (4*c*d^3)
/(7*b)) + x^4*((a*((a*d^4)/b^2 - (4*c*d^3)/b))/(4*b) + (3*c^2*d^2)/(2*b)) + (d^4*x^10)/(10*b) + (log(b^(1/3)*x
+ a^(1/3))*(a*d - b*c)^4)/(3*a^(2/3)*b^(13/3)) + (log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1
i)/6 - 1/6)*(a*d - b*c)^4)/(a^(2/3)*b^(13/3)) - (log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)
/2 + 1/2)*(a*d - b*c)^4)/(3*a^(2/3)*b^(13/3))

________________________________________________________________________________________