3.1.13 \(\int \frac {(a+b x^3)^2}{(c+d x^3)^3} \, dx\) [13]

Optimal. Leaf size=258 \[ -\frac {(b c-a d) x \left (a+b x^3\right )}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (4 b c+5 a d) x}{18 c^2 d^2 \left (c+d x^3\right )}-\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{9 \sqrt {3} c^{8/3} d^{7/3}}+\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{7/3}}-\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{7/3}} \]

[Out]

-1/6*(-a*d+b*c)*x*(b*x^3+a)/c/d/(d*x^3+c)^2-1/18*(-a*d+b*c)*(5*a*d+4*b*c)*x/c^2/d^2/(d*x^3+c)+1/27*(5*a^2*d^2+
2*a*b*c*d+2*b^2*c^2)*ln(c^(1/3)+d^(1/3)*x)/c^(8/3)/d^(7/3)-1/54*(5*a^2*d^2+2*a*b*c*d+2*b^2*c^2)*ln(c^(2/3)-c^(
1/3)*d^(1/3)*x+d^(2/3)*x^2)/c^(8/3)/d^(7/3)-1/27*(5*a^2*d^2+2*a*b*c*d+2*b^2*c^2)*arctan(1/3*(c^(1/3)-2*d^(1/3)
*x)/c^(1/3)*3^(1/2))/c^(8/3)/d^(7/3)*3^(1/2)

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Rubi [A]
time = 0.15, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {424, 393, 206, 31, 648, 631, 210, 642} \begin {gather*} -\frac {\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \text {ArcTan}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{9 \sqrt {3} c^{8/3} d^{7/3}}-\frac {\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{7/3}}+\frac {\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{7/3}}-\frac {x (b c-a d) (5 a d+4 b c)}{18 c^2 d^2 \left (c+d x^3\right )}-\frac {x \left (a+b x^3\right ) (b c-a d)}{6 c d \left (c+d x^3\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2/(c + d*x^3)^3,x]

[Out]

-1/6*((b*c - a*d)*x*(a + b*x^3))/(c*d*(c + d*x^3)^2) - ((b*c - a*d)*(4*b*c + 5*a*d)*x)/(18*c^2*d^2*(c + d*x^3)
) - ((2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(9*Sqrt[3]*c^(8/3)
*d^(7/3)) + ((2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*Log[c^(1/3) + d^(1/3)*x])/(27*c^(8/3)*d^(7/3)) - ((2*b^2*c^2
+ 2*a*b*c*d + 5*a^2*d^2)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(54*c^(8/3)*d^(7/3))

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 393

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 424

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a*d - c*b)*x*(a + b*x^n)^(
p + 1)*((c + d*x^n)^(q - 1)/(a*b*n*(p + 1))), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q
, x]

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 (a+b x^3\right )^2}{\left (c+d x^3\right )^3} \, dx &=-\frac {(b c-a d) x \left (a+b x^3\right )}{6 c d \left (c+d x^3\right )^2}+\frac {\int \frac {a (b c+5 a d)+2 b (2 b c+a d) x^3}{\left (c+d x^3\right )^2} \, dx}{6 c d}\\ &=-\frac {(b c-a d) x \left (a+b x^3\right )}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (4 b c+5 a d) x}{18 c^2 d^2 \left (c+d x^3\right )}+\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \int \frac {1}{c+d x^3} \, dx}{9 c^2 d^2}\\ &=-\frac {(b c-a d) x \left (a+b x^3\right )}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (4 b c+5 a d) x}{18 c^2 d^2 \left (c+d x^3\right )}+\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \int \frac {1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{27 c^{8/3} d^2}+\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \int \frac {2 \sqrt [3]{c}-\sqrt [3]{d} x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{27 c^{8/3} d^2}\\ &=-\frac {(b c-a d) x \left (a+b x^3\right )}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (4 b c+5 a d) x}{18 c^2 d^2 \left (c+d x^3\right )}+\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{7/3}}-\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \int \frac {-\sqrt [3]{c} \sqrt [3]{d}+2 d^{2/3} x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{54 c^{8/3} d^{7/3}}+\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \int \frac {1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{18 c^{7/3} d^2}\\ &=-\frac {(b c-a d) x \left (a+b x^3\right )}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (4 b c+5 a d) x}{18 c^2 d^2 \left (c+d x^3\right )}+\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{7/3}}-\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{7/3}}+\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{9 c^{8/3} d^{7/3}}\\ &=-\frac {(b c-a d) x \left (a+b x^3\right )}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (4 b c+5 a d) x}{18 c^2 d^2 \left (c+d x^3\right )}-\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{9 \sqrt {3} c^{8/3} d^{7/3}}+\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{7/3}}-\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{7/3}}\\ \end {align*}

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Mathematica [A]
time = 0.20, size = 234, normalized size = 0.91 \begin {gather*} \frac {-\frac {3 c^{2/3} \sqrt [3]{d} x \left (2 a b c d \left (2 c-d x^3\right )-a^2 d^2 \left (8 c+5 d x^3\right )+b^2 c^2 \left (4 c+7 d x^3\right )\right )}{\left (c+d x^3\right )^2}-2 \sqrt {3} \left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )+2 \left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )-\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{7/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2/(c + d*x^3)^3,x]

[Out]

((-3*c^(2/3)*d^(1/3)*x*(2*a*b*c*d*(2*c - d*x^3) - a^2*d^2*(8*c + 5*d*x^3) + b^2*c^2*(4*c + 7*d*x^3)))/(c + d*x
^3)^2 - 2*Sqrt[3]*(2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]] + 2*(2*b^2*c
^2 + 2*a*b*c*d + 5*a^2*d^2)*Log[c^(1/3) + d^(1/3)*x] - (2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*Log[c^(2/3) - c^(1/
3)*d^(1/3)*x + d^(2/3)*x^2])/(54*c^(8/3)*d^(7/3))

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Maple [A]
time = 0.27, size = 200, normalized size = 0.78

method	result	size
risch	\(\frac {\frac {\left (5 a^{2} d^{2}+2 a b c d -7 b^{2} c^{2}\right ) x^{4}}{18 c^{2} d}+\frac {2 \left (2 a^{2} d^{2}-a b c d -b^{2} c^{2}\right ) x}{9 d^{2} c}}{\left (d \,x^{3}+c \right )^{2}}+\frac {\munderset {\textit {\_R} =\RootOf \left (d \,\textit {\_Z}^{3}+c \right )}{\sum }\frac {\left (5 a^{2} d^{2}+2 a b c d +2 b^{2} c^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{27 c^{2} d^{3}}\)	\(131\)
default	\(\frac {\frac {\left (5 a^{2} d^{2}+2 a b c d -7 b^{2} c^{2}\right ) x^{4}}{18 c^{2} d}+\frac {2 \left (2 a^{2} d^{2}-a b c d -b^{2} c^{2}\right ) x}{9 d^{2} c}}{\left (d \,x^{3}+c \right )^{2}}+\frac {\left (5 a^{2} d^{2}+2 a b c d +2 b^{2} c^{2}\right ) \left (\frac {\ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}\right )}{9 c^{2} d^{2}}\)	\(200\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2/(d*x^3+c)^3,x,method=_RETURNVERBOSE)

[Out]

(1/18*(5*a^2*d^2+2*a*b*c*d-7*b^2*c^2)/c^2/d*x^4+2/9*(2*a^2*d^2-a*b*c*d-b^2*c^2)/d^2/c*x)/(d*x^3+c)^2+1/9*(5*a^
2*d^2+2*a*b*c*d+2*b^2*c^2)/c^2/d^2*(1/3/d/(c/d)^(2/3)*ln(x+(c/d)^(1/3))-1/6/d/(c/d)^(2/3)*ln(x^2-(c/d)^(1/3)*x
+(c/d)^(2/3))+1/3/d/(c/d)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1)))

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Maxima [A]
time = 0.52, size = 267, normalized size = 1.03 \begin {gather*} -\frac {{\left (7 \, b^{2} c^{2} d - 2 \, a b c d^{2} - 5 \, a^{2} d^{3}\right )} x^{4} + 4 \, {\left (b^{2} c^{3} + a b c^{2} d - 2 \, a^{2} c d^{2}\right )} x}{18 \, {\left (c^{2} d^{4} x^{6} + 2 \, c^{3} d^{3} x^{3} + c^{4} d^{2}\right )}} + \frac {\sqrt {3} {\left (2 \, b^{2} c^{2} + 2 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{27 \, c^{2} d^{3} \left (\frac {c}{d}\right )^{\frac {2}{3}}} - \frac {{\left (2 \, b^{2} c^{2} + 2 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{54 \, c^{2} d^{3} \left (\frac {c}{d}\right )^{\frac {2}{3}}} + \frac {{\left (2 \, b^{2} c^{2} + 2 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{27 \, c^{2} d^{3} \left (\frac {c}{d}\right )^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/(d*x^3+c)^3,x, algorithm="maxima")

[Out]

-1/18*((7*b^2*c^2*d - 2*a*b*c*d^2 - 5*a^2*d^3)*x^4 + 4*(b^2*c^3 + a*b*c^2*d - 2*a^2*c*d^2)*x)/(c^2*d^4*x^6 + 2
*c^3*d^3*x^3 + c^4*d^2) + 1/27*sqrt(3)*(2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*arctan(1/3*sqrt(3)*(2*x - (c/d)^(1/
3))/(c/d)^(1/3))/(c^2*d^3*(c/d)^(2/3)) - 1/54*(2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*log(x^2 - x*(c/d)^(1/3) + (c
/d)^(2/3))/(c^2*d^3*(c/d)^(2/3)) + 1/27*(2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*log(x + (c/d)^(1/3))/(c^2*d^3*(c/d
)^(2/3))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 513 vs. \(2 (217) = 434\).
time = 3.45, size = 1067, normalized size = 4.14 \begin {gather*} \left [-\frac {3 \, {\left (7 \, b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} - 5 \, a^{2} c^{2} d^{4}\right )} x^{4} - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, b^{2} c^{5} d + 2 \, a b c^{4} d^{2} + 5 \, a^{2} c^{3} d^{3} + {\left (2 \, b^{2} c^{3} d^{3} + 2 \, a b c^{2} d^{4} + 5 \, a^{2} c d^{5}\right )} x^{6} + 2 \, {\left (2 \, b^{2} c^{4} d^{2} + 2 \, a b c^{3} d^{3} + 5 \, a^{2} c^{2} d^{4}\right )} x^{3}\right )} \sqrt {-\frac {\left (c^{2} d\right )^{\frac {1}{3}}}{d}} \log \left (\frac {2 \, c d x^{3} - 3 \, \left (c^{2} d\right )^{\frac {1}{3}} c x - c^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, c d x^{2} + \left (c^{2} d\right )^{\frac {2}{3}} x - \left (c^{2} d\right )^{\frac {1}{3}} c\right )} \sqrt {-\frac {\left (c^{2} d\right )^{\frac {1}{3}}}{d}}}{d x^{3} + c}\right ) + {\left ({\left (2 \, b^{2} c^{2} d^{2} + 2 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} x^{6} + 2 \, b^{2} c^{4} + 2 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2} + 2 \, {\left (2 \, b^{2} c^{3} d + 2 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{3}\right )} \left (c^{2} d\right )^{\frac {2}{3}} \log \left (c d x^{2} - \left (c^{2} d\right )^{\frac {2}{3}} x + \left (c^{2} d\right )^{\frac {1}{3}} c\right ) - 2 \, {\left ({\left (2 \, b^{2} c^{2} d^{2} + 2 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} x^{6} + 2 \, b^{2} c^{4} + 2 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2} + 2 \, {\left (2 \, b^{2} c^{3} d + 2 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{3}\right )} \left (c^{2} d\right )^{\frac {2}{3}} \log \left (c d x + \left (c^{2} d\right )^{\frac {2}{3}}\right ) + 12 \, {\left (b^{2} c^{5} d + a b c^{4} d^{2} - 2 \, a^{2} c^{3} d^{3}\right )} x}{54 \, {\left (c^{4} d^{5} x^{6} + 2 \, c^{5} d^{4} x^{3} + c^{6} d^{3}\right )}}, -\frac {3 \, {\left (7 \, b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} - 5 \, a^{2} c^{2} d^{4}\right )} x^{4} - 6 \, \sqrt {\frac {1}{3}} {\left (2 \, b^{2} c^{5} d + 2 \, a b c^{4} d^{2} + 5 \, a^{2} c^{3} d^{3} + {\left (2 \, b^{2} c^{3} d^{3} + 2 \, a b c^{2} d^{4} + 5 \, a^{2} c d^{5}\right )} x^{6} + 2 \, {\left (2 \, b^{2} c^{4} d^{2} + 2 \, a b c^{3} d^{3} + 5 \, a^{2} c^{2} d^{4}\right )} x^{3}\right )} \sqrt {\frac {\left (c^{2} d\right )^{\frac {1}{3}}}{d}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (c^{2} d\right )^{\frac {2}{3}} x - \left (c^{2} d\right )^{\frac {1}{3}} c\right )} \sqrt {\frac {\left (c^{2} d\right )^{\frac {1}{3}}}{d}}}{c^{2}}\right ) + {\left ({\left (2 \, b^{2} c^{2} d^{2} + 2 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} x^{6} + 2 \, b^{2} c^{4} + 2 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2} + 2 \, {\left (2 \, b^{2} c^{3} d + 2 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{3}\right )} \left (c^{2} d\right )^{\frac {2}{3}} \log \left (c d x^{2} - \left (c^{2} d\right )^{\frac {2}{3}} x + \left (c^{2} d\right )^{\frac {1}{3}} c\right ) - 2 \, {\left ({\left (2 \, b^{2} c^{2} d^{2} + 2 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} x^{6} + 2 \, b^{2} c^{4} + 2 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2} + 2 \, {\left (2 \, b^{2} c^{3} d + 2 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{3}\right )} \left (c^{2} d\right )^{\frac {2}{3}} \log \left (c d x + \left (c^{2} d\right )^{\frac {2}{3}}\right ) + 12 \, {\left (b^{2} c^{5} d + a b c^{4} d^{2} - 2 \, a^{2} c^{3} d^{3}\right )} x}{54 \, {\left (c^{4} d^{5} x^{6} + 2 \, c^{5} d^{4} x^{3} + c^{6} d^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/(d*x^3+c)^3,x, algorithm="fricas")

[Out]

[-1/54*(3*(7*b^2*c^4*d^2 - 2*a*b*c^3*d^3 - 5*a^2*c^2*d^4)*x^4 - 3*sqrt(1/3)*(2*b^2*c^5*d + 2*a*b*c^4*d^2 + 5*a
^2*c^3*d^3 + (2*b^2*c^3*d^3 + 2*a*b*c^2*d^4 + 5*a^2*c*d^5)*x^6 + 2*(2*b^2*c^4*d^2 + 2*a*b*c^3*d^3 + 5*a^2*c^2*
d^4)*x^3)*sqrt(-(c^2*d)^(1/3)/d)*log((2*c*d*x^3 - 3*(c^2*d)^(1/3)*c*x - c^2 + 3*sqrt(1/3)*(2*c*d*x^2 + (c^2*d)
^(2/3)*x - (c^2*d)^(1/3)*c)*sqrt(-(c^2*d)^(1/3)/d))/(d*x^3 + c)) + ((2*b^2*c^2*d^2 + 2*a*b*c*d^3 + 5*a^2*d^4)*
x^6 + 2*b^2*c^4 + 2*a*b*c^3*d + 5*a^2*c^2*d^2 + 2*(2*b^2*c^3*d + 2*a*b*c^2*d^2 + 5*a^2*c*d^3)*x^3)*(c^2*d)^(2/
3)*log(c*d*x^2 - (c^2*d)^(2/3)*x + (c^2*d)^(1/3)*c) - 2*((2*b^2*c^2*d^2 + 2*a*b*c*d^3 + 5*a^2*d^4)*x^6 + 2*b^2
*c^4 + 2*a*b*c^3*d + 5*a^2*c^2*d^2 + 2*(2*b^2*c^3*d + 2*a*b*c^2*d^2 + 5*a^2*c*d^3)*x^3)*(c^2*d)^(2/3)*log(c*d*
x + (c^2*d)^(2/3)) + 12*(b^2*c^5*d + a*b*c^4*d^2 - 2*a^2*c^3*d^3)*x)/(c^4*d^5*x^6 + 2*c^5*d^4*x^3 + c^6*d^3),
-1/54*(3*(7*b^2*c^4*d^2 - 2*a*b*c^3*d^3 - 5*a^2*c^2*d^4)*x^4 - 6*sqrt(1/3)*(2*b^2*c^5*d + 2*a*b*c^4*d^2 + 5*a^
2*c^3*d^3 + (2*b^2*c^3*d^3 + 2*a*b*c^2*d^4 + 5*a^2*c*d^5)*x^6 + 2*(2*b^2*c^4*d^2 + 2*a*b*c^3*d^3 + 5*a^2*c^2*d
^4)*x^3)*sqrt((c^2*d)^(1/3)/d)*arctan(sqrt(1/3)*(2*(c^2*d)^(2/3)*x - (c^2*d)^(1/3)*c)*sqrt((c^2*d)^(1/3)/d)/c^
2) + ((2*b^2*c^2*d^2 + 2*a*b*c*d^3 + 5*a^2*d^4)*x^6 + 2*b^2*c^4 + 2*a*b*c^3*d + 5*a^2*c^2*d^2 + 2*(2*b^2*c^3*d
+ 2*a*b*c^2*d^2 + 5*a^2*c*d^3)*x^3)*(c^2*d)^(2/3)*log(c*d*x^2 - (c^2*d)^(2/3)*x + (c^2*d)^(1/3)*c) - 2*((2*b^
2*c^2*d^2 + 2*a*b*c*d^3 + 5*a^2*d^4)*x^6 + 2*b^2*c^4 + 2*a*b*c^3*d + 5*a^2*c^2*d^2 + 2*(2*b^2*c^3*d + 2*a*b*c^
2*d^2 + 5*a^2*c*d^3)*x^3)*(c^2*d)^(2/3)*log(c*d*x + (c^2*d)^(2/3)) + 12*(b^2*c^5*d + a*b*c^4*d^2 - 2*a^2*c^3*d
^3)*x)/(c^4*d^5*x^6 + 2*c^5*d^4*x^3 + c^6*d^3)]

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Sympy [A]
time = 1.18, size = 233, normalized size = 0.90 \begin {gather*} \frac {x^{4} \cdot \left (5 a^{2} d^{3} + 2 a b c d^{2} - 7 b^{2} c^{2} d\right ) + x \left (8 a^{2} c d^{2} - 4 a b c^{2} d - 4 b^{2} c^{3}\right )}{18 c^{4} d^{2} + 36 c^{3} d^{3} x^{3} + 18 c^{2} d^{4} x^{6}} + \operatorname {RootSum} {\left (19683 t^{3} c^{8} d^{7} - 125 a^{6} d^{6} - 150 a^{5} b c d^{5} - 210 a^{4} b^{2} c^{2} d^{4} - 128 a^{3} b^{3} c^{3} d^{3} - 84 a^{2} b^{4} c^{4} d^{2} - 24 a b^{5} c^{5} d - 8 b^{6} c^{6}, \left ( t \mapsto t \log {\left (\frac {27 t c^{3} d^{2}}{5 a^{2} d^{2} + 2 a b c d + 2 b^{2} c^{2}} + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x**4*(5*a**2*d**3 + 2*a*b*c*d**2 - 7*b**2*c**2*d) + x*(8*a**2*c*d**2 - 4*a*b*c**2*d - 4*b**2*c**3))/(18*c**4*
d**2 + 36*c**3*d**3*x**3 + 18*c**2*d**4*x**6) + RootSum(19683*_t**3*c**8*d**7 - 125*a**6*d**6 - 150*a**5*b*c*d
**5 - 210*a**4*b**2*c**2*d**4 - 128*a**3*b**3*c**3*d**3 - 84*a**2*b**4*c**4*d**2 - 24*a*b**5*c**5*d - 8*b**6*c
**6, Lambda(_t, _t*log(27*_t*c**3*d**2/(5*a**2*d**2 + 2*a*b*c*d + 2*b**2*c**2) + x)))

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Giac [A]
time = 0.69, size = 264, normalized size = 1.02 \begin {gather*} -\frac {\sqrt {3} {\left (2 \, b^{2} c^{2} + 2 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-c d^{2}\right )^{\frac {2}{3}} c^{2} d} - \frac {{\left (2 \, b^{2} c^{2} + 2 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right )}{54 \, \left (-c d^{2}\right )^{\frac {2}{3}} c^{2} d} - \frac {{\left (2 \, b^{2} c^{2} + 2 \, a b c d + 5 \, a^{2} d^{2}\right )} \left (-\frac {c}{d}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \right |}\right )}{27 \, c^{3} d^{2}} - \frac {7 \, b^{2} c^{2} d x^{4} - 2 \, a b c d^{2} x^{4} - 5 \, a^{2} d^{3} x^{4} + 4 \, b^{2} c^{3} x + 4 \, a b c^{2} d x - 8 \, a^{2} c d^{2} x}{18 \, {\left (d x^{3} + c\right )}^{2} c^{2} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/(d*x^3+c)^3,x, algorithm="giac")

[Out]

-1/27*sqrt(3)*(2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*arctan(1/3*sqrt(3)*(2*x + (-c/d)^(1/3))/(-c/d)^(1/3))/((-c*d
^2)^(2/3)*c^2*d) - 1/54*(2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*log(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/((-c*d^2)
^(2/3)*c^2*d) - 1/27*(2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*(-c/d)^(1/3)*log(abs(x - (-c/d)^(1/3)))/(c^3*d^2) - 1
/18*(7*b^2*c^2*d*x^4 - 2*a*b*c*d^2*x^4 - 5*a^2*d^3*x^4 + 4*b^2*c^3*x + 4*a*b*c^2*d*x - 8*a^2*c*d^2*x)/((d*x^3
+ c)^2*c^2*d^2)

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Mupad [B]
time = 1.43, size = 249, normalized size = 0.97 \begin {gather*} \frac {\ln \left (d^{1/3}\,x+c^{1/3}\right )\,\left (5\,a^2\,d^2+2\,a\,b\,c\,d+2\,b^2\,c^2\right )}{27\,c^{8/3}\,d^{7/3}}-\frac {\frac {2\,x\,\left (-2\,a^2\,d^2+a\,b\,c\,d+b^2\,c^2\right )}{9\,c\,d^2}-\frac {x^4\,\left (5\,a^2\,d^2+2\,a\,b\,c\,d-7\,b^2\,c^2\right )}{18\,c^2\,d}}{c^2+2\,c\,d\,x^3+d^2\,x^6}+\frac {\ln \left (2\,d^{1/3}\,x-c^{1/3}+\sqrt {3}\,c^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5\,a^2\,d^2+2\,a\,b\,c\,d+2\,b^2\,c^2\right )}{27\,c^{8/3}\,d^{7/3}}-\frac {\ln \left (c^{1/3}-2\,d^{1/3}\,x+\sqrt {3}\,c^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5\,a^2\,d^2+2\,a\,b\,c\,d+2\,b^2\,c^2\right )}{27\,c^{8/3}\,d^{7/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^2/(c + d*x^3)^3,x)

[Out]

(log(d^(1/3)*x + c^(1/3))*(5*a^2*d^2 + 2*b^2*c^2 + 2*a*b*c*d))/(27*c^(8/3)*d^(7/3)) - ((2*x*(b^2*c^2 - 2*a^2*d
^2 + a*b*c*d))/(9*c*d^2) - (x^4*(5*a^2*d^2 - 7*b^2*c^2 + 2*a*b*c*d))/(18*c^2*d))/(c^2 + d^2*x^6 + 2*c*d*x^3) +
(log(3^(1/2)*c^(1/3)*1i + 2*d^(1/3)*x - c^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(5*a^2*d^2 + 2*b^2*c^2 + 2*a*b*c*d))/
(27*c^(8/3)*d^(7/3)) - (log(3^(1/2)*c^(1/3)*1i - 2*d^(1/3)*x + c^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(5*a^2*d^2 + 2*
b^2*c^2 + 2*a*b*c*d))/(27*c^(8/3)*d^(7/3))

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