3.23 \(\int \frac {(c+d x^3)^2}{(a+b x^3)^2} \, dx\)

Optimal. Leaf size=203 \[ -\frac {(b c-a d) (2 a d+b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} b^{7/3}}+\frac {2 (b c-a d) (2 a d+b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{7/3}}-\frac {2 (b c-a d) (2 a d+b c) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{7/3}}+\frac {x (b c-a d)^2}{3 a b^2 \left (a+b x^3\right )}+\frac {d^2 x}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.23, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {390, 385, 200, 31, 634, 617, 204, 628} \[ -\frac {(b c-a d) (2 a d+b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} b^{7/3}}+\frac {2 (b c-a d) (2 a d+b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{7/3}}-\frac {2 (b c-a d) (2 a d+b c) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{7/3}}+\frac {x (b c-a d)^2}{3 a b^2 \left (a+b x^3\right )}+\frac {d^2 x}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3)^2/(a + b*x^3)^2,x]

[Out]

(d^2*x)/b^2 + ((b*c - a*d)^2*x)/(3*a*b^2*(a + b*x^3)) - (2*(b*c - a*d)*(b*c + 2*a*d)*ArcTan[(a^(1/3) - 2*b^(1/
3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*b^(7/3)) + (2*(b*c - a*d)*(b*c + 2*a*d)*Log[a^(1/3) + b^(1/3)*x])
/(9*a^(5/3)*b^(7/3)) - ((b*c - a*d)*(b*c + 2*a*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(9*a^(5/3)*b
^(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 200

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 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 385

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 390

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 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 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 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]

Rubi steps

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

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Mathematica [A]  time = 0.20, size = 205, normalized size = 1.01 \[ \frac {\frac {2 \left (-2 a^2 d^2+a b c d+b^2 c^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}-\frac {2 \sqrt {3} \left (-2 a^2 d^2+a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3}}-\frac {\left (-2 a^2 d^2+a b c d+b^2 c^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}+\frac {3 \sqrt [3]{b} x (b c-a d)^2}{a \left (a+b x^3\right )}+9 \sqrt [3]{b} d^2 x}{9 b^{7/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9*b^(1/3)*d^2*x + (3*b^(1/3)*(b*c - a*d)^2*x)/(a*(a + b*x^3)) - (2*Sqrt[3]*(b^2*c^2 + a*b*c*d - 2*a^2*d^2)*Ar
cTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(5/3) + (2*(b^2*c^2 + a*b*c*d - 2*a^2*d^2)*Log[a^(1/3) + b^(1/3)*
x])/a^(5/3) - ((b^2*c^2 + a*b*c*d - 2*a^2*d^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3))/(9*b^(
7/3))

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fricas [B]  time = 0.47, size = 768, normalized size = 3.78 \[ \left [\frac {9 \, a^{3} b^{2} d^{2} x^{4} - 3 \, \sqrt {\frac {1}{3}} {\left (a^{2} b^{3} c^{2} + a^{3} b^{2} c d - 2 \, a^{4} b d^{2} + {\left (a b^{4} c^{2} + a^{2} b^{3} c d - 2 \, a^{3} b^{2} d^{2}\right )} x^{3}\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 ) - {\left (a b^{2} c^{2} + a^{2} b c d - 2 \, a^{3} d^{2} + {\left (b^{3} c^{2} + a b^{2} c d - 2 \, a^{2} b d^{2}\right )} x^{3}\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 ) + 2 \, {\left (a b^{2} c^{2} + a^{2} b c d - 2 \, a^{3} d^{2} + {\left (b^{3} c^{2} + a b^{2} c d - 2 \, a^{2} b d^{2}\right )} x^{3}\right )} \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (-a^{2} b\right )^{\frac {2}{3}}\right ) + 3 \, {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + 4 \, a^{4} b d^{2}\right )} x}{9 \, {\left (a^{3} b^{4} x^{3} + a^{4} b^{3}\right )}}, \frac {9 \, a^{3} b^{2} d^{2} x^{4} + 6 \, \sqrt {\frac {1}{3}} {\left (a^{2} b^{3} c^{2} + a^{3} b^{2} c d - 2 \, a^{4} b d^{2} + {\left (a b^{4} c^{2} + a^{2} b^{3} c d - 2 \, a^{3} b^{2} d^{2}\right )} x^{3}\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 ) - {\left (a b^{2} c^{2} + a^{2} b c d - 2 \, a^{3} d^{2} + {\left (b^{3} c^{2} + a b^{2} c d - 2 \, a^{2} b d^{2}\right )} x^{3}\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 ) + 2 \, {\left (a b^{2} c^{2} + a^{2} b c d - 2 \, a^{3} d^{2} + {\left (b^{3} c^{2} + a b^{2} c d - 2 \, a^{2} b d^{2}\right )} x^{3}\right )} \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (-a^{2} b\right )^{\frac {2}{3}}\right ) + 3 \, {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + 4 \, a^{4} b d^{2}\right )} x}{9 \, {\left (a^{3} b^{4} x^{3} + a^{4} b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/9*(9*a^3*b^2*d^2*x^4 - 3*sqrt(1/3)*(a^2*b^3*c^2 + a^3*b^2*c*d - 2*a^4*b*d^2 + (a*b^4*c^2 + a^2*b^3*c*d - 2*
a^3*b^2*d^2)*x^3)*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)) - (a*b^2*c^2 + a^2*b*c*d - 2*a^3*d
^2 + (b^3*c^2 + a*b^2*c*d - 2*a^2*b*d^2)*x^3)*(-a^2*b)^(2/3)*log(a*b*x^2 - (-a^2*b)^(2/3)*x - (-a^2*b)^(1/3)*a
) + 2*(a*b^2*c^2 + a^2*b*c*d - 2*a^3*d^2 + (b^3*c^2 + a*b^2*c*d - 2*a^2*b*d^2)*x^3)*(-a^2*b)^(2/3)*log(a*b*x +
(-a^2*b)^(2/3)) + 3*(a^2*b^3*c^2 - 2*a^3*b^2*c*d + 4*a^4*b*d^2)*x)/(a^3*b^4*x^3 + a^4*b^3), 1/9*(9*a^3*b^2*d^
2*x^4 + 6*sqrt(1/3)*(a^2*b^3*c^2 + a^3*b^2*c*d - 2*a^4*b*d^2 + (a*b^4*c^2 + a^2*b^3*c*d - 2*a^3*b^2*d^2)*x^3)*
sqrt(-(-a^2*b)^(1/3)/b)*arctan(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)
- (a*b^2*c^2 + a^2*b*c*d - 2*a^3*d^2 + (b^3*c^2 + a*b^2*c*d - 2*a^2*b*d^2)*x^3)*(-a^2*b)^(2/3)*log(a*b*x^2 - (
-a^2*b)^(2/3)*x - (-a^2*b)^(1/3)*a) + 2*(a*b^2*c^2 + a^2*b*c*d - 2*a^3*d^2 + (b^3*c^2 + a*b^2*c*d - 2*a^2*b*d^
2)*x^3)*(-a^2*b)^(2/3)*log(a*b*x + (-a^2*b)^(2/3)) + 3*(a^2*b^3*c^2 - 2*a^3*b^2*c*d + 4*a^4*b*d^2)*x)/(a^3*b^4
*x^3 + a^4*b^3)]

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giac [A]  time = 0.21, size = 227, normalized size = 1.12 \[ \frac {d^{2} x}{b^{2}} - \frac {2 \, \sqrt {3} {\left (b^{2} c^{2} + a b c d - 2 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b} - \frac {{\left (b^{2} c^{2} + a b c d - 2 \, a^{2} d^{2}\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b} - \frac {2 \, {\left (b^{2} c^{2} + a b c d - 2 \, a^{2} d^{2}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{2} b^{2}} + \frac {b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{3 \, {\left (b x^{3} + a\right )} a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^2*x/b^2 - 2/9*sqrt(3)*(b^2*c^2 + a*b*c*d - 2*a^2*d^2)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/
((-a*b^2)^(2/3)*a*b) - 1/9*(b^2*c^2 + a*b*c*d - 2*a^2*d^2)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^
(2/3)*a*b) - 2/9*(b^2*c^2 + a*b*c*d - 2*a^2*d^2)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^2*b^2) + 1/3*(b^2*
c^2*x - 2*a*b*c*d*x + a^2*d^2*x)/((b*x^3 + a)*a*b^2)

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maple [B]  time = 0.05, size = 367, normalized size = 1.81 \[ \frac {a \,d^{2} x}{3 \left (b \,x^{3}+a \right ) b^{2}}+\frac {c^{2} x}{3 \left (b \,x^{3}+a \right ) a}-\frac {2 c d x}{3 \left (b \,x^{3}+a \right ) b}-\frac {4 \sqrt {3}\, a \,d^{2} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}-\frac {4 a \,d^{2} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}+\frac {2 a \,d^{2} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}+\frac {2 \sqrt {3}\, c^{2} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a b}+\frac {2 c^{2} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a b}-\frac {c^{2} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a b}+\frac {2 \sqrt {3}\, c d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}+\frac {2 c d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}-\frac {c d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}+\frac {d^{2} x}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.39, size = 220, normalized size = 1.08 \[ \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{3 \, {\left (a b^{3} x^{3} + a^{2} b^{2}\right )}} + \frac {d^{2} x}{b^{2}} + \frac {2 \, \sqrt {3} {\left (b^{2} c^{2} + a b c d - 2 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (b^{2} c^{2} + a b c d - 2 \, a^{2} d^{2}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {2 \, {\left (b^{2} c^{2} + a b c d - 2 \, a^{2} d^{2}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x/(a*b^3*x^3 + a^2*b^2) + d^2*x/b^2 + 2/9*sqrt(3)*(b^2*c^2 + a*b*c*d - 2*a
^2*d^2)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b^3*(a/b)^(2/3)) - 1/9*(b^2*c^2 + a*b*c*d - 2*a
^2*d^2)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a*b^3*(a/b)^(2/3)) + 2/9*(b^2*c^2 + a*b*c*d - 2*a^2*d^2)*log(x
+ (a/b)^(1/3))/(a*b^3*(a/b)^(2/3))

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mupad [B]  time = 1.47, size = 191, normalized size = 0.94 \[ \frac {d^2\,x}{b^2}+\frac {x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{3\,a\,\left (b^3\,x^3+a\,b^2\right )}-\frac {2\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (a\,d-b\,c\right )\,\left (2\,a\,d+b\,c\right )}{9\,a^{5/3}\,b^{7/3}}-\frac {2\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (a\,d-b\,c\right )\,\left (2\,a\,d+b\,c\right )}{9\,a^{5/3}\,b^{7/3}}+\frac {2\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (a\,d-b\,c\right )\,\left (2\,a\,d+b\,c\right )}{9\,a^{5/3}\,b^{7/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^3)^2/(a + b*x^3)^2,x)

[Out]

(d^2*x)/b^2 + (x*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(3*a*(a*b^2 + b^3*x^3)) - (2*log(b^(1/3)*x + a^(1/3))*(a*d -
b*c)*(2*a*d + b*c))/(9*a^(5/3)*b^(7/3)) - (2*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)*(2*a*d + b*c))/(9*a^(5/3)*b^(7/3)) + (2*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)*(2*a*d + b*c))/(9*a^(5/3)*b^(7/3))

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sympy [A]  time = 2.56, size = 189, normalized size = 0.93 \[ \frac {x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{3 a^{2} b^{2} + 3 a b^{3} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} a^{5} b^{7} + 64 a^{6} d^{6} - 96 a^{5} b c d^{5} - 48 a^{4} b^{2} c^{2} d^{4} + 88 a^{3} b^{3} c^{3} d^{3} + 24 a^{2} b^{4} c^{4} d^{2} - 24 a b^{5} c^{5} d - 8 b^{6} c^{6}, \left (t \mapsto t \log {\left (- \frac {9 t a^{2} b^{2}}{4 a^{2} d^{2} - 2 a b c d - 2 b^{2} c^{2}} + x \right )} \right )\right )} + \frac {d^{2} x}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(a**2*d**2 - 2*a*b*c*d + b**2*c**2)/(3*a**2*b**2 + 3*a*b**3*x**3) + RootSum(729*_t**3*a**5*b**7 + 64*a**6*d*
*6 - 96*a**5*b*c*d**5 - 48*a**4*b**2*c**2*d**4 + 88*a**3*b**3*c**3*d**3 + 24*a**2*b**4*c**4*d**2 - 24*a*b**5*c
**5*d - 8*b**6*c**6, Lambda(_t, _t*log(-9*_t*a**2*b**2/(4*a**2*d**2 - 2*a*b*c*d - 2*b**2*c**2) + x))) + d**2*x
/b**2

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