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

Optimal. Leaf size=234 \[ -\frac {(b c-a d)^2 (7 a d+2 b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{10/3}}+\frac {(b c-a d)^2 (7 a d+2 b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{10/3}}-\frac {(b c-a d)^2 (7 a d+2 b c) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{10/3}}+\frac {d^2 x (3 b c-2 a d)}{b^3}+\frac {x (b c-a d)^3}{3 a b^3 \left (a+b x^3\right )}+\frac {d^3 x^4}{4 b^2} \]

[Out]

d^2*(-2*a*d+3*b*c)*x/b^3+1/4*d^3*x^4/b^2+1/3*(-a*d+b*c)^3*x/a/b^3/(b*x^3+a)+1/9*(-a*d+b*c)^2*(7*a*d+2*b*c)*ln(
a^(1/3)+b^(1/3)*x)/a^(5/3)/b^(10/3)-1/18*(-a*d+b*c)^2*(7*a*d+2*b*c)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/
a^(5/3)/b^(10/3)-1/9*(-a*d+b*c)^2*(7*a*d+2*b*c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(5/3)/b^(1
0/3)*3^(1/2)

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Rubi [A]  time = 0.22, antiderivative size = 234, 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} \[ -\frac {(b c-a d)^2 (7 a d+2 b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{10/3}}+\frac {(b c-a d)^2 (7 a d+2 b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{10/3}}-\frac {(b c-a d)^2 (7 a d+2 b c) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{10/3}}+\frac {d^2 x (3 b c-2 a d)}{b^3}+\frac {x (b c-a d)^3}{3 a b^3 \left (a+b x^3\right )}+\frac {d^3 x^4}{4 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3)^3/(a + b*x^3)^2,x]

[Out]

(d^2*(3*b*c - 2*a*d)*x)/b^3 + (d^3*x^4)/(4*b^2) + ((b*c - a*d)^3*x)/(3*a*b^3*(a + b*x^3)) - ((b*c - a*d)^2*(2*
b*c + 7*a*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*b^(10/3)) + ((b*c - a*d)^2*
(2*b*c + 7*a*d)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(5/3)*b^(10/3)) - ((b*c - a*d)^2*(2*b*c + 7*a*d)*Log[a^(2/3) -
a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(5/3)*b^(10/3))

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 )^3}{\left (a+b x^3\right )^2} \, dx &=\int \left (\frac {d^2 (3 b c-2 a d)}{b^3}+\frac {d^3 x^3}{b^2}+\frac {(b c-a d)^2 (b c+2 a d)+3 b d (b c-a d)^2 x^3}{b^3 \left (a+b x^3\right )^2}\right ) \, dx\\ &=\frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^4}{4 b^2}+\frac {\int \frac {(b c-a d)^2 (b c+2 a d)+3 b d (b c-a d)^2 x^3}{\left (a+b x^3\right )^2} \, dx}{b^3}\\ &=\frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^4}{4 b^2}+\frac {(b c-a d)^3 x}{3 a b^3 \left (a+b x^3\right )}+\frac {\left ((b c-a d)^2 (2 b c+7 a d)\right ) \int \frac {1}{a+b x^3} \, dx}{3 a b^3}\\ &=\frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^4}{4 b^2}+\frac {(b c-a d)^3 x}{3 a b^3 \left (a+b x^3\right )}+\frac {\left ((b c-a d)^2 (2 b c+7 a d)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{5/3} b^3}+\frac {\left ((b c-a d)^2 (2 b c+7 a d)\right ) \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^3}\\ &=\frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^4}{4 b^2}+\frac {(b c-a d)^3 x}{3 a b^3 \left (a+b x^3\right )}+\frac {(b c-a d)^2 (2 b c+7 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{10/3}}-\frac {\left ((b c-a d)^2 (2 b c+7 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}{18 a^{5/3} b^{10/3}}+\frac {\left ((b c-a d)^2 (2 b c+7 a d)\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{4/3} b^3}\\ &=\frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^4}{4 b^2}+\frac {(b c-a d)^3 x}{3 a b^3 \left (a+b x^3\right )}+\frac {(b c-a d)^2 (2 b c+7 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{10/3}}-\frac {(b c-a d)^2 (2 b c+7 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{10/3}}+\frac {\left ((b c-a d)^2 (2 b c+7 a d)\right ) \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^{10/3}}\\ &=\frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^4}{4 b^2}+\frac {(b c-a d)^3 x}{3 a b^3 \left (a+b x^3\right )}-\frac {(b c-a d)^2 (2 b c+7 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^{10/3}}+\frac {(b c-a d)^2 (2 b c+7 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{10/3}}-\frac {(b c-a d)^2 (2 b c+7 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{10/3}}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 227, normalized size = 0.97 \[ \frac {-\frac {2 (b c-a d)^2 (7 a d+2 b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}+\frac {4 (b c-a d)^2 (7 a d+2 b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}+\frac {4 \sqrt {3} (b c-a d)^2 (7 a d+2 b c) \tan ^{-1}\left (\frac {2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{5/3}}+36 \sqrt [3]{b} d^2 x (3 b c-2 a d)+\frac {12 \sqrt [3]{b} x (b c-a d)^3}{a \left (a+b x^3\right )}+9 b^{4/3} d^3 x^4}{36 b^{10/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(36*b^(1/3)*d^2*(3*b*c - 2*a*d)*x + 9*b^(4/3)*d^3*x^4 + (12*b^(1/3)*(b*c - a*d)^3*x)/(a*(a + b*x^3)) + (4*Sqrt
[3]*(b*c - a*d)^2*(2*b*c + 7*a*d)*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/a^(5/3) + (4*(b*c - a*d)
^2*(2*b*c + 7*a*d)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) - (2*(b*c - a*d)^2*(2*b*c + 7*a*d)*Log[a^(2/3) - a^(1/3)*
b^(1/3)*x + b^(2/3)*x^2])/a^(5/3))/(36*b^(10/3))

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fricas [B]  time = 0.46, size = 1027, normalized size = 4.39 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/36*(9*a^3*b^3*d^3*x^7 + 9*(12*a^3*b^3*c*d^2 - 7*a^4*b^2*d^3)*x^4 + 6*sqrt(1/3)*(2*a^2*b^4*c^3 + 3*a^3*b^3*c
^2*d - 12*a^4*b^2*c*d^2 + 7*a^5*b*d^3 + (2*a*b^5*c^3 + 3*a^2*b^4*c^2*d - 12*a^3*b^3*c*d^2 + 7*a^4*b^2*d^3)*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)) - 2*(2*a*b^3*c^3 + 3*a^2*b^2*c^2*d - 12*a^3*b*c*d^2 +
7*a^4*d^3 + (2*b^4*c^3 + 3*a*b^3*c^2*d - 12*a^2*b^2*c*d^2 + 7*a^3*b*d^3)*x^3)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^
2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 4*(2*a*b^3*c^3 + 3*a^2*b^2*c^2*d - 12*a^3*b*c*d^2 + 7*a^4*d^3 + (2*b^4*c^3 +
3*a*b^3*c^2*d - 12*a^2*b^2*c*d^2 + 7*a^3*b*d^3)*x^3)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)) + 12*(a^2*b^4*c
^3 - 3*a^3*b^3*c^2*d + 12*a^4*b^2*c*d^2 - 7*a^5*b*d^3)*x)/(a^3*b^5*x^3 + a^4*b^4), 1/36*(9*a^3*b^3*d^3*x^7 + 9
*(12*a^3*b^3*c*d^2 - 7*a^4*b^2*d^3)*x^4 + 12*sqrt(1/3)*(2*a^2*b^4*c^3 + 3*a^3*b^3*c^2*d - 12*a^4*b^2*c*d^2 + 7
*a^5*b*d^3 + (2*a*b^5*c^3 + 3*a^2*b^4*c^2*d - 12*a^3*b^3*c*d^2 + 7*a^4*b^2*d^3)*x^3)*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) - 2*(2*a*b^3*c^3 + 3*a^2*b^2*c^
2*d - 12*a^3*b*c*d^2 + 7*a^4*d^3 + (2*b^4*c^3 + 3*a*b^3*c^2*d - 12*a^2*b^2*c*d^2 + 7*a^3*b*d^3)*x^3)*(a^2*b)^(
2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 4*(2*a*b^3*c^3 + 3*a^2*b^2*c^2*d - 12*a^3*b*c*d^2 + 7*
a^4*d^3 + (2*b^4*c^3 + 3*a*b^3*c^2*d - 12*a^2*b^2*c*d^2 + 7*a^3*b*d^3)*x^3)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^
(2/3)) + 12*(a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 12*a^4*b^2*c*d^2 - 7*a^5*b*d^3)*x)/(a^3*b^5*x^3 + a^4*b^4)]

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giac [A]  time = 0.20, size = 319, normalized size = 1.36 \[ -\frac {\sqrt {3} {\left (2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 12 \, a^{2} b c d^{2} + 7 \, a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2}} - \frac {{\left (2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 12 \, a^{2} b c d^{2} + 7 \, a^{3} d^{3}\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2}} - \frac {{\left (2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 12 \, a^{2} b c d^{2} + 7 \, a^{3} d^{3}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{2} b^{3}} + \frac {b^{3} c^{3} x - 3 \, a b^{2} c^{2} d x + 3 \, a^{2} b c d^{2} x - a^{3} d^{3} x}{3 \, {\left (b x^{3} + a\right )} a b^{3}} + \frac {b^{6} d^{3} x^{4} + 12 \, b^{6} c d^{2} x - 8 \, a b^{5} d^{3} x}{4 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*sqrt(3)*(2*b^3*c^3 + 3*a*b^2*c^2*d - 12*a^2*b*c*d^2 + 7*a^3*d^3)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/
(-a/b)^(1/3))/((-a*b^2)^(2/3)*a*b^2) - 1/18*(2*b^3*c^3 + 3*a*b^2*c^2*d - 12*a^2*b*c*d^2 + 7*a^3*d^3)*log(x^2 +
x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(2/3)*a*b^2) - 1/9*(2*b^3*c^3 + 3*a*b^2*c^2*d - 12*a^2*b*c*d^2 + 7*a
^3*d^3)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^2*b^3) + 1/3*(b^3*c^3*x - 3*a*b^2*c^2*d*x + 3*a^2*b*c*d^2*x
- a^3*d^3*x)/((b*x^3 + a)*a*b^3) + 1/4*(b^6*d^3*x^4 + 12*b^6*c*d^2*x - 8*a*b^5*d^3*x)/b^8

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maple [B]  time = 0.05, size = 529, normalized size = 2.26 \[ \frac {d^{3} x^{4}}{4 b^{2}}-\frac {a^{2} d^{3} x}{3 \left (b \,x^{3}+a \right ) b^{3}}+\frac {a c \,d^{2} x}{\left (b \,x^{3}+a \right ) b^{2}}+\frac {c^{3} x}{3 \left (b \,x^{3}+a \right ) a}-\frac {c^{2} d x}{\left (b \,x^{3}+a \right ) b}+\frac {7 \sqrt {3}\, a^{2} d^{3} \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^{4}}+\frac {7 a^{2} d^{3} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}-\frac {7 a^{2} d^{3} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}-\frac {4 \sqrt {3}\, a c \,d^{2} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}-\frac {4 a c \,d^{2} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}+\frac {2 a c \,d^{2} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}-\frac {2 a \,d^{3} x}{b^{3}}+\frac {2 \sqrt {3}\, c^{3} \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^{3} \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^{3} \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 {\sqrt {3}\, c^{2} d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}+\frac {c^{2} d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}-\frac {c^{2} d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}+\frac {3 c \,d^{2} x}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.22, size = 306, normalized size = 1.31 \[ \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{3 \, {\left (a b^{4} x^{3} + a^{2} b^{3}\right )}} + \frac {b d^{3} x^{4} + 4 \, {\left (3 \, b c d^{2} - 2 \, a d^{3}\right )} x}{4 \, b^{3}} + \frac {\sqrt {3} {\left (2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 12 \, a^{2} b c d^{2} + 7 \, a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 12 \, a^{2} b c d^{2} + 7 \, a^{3} d^{3}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 12 \, a^{2} b c d^{2} + 7 \, a^{3} d^{3}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x/(a*b^4*x^3 + a^2*b^3) + 1/4*(b*d^3*x^4 + 4*(3*b*c*d^
2 - 2*a*d^3)*x)/b^3 + 1/9*sqrt(3)*(2*b^3*c^3 + 3*a*b^2*c^2*d - 12*a^2*b*c*d^2 + 7*a^3*d^3)*arctan(1/3*sqrt(3)*
(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b^4*(a/b)^(2/3)) - 1/18*(2*b^3*c^3 + 3*a*b^2*c^2*d - 12*a^2*b*c*d^2 + 7*a^
3*d^3)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a*b^4*(a/b)^(2/3)) + 1/9*(2*b^3*c^3 + 3*a*b^2*c^2*d - 12*a^2*b*
c*d^2 + 7*a^3*d^3)*log(x + (a/b)^(1/3))/(a*b^4*(a/b)^(2/3))

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mupad [B]  time = 0.30, size = 240, normalized size = 1.03 \[ \frac {d^3\,x^4}{4\,b^2}-x\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )-\frac {x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{3\,a\,\left (b^4\,x^3+a\,b^3\right )}+\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (7\,a\,d+2\,b\,c\right )}{9\,a^{5/3}\,b^{10/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (7\,a\,d+2\,b\,c\right )}{9\,a^{5/3}\,b^{10/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (7\,a\,d+2\,b\,c\right )}{9\,a^{5/3}\,b^{10/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^3)^3/(a + b*x^3)^2,x)

[Out]

(d^3*x^4)/(4*b^2) - x*((2*a*d^3)/b^3 - (3*c*d^2)/b^2) - (x*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)
)/(3*a*(a*b^3 + b^4*x^3)) + (log(b^(1/3)*x + a^(1/3))*(a*d - b*c)^2*(7*a*d + 2*b*c))/(9*a^(5/3)*b^(10/3)) - (l
og(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*(7*a*d + 2*b*c))/(9*a^(5/3
)*b^(10/3)) + (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)^2*(7*a*d + 2
*b*c))/(9*a^(5/3)*b^(10/3))

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sympy [A]  time = 4.33, size = 291, normalized size = 1.24 \[ x \left (- \frac {2 a d^{3}}{b^{3}} + \frac {3 c d^{2}}{b^{2}}\right ) + \frac {x \left (- a^{3} d^{3} + 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + b^{3} c^{3}\right )}{3 a^{2} b^{3} + 3 a b^{4} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} a^{5} b^{10} - 343 a^{9} d^{9} + 1764 a^{8} b c d^{8} - 3465 a^{7} b^{2} c^{2} d^{7} + 2946 a^{6} b^{3} c^{3} d^{6} - 477 a^{5} b^{4} c^{4} d^{5} - 792 a^{4} b^{5} c^{5} d^{4} + 321 a^{3} b^{6} c^{6} d^{3} + 90 a^{2} b^{7} c^{7} d^{2} - 36 a b^{8} c^{8} d - 8 b^{9} c^{9}, \left (t \mapsto t \log {\left (\frac {9 t a^{2} b^{3}}{7 a^{3} d^{3} - 12 a^{2} b c d^{2} + 3 a b^{2} c^{2} d + 2 b^{3} c^{3}} + x \right )} \right )\right )} + \frac {d^{3} x^{4}}{4 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(-2*a*d**3/b**3 + 3*c*d**2/b**2) + x*(-a**3*d**3 + 3*a**2*b*c*d**2 - 3*a*b**2*c**2*d + b**3*c**3)/(3*a**2*b*
*3 + 3*a*b**4*x**3) + RootSum(729*_t**3*a**5*b**10 - 343*a**9*d**9 + 1764*a**8*b*c*d**8 - 3465*a**7*b**2*c**2*
d**7 + 2946*a**6*b**3*c**3*d**6 - 477*a**5*b**4*c**4*d**5 - 792*a**4*b**5*c**5*d**4 + 321*a**3*b**6*c**6*d**3
+ 90*a**2*b**7*c**7*d**2 - 36*a*b**8*c**8*d - 8*b**9*c**9, Lambda(_t, _t*log(9*_t*a**2*b**3/(7*a**3*d**3 - 12*
a**2*b*c*d**2 + 3*a*b**2*c**2*d + 2*b**3*c**3) + x))) + d**3*x**4/(4*b**2)

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