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

Optimal. Leaf size=208 \[ \frac{d x \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}-\frac{(b c-a d)^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{10/3}}+\frac{(b c-a d)^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{10/3}}-\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{10/3}}+\frac{d^2 x^4 (3 b c-a d)}{4 b^2}+\frac{d^3 x^7}{7 b} \]

[Out]

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

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Rubi [A]  time = 0.147699, antiderivative size = 208, normalized size of antiderivative = 1., 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 = {390, 200, 31, 634, 617, 204, 628} \[ \frac{d x \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}-\frac{(b c-a d)^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{10/3}}+\frac{(b c-a d)^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{10/3}}-\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{10/3}}+\frac{d^2 x^4 (3 b c-a d)}{4 b^2}+\frac{d^3 x^7}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3)^3/(a + b*x^3),x]

[Out]

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

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 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 31

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

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]

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 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 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]

Rubi steps

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

Mathematica [A]  time = 0.0911959, size = 203, normalized size = 0.98 \[ \frac{84 \sqrt [3]{b} d x \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )+\frac{14 (a d-b c)^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}+\frac{28 (b c-a d)^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+\frac{28 \sqrt{3} (b c-a d)^3 \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{2/3}}+21 b^{4/3} d^2 x^4 (3 b c-a d)+12 b^{7/3} d^3 x^7}{84 b^{10/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(84*b^(1/3)*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x + 21*b^(4/3)*d^2*(3*b*c - a*d)*x^4 + 12*b^(7/3)*d^3*x^7 + (2
8*Sqrt[3]*(b*c - a*d)^3*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/a^(2/3) + (28*(b*c - a*d)^3*Log[a^
(1/3) + b^(1/3)*x])/a^(2/3) + (14*(-(b*c) + a*d)^3*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3))/(8
4*b^(10/3))

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Maple [B]  time = 0.003, size = 486, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.78899, size = 1544, normalized size = 7.42 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/84*(12*a^2*b^3*d^3*x^7 + 21*(3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*x^4 - 42*sqrt(1/3)*(a*b^4*c^3 - 3*a^2*b^3*c^2*d
+ 3*a^3*b^2*c*d^2 - a^4*b*d^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)) - 14*(b^3*c^3 - 3*a*
b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(-a^2*b)^(2/3)*log(a*b*x^2 - (-a^2*b)^(2/3)*x - (-a^2*b)^(1/3)*a) + 28*(b
^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(-a^2*b)^(2/3)*log(a*b*x + (-a^2*b)^(2/3)) + 84*(3*a^2*b^3*c
^2*d - 3*a^3*b^2*c*d^2 + a^4*b*d^3)*x)/(a^2*b^4), 1/84*(12*a^2*b^3*d^3*x^7 + 21*(3*a^2*b^3*c*d^2 - a^3*b^2*d^3
)*x^4 + 84*sqrt(1/3)*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3*b^2*c*d^2 - a^4*b*d^3)*sqrt(-(-a^2*b)^(1/3)/b)*arcta
n(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) - 14*(b^3*c^3 - 3*a*b^2*c^2*d
+ 3*a^2*b*c*d^2 - a^3*d^3)*(-a^2*b)^(2/3)*log(a*b*x^2 - (-a^2*b)^(2/3)*x - (-a^2*b)^(1/3)*a) + 28*(b^3*c^3 -
3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(-a^2*b)^(2/3)*log(a*b*x + (-a^2*b)^(2/3)) + 84*(3*a^2*b^3*c^2*d - 3*
a^3*b^2*c*d^2 + a^4*b*d^3)*x)/(a^2*b^4)]

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Sympy [A]  time = 2.03412, size = 255, normalized size = 1.23 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} a^{2} b^{10} + a^{9} d^{9} - 9 a^{8} b c d^{8} + 36 a^{7} b^{2} c^{2} d^{7} - 84 a^{6} b^{3} c^{3} d^{6} + 126 a^{5} b^{4} c^{4} d^{5} - 126 a^{4} b^{5} c^{5} d^{4} + 84 a^{3} b^{6} c^{6} d^{3} - 36 a^{2} b^{7} c^{7} d^{2} + 9 a b^{8} c^{8} d - b^{9} c^{9}, \left ( t \mapsto t \log{\left (- \frac{3 t a b^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )} \right )\right )} + \frac{d^{3} x^{7}}{7 b} - \frac{x^{4} \left (a d^{3} - 3 b c d^{2}\right )}{4 b^{2}} + \frac{x \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(27*_t**3*a**2*b**10 + a**9*d**9 - 9*a**8*b*c*d**8 + 36*a**7*b**2*c**2*d**7 - 84*a**6*b**3*c**3*d**6 +
126*a**5*b**4*c**4*d**5 - 126*a**4*b**5*c**5*d**4 + 84*a**3*b**6*c**6*d**3 - 36*a**2*b**7*c**7*d**2 + 9*a*b**8
*c**8*d - b**9*c**9, Lambda(_t, _t*log(-3*_t*a*b**3/(a**3*d**3 - 3*a**2*b*c*d**2 + 3*a*b**2*c**2*d - b**3*c**3
) + x))) + d**3*x**7/(7*b) - x**4*(a*d**3 - 3*b*c*d**2)/(4*b**2) + x*(a**2*d**3 - 3*a*b*c*d**2 + 3*b**2*c**2*d
)/b**3

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Giac [B]  time = 1.10148, size = 473, normalized size = 2.27 \begin{align*} \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c^{3} - 3 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} c^{2} d + 3 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b c d^{2} - \left (-a b^{2}\right )^{\frac{1}{3}} 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 )}{3 \, a b^{4}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c^{3} - 3 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} c^{2} d + 3 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b c d^{2} - \left (-a b^{2}\right )^{\frac{1}{3}} 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 )}{6 \, a b^{4}} - \frac{{\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} 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 )}{3 \, a b^{7}} + \frac{4 \, b^{6} d^{3} x^{7} + 21 \, b^{6} c d^{2} x^{4} - 7 \, a b^{5} d^{3} x^{4} + 84 \, b^{6} c^{2} d x - 84 \, a b^{5} c d^{2} x + 28 \, a^{2} b^{4} d^{3} x}{28 \, b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(3)*((-a*b^2)^(1/3)*b^3*c^3 - 3*(-a*b^2)^(1/3)*a*b^2*c^2*d + 3*(-a*b^2)^(1/3)*a^2*b*c*d^2 - (-a*b^2)^(
1/3)*a^3*d^3)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^4) + 1/6*((-a*b^2)^(1/3)*b^3*c^3 - 3*
(-a*b^2)^(1/3)*a*b^2*c^2*d + 3*(-a*b^2)^(1/3)*a^2*b*c*d^2 - (-a*b^2)^(1/3)*a^3*d^3)*log(x^2 + x*(-a/b)^(1/3) +
(-a/b)^(2/3))/(a*b^4) - 1/3*(b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*(-a/b)^(1/3)*log(abs(x
- (-a/b)^(1/3)))/(a*b^7) + 1/28*(4*b^6*d^3*x^7 + 21*b^6*c*d^2*x^4 - 7*a*b^5*d^3*x^4 + 84*b^6*c^2*d*x - 84*a*b^
5*c*d^2*x + 28*a^2*b^4*d^3*x)/b^7