3.62 \(\int \frac{c+d x^3}{(a+b x^3)^{13/3}} \, dx\)

Optimal. Leaf size=121 \[ \frac{9 x (a d+9 b c)}{140 a^4 b \sqrt [3]{a+b x^3}}+\frac{3 x (a d+9 b c)}{140 a^3 b \left (a+b x^3\right )^{4/3}}+\frac{x (a d+9 b c)}{70 a^2 b \left (a+b x^3\right )^{7/3}}+\frac{x (b c-a d)}{10 a b \left (a+b x^3\right )^{10/3}} \]

[Out]

((b*c - a*d)*x)/(10*a*b*(a + b*x^3)^(10/3)) + ((9*b*c + a*d)*x)/(70*a^2*b*(a + b*x^3)^(7/3)) + (3*(9*b*c + a*d
)*x)/(140*a^3*b*(a + b*x^3)^(4/3)) + (9*(9*b*c + a*d)*x)/(140*a^4*b*(a + b*x^3)^(1/3))

________________________________________________________________________________________

Rubi [A]  time = 0.0352966, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {385, 192, 191} \[ \frac{9 x (a d+9 b c)}{140 a^4 b \sqrt [3]{a+b x^3}}+\frac{3 x (a d+9 b c)}{140 a^3 b \left (a+b x^3\right )^{4/3}}+\frac{x (a d+9 b c)}{70 a^2 b \left (a+b x^3\right )^{7/3}}+\frac{x (b c-a d)}{10 a b \left (a+b x^3\right )^{10/3}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3)/(a + b*x^3)^(13/3),x]

[Out]

((b*c - a*d)*x)/(10*a*b*(a + b*x^3)^(10/3)) + ((9*b*c + a*d)*x)/(70*a^2*b*(a + b*x^3)^(7/3)) + (3*(9*b*c + a*d
)*x)/(140*a^3*b*(a + b*x^3)^(4/3)) + (9*(9*b*c + a*d)*x)/(140*a^4*b*(a + b*x^3)^(1/3))

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 192

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

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{c+d x^3}{\left (a+b x^3\right )^{13/3}} \, dx &=\frac{(b c-a d) x}{10 a b \left (a+b x^3\right )^{10/3}}+\frac{(9 b c+a d) \int \frac{1}{\left (a+b x^3\right )^{10/3}} \, dx}{10 a b}\\ &=\frac{(b c-a d) x}{10 a b \left (a+b x^3\right )^{10/3}}+\frac{(9 b c+a d) x}{70 a^2 b \left (a+b x^3\right )^{7/3}}+\frac{(3 (9 b c+a d)) \int \frac{1}{\left (a+b x^3\right )^{7/3}} \, dx}{35 a^2 b}\\ &=\frac{(b c-a d) x}{10 a b \left (a+b x^3\right )^{10/3}}+\frac{(9 b c+a d) x}{70 a^2 b \left (a+b x^3\right )^{7/3}}+\frac{3 (9 b c+a d) x}{140 a^3 b \left (a+b x^3\right )^{4/3}}+\frac{(9 (9 b c+a d)) \int \frac{1}{\left (a+b x^3\right )^{4/3}} \, dx}{140 a^3 b}\\ &=\frac{(b c-a d) x}{10 a b \left (a+b x^3\right )^{10/3}}+\frac{(9 b c+a d) x}{70 a^2 b \left (a+b x^3\right )^{7/3}}+\frac{3 (9 b c+a d) x}{140 a^3 b \left (a+b x^3\right )^{4/3}}+\frac{9 (9 b c+a d) x}{140 a^4 b \sqrt [3]{a+b x^3}}\\ \end{align*}

Mathematica [A]  time = 0.0314036, size = 80, normalized size = 0.66 \[ \frac{x \left (15 a^2 b x^3 \left (21 c+2 d x^3\right )+35 a^3 \left (4 c+d x^3\right )+9 a b^2 x^6 \left (30 c+d x^3\right )+81 b^3 c x^9\right )}{140 a^4 \left (a+b x^3\right )^{10/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3)/(a + b*x^3)^(13/3),x]

[Out]

(x*(81*b^3*c*x^9 + 35*a^3*(4*c + d*x^3) + 9*a*b^2*x^6*(30*c + d*x^3) + 15*a^2*b*x^3*(21*c + 2*d*x^3)))/(140*a^
4*(a + b*x^3)^(10/3))

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 81, normalized size = 0.7 \begin{align*}{\frac{x \left ( 9\,a{b}^{2}d{x}^{9}+81\,{b}^{3}c{x}^{9}+30\,{a}^{2}bd{x}^{6}+270\,a{b}^{2}c{x}^{6}+35\,{a}^{3}d{x}^{3}+315\,{a}^{2}bc{x}^{3}+140\,c{a}^{3} \right ) }{140\,{a}^{4}} \left ( b{x}^{3}+a \right ) ^{-{\frac{10}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^3+c)/(b*x^3+a)^(13/3),x)

[Out]

1/140*x*(9*a*b^2*d*x^9+81*b^3*c*x^9+30*a^2*b*d*x^6+270*a*b^2*c*x^6+35*a^3*d*x^3+315*a^2*b*c*x^3+140*a^3*c)/(b*
x^3+a)^(10/3)/a^4

________________________________________________________________________________________

Maxima [A]  time = 0.94365, size = 162, normalized size = 1.34 \begin{align*} \frac{{\left (14 \, b^{2} - \frac{40 \,{\left (b x^{3} + a\right )} b}{x^{3}} + \frac{35 \,{\left (b x^{3} + a\right )}^{2}}{x^{6}}\right )} d x^{10}}{140 \,{\left (b x^{3} + a\right )}^{\frac{10}{3}} a^{3}} - \frac{{\left (14 \, b^{3} - \frac{60 \,{\left (b x^{3} + a\right )} b^{2}}{x^{3}} + \frac{105 \,{\left (b x^{3} + a\right )}^{2} b}{x^{6}} - \frac{140 \,{\left (b x^{3} + a\right )}^{3}}{x^{9}}\right )} c x^{10}}{140 \,{\left (b x^{3} + a\right )}^{\frac{10}{3}} a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)/(b*x^3+a)^(13/3),x, algorithm="maxima")

[Out]

1/140*(14*b^2 - 40*(b*x^3 + a)*b/x^3 + 35*(b*x^3 + a)^2/x^6)*d*x^10/((b*x^3 + a)^(10/3)*a^3) - 1/140*(14*b^3 -
60*(b*x^3 + a)*b^2/x^3 + 105*(b*x^3 + a)^2*b/x^6 - 140*(b*x^3 + a)^3/x^9)*c*x^10/((b*x^3 + a)^(10/3)*a^4)

________________________________________________________________________________________

Fricas [A]  time = 1.65692, size = 263, normalized size = 2.17 \begin{align*} \frac{{\left (9 \,{\left (9 \, b^{3} c + a b^{2} d\right )} x^{10} + 30 \,{\left (9 \, a b^{2} c + a^{2} b d\right )} x^{7} + 140 \, a^{3} c x + 35 \,{\left (9 \, a^{2} b c + a^{3} d\right )} x^{4}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{140 \,{\left (a^{4} b^{4} x^{12} + 4 \, a^{5} b^{3} x^{9} + 6 \, a^{6} b^{2} x^{6} + 4 \, a^{7} b x^{3} + a^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)/(b*x^3+a)^(13/3),x, algorithm="fricas")

[Out]

1/140*(9*(9*b^3*c + a*b^2*d)*x^10 + 30*(9*a*b^2*c + a^2*b*d)*x^7 + 140*a^3*c*x + 35*(9*a^2*b*c + a^3*d)*x^4)*(
b*x^3 + a)^(2/3)/(a^4*b^4*x^12 + 4*a^5*b^3*x^9 + 6*a^6*b^2*x^6 + 4*a^7*b*x^3 + a^8)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**3+c)/(b*x**3+a)**(13/3),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d x^{3} + c}{{\left (b x^{3} + a\right )}^{\frac{13}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)/(b*x^3+a)^(13/3),x, algorithm="giac")

[Out]

integrate((d*x^3 + c)/(b*x^3 + a)^(13/3), x)