3.76 \(\int \frac{(c+d x^3)^2}{(a+b x^3)^{13/3}} \, dx\)

Optimal. Leaf size=174 \[ \frac{9 c^2 x (9 b c-10 a d)}{140 a^4 \sqrt [3]{a+b x^3} (b c-a d)}+\frac{x \left (c+d x^3\right )^2 (9 b c-10 a d)}{70 a^2 \left (a+b x^3\right )^{7/3} (b c-a d)}+\frac{3 c x \left (c+d x^3\right ) (9 b c-10 a d)}{140 a^3 \left (a+b x^3\right )^{4/3} (b c-a d)}+\frac{b x \left (c+d x^3\right )^3}{10 a \left (a+b x^3\right )^{10/3} (b c-a d)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0726912, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {382, 378, 191} \[ \frac{9 c^2 x (9 b c-10 a d)}{140 a^4 \sqrt [3]{a+b x^3} (b c-a d)}+\frac{x \left (c+d x^3\right )^2 (9 b c-10 a d)}{70 a^2 \left (a+b x^3\right )^{7/3} (b c-a d)}+\frac{3 c x \left (c+d x^3\right ) (9 b c-10 a d)}{140 a^3 \left (a+b x^3\right )^{4/3} (b c-a d)}+\frac{b x \left (c+d x^3\right )^3}{10 a \left (a+b x^3\right )^{10/3} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3)^2/(a + b*x^3)^(13/3),x]

[Out]

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

Rule 382

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

Rule 378

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^q)/(a*n*(p + 1)), x] - Dist[(c*q)/(a*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && GtQ[q, 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{\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{13/3}} \, dx &=\frac{b x \left (c+d x^3\right )^3}{10 a (b c-a d) \left (a+b x^3\right )^{10/3}}+\frac{(9 b c-10 a d) \int \frac{\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{10/3}} \, dx}{10 a (b c-a d)}\\ &=\frac{(9 b c-10 a d) x \left (c+d x^3\right )^2}{70 a^2 (b c-a d) \left (a+b x^3\right )^{7/3}}+\frac{b x \left (c+d x^3\right )^3}{10 a (b c-a d) \left (a+b x^3\right )^{10/3}}+\frac{(3 c (9 b c-10 a d)) \int \frac{c+d x^3}{\left (a+b x^3\right )^{7/3}} \, dx}{35 a^2 (b c-a d)}\\ &=\frac{3 c (9 b c-10 a d) x \left (c+d x^3\right )}{140 a^3 (b c-a d) \left (a+b x^3\right )^{4/3}}+\frac{(9 b c-10 a d) x \left (c+d x^3\right )^2}{70 a^2 (b c-a d) \left (a+b x^3\right )^{7/3}}+\frac{b x \left (c+d x^3\right )^3}{10 a (b c-a d) \left (a+b x^3\right )^{10/3}}+\frac{\left (9 c^2 (9 b c-10 a d)\right ) \int \frac{1}{\left (a+b x^3\right )^{4/3}} \, dx}{140 a^3 (b c-a d)}\\ &=\frac{9 c^2 (9 b c-10 a d) x}{140 a^4 (b c-a d) \sqrt [3]{a+b x^3}}+\frac{3 c (9 b c-10 a d) x \left (c+d x^3\right )}{140 a^3 (b c-a d) \left (a+b x^3\right )^{4/3}}+\frac{(9 b c-10 a d) x \left (c+d x^3\right )^2}{70 a^2 (b c-a d) \left (a+b x^3\right )^{7/3}}+\frac{b x \left (c+d x^3\right )^3}{10 a (b c-a d) \left (a+b x^3\right )^{10/3}}\\ \end{align*}

Mathematica [A]  time = 5.06123, size = 106, normalized size = 0.61 \[ \frac{x \left (3 a^2 b x^3 \left (105 c^2+20 c d x^3+2 d^2 x^6\right )+10 a^3 \left (14 c^2+7 c d x^3+2 d^2 x^6\right )+18 a b^2 c x^6 \left (15 c+d x^3\right )+81 b^3 c^2 x^9\right )}{140 a^4 \left (a+b x^3\right )^{10/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3)^2/(a + b*x^3)^(13/3),x]

[Out]

(x*(81*b^3*c^2*x^9 + 18*a*b^2*c*x^6*(15*c + d*x^3) + 10*a^3*(14*c^2 + 7*c*d*x^3 + 2*d^2*x^6) + 3*a^2*b*x^3*(10
5*c^2 + 20*c*d*x^3 + 2*d^2*x^6)))/(140*a^4*(a + b*x^3)^(10/3))

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 115, normalized size = 0.7 \begin{align*}{\frac{x \left ( 6\,{a}^{2}b{d}^{2}{x}^{9}+18\,a{b}^{2}cd{x}^{9}+81\,{b}^{3}{c}^{2}{x}^{9}+20\,{a}^{3}{d}^{2}{x}^{6}+60\,{a}^{2}bcd{x}^{6}+270\,a{b}^{2}{c}^{2}{x}^{6}+70\,{a}^{3}cd{x}^{3}+315\,{a}^{2}b{c}^{2}{x}^{3}+140\,{c}^{2}{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)^2/(b*x^3+a)^(13/3),x)

[Out]

1/140*x*(6*a^2*b*d^2*x^9+18*a*b^2*c*d*x^9+81*b^3*c^2*x^9+20*a^3*d^2*x^6+60*a^2*b*c*d*x^6+270*a*b^2*c^2*x^6+70*
a^3*c*d*x^3+315*a^2*b*c^2*x^3+140*a^3*c^2)/(b*x^3+a)^(10/3)/a^4

________________________________________________________________________________________

Maxima [A]  time = 0.971651, size = 215, normalized size = 1.24 \begin{align*} -\frac{{\left (7 \, b - \frac{10 \,{\left (b x^{3} + a\right )}}{x^{3}}\right )} d^{2} x^{10}}{70 \,{\left (b x^{3} + a\right )}^{\frac{10}{3}} a^{2}} + \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 )} c d x^{10}}{70 \,{\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^{2} 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)^2/(b*x^3+a)^(13/3),x, algorithm="maxima")

[Out]

-1/70*(7*b - 10*(b*x^3 + a)/x^3)*d^2*x^10/((b*x^3 + a)^(10/3)*a^2) + 1/70*(14*b^2 - 40*(b*x^3 + a)*b/x^3 + 35*
(b*x^3 + a)^2/x^6)*c*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^2*x^10/((b*x^3 + a)^(10/3)*a^4)

________________________________________________________________________________________

Fricas [A]  time = 1.73703, size = 328, normalized size = 1.89 \begin{align*} \frac{{\left (3 \,{\left (27 \, b^{3} c^{2} + 6 \, a b^{2} c d + 2 \, a^{2} b d^{2}\right )} x^{10} + 10 \,{\left (27 \, a b^{2} c^{2} + 6 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{7} + 140 \, a^{3} c^{2} x + 35 \,{\left (9 \, a^{2} b c^{2} + 2 \, a^{3} c 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)^2/(b*x^3+a)^(13/3),x, algorithm="fricas")

[Out]

1/140*(3*(27*b^3*c^2 + 6*a*b^2*c*d + 2*a^2*b*d^2)*x^10 + 10*(27*a*b^2*c^2 + 6*a^2*b*c*d + 2*a^3*d^2)*x^7 + 140
*a^3*c^2*x + 35*(9*a^2*b*c^2 + 2*a^3*c*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)**2/(b*x**3+a)**(13/3),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x^{3} + c\right )}^{2}}{{\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)^2/(b*x^3+a)^(13/3),x, algorithm="giac")

[Out]

integrate((d*x^3 + c)^2/(b*x^3 + a)^(13/3), x)