3.77 \(\int \frac{(c+d x^3)^2}{(a+b x^3)^{16/3}} \, dx\)

Optimal. Leaf size=211 \[ \frac{9 x \left (2 a^2 d^2+9 a b c d+54 b^2 c^2\right )}{910 a^5 b^2 \sqrt [3]{a+b x^3}}+\frac{3 x \left (2 a^2 d^2+9 a b c d+54 b^2 c^2\right )}{910 a^4 b^2 \left (a+b x^3\right )^{4/3}}+\frac{x \left (2 a^2 d^2+9 a b c d+54 b^2 c^2\right )}{455 a^3 b^2 \left (a+b x^3\right )^{7/3}}+\frac{2 x (b c-a d) (a d+3 b c)}{65 a^2 b^2 \left (a+b x^3\right )^{10/3}}+\frac{x \left (c+d x^3\right ) (b c-a d)}{13 a b \left (a+b x^3\right )^{13/3}} \]

[Out]

(2*(b*c - a*d)*(3*b*c + a*d)*x)/(65*a^2*b^2*(a + b*x^3)^(10/3)) + ((54*b^2*c^2 + 9*a*b*c*d + 2*a^2*d^2)*x)/(45
5*a^3*b^2*(a + b*x^3)^(7/3)) + (3*(54*b^2*c^2 + 9*a*b*c*d + 2*a^2*d^2)*x)/(910*a^4*b^2*(a + b*x^3)^(4/3)) + (9
*(54*b^2*c^2 + 9*a*b*c*d + 2*a^2*d^2)*x)/(910*a^5*b^2*(a + b*x^3)^(1/3)) + ((b*c - a*d)*x*(c + d*x^3))/(13*a*b
*(a + b*x^3)^(13/3))

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Rubi [A]  time = 0.127353, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {413, 385, 192, 191} \[ \frac{9 x \left (2 a^2 d^2+9 a b c d+54 b^2 c^2\right )}{910 a^5 b^2 \sqrt [3]{a+b x^3}}+\frac{3 x \left (2 a^2 d^2+9 a b c d+54 b^2 c^2\right )}{910 a^4 b^2 \left (a+b x^3\right )^{4/3}}+\frac{x \left (2 a^2 d^2+9 a b c d+54 b^2 c^2\right )}{455 a^3 b^2 \left (a+b x^3\right )^{7/3}}+\frac{2 x (b c-a d) (a d+3 b c)}{65 a^2 b^2 \left (a+b x^3\right )^{10/3}}+\frac{x \left (c+d x^3\right ) (b c-a d)}{13 a b \left (a+b x^3\right )^{13/3}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3)^2/(a + b*x^3)^(16/3),x]

[Out]

(2*(b*c - a*d)*(3*b*c + a*d)*x)/(65*a^2*b^2*(a + b*x^3)^(10/3)) + ((54*b^2*c^2 + 9*a*b*c*d + 2*a^2*d^2)*x)/(45
5*a^3*b^2*(a + b*x^3)^(7/3)) + (3*(54*b^2*c^2 + 9*a*b*c*d + 2*a^2*d^2)*x)/(910*a^4*b^2*(a + b*x^3)^(4/3)) + (9
*(54*b^2*c^2 + 9*a*b*c*d + 2*a^2*d^2)*x)/(910*a^5*b^2*(a + b*x^3)^(1/3)) + ((b*c - a*d)*x*(c + d*x^3))/(13*a*b
*(a + b*x^3)^(13/3))

Rule 413

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

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{\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{16/3}} \, dx &=\frac{(b c-a d) x \left (c+d x^3\right )}{13 a b \left (a+b x^3\right )^{13/3}}+\frac{\int \frac{c (12 b c+a d)+d (9 b c+4 a d) x^3}{\left (a+b x^3\right )^{13/3}} \, dx}{13 a b}\\ &=\frac{2 (b c-a d) (3 b c+a d) x}{65 a^2 b^2 \left (a+b x^3\right )^{10/3}}+\frac{(b c-a d) x \left (c+d x^3\right )}{13 a b \left (a+b x^3\right )^{13/3}}+\frac{\left (54 b^2 c^2+9 a b c d+2 a^2 d^2\right ) \int \frac{1}{\left (a+b x^3\right )^{10/3}} \, dx}{65 a^2 b^2}\\ &=\frac{2 (b c-a d) (3 b c+a d) x}{65 a^2 b^2 \left (a+b x^3\right )^{10/3}}+\frac{\left (54 b^2 c^2+9 a b c d+2 a^2 d^2\right ) x}{455 a^3 b^2 \left (a+b x^3\right )^{7/3}}+\frac{(b c-a d) x \left (c+d x^3\right )}{13 a b \left (a+b x^3\right )^{13/3}}+\frac{\left (6 \left (54 b^2 c^2+9 a b c d+2 a^2 d^2\right )\right ) \int \frac{1}{\left (a+b x^3\right )^{7/3}} \, dx}{455 a^3 b^2}\\ &=\frac{2 (b c-a d) (3 b c+a d) x}{65 a^2 b^2 \left (a+b x^3\right )^{10/3}}+\frac{\left (54 b^2 c^2+9 a b c d+2 a^2 d^2\right ) x}{455 a^3 b^2 \left (a+b x^3\right )^{7/3}}+\frac{3 \left (54 b^2 c^2+9 a b c d+2 a^2 d^2\right ) x}{910 a^4 b^2 \left (a+b x^3\right )^{4/3}}+\frac{(b c-a d) x \left (c+d x^3\right )}{13 a b \left (a+b x^3\right )^{13/3}}+\frac{\left (9 \left (54 b^2 c^2+9 a b c d+2 a^2 d^2\right )\right ) \int \frac{1}{\left (a+b x^3\right )^{4/3}} \, dx}{910 a^4 b^2}\\ &=\frac{2 (b c-a d) (3 b c+a d) x}{65 a^2 b^2 \left (a+b x^3\right )^{10/3}}+\frac{\left (54 b^2 c^2+9 a b c d+2 a^2 d^2\right ) x}{455 a^3 b^2 \left (a+b x^3\right )^{7/3}}+\frac{3 \left (54 b^2 c^2+9 a b c d+2 a^2 d^2\right ) x}{910 a^4 b^2 \left (a+b x^3\right )^{4/3}}+\frac{9 \left (54 b^2 c^2+9 a b c d+2 a^2 d^2\right ) x}{910 a^5 b^2 \sqrt [3]{a+b x^3}}+\frac{(b c-a d) x \left (c+d x^3\right )}{13 a b \left (a+b x^3\right )^{13/3}}\\ \end{align*}

Mathematica [A]  time = 5.07386, size = 138, normalized size = 0.65 \[ \frac{x \left (9 a^2 b^2 x^6 \left (390 c^2+39 c d x^3+2 d^2 x^6\right )+39 a^3 b x^3 \left (70 c^2+15 c d x^3+2 d^2 x^6\right )+65 a^4 \left (14 c^2+7 c d x^3+2 d^2 x^6\right )+81 a b^3 c x^9 \left (26 c+d x^3\right )+486 b^4 c^2 x^{12}\right )}{910 a^5 \left (a+b x^3\right )^{13/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3)^2/(a + b*x^3)^(16/3),x]

[Out]

(x*(486*b^4*c^2*x^12 + 81*a*b^3*c*x^9*(26*c + d*x^3) + 65*a^4*(14*c^2 + 7*c*d*x^3 + 2*d^2*x^6) + 39*a^3*b*x^3*
(70*c^2 + 15*c*d*x^3 + 2*d^2*x^6) + 9*a^2*b^2*x^6*(390*c^2 + 39*c*d*x^3 + 2*d^2*x^6)))/(910*a^5*(a + b*x^3)^(1
3/3))

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Maple [A]  time = 0.007, size = 156, normalized size = 0.7 \begin{align*}{\frac{x \left ( 18\,{a}^{2}{b}^{2}{d}^{2}{x}^{12}+81\,a{b}^{3}cd{x}^{12}+486\,{b}^{4}{c}^{2}{x}^{12}+78\,{a}^{3}b{d}^{2}{x}^{9}+351\,{a}^{2}{b}^{2}cd{x}^{9}+2106\,a{b}^{3}{c}^{2}{x}^{9}+130\,{a}^{4}{d}^{2}{x}^{6}+585\,{a}^{3}bcd{x}^{6}+3510\,{a}^{2}{b}^{2}{c}^{2}{x}^{6}+455\,{a}^{4}cd{x}^{3}+2730\,{a}^{3}b{c}^{2}{x}^{3}+910\,{c}^{2}{a}^{4} \right ) }{910\,{a}^{5}} \left ( b{x}^{3}+a \right ) ^{-{\frac{13}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^3+c)^2/(b*x^3+a)^(16/3),x)

[Out]

1/910*x*(18*a^2*b^2*d^2*x^12+81*a*b^3*c*d*x^12+486*b^4*c^2*x^12+78*a^3*b*d^2*x^9+351*a^2*b^2*c*d*x^9+2106*a*b^
3*c^2*x^9+130*a^4*d^2*x^6+585*a^3*b*c*d*x^6+3510*a^2*b^2*c^2*x^6+455*a^4*c*d*x^3+2730*a^3*b*c^2*x^3+910*a^4*c^
2)/(b*x^3+a)^(13/3)/a^5

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Maxima [A]  time = 0.977636, size = 284, normalized size = 1.35 \begin{align*} \frac{{\left (35 \, b^{2} - \frac{91 \,{\left (b x^{3} + a\right )} b}{x^{3}} + \frac{65 \,{\left (b x^{3} + a\right )}^{2}}{x^{6}}\right )} d^{2} x^{13}}{455 \,{\left (b x^{3} + a\right )}^{\frac{13}{3}} a^{3}} - \frac{{\left (140 \, b^{3} - \frac{546 \,{\left (b x^{3} + a\right )} b^{2}}{x^{3}} + \frac{780 \,{\left (b x^{3} + a\right )}^{2} b}{x^{6}} - \frac{455 \,{\left (b x^{3} + a\right )}^{3}}{x^{9}}\right )} c d x^{13}}{910 \,{\left (b x^{3} + a\right )}^{\frac{13}{3}} a^{4}} + \frac{{\left (35 \, b^{4} - \frac{182 \,{\left (b x^{3} + a\right )} b^{3}}{x^{3}} + \frac{390 \,{\left (b x^{3} + a\right )}^{2} b^{2}}{x^{6}} - \frac{455 \,{\left (b x^{3} + a\right )}^{3} b}{x^{9}} + \frac{455 \,{\left (b x^{3} + a\right )}^{4}}{x^{12}}\right )} c^{2} x^{13}}{455 \,{\left (b x^{3} + a\right )}^{\frac{13}{3}} a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^2/(b*x^3+a)^(16/3),x, algorithm="maxima")

[Out]

1/455*(35*b^2 - 91*(b*x^3 + a)*b/x^3 + 65*(b*x^3 + a)^2/x^6)*d^2*x^13/((b*x^3 + a)^(13/3)*a^3) - 1/910*(140*b^
3 - 546*(b*x^3 + a)*b^2/x^3 + 780*(b*x^3 + a)^2*b/x^6 - 455*(b*x^3 + a)^3/x^9)*c*d*x^13/((b*x^3 + a)^(13/3)*a^
4) + 1/455*(35*b^4 - 182*(b*x^3 + a)*b^3/x^3 + 390*(b*x^3 + a)^2*b^2/x^6 - 455*(b*x^3 + a)^3*b/x^9 + 455*(b*x^
3 + a)^4/x^12)*c^2*x^13/((b*x^3 + a)^(13/3)*a^5)

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Fricas [A]  time = 1.84741, size = 433, normalized size = 2.05 \begin{align*} \frac{{\left (9 \,{\left (54 \, b^{4} c^{2} + 9 \, a b^{3} c d + 2 \, a^{2} b^{2} d^{2}\right )} x^{13} + 39 \,{\left (54 \, a b^{3} c^{2} + 9 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{10} + 65 \,{\left (54 \, a^{2} b^{2} c^{2} + 9 \, a^{3} b c d + 2 \, a^{4} d^{2}\right )} x^{7} + 910 \, a^{4} c^{2} x + 455 \,{\left (6 \, a^{3} b c^{2} + a^{4} c d\right )} x^{4}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{910 \,{\left (a^{5} b^{5} x^{15} + 5 \, a^{6} b^{4} x^{12} + 10 \, a^{7} b^{3} x^{9} + 10 \, a^{8} b^{2} x^{6} + 5 \, a^{9} b x^{3} + a^{10}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^2/(b*x^3+a)^(16/3),x, algorithm="fricas")

[Out]

1/910*(9*(54*b^4*c^2 + 9*a*b^3*c*d + 2*a^2*b^2*d^2)*x^13 + 39*(54*a*b^3*c^2 + 9*a^2*b^2*c*d + 2*a^3*b*d^2)*x^1
0 + 65*(54*a^2*b^2*c^2 + 9*a^3*b*c*d + 2*a^4*d^2)*x^7 + 910*a^4*c^2*x + 455*(6*a^3*b*c^2 + a^4*c*d)*x^4)*(b*x^
3 + a)^(2/3)/(a^5*b^5*x^15 + 5*a^6*b^4*x^12 + 10*a^7*b^3*x^9 + 10*a^8*b^2*x^6 + 5*a^9*b*x^3 + a^10)

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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)**(16/3),x)

[Out]

Timed out

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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{16}{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)^(16/3),x, algorithm="giac")

[Out]

integrate((d*x^3 + c)^2/(b*x^3 + a)^(16/3), x)