3.78 \(\int \frac{(c+d x^3)^2}{(a+b x^3)^{19/3}} \, dx\)

Optimal. Leaf size=253 \[ \frac{81 x \left (a^2 d^2+6 a b c d+45 b^2 c^2\right )}{7280 a^6 b^2 \sqrt [3]{a+b x^3}}+\frac{27 x \left (a^2 d^2+6 a b c d+45 b^2 c^2\right )}{7280 a^5 b^2 \left (a+b x^3\right )^{4/3}}+\frac{9 x \left (a^2 d^2+6 a b c d+45 b^2 c^2\right )}{3640 a^4 b^2 \left (a+b x^3\right )^{7/3}}+\frac{x \left (a^2 d^2+6 a b c d+45 b^2 c^2\right )}{520 a^3 b^2 \left (a+b x^3\right )^{10/3}}+\frac{x (b c-a d) (4 a d+15 b c)}{208 a^2 b^2 \left (a+b x^3\right )^{13/3}}+\frac{x \left (c+d x^3\right ) (b c-a d)}{16 a b \left (a+b x^3\right )^{16/3}} \]

[Out]

((b*c - a*d)*(15*b*c + 4*a*d)*x)/(208*a^2*b^2*(a + b*x^3)^(13/3)) + ((45*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x)/(52
0*a^3*b^2*(a + b*x^3)^(10/3)) + (9*(45*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x)/(3640*a^4*b^2*(a + b*x^3)^(7/3)) + (2
7*(45*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x)/(7280*a^5*b^2*(a + b*x^3)^(4/3)) + (81*(45*b^2*c^2 + 6*a*b*c*d + a^2*d
^2)*x)/(7280*a^6*b^2*(a + b*x^3)^(1/3)) + ((b*c - a*d)*x*(c + d*x^3))/(16*a*b*(a + b*x^3)^(16/3))

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Rubi [A]  time = 0.207149, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {413, 385, 192, 191} \[ \frac{81 x \left (a^2 d^2+6 a b c d+45 b^2 c^2\right )}{7280 a^6 b^2 \sqrt [3]{a+b x^3}}+\frac{27 x \left (a^2 d^2+6 a b c d+45 b^2 c^2\right )}{7280 a^5 b^2 \left (a+b x^3\right )^{4/3}}+\frac{9 x \left (a^2 d^2+6 a b c d+45 b^2 c^2\right )}{3640 a^4 b^2 \left (a+b x^3\right )^{7/3}}+\frac{x \left (a^2 d^2+6 a b c d+45 b^2 c^2\right )}{520 a^3 b^2 \left (a+b x^3\right )^{10/3}}+\frac{x (b c-a d) (4 a d+15 b c)}{208 a^2 b^2 \left (a+b x^3\right )^{13/3}}+\frac{x \left (c+d x^3\right ) (b c-a d)}{16 a b \left (a+b x^3\right )^{16/3}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3)^2/(a + b*x^3)^(19/3),x]

[Out]

((b*c - a*d)*(15*b*c + 4*a*d)*x)/(208*a^2*b^2*(a + b*x^3)^(13/3)) + ((45*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x)/(52
0*a^3*b^2*(a + b*x^3)^(10/3)) + (9*(45*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x)/(3640*a^4*b^2*(a + b*x^3)^(7/3)) + (2
7*(45*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x)/(7280*a^5*b^2*(a + b*x^3)^(4/3)) + (81*(45*b^2*c^2 + 6*a*b*c*d + a^2*d
^2)*x)/(7280*a^6*b^2*(a + b*x^3)^(1/3)) + ((b*c - a*d)*x*(c + d*x^3))/(16*a*b*(a + b*x^3)^(16/3))

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 )^{19/3}} \, dx &=\frac{(b c-a d) x \left (c+d x^3\right )}{16 a b \left (a+b x^3\right )^{16/3}}+\frac{\int \frac{c (15 b c+a d)+4 d (3 b c+a d) x^3}{\left (a+b x^3\right )^{16/3}} \, dx}{16 a b}\\ &=\frac{(b c-a d) (15 b c+4 a d) x}{208 a^2 b^2 \left (a+b x^3\right )^{13/3}}+\frac{(b c-a d) x \left (c+d x^3\right )}{16 a b \left (a+b x^3\right )^{16/3}}+\frac{\left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) \int \frac{1}{\left (a+b x^3\right )^{13/3}} \, dx}{52 a^2 b^2}\\ &=\frac{(b c-a d) (15 b c+4 a d) x}{208 a^2 b^2 \left (a+b x^3\right )^{13/3}}+\frac{\left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{520 a^3 b^2 \left (a+b x^3\right )^{10/3}}+\frac{(b c-a d) x \left (c+d x^3\right )}{16 a b \left (a+b x^3\right )^{16/3}}+\frac{\left (9 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right )\right ) \int \frac{1}{\left (a+b x^3\right )^{10/3}} \, dx}{520 a^3 b^2}\\ &=\frac{(b c-a d) (15 b c+4 a d) x}{208 a^2 b^2 \left (a+b x^3\right )^{13/3}}+\frac{\left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{520 a^3 b^2 \left (a+b x^3\right )^{10/3}}+\frac{9 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{3640 a^4 b^2 \left (a+b x^3\right )^{7/3}}+\frac{(b c-a d) x \left (c+d x^3\right )}{16 a b \left (a+b x^3\right )^{16/3}}+\frac{\left (27 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right )\right ) \int \frac{1}{\left (a+b x^3\right )^{7/3}} \, dx}{1820 a^4 b^2}\\ &=\frac{(b c-a d) (15 b c+4 a d) x}{208 a^2 b^2 \left (a+b x^3\right )^{13/3}}+\frac{\left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{520 a^3 b^2 \left (a+b x^3\right )^{10/3}}+\frac{9 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{3640 a^4 b^2 \left (a+b x^3\right )^{7/3}}+\frac{27 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{7280 a^5 b^2 \left (a+b x^3\right )^{4/3}}+\frac{(b c-a d) x \left (c+d x^3\right )}{16 a b \left (a+b x^3\right )^{16/3}}+\frac{\left (81 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right )\right ) \int \frac{1}{\left (a+b x^3\right )^{4/3}} \, dx}{7280 a^5 b^2}\\ &=\frac{(b c-a d) (15 b c+4 a d) x}{208 a^2 b^2 \left (a+b x^3\right )^{13/3}}+\frac{\left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{520 a^3 b^2 \left (a+b x^3\right )^{10/3}}+\frac{9 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{3640 a^4 b^2 \left (a+b x^3\right )^{7/3}}+\frac{27 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{7280 a^5 b^2 \left (a+b x^3\right )^{4/3}}+\frac{81 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{7280 a^6 b^2 \sqrt [3]{a+b x^3}}+\frac{(b c-a d) x \left (c+d x^3\right )}{16 a b \left (a+b x^3\right )^{16/3}}\\ \end{align*}

Mathematica [A]  time = 5.08619, size = 169, normalized size = 0.67 \[ \frac{x \left (81 a^2 b^3 x^9 \left (520 c^2+32 c d x^3+d^2 x^6\right )+144 a^3 b^2 x^6 \left (325 c^2+39 c d x^3+3 d^2 x^6\right )+156 a^4 b x^3 \left (175 c^2+40 c d x^3+6 d^2 x^6\right )+520 a^5 \left (14 c^2+7 c d x^3+2 d^2 x^6\right )+486 a b^4 c x^{12} \left (40 c+d x^3\right )+3645 b^5 c^2 x^{15}\right )}{7280 a^6 \left (a+b x^3\right )^{16/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3)^2/(a + b*x^3)^(19/3),x]

[Out]

(x*(3645*b^5*c^2*x^15 + 486*a*b^4*c*x^12*(40*c + d*x^3) + 81*a^2*b^3*x^9*(520*c^2 + 32*c*d*x^3 + d^2*x^6) + 52
0*a^5*(14*c^2 + 7*c*d*x^3 + 2*d^2*x^6) + 144*a^3*b^2*x^6*(325*c^2 + 39*c*d*x^3 + 3*d^2*x^6) + 156*a^4*b*x^3*(1
75*c^2 + 40*c*d*x^3 + 6*d^2*x^6)))/(7280*a^6*(a + b*x^3)^(16/3))

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Maple [A]  time = 0.008, size = 197, normalized size = 0.8 \begin{align*}{\frac{x \left ( 81\,{a}^{2}{b}^{3}{d}^{2}{x}^{15}+486\,a{b}^{4}cd{x}^{15}+3645\,{b}^{5}{c}^{2}{x}^{15}+432\,{a}^{3}{b}^{2}{d}^{2}{x}^{12}+2592\,{a}^{2}{b}^{3}cd{x}^{12}+19440\,a{b}^{4}{c}^{2}{x}^{12}+936\,{a}^{4}b{d}^{2}{x}^{9}+5616\,{a}^{3}{b}^{2}cd{x}^{9}+42120\,{a}^{2}{b}^{3}{c}^{2}{x}^{9}+1040\,{a}^{5}{d}^{2}{x}^{6}+6240\,{a}^{4}bcd{x}^{6}+46800\,{a}^{3}{b}^{2}{c}^{2}{x}^{6}+3640\,{a}^{5}cd{x}^{3}+27300\,{a}^{4}b{c}^{2}{x}^{3}+7280\,{c}^{2}{a}^{5} \right ) }{7280\,{a}^{6}} \left ( b{x}^{3}+a \right ) ^{-{\frac{16}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^3+c)^2/(b*x^3+a)^(19/3),x)

[Out]

1/7280*x*(81*a^2*b^3*d^2*x^15+486*a*b^4*c*d*x^15+3645*b^5*c^2*x^15+432*a^3*b^2*d^2*x^12+2592*a^2*b^3*c*d*x^12+
19440*a*b^4*c^2*x^12+936*a^4*b*d^2*x^9+5616*a^3*b^2*c*d*x^9+42120*a^2*b^3*c^2*x^9+1040*a^5*d^2*x^6+6240*a^4*b*
c*d*x^6+46800*a^3*b^2*c^2*x^6+3640*a^5*c*d*x^3+27300*a^4*b*c^2*x^3+7280*a^5*c^2)/(b*x^3+a)^(16/3)/a^6

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Maxima [A]  time = 0.969908, size = 352, normalized size = 1.39 \begin{align*} -\frac{{\left (455 \, b^{3} - \frac{1680 \,{\left (b x^{3} + a\right )} b^{2}}{x^{3}} + \frac{2184 \,{\left (b x^{3} + a\right )}^{2} b}{x^{6}} - \frac{1040 \,{\left (b x^{3} + a\right )}^{3}}{x^{9}}\right )} d^{2} x^{16}}{7280 \,{\left (b x^{3} + a\right )}^{\frac{16}{3}} a^{4}} + \frac{{\left (455 \, b^{4} - \frac{2240 \,{\left (b x^{3} + a\right )} b^{3}}{x^{3}} + \frac{4368 \,{\left (b x^{3} + a\right )}^{2} b^{2}}{x^{6}} - \frac{4160 \,{\left (b x^{3} + a\right )}^{3} b}{x^{9}} + \frac{1820 \,{\left (b x^{3} + a\right )}^{4}}{x^{12}}\right )} c d x^{16}}{3640 \,{\left (b x^{3} + a\right )}^{\frac{16}{3}} a^{5}} - \frac{{\left (91 \, b^{5} - \frac{560 \,{\left (b x^{3} + a\right )} b^{4}}{x^{3}} + \frac{1456 \,{\left (b x^{3} + a\right )}^{2} b^{3}}{x^{6}} - \frac{2080 \,{\left (b x^{3} + a\right )}^{3} b^{2}}{x^{9}} + \frac{1820 \,{\left (b x^{3} + a\right )}^{4} b}{x^{12}} - \frac{1456 \,{\left (b x^{3} + a\right )}^{5}}{x^{15}}\right )} c^{2} x^{16}}{1456 \,{\left (b x^{3} + a\right )}^{\frac{16}{3}} a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^2/(b*x^3+a)^(19/3),x, algorithm="maxima")

[Out]

-1/7280*(455*b^3 - 1680*(b*x^3 + a)*b^2/x^3 + 2184*(b*x^3 + a)^2*b/x^6 - 1040*(b*x^3 + a)^3/x^9)*d^2*x^16/((b*
x^3 + a)^(16/3)*a^4) + 1/3640*(455*b^4 - 2240*(b*x^3 + a)*b^3/x^3 + 4368*(b*x^3 + a)^2*b^2/x^6 - 4160*(b*x^3 +
a)^3*b/x^9 + 1820*(b*x^3 + a)^4/x^12)*c*d*x^16/((b*x^3 + a)^(16/3)*a^5) - 1/1456*(91*b^5 - 560*(b*x^3 + a)*b^
4/x^3 + 1456*(b*x^3 + a)^2*b^3/x^6 - 2080*(b*x^3 + a)^3*b^2/x^9 + 1820*(b*x^3 + a)^4*b/x^12 - 1456*(b*x^3 + a)
^5/x^15)*c^2*x^16/((b*x^3 + a)^(16/3)*a^6)

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Fricas [A]  time = 2.06553, size = 544, normalized size = 2.15 \begin{align*} \frac{{\left (81 \,{\left (45 \, b^{5} c^{2} + 6 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{16} + 432 \,{\left (45 \, a b^{4} c^{2} + 6 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x^{13} + 936 \,{\left (45 \, a^{2} b^{3} c^{2} + 6 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x^{10} + 7280 \, a^{5} c^{2} x + 1040 \,{\left (45 \, a^{3} b^{2} c^{2} + 6 \, a^{4} b c d + a^{5} d^{2}\right )} x^{7} + 1820 \,{\left (15 \, a^{4} b c^{2} + 2 \, a^{5} c d\right )} x^{4}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{7280 \,{\left (a^{6} b^{6} x^{18} + 6 \, a^{7} b^{5} x^{15} + 15 \, a^{8} b^{4} x^{12} + 20 \, a^{9} b^{3} x^{9} + 15 \, a^{10} b^{2} x^{6} + 6 \, a^{11} b x^{3} + a^{12}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^2/(b*x^3+a)^(19/3),x, algorithm="fricas")

[Out]

1/7280*(81*(45*b^5*c^2 + 6*a*b^4*c*d + a^2*b^3*d^2)*x^16 + 432*(45*a*b^4*c^2 + 6*a^2*b^3*c*d + a^3*b^2*d^2)*x^
13 + 936*(45*a^2*b^3*c^2 + 6*a^3*b^2*c*d + a^4*b*d^2)*x^10 + 7280*a^5*c^2*x + 1040*(45*a^3*b^2*c^2 + 6*a^4*b*c
*d + a^5*d^2)*x^7 + 1820*(15*a^4*b*c^2 + 2*a^5*c*d)*x^4)*(b*x^3 + a)^(2/3)/(a^6*b^6*x^18 + 6*a^7*b^5*x^15 + 15
*a^8*b^4*x^12 + 20*a^9*b^3*x^9 + 15*a^10*b^2*x^6 + 6*a^11*b*x^3 + a^12)

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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)**(19/3),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{19}{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)^(19/3),x, algorithm="giac")

[Out]

integrate((d*x^3 + c)^2/(b*x^3 + a)^(19/3), x)