Integrand size = 22, antiderivative size = 117 \[ \int \frac {\sqrt [3]{a+b x^3} \left (A+B x^3\right )}{x^{14}} \, dx=-\frac {A \left (a+b x^3\right )^{4/3}}{13 a x^{13}}+\frac {(9 A b-13 a B) \left (a+b x^3\right )^{4/3}}{130 a^2 x^{10}}-\frac {3 b (9 A b-13 a B) \left (a+b x^3\right )^{4/3}}{455 a^3 x^7}+\frac {9 b^2 (9 A b-13 a B) \left (a+b x^3\right )^{4/3}}{1820 a^4 x^4} \] Output:
-1/13*A*(b*x^3+a)^(4/3)/a/x^13+1/130*(9*A*b-13*B*a)*(b*x^3+a)^(4/3)/a^2/x^ 10-3/455*b*(9*A*b-13*B*a)*(b*x^3+a)^(4/3)/a^3/x^7+9/1820*b^2*(9*A*b-13*B*a )*(b*x^3+a)^(4/3)/a^4/x^4
Time = 0.31 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt [3]{a+b x^3} \left (A+B x^3\right )}{x^{14}} \, dx=\frac {\left (a+b x^3\right )^{4/3} \left (-140 a^3 A+126 a^2 A b x^3-182 a^3 B x^3-108 a A b^2 x^6+156 a^2 b B x^6+81 A b^3 x^9-117 a b^2 B x^9\right )}{1820 a^4 x^{13}} \] Input:
Integrate[((a + b*x^3)^(1/3)*(A + B*x^3))/x^14,x]
Output:
((a + b*x^3)^(4/3)*(-140*a^3*A + 126*a^2*A*b*x^3 - 182*a^3*B*x^3 - 108*a*A *b^2*x^6 + 156*a^2*b*B*x^6 + 81*A*b^3*x^9 - 117*a*b^2*B*x^9))/(1820*a^4*x^ 13)
Time = 0.40 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {955, 803, 803, 796}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt [3]{a+b x^3} \left (A+B x^3\right )}{x^{14}} \, dx\) |
\(\Big \downarrow \) 955 |
\(\displaystyle -\frac {(9 A b-13 a B) \int \frac {\sqrt [3]{b x^3+a}}{x^{11}}dx}{13 a}-\frac {A \left (a+b x^3\right )^{4/3}}{13 a x^{13}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {(9 A b-13 a B) \left (-\frac {3 b \int \frac {\sqrt [3]{b x^3+a}}{x^8}dx}{5 a}-\frac {\left (a+b x^3\right )^{4/3}}{10 a x^{10}}\right )}{13 a}-\frac {A \left (a+b x^3\right )^{4/3}}{13 a x^{13}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {(9 A b-13 a B) \left (-\frac {3 b \left (-\frac {3 b \int \frac {\sqrt [3]{b x^3+a}}{x^5}dx}{7 a}-\frac {\left (a+b x^3\right )^{4/3}}{7 a x^7}\right )}{5 a}-\frac {\left (a+b x^3\right )^{4/3}}{10 a x^{10}}\right )}{13 a}-\frac {A \left (a+b x^3\right )^{4/3}}{13 a x^{13}}\) |
\(\Big \downarrow \) 796 |
\(\displaystyle -\frac {\left (-\frac {3 b \left (\frac {3 b \left (a+b x^3\right )^{4/3}}{28 a^2 x^4}-\frac {\left (a+b x^3\right )^{4/3}}{7 a x^7}\right )}{5 a}-\frac {\left (a+b x^3\right )^{4/3}}{10 a x^{10}}\right ) (9 A b-13 a B)}{13 a}-\frac {A \left (a+b x^3\right )^{4/3}}{13 a x^{13}}\) |
Input:
Int[((a + b*x^3)^(1/3)*(A + B*x^3))/x^14,x]
Output:
-1/13*(A*(a + b*x^3)^(4/3))/(a*x^13) - ((9*A*b - 13*a*B)*(-1/10*(a + b*x^3 )^(4/3)/(a*x^10) - (3*b*(-1/7*(a + b*x^3)^(4/3)/(a*x^7) + (3*b*(a + b*x^3) ^(4/3))/(28*a^2*x^4)))/(5*a)))/(13*a)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*(( a + b*x^n)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*(m + 1 ))) Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x] && I LtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)) Int[(e *x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b* c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
Time = 1.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.63
method | result | size |
pseudoelliptic | \(-\frac {\left (\left (\frac {13 B \,x^{3}}{10}+A \right ) a^{3}-\frac {9 b \left (\frac {26 B \,x^{3}}{21}+A \right ) x^{3} a^{2}}{10}+\frac {27 \left (\frac {13 B \,x^{3}}{12}+A \right ) b^{2} x^{6} a}{35}-\frac {81 A \,x^{9} b^{3}}{140}\right ) \left (b \,x^{3}+a \right )^{\frac {4}{3}}}{13 x^{13} a^{4}}\) | \(74\) |
gosper | \(-\frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (-81 A \,x^{9} b^{3}+117 B \,x^{9} a \,b^{2}+108 A \,x^{6} a \,b^{2}-156 B \,x^{6} a^{2} b -126 a^{2} A b \,x^{3}+182 B \,x^{3} a^{3}+140 a^{3} A \right )}{1820 x^{13} a^{4}}\) | \(83\) |
orering | \(-\frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (-81 A \,x^{9} b^{3}+117 B \,x^{9} a \,b^{2}+108 A \,x^{6} a \,b^{2}-156 B \,x^{6} a^{2} b -126 a^{2} A b \,x^{3}+182 B \,x^{3} a^{3}+140 a^{3} A \right )}{1820 x^{13} a^{4}}\) | \(83\) |
trager | \(-\frac {\left (-81 A \,b^{4} x^{12}+117 B a \,b^{3} x^{12}+27 A a \,b^{3} x^{9}-39 B \,a^{2} b^{2} x^{9}-18 A \,a^{2} b^{2} x^{6}+26 B \,a^{3} b \,x^{6}+14 A \,a^{3} b \,x^{3}+182 B \,a^{4} x^{3}+140 A \,a^{4}\right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{1820 x^{13} a^{4}}\) | \(107\) |
risch | \(-\frac {\left (-81 A \,b^{4} x^{12}+117 B a \,b^{3} x^{12}+27 A a \,b^{3} x^{9}-39 B \,a^{2} b^{2} x^{9}-18 A \,a^{2} b^{2} x^{6}+26 B \,a^{3} b \,x^{6}+14 A \,a^{3} b \,x^{3}+182 B \,a^{4} x^{3}+140 A \,a^{4}\right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{1820 x^{13} a^{4}}\) | \(107\) |
Input:
int((b*x^3+a)^(1/3)*(B*x^3+A)/x^14,x,method=_RETURNVERBOSE)
Output:
-1/13*((13/10*B*x^3+A)*a^3-9/10*b*(26/21*B*x^3+A)*x^3*a^2+27/35*(13/12*B*x ^3+A)*b^2*x^6*a-81/140*A*x^9*b^3)*(b*x^3+a)^(4/3)/x^13/a^4
Time = 0.12 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt [3]{a+b x^3} \left (A+B x^3\right )}{x^{14}} \, dx=-\frac {{\left (9 \, {\left (13 \, B a b^{3} - 9 \, A b^{4}\right )} x^{12} - 3 \, {\left (13 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{9} + 2 \, {\left (13 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{6} + 140 \, A a^{4} + 14 \, {\left (13 \, B a^{4} + A a^{3} b\right )} x^{3}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{1820 \, a^{4} x^{13}} \] Input:
integrate((b*x^3+a)^(1/3)*(B*x^3+A)/x^14,x, algorithm="fricas")
Output:
-1/1820*(9*(13*B*a*b^3 - 9*A*b^4)*x^12 - 3*(13*B*a^2*b^2 - 9*A*a*b^3)*x^9 + 2*(13*B*a^3*b - 9*A*a^2*b^2)*x^6 + 140*A*a^4 + 14*(13*B*a^4 + A*a^3*b)*x ^3)*(b*x^3 + a)^(1/3)/(a^4*x^13)
Leaf count of result is larger than twice the leaf count of optimal. 1392 vs. \(2 (112) = 224\).
Time = 2.38 (sec) , antiderivative size = 1392, normalized size of antiderivative = 11.90 \[ \int \frac {\sqrt [3]{a+b x^3} \left (A+B x^3\right )}{x^{14}} \, dx=\text {Too large to display} \] Input:
integrate((b*x**3+a)**(1/3)*(B*x**3+A)/x**14,x)
Output:
-280*A*a**7*b**(28/3)*(a/(b*x**3) + 1)**(1/3)*gamma(-13/3)/(81*a**7*b**9*x **12*gamma(-1/3) + 243*a**6*b**10*x**15*gamma(-1/3) + 243*a**5*b**11*x**18 *gamma(-1/3) + 81*a**4*b**12*x**21*gamma(-1/3)) - 868*A*a**6*b**(31/3)*x** 3*(a/(b*x**3) + 1)**(1/3)*gamma(-13/3)/(81*a**7*b**9*x**12*gamma(-1/3) + 2 43*a**6*b**10*x**15*gamma(-1/3) + 243*a**5*b**11*x**18*gamma(-1/3) + 81*a* *4*b**12*x**21*gamma(-1/3)) - 888*A*a**5*b**(34/3)*x**6*(a/(b*x**3) + 1)** (1/3)*gamma(-13/3)/(81*a**7*b**9*x**12*gamma(-1/3) + 243*a**6*b**10*x**15* gamma(-1/3) + 243*a**5*b**11*x**18*gamma(-1/3) + 81*a**4*b**12*x**21*gamma (-1/3)) - 310*A*a**4*b**(37/3)*x**9*(a/(b*x**3) + 1)**(1/3)*gamma(-13/3)/( 81*a**7*b**9*x**12*gamma(-1/3) + 243*a**6*b**10*x**15*gamma(-1/3) + 243*a* *5*b**11*x**18*gamma(-1/3) + 81*a**4*b**12*x**21*gamma(-1/3)) + 80*A*a**3* b**(40/3)*x**12*(a/(b*x**3) + 1)**(1/3)*gamma(-13/3)/(81*a**7*b**9*x**12*g amma(-1/3) + 243*a**6*b**10*x**15*gamma(-1/3) + 243*a**5*b**11*x**18*gamma (-1/3) + 81*a**4*b**12*x**21*gamma(-1/3)) + 360*A*a**2*b**(43/3)*x**15*(a/ (b*x**3) + 1)**(1/3)*gamma(-13/3)/(81*a**7*b**9*x**12*gamma(-1/3) + 243*a* *6*b**10*x**15*gamma(-1/3) + 243*a**5*b**11*x**18*gamma(-1/3) + 81*a**4*b* *12*x**21*gamma(-1/3)) + 432*A*a*b**(46/3)*x**18*(a/(b*x**3) + 1)**(1/3)*g amma(-13/3)/(81*a**7*b**9*x**12*gamma(-1/3) + 243*a**6*b**10*x**15*gamma(- 1/3) + 243*a**5*b**11*x**18*gamma(-1/3) + 81*a**4*b**12*x**21*gamma(-1/3)) + 162*A*b**(49/3)*x**21*(a/(b*x**3) + 1)**(1/3)*gamma(-13/3)/(81*a**7*...
Time = 0.03 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt [3]{a+b x^3} \left (A+B x^3\right )}{x^{14}} \, dx=-\frac {B {\left (\frac {35 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} b^{2}}{x^{4}} - \frac {40 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} b}{x^{7}} + \frac {14 \, {\left (b x^{3} + a\right )}^{\frac {10}{3}}}{x^{10}}\right )}}{140 \, a^{3}} + \frac {A {\left (\frac {455 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} b^{3}}{x^{4}} - \frac {780 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} b^{2}}{x^{7}} + \frac {546 \, {\left (b x^{3} + a\right )}^{\frac {10}{3}} b}{x^{10}} - \frac {140 \, {\left (b x^{3} + a\right )}^{\frac {13}{3}}}{x^{13}}\right )}}{1820 \, a^{4}} \] Input:
integrate((b*x^3+a)^(1/3)*(B*x^3+A)/x^14,x, algorithm="maxima")
Output:
-1/140*B*(35*(b*x^3 + a)^(4/3)*b^2/x^4 - 40*(b*x^3 + a)^(7/3)*b/x^7 + 14*( b*x^3 + a)^(10/3)/x^10)/a^3 + 1/1820*A*(455*(b*x^3 + a)^(4/3)*b^3/x^4 - 78 0*(b*x^3 + a)^(7/3)*b^2/x^7 + 546*(b*x^3 + a)^(10/3)*b/x^10 - 140*(b*x^3 + a)^(13/3)/x^13)/a^4
\[ \int \frac {\sqrt [3]{a+b x^3} \left (A+B x^3\right )}{x^{14}} \, dx=\int { \frac {{\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x^{14}} \,d x } \] Input:
integrate((b*x^3+a)^(1/3)*(B*x^3+A)/x^14,x, algorithm="giac")
Output:
integrate((B*x^3 + A)*(b*x^3 + a)^(1/3)/x^14, x)
Time = 1.90 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.49 \[ \int \frac {\sqrt [3]{a+b x^3} \left (A+B x^3\right )}{x^{14}} \, dx=\frac {81\,A\,b^4\,{\left (b\,x^3+a\right )}^{1/3}}{1820\,a^4\,x}-\frac {B\,{\left (b\,x^3+a\right )}^{1/3}}{10\,x^{10}}-\frac {A\,b\,{\left (b\,x^3+a\right )}^{1/3}}{130\,a\,x^{10}}-\frac {B\,b\,{\left (b\,x^3+a\right )}^{1/3}}{70\,a\,x^7}-\frac {A\,{\left (b\,x^3+a\right )}^{1/3}}{13\,x^{13}}-\frac {27\,A\,b^3\,{\left (b\,x^3+a\right )}^{1/3}}{1820\,a^3\,x^4}+\frac {9\,A\,b^2\,{\left (b\,x^3+a\right )}^{1/3}}{910\,a^2\,x^7}-\frac {9\,B\,b^3\,{\left (b\,x^3+a\right )}^{1/3}}{140\,a^3\,x}+\frac {3\,B\,b^2\,{\left (b\,x^3+a\right )}^{1/3}}{140\,a^2\,x^4} \] Input:
int(((A + B*x^3)*(a + b*x^3)^(1/3))/x^14,x)
Output:
(81*A*b^4*(a + b*x^3)^(1/3))/(1820*a^4*x) - (B*(a + b*x^3)^(1/3))/(10*x^10 ) - (A*b*(a + b*x^3)^(1/3))/(130*a*x^10) - (B*b*(a + b*x^3)^(1/3))/(70*a*x ^7) - (A*(a + b*x^3)^(1/3))/(13*x^13) - (27*A*b^3*(a + b*x^3)^(1/3))/(1820 *a^3*x^4) + (9*A*b^2*(a + b*x^3)^(1/3))/(910*a^2*x^7) - (9*B*b^3*(a + b*x^ 3)^(1/3))/(140*a^3*x) + (3*B*b^2*(a + b*x^3)^(1/3))/(140*a^2*x^4)
Time = 0.22 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.51 \[ \int \frac {\sqrt [3]{a+b x^3} \left (A+B x^3\right )}{x^{14}} \, dx=\frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (-9 b^{4} x^{12}+3 a \,b^{3} x^{9}-2 a^{2} b^{2} x^{6}-49 a^{3} b \,x^{3}-35 a^{4}\right )}{455 a^{3} x^{13}} \] Input:
int((b*x^3+a)^(1/3)*(B*x^3+A)/x^14,x)
Output:
((a + b*x**3)**(1/3)*( - 35*a**4 - 49*a**3*b*x**3 - 2*a**2*b**2*x**6 + 3*a *b**3*x**9 - 9*b**4*x**12))/(455*a**3*x**13)