Integrand size = 22, antiderivative size = 83 \[ \int \frac {\left (a+b x^3\right )^3 \left (A+B x^3\right )}{\sqrt {x}} \, dx=2 a^3 A \sqrt {x}+\frac {2}{7} a^2 (3 A b+a B) x^{7/2}+\frac {6}{13} a b (A b+a B) x^{13/2}+\frac {2}{19} b^2 (A b+3 a B) x^{19/2}+\frac {2}{25} b^3 B x^{25/2} \] Output:
2*a^3*A*x^(1/2)+2/7*a^2*(3*A*b+B*a)*x^(7/2)+6/13*a*b*(A*b+B*a)*x^(13/2)+2/ 19*b^2*(A*b+3*B*a)*x^(19/2)+2/25*b^3*B*x^(25/2)
Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^3\right )^3 \left (A+B x^3\right )}{\sqrt {x}} \, dx=\frac {2 \sqrt {x} \left (6175 a^3 \left (7 A+B x^3\right )+1425 a^2 b x^3 \left (13 A+7 B x^3\right )+525 a b^2 x^6 \left (19 A+13 B x^3\right )+91 b^3 x^9 \left (25 A+19 B x^3\right )\right )}{43225} \] Input:
Integrate[((a + b*x^3)^3*(A + B*x^3))/Sqrt[x],x]
Output:
(2*Sqrt[x]*(6175*a^3*(7*A + B*x^3) + 1425*a^2*b*x^3*(13*A + 7*B*x^3) + 525 *a*b^2*x^6*(19*A + 13*B*x^3) + 91*b^3*x^9*(25*A + 19*B*x^3)))/43225
Time = 0.36 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {950, 2009}
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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^3 \left (A+B x^3\right )}{\sqrt {x}} \, dx\) |
\(\Big \downarrow \) 950 |
\(\displaystyle \int \left (\frac {a^3 A}{\sqrt {x}}+a^2 x^{5/2} (a B+3 A b)+b^2 x^{17/2} (3 a B+A b)+3 a b x^{11/2} (a B+A b)+b^3 B x^{23/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 a^3 A \sqrt {x}+\frac {2}{7} a^2 x^{7/2} (a B+3 A b)+\frac {2}{19} b^2 x^{19/2} (3 a B+A b)+\frac {6}{13} a b x^{13/2} (a B+A b)+\frac {2}{25} b^3 B x^{25/2}\) |
Input:
Int[((a + b*x^3)^3*(A + B*x^3))/Sqrt[x],x]
Output:
2*a^3*A*Sqrt[x] + (2*a^2*(3*A*b + a*B)*x^(7/2))/7 + (6*a*b*(A*b + a*B)*x^( 13/2))/13 + (2*b^2*(A*b + 3*a*B)*x^(19/2))/19 + (2*b^3*B*x^(25/2))/25
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^ n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && IGt Q[p, 0] && IGtQ[q, 0]
Time = 0.59 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(\frac {2 b^{3} B \,x^{\frac {25}{2}}}{25}+\frac {2 \left (b^{3} A +3 a \,b^{2} B \right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (3 a^{2} b A +a^{3} B \right ) x^{\frac {7}{2}}}{7}+2 a^{3} A \sqrt {x}\) | \(76\) |
default | \(\frac {2 b^{3} B \,x^{\frac {25}{2}}}{25}+\frac {2 \left (b^{3} A +3 a \,b^{2} B \right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (3 a^{2} b A +a^{3} B \right ) x^{\frac {7}{2}}}{7}+2 a^{3} A \sqrt {x}\) | \(76\) |
trager | \(\left (\frac {2}{25} B \,b^{3} x^{12}+\frac {2}{19} A \,x^{9} b^{3}+\frac {6}{19} B \,x^{9} a \,b^{2}+\frac {6}{13} A \,x^{6} a \,b^{2}+\frac {6}{13} B \,x^{6} a^{2} b +\frac {6}{7} a^{2} A b \,x^{3}+\frac {2}{7} B \,x^{3} a^{3}+2 a^{3} A \right ) \sqrt {x}\) | \(79\) |
gosper | \(\frac {2 \sqrt {x}\, \left (1729 B \,b^{3} x^{12}+2275 A \,x^{9} b^{3}+6825 B \,x^{9} a \,b^{2}+9975 A \,x^{6} a \,b^{2}+9975 B \,x^{6} a^{2} b +18525 a^{2} A b \,x^{3}+6175 B \,x^{3} a^{3}+43225 a^{3} A \right )}{43225}\) | \(80\) |
risch | \(\frac {2 \sqrt {x}\, \left (1729 B \,b^{3} x^{12}+2275 A \,x^{9} b^{3}+6825 B \,x^{9} a \,b^{2}+9975 A \,x^{6} a \,b^{2}+9975 B \,x^{6} a^{2} b +18525 a^{2} A b \,x^{3}+6175 B \,x^{3} a^{3}+43225 a^{3} A \right )}{43225}\) | \(80\) |
orering | \(\frac {2 \sqrt {x}\, \left (1729 B \,b^{3} x^{12}+2275 A \,x^{9} b^{3}+6825 B \,x^{9} a \,b^{2}+9975 A \,x^{6} a \,b^{2}+9975 B \,x^{6} a^{2} b +18525 a^{2} A b \,x^{3}+6175 B \,x^{3} a^{3}+43225 a^{3} A \right )}{43225}\) | \(80\) |
Input:
int((b*x^3+a)^3*(B*x^3+A)/x^(1/2),x,method=_RETURNVERBOSE)
Output:
2/25*b^3*B*x^(25/2)+2/19*(A*b^3+3*B*a*b^2)*x^(19/2)+2/13*(3*A*a*b^2+3*B*a^ 2*b)*x^(13/2)+2/7*(3*A*a^2*b+B*a^3)*x^(7/2)+2*a^3*A*x^(1/2)
Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^3 \left (A+B x^3\right )}{\sqrt {x}} \, dx=\frac {2}{43225} \, {\left (1729 \, B b^{3} x^{12} + 2275 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{9} + 9975 \, {\left (B a^{2} b + A a b^{2}\right )} x^{6} + 43225 \, A a^{3} + 6175 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3}\right )} \sqrt {x} \] Input:
integrate((b*x^3+a)^3*(B*x^3+A)/x^(1/2),x, algorithm="fricas")
Output:
2/43225*(1729*B*b^3*x^12 + 2275*(3*B*a*b^2 + A*b^3)*x^9 + 9975*(B*a^2*b + A*a*b^2)*x^6 + 43225*A*a^3 + 6175*(B*a^3 + 3*A*a^2*b)*x^3)*sqrt(x)
Time = 1.11 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.35 \[ \int \frac {\left (a+b x^3\right )^3 \left (A+B x^3\right )}{\sqrt {x}} \, dx=2 A a^{3} \sqrt {x} + \frac {6 A a^{2} b x^{\frac {7}{2}}}{7} + \frac {6 A a b^{2} x^{\frac {13}{2}}}{13} + \frac {2 A b^{3} x^{\frac {19}{2}}}{19} + \frac {2 B a^{3} x^{\frac {7}{2}}}{7} + \frac {6 B a^{2} b x^{\frac {13}{2}}}{13} + \frac {6 B a b^{2} x^{\frac {19}{2}}}{19} + \frac {2 B b^{3} x^{\frac {25}{2}}}{25} \] Input:
integrate((b*x**3+a)**3*(B*x**3+A)/x**(1/2),x)
Output:
2*A*a**3*sqrt(x) + 6*A*a**2*b*x**(7/2)/7 + 6*A*a*b**2*x**(13/2)/13 + 2*A*b **3*x**(19/2)/19 + 2*B*a**3*x**(7/2)/7 + 6*B*a**2*b*x**(13/2)/13 + 6*B*a*b **2*x**(19/2)/19 + 2*B*b**3*x**(25/2)/25
Time = 0.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^3\right )^3 \left (A+B x^3\right )}{\sqrt {x}} \, dx=\frac {2}{25} \, B b^{3} x^{\frac {25}{2}} + \frac {2}{19} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {19}{2}} + \frac {6}{13} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {13}{2}} + 2 \, A a^{3} \sqrt {x} + \frac {2}{7} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {7}{2}} \] Input:
integrate((b*x^3+a)^3*(B*x^3+A)/x^(1/2),x, algorithm="maxima")
Output:
2/25*B*b^3*x^(25/2) + 2/19*(3*B*a*b^2 + A*b^3)*x^(19/2) + 6/13*(B*a^2*b + A*a*b^2)*x^(13/2) + 2*A*a^3*sqrt(x) + 2/7*(B*a^3 + 3*A*a^2*b)*x^(7/2)
Time = 0.12 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^3\right )^3 \left (A+B x^3\right )}{\sqrt {x}} \, dx=\frac {2}{25} \, B b^{3} x^{\frac {25}{2}} + \frac {6}{19} \, B a b^{2} x^{\frac {19}{2}} + \frac {2}{19} \, A b^{3} x^{\frac {19}{2}} + \frac {6}{13} \, B a^{2} b x^{\frac {13}{2}} + \frac {6}{13} \, A a b^{2} x^{\frac {13}{2}} + \frac {2}{7} \, B a^{3} x^{\frac {7}{2}} + \frac {6}{7} \, A a^{2} b x^{\frac {7}{2}} + 2 \, A a^{3} \sqrt {x} \] Input:
integrate((b*x^3+a)^3*(B*x^3+A)/x^(1/2),x, algorithm="giac")
Output:
2/25*B*b^3*x^(25/2) + 6/19*B*a*b^2*x^(19/2) + 2/19*A*b^3*x^(19/2) + 6/13*B *a^2*b*x^(13/2) + 6/13*A*a*b^2*x^(13/2) + 2/7*B*a^3*x^(7/2) + 6/7*A*a^2*b* x^(7/2) + 2*A*a^3*sqrt(x)
Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b x^3\right )^3 \left (A+B x^3\right )}{\sqrt {x}} \, dx=x^{7/2}\,\left (\frac {2\,B\,a^3}{7}+\frac {6\,A\,b\,a^2}{7}\right )+x^{19/2}\,\left (\frac {2\,A\,b^3}{19}+\frac {6\,B\,a\,b^2}{19}\right )+2\,A\,a^3\,\sqrt {x}+\frac {2\,B\,b^3\,x^{25/2}}{25}+\frac {6\,a\,b\,x^{13/2}\,\left (A\,b+B\,a\right )}{13} \] Input:
int(((A + B*x^3)*(a + b*x^3)^3)/x^(1/2),x)
Output:
x^(7/2)*((2*B*a^3)/7 + (6*A*a^2*b)/7) + x^(19/2)*((2*A*b^3)/19 + (6*B*a*b^ 2)/19) + 2*A*a^3*x^(1/2) + (2*B*b^3*x^(25/2))/25 + (6*a*b*x^(13/2)*(A*b + B*a))/13
Time = 0.21 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.57 \[ \int \frac {\left (a+b x^3\right )^3 \left (A+B x^3\right )}{\sqrt {x}} \, dx=\frac {2 \sqrt {x}\, \left (1729 b^{4} x^{12}+9100 a \,b^{3} x^{9}+19950 a^{2} b^{2} x^{6}+24700 a^{3} b \,x^{3}+43225 a^{4}\right )}{43225} \] Input:
int((b*x^3+a)^3*(B*x^3+A)/x^(1/2),x)
Output:
(2*sqrt(x)*(43225*a**4 + 24700*a**3*b*x**3 + 19950*a**2*b**2*x**6 + 9100*a *b**3*x**9 + 1729*b**4*x**12))/43225