Integrand size = 22, antiderivative size = 121 \[ \int \sqrt {x} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\frac {2}{3} a^3 A x^{3/2}+\frac {2 a^2 (3 A b+a B) x^{\frac {3}{2}+n}}{3+2 n}+\frac {6 a b (A b+a B) x^{\frac {3}{2}+2 n}}{3+4 n}+\frac {2 b^2 (A b+3 a B) x^{\frac {3}{2}+3 n}}{3 (1+2 n)}+\frac {2 b^3 B x^{\frac {3}{2}+4 n}}{3+8 n} \] Output:
2/3*a^3*A*x^(3/2)+2*a^2*(3*A*b+B*a)*x^(3/2+n)/(3+2*n)+6*a*b*(A*b+B*a)*x^(3 /2+2*n)/(3+4*n)+2*b^2*(A*b+3*B*a)*x^(3/2+3*n)/(3+6*n)+2*b^3*B*x^(3/2+4*n)/ (3+8*n)
Time = 0.32 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.98 \[ \int \sqrt {x} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=2 \left (\frac {1}{3} a^3 A x^{3/2}+\frac {a^2 (3 A b+a B) x^{\frac {3}{2}+n}}{3+2 n}+\frac {3 a b (A b+a B) x^{\frac {3}{2}+2 n}}{3+4 n}+\frac {b^2 (A b+3 a B) x^{\frac {3}{2}+3 n}}{3+6 n}+\frac {b^3 B x^{\frac {3}{2}+4 n}}{3+8 n}\right ) \] Input:
Integrate[Sqrt[x]*(a + b*x^n)^3*(A + B*x^n),x]
Output:
2*((a^3*A*x^(3/2))/3 + (a^2*(3*A*b + a*B)*x^(3/2 + n))/(3 + 2*n) + (3*a*b* (A*b + a*B)*x^(3/2 + 2*n))/(3 + 4*n) + (b^2*(A*b + 3*a*B)*x^(3/2 + 3*n))/( 3 + 6*n) + (b^3*B*x^(3/2 + 4*n))/(3 + 8*n))
Time = 0.42 (sec) , antiderivative size = 121, 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.
\(\displaystyle \int \sqrt {x} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx\) |
\(\Big \downarrow \) 950 |
\(\displaystyle \int \left (a^3 A \sqrt {x}+a^2 x^{n+\frac {1}{2}} (a B+3 A b)+b^2 x^{3 n+\frac {1}{2}} (3 a B+A b)+3 a b x^{2 n+\frac {1}{2}} (a B+A b)+b^3 B x^{4 n+\frac {1}{2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{3} a^3 A x^{3/2}+\frac {2 a^2 x^{n+\frac {3}{2}} (a B+3 A b)}{2 n+3}+\frac {2 b^2 x^{3 n+\frac {3}{2}} (3 a B+A b)}{3 (2 n+1)}+\frac {6 a b x^{2 n+\frac {3}{2}} (a B+A b)}{4 n+3}+\frac {2 b^3 B x^{4 n+\frac {3}{2}}}{8 n+3}\) |
Input:
Int[Sqrt[x]*(a + b*x^n)^3*(A + B*x^n),x]
Output:
(2*a^3*A*x^(3/2))/3 + (2*a^2*(3*A*b + a*B)*x^(3/2 + n))/(3 + 2*n) + (6*a*b *(A*b + a*B)*x^(3/2 + 2*n))/(3 + 4*n) + (2*b^2*(A*b + 3*a*B)*x^(3/2 + 3*n) )/(3*(1 + 2*n)) + (2*b^3*B*x^(3/2 + 4*n))/(3 + 8*n)
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1189\) vs. \(2(107)=214\).
Time = 0.56 (sec) , antiderivative size = 1190, normalized size of antiderivative = 9.83
Input:
int(x^(1/2)*(a+b*x^n)^3*(A+B*x^n),x,method=_RETURNVERBOSE)
Output:
2/9*(192*n^3+704*n^2+558*n+121)*x^(3/2)/(3+4*n)/(16*n^2+30*n+9)*(a+b*x^n)^ 3*(A+B*x^n)-40/9*x^2*(40*n^3+42*n^2+20*n+3)/(32*n^3+76*n^2+48*n+9)/(3+4*n) *(1/2*(a+b*x^n)^3*(A+B*x^n)/x^(1/2)+3/x^(1/2)*(a+b*x^n)^2*(A+B*x^n)*b*x^n* n+1/x^(1/2)*(a+b*x^n)^3*B*x^n*n)+80/9*x^3*(14*n^2+1)/(32*n^3+76*n^2+48*n+9 )/(3+4*n)*(-1/4*(a+b*x^n)^3*(A+B*x^n)/x^(3/2)+6/x^(3/2)*(a+b*x^n)*(A+B*x^n )*b^2*(x^n)^2*n^2+6/x^(3/2)*(a+b*x^n)^2*B*(x^n)^2*n^2*b+3/x^(3/2)*(a+b*x^n )^2*(A+B*x^n)*b*x^n*n^2+1/x^(3/2)*(a+b*x^n)^3*B*x^n*n^2)-80/9*x^4*(-1+4*n) /(128*n^4+400*n^3+420*n^2+180*n+27)*(-3/4*(a+b*x^n)^2*(A+B*x^n)/x^(5/2)*b* x^n*n-1/4*(a+b*x^n)^3*B*x^n*n/x^(5/2)+3/8*(a+b*x^n)^3*(A+B*x^n)/x^(5/2)-9* (a+b*x^n)*(A+B*x^n)/x^(5/2)*b^2*(x^n)^2*n^2+6/x^(5/2)*b^3*(x^n)^3*n^3*(A+B *x^n)+18/x^(5/2)*(a+b*x^n)*B*(x^n)^3*n^3*b^2+18/x^(5/2)*(a+b*x^n)*(A+B*x^n )*b^2*(x^n)^2*n^3-9*(a+b*x^n)^2*B*(x^n)^2*n^2/x^(5/2)*b+18/x^(5/2)*(a+b*x^ n)^2*B*(x^n)^2*n^3*b-9/2*(a+b*x^n)^2*(A+B*x^n)/x^(5/2)*b*x^n*n^2+3/x^(5/2) *(a+b*x^n)^2*(A+B*x^n)*b*x^n*n^3-3/2*(a+b*x^n)^3*B*x^n*n^2/x^(5/2)+1/x^(5/ 2)*(a+b*x^n)^3*B*x^n*n^3)+32/9/(128*n^4+400*n^3+420*n^2+180*n+27)*x^5*(-72 *(a+b*x^n)^2*B*(x^n)^2*n^3/x^(7/2)*b+21*(a+b*x^n)*(A+B*x^n)/x^(7/2)*b^2*(x ^n)^2*n^2+108/x^(7/2)*(a+b*x^n)*B*(x^n)^3*n^4*b^2+42/x^(7/2)*(a+b*x^n)*(A+ B*x^n)*b^2*(x^n)^2*n^4-24*b^3*(x^n)^3*n^3/x^(7/2)*(A+B*x^n)-72*(a+b*x^n)*B *(x^n)^3*n^3/x^(7/2)*b^2-72*(a+b*x^n)*(A+B*x^n)/x^(7/2)*b^2*(x^n)^2*n^3+42 /x^(7/2)*(a+b*x^n)^2*B*(x^n)^2*n^4*b+21*(a+b*x^n)^2*B*(x^n)^2*n^2/x^(7/...
Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (107) = 214\).
Time = 0.12 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.79 \[ \int \sqrt {x} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\frac {2 \, {\left (3 \, {\left (16 \, B b^{3} n^{3} + 44 \, B b^{3} n^{2} + 36 \, B b^{3} n + 9 \, B b^{3}\right )} x^{\frac {3}{2}} x^{4 \, n} + {\left (81 \, B a b^{2} + 27 \, A b^{3} + 64 \, {\left (3 \, B a b^{2} + A b^{3}\right )} n^{3} + 168 \, {\left (3 \, B a b^{2} + A b^{3}\right )} n^{2} + 126 \, {\left (3 \, B a b^{2} + A b^{3}\right )} n\right )} x^{\frac {3}{2}} x^{3 \, n} + 9 \, {\left (9 \, B a^{2} b + 9 \, A a b^{2} + 32 \, {\left (B a^{2} b + A a b^{2}\right )} n^{3} + 76 \, {\left (B a^{2} b + A a b^{2}\right )} n^{2} + 48 \, {\left (B a^{2} b + A a b^{2}\right )} n\right )} x^{\frac {3}{2}} x^{2 \, n} + 3 \, {\left (9 \, B a^{3} + 27 \, A a^{2} b + 64 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} n^{3} + 104 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} n^{2} + 54 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} n\right )} x^{\frac {3}{2}} x^{n} + {\left (128 \, A a^{3} n^{4} + 400 \, A a^{3} n^{3} + 420 \, A a^{3} n^{2} + 180 \, A a^{3} n + 27 \, A a^{3}\right )} x^{\frac {3}{2}}\right )}}{3 \, {\left (128 \, n^{4} + 400 \, n^{3} + 420 \, n^{2} + 180 \, n + 27\right )}} \] Input:
integrate(x^(1/2)*(a+b*x^n)^3*(A+B*x^n),x, algorithm="fricas")
Output:
2/3*(3*(16*B*b^3*n^3 + 44*B*b^3*n^2 + 36*B*b^3*n + 9*B*b^3)*x^(3/2)*x^(4*n ) + (81*B*a*b^2 + 27*A*b^3 + 64*(3*B*a*b^2 + A*b^3)*n^3 + 168*(3*B*a*b^2 + A*b^3)*n^2 + 126*(3*B*a*b^2 + A*b^3)*n)*x^(3/2)*x^(3*n) + 9*(9*B*a^2*b + 9*A*a*b^2 + 32*(B*a^2*b + A*a*b^2)*n^3 + 76*(B*a^2*b + A*a*b^2)*n^2 + 48*( B*a^2*b + A*a*b^2)*n)*x^(3/2)*x^(2*n) + 3*(9*B*a^3 + 27*A*a^2*b + 64*(B*a^ 3 + 3*A*a^2*b)*n^3 + 104*(B*a^3 + 3*A*a^2*b)*n^2 + 54*(B*a^3 + 3*A*a^2*b)* n)*x^(3/2)*x^n + (128*A*a^3*n^4 + 400*A*a^3*n^3 + 420*A*a^3*n^2 + 180*A*a^ 3*n + 27*A*a^3)*x^(3/2))/(128*n^4 + 400*n^3 + 420*n^2 + 180*n + 27)
Time = 2.15 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.55 \[ \int \sqrt {x} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\frac {2 A a^{3} x^{\frac {3}{2}}}{3} + 6 A a^{2} b \left (\begin {cases} \frac {x^{\frac {3}{2}} x^{n}}{2 n + 3} & \text {for}\: n \neq - \frac {3}{2} \\x^{\frac {3}{2}} x^{n} \log {\left (\sqrt {x} \right )} & \text {otherwise} \end {cases}\right ) + 6 A a b^{2} \left (\begin {cases} \frac {x^{\frac {3}{2}} x^{2 n}}{4 n + 3} & \text {for}\: n \neq - \frac {3}{4} \\x^{\frac {3}{2}} x^{2 n} \log {\left (\sqrt {x} \right )} & \text {otherwise} \end {cases}\right ) + 2 A b^{3} \left (\begin {cases} \frac {x^{\frac {3}{2}} x^{3 n}}{6 n + 3} & \text {for}\: n \neq - \frac {1}{2} \\x^{\frac {3}{2}} x^{3 n} \log {\left (\sqrt {x} \right )} & \text {otherwise} \end {cases}\right ) + 2 B a^{3} \left (\begin {cases} \frac {x^{\frac {3}{2}} x^{n}}{2 n + 3} & \text {for}\: n \neq - \frac {3}{2} \\x^{\frac {3}{2}} x^{n} \log {\left (\sqrt {x} \right )} & \text {otherwise} \end {cases}\right ) + 6 B a^{2} b \left (\begin {cases} \frac {x^{\frac {3}{2}} x^{2 n}}{4 n + 3} & \text {for}\: n \neq - \frac {3}{4} \\x^{\frac {3}{2}} x^{2 n} \log {\left (\sqrt {x} \right )} & \text {otherwise} \end {cases}\right ) + 6 B a b^{2} \left (\begin {cases} \frac {x^{\frac {3}{2}} x^{3 n}}{6 n + 3} & \text {for}\: n \neq - \frac {1}{2} \\x^{\frac {3}{2}} x^{3 n} \log {\left (\sqrt {x} \right )} & \text {otherwise} \end {cases}\right ) + 2 B b^{3} \left (\begin {cases} \frac {x^{\frac {3}{2}} x^{4 n}}{8 n + 3} & \text {for}\: n \neq - \frac {3}{8} \\x^{\frac {3}{2}} x^{4 n} \log {\left (\sqrt {x} \right )} & \text {otherwise} \end {cases}\right ) \] Input:
integrate(x**(1/2)*(a+b*x**n)**3*(A+B*x**n),x)
Output:
2*A*a**3*x**(3/2)/3 + 6*A*a**2*b*Piecewise((x**(3/2)*x**n/(2*n + 3), Ne(n, -3/2)), (x**(3/2)*x**n*log(sqrt(x)), True)) + 6*A*a*b**2*Piecewise((x**(3 /2)*x**(2*n)/(4*n + 3), Ne(n, -3/4)), (x**(3/2)*x**(2*n)*log(sqrt(x)), Tru e)) + 2*A*b**3*Piecewise((x**(3/2)*x**(3*n)/(6*n + 3), Ne(n, -1/2)), (x**( 3/2)*x**(3*n)*log(sqrt(x)), True)) + 2*B*a**3*Piecewise((x**(3/2)*x**n/(2* n + 3), Ne(n, -3/2)), (x**(3/2)*x**n*log(sqrt(x)), True)) + 6*B*a**2*b*Pie cewise((x**(3/2)*x**(2*n)/(4*n + 3), Ne(n, -3/4)), (x**(3/2)*x**(2*n)*log( sqrt(x)), True)) + 6*B*a*b**2*Piecewise((x**(3/2)*x**(3*n)/(6*n + 3), Ne(n , -1/2)), (x**(3/2)*x**(3*n)*log(sqrt(x)), True)) + 2*B*b**3*Piecewise((x* *(3/2)*x**(4*n)/(8*n + 3), Ne(n, -3/8)), (x**(3/2)*x**(4*n)*log(sqrt(x)), True))
Time = 0.03 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.24 \[ \int \sqrt {x} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\frac {2}{3} \, A a^{3} x^{\frac {3}{2}} + \frac {2 \, B b^{3} x^{4 \, n + \frac {3}{2}}}{8 \, n + 3} + \frac {2 \, B a b^{2} x^{3 \, n + \frac {3}{2}}}{2 \, n + 1} + \frac {2 \, A b^{3} x^{3 \, n + \frac {3}{2}}}{3 \, {\left (2 \, n + 1\right )}} + \frac {6 \, B a^{2} b x^{2 \, n + \frac {3}{2}}}{4 \, n + 3} + \frac {6 \, A a b^{2} x^{2 \, n + \frac {3}{2}}}{4 \, n + 3} + \frac {2 \, B a^{3} x^{n + \frac {3}{2}}}{2 \, n + 3} + \frac {6 \, A a^{2} b x^{n + \frac {3}{2}}}{2 \, n + 3} \] Input:
integrate(x^(1/2)*(a+b*x^n)^3*(A+B*x^n),x, algorithm="maxima")
Output:
2/3*A*a^3*x^(3/2) + 2*B*b^3*x^(4*n + 3/2)/(8*n + 3) + 2*B*a*b^2*x^(3*n + 3 /2)/(2*n + 1) + 2/3*A*b^3*x^(3*n + 3/2)/(2*n + 1) + 6*B*a^2*b*x^(2*n + 3/2 )/(4*n + 3) + 6*A*a*b^2*x^(2*n + 3/2)/(4*n + 3) + 2*B*a^3*x^(n + 3/2)/(2*n + 3) + 6*A*a^2*b*x^(n + 3/2)/(2*n + 3)
Time = 0.14 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.45 \[ \int \sqrt {x} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\frac {2}{3} \, A a^{3} x^{\frac {3}{2}} + \frac {2 \, B b^{3} x^{\frac {3}{2}} \sqrt {x}^{8 \, n}}{8 \, n + 3} + \frac {2 \, B a b^{2} x^{\frac {3}{2}} \sqrt {x}^{6 \, n}}{2 \, n + 1} + \frac {2 \, A b^{3} x^{\frac {3}{2}} \sqrt {x}^{6 \, n}}{3 \, {\left (2 \, n + 1\right )}} + \frac {6 \, B a^{2} b x^{\frac {3}{2}} \sqrt {x}^{4 \, n}}{4 \, n + 3} + \frac {6 \, A a b^{2} x^{\frac {3}{2}} \sqrt {x}^{4 \, n}}{4 \, n + 3} + \frac {2 \, B a^{3} x^{\frac {3}{2}} \sqrt {x}^{2 \, n}}{2 \, n + 3} + \frac {6 \, A a^{2} b x^{\frac {3}{2}} \sqrt {x}^{2 \, n}}{2 \, n + 3} \] Input:
integrate(x^(1/2)*(a+b*x^n)^3*(A+B*x^n),x, algorithm="giac")
Output:
2/3*A*a^3*x^(3/2) + 2*B*b^3*x^(3/2)*sqrt(x)^(8*n)/(8*n + 3) + 2*B*a*b^2*x^ (3/2)*sqrt(x)^(6*n)/(2*n + 1) + 2/3*A*b^3*x^(3/2)*sqrt(x)^(6*n)/(2*n + 1) + 6*B*a^2*b*x^(3/2)*sqrt(x)^(4*n)/(4*n + 3) + 6*A*a*b^2*x^(3/2)*sqrt(x)^(4 *n)/(4*n + 3) + 2*B*a^3*x^(3/2)*sqrt(x)^(2*n)/(2*n + 3) + 6*A*a^2*b*x^(3/2 )*sqrt(x)^(2*n)/(2*n + 3)
Time = 4.24 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.95 \[ \int \sqrt {x} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\frac {2\,A\,a^3\,x^{3/2}}{3}+\frac {x^n\,x^{3/2}\,\left (2\,B\,a^3+6\,A\,b\,a^2\right )}{2\,n+3}+\frac {x^{3\,n}\,x^{3/2}\,\left (2\,A\,b^3+6\,B\,a\,b^2\right )}{6\,n+3}+\frac {2\,B\,b^3\,x^{4\,n}\,x^{3/2}}{8\,n+3}+\frac {6\,a\,b\,x^{2\,n}\,x^{3/2}\,\left (A\,b+B\,a\right )}{4\,n+3} \] Input:
int(x^(1/2)*(A + B*x^n)*(a + b*x^n)^3,x)
Output:
(2*A*a^3*x^(3/2))/3 + (x^n*x^(3/2)*(2*B*a^3 + 6*A*a^2*b))/(2*n + 3) + (x^( 3*n)*x^(3/2)*(2*A*b^3 + 6*B*a*b^2))/(6*n + 3) + (2*B*b^3*x^(4*n)*x^(3/2))/ (8*n + 3) + (6*a*b*x^(2*n)*x^(3/2)*(A*b + B*a))/(4*n + 3)
Time = 0.21 (sec) , antiderivative size = 263, normalized size of antiderivative = 2.17 \[ \int \sqrt {x} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\frac {2 \sqrt {x}\, x \left (48 x^{4 n} b^{4} n^{3}+132 x^{4 n} b^{4} n^{2}+108 x^{4 n} b^{4} n +27 x^{4 n} b^{4}+256 x^{3 n} a \,b^{3} n^{3}+672 x^{3 n} a \,b^{3} n^{2}+504 x^{3 n} a \,b^{3} n +108 x^{3 n} a \,b^{3}+576 x^{2 n} a^{2} b^{2} n^{3}+1368 x^{2 n} a^{2} b^{2} n^{2}+864 x^{2 n} a^{2} b^{2} n +162 x^{2 n} a^{2} b^{2}+768 x^{n} a^{3} b \,n^{3}+1248 x^{n} a^{3} b \,n^{2}+648 x^{n} a^{3} b n +108 x^{n} a^{3} b +128 a^{4} n^{4}+400 a^{4} n^{3}+420 a^{4} n^{2}+180 a^{4} n +27 a^{4}\right )}{384 n^{4}+1200 n^{3}+1260 n^{2}+540 n +81} \] Input:
int(x^(1/2)*(a+b*x^n)^3*(A+B*x^n),x)
Output:
(2*sqrt(x)*x*(48*x**(4*n)*b**4*n**3 + 132*x**(4*n)*b**4*n**2 + 108*x**(4*n )*b**4*n + 27*x**(4*n)*b**4 + 256*x**(3*n)*a*b**3*n**3 + 672*x**(3*n)*a*b* *3*n**2 + 504*x**(3*n)*a*b**3*n + 108*x**(3*n)*a*b**3 + 576*x**(2*n)*a**2* b**2*n**3 + 1368*x**(2*n)*a**2*b**2*n**2 + 864*x**(2*n)*a**2*b**2*n + 162* x**(2*n)*a**2*b**2 + 768*x**n*a**3*b*n**3 + 1248*x**n*a**3*b*n**2 + 648*x* *n*a**3*b*n + 108*x**n*a**3*b + 128*a**4*n**4 + 400*a**4*n**3 + 420*a**4*n **2 + 180*a**4*n + 27*a**4))/(3*(128*n**4 + 400*n**3 + 420*n**2 + 180*n + 27))