Integrand size = 22, antiderivative size = 117 \[ \int \frac {\left (a+b x^n\right )^3 \left (A+B x^n\right )}{\sqrt {x}} \, dx=2 a^3 A \sqrt {x}+\frac {2 a^2 (3 A b+a B) x^{\frac {1}{2}+n}}{1+2 n}+\frac {6 a b (A b+a B) x^{\frac {1}{2}+2 n}}{1+4 n}+\frac {2 b^2 (A b+3 a B) x^{\frac {1}{2}+3 n}}{1+6 n}+\frac {2 b^3 B x^{\frac {1}{2}+4 n}}{1+8 n} \] Output:
2*a^3*A*x^(1/2)+2*a^2*(3*A*b+B*a)*x^(1/2+n)/(1+2*n)+6*a*b*(A*b+B*a)*x^(1/2 +2*n)/(1+4*n)+2*b^2*(A*b+3*B*a)*x^(1/2+3*n)/(1+6*n)+2*b^3*B*x^(1/2+4*n)/(1 +8*n)
Time = 0.26 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^n\right )^3 \left (A+B x^n\right )}{\sqrt {x}} \, dx=2 \left (a^3 A \sqrt {x}+\frac {a^2 (3 A b+a B) x^{\frac {1}{2}+n}}{1+2 n}+\frac {3 a b (A b+a B) x^{\frac {1}{2}+2 n}}{1+4 n}+\frac {b^2 (A b+3 a B) x^{\frac {1}{2}+3 n}}{1+6 n}+\frac {b^3 B x^{\frac {1}{2}+4 n}}{1+8 n}\right ) \] Input:
Integrate[((a + b*x^n)^3*(A + B*x^n))/Sqrt[x],x]
Output:
2*(a^3*A*Sqrt[x] + (a^2*(3*A*b + a*B)*x^(1/2 + n))/(1 + 2*n) + (3*a*b*(A*b + a*B)*x^(1/2 + 2*n))/(1 + 4*n) + (b^2*(A*b + 3*a*B)*x^(1/2 + 3*n))/(1 + 6*n) + (b^3*B*x^(1/2 + 4*n))/(1 + 8*n))
Time = 0.43 (sec) , antiderivative size = 117, 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.
\(\displaystyle \int \frac {\left (a+b x^n\right )^3 \left (A+B x^n\right )}{\sqrt {x}} \, dx\) |
\(\Big \downarrow \) 950 |
\(\displaystyle \int \left (\frac {a^3 A}{\sqrt {x}}+a^2 x^{n-\frac {1}{2}} (a B+3 A b)+b^2 x^{3 n-\frac {1}{2}} (3 a B+A b)+3 a b x^{2 n-\frac {1}{2}} (a B+A b)+b^3 B x^{4 n-\frac {1}{2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 a^3 A \sqrt {x}+\frac {2 a^2 x^{n+\frac {1}{2}} (a B+3 A b)}{2 n+1}+\frac {2 b^2 x^{3 n+\frac {1}{2}} (3 a B+A b)}{6 n+1}+\frac {6 a b x^{2 n+\frac {1}{2}} (a B+A b)}{4 n+1}+\frac {2 b^3 B x^{4 n+\frac {1}{2}}}{8 n+1}\) |
Input:
Int[((a + b*x^n)^3*(A + B*x^n))/Sqrt[x],x]
Output:
2*a^3*A*Sqrt[x] + (2*a^2*(3*A*b + a*B)*x^(1/2 + n))/(1 + 2*n) + (6*a*b*(A* b + a*B)*x^(1/2 + 2*n))/(1 + 4*n) + (2*b^2*(A*b + 3*a*B)*x^(1/2 + 3*n))/(1 + 6*n) + (2*b^3*B*x^(1/2 + 4*n))/(1 + 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. \(1240\) vs. \(2(107)=214\).
Time = 0.52 (sec) , antiderivative size = 1241, normalized size of antiderivative = 10.61
Input:
int((a+b*x^n)^3*(A+B*x^n)/x^(1/2),x,method=_RETURNVERBOSE)
Output:
2*x^(1/2)*(384*n^4+140*n^2+1)/(96*n^3+76*n^2+16*n+1)/(1+4*n)*(a+b*x^n)^3*( A+B*x^n)-40*x^2*(40*n^3-42*n^2+20*n-3)/(96*n^3+76*n^2+16*n+1)/(1+4*n)*(3*( a+b*x^n)^2*(A+B*x^n)/x^(3/2)*b*x^n*n+(a+b*x^n)^3*B*x^n*n/x^(3/2)-1/2*(a+b* x^n)^3*(A+B*x^n)/x^(3/2))+80*x^3*(14*n^2-16*n+5)/(96*n^3+76*n^2+16*n+1)/(1 +4*n)*(6*(a+b*x^n)*(A+B*x^n)/x^(5/2)*b^2*(x^n)^2*n^2+6*(a+b*x^n)^2*B*(x^n) ^2*n^2/x^(5/2)*b-6*(a+b*x^n)^2*(A+B*x^n)/x^(5/2)*b*x^n*n+3*(a+b*x^n)^2*(A+ B*x^n)/x^(5/2)*b*x^n*n^2+(a+b*x^n)^3*B*x^n*n^2/x^(5/2)-2*(a+b*x^n)^3*B*x^n *n/x^(5/2)+3/4*(a+b*x^n)^3*(A+B*x^n)/x^(5/2))-80*x^4*(-3+4*n)/(384*n^4+400 *n^3+140*n^2+20*n+1)*(6*b^3*(x^n)^3*n^3/x^(7/2)*(A+B*x^n)+18*(a+b*x^n)*B*( x^n)^3*n^3/x^(7/2)*b^2-27*(a+b*x^n)*(A+B*x^n)/x^(7/2)*b^2*(x^n)^2*n^2+18*( a+b*x^n)*(A+B*x^n)/x^(7/2)*b^2*(x^n)^2*n^3+18*(a+b*x^n)^2*B*(x^n)^2*n^3/x^ (7/2)*b-27*(a+b*x^n)^2*B*(x^n)^2*n^2/x^(7/2)*b+69/4*(a+b*x^n)^2*(A+B*x^n)/ x^(7/2)*b*x^n*n-27/2*(a+b*x^n)^2*(A+B*x^n)/x^(7/2)*b*x^n*n^2+3*(a+b*x^n)^2 *(A+B*x^n)/x^(7/2)*b*x^n*n^3+(a+b*x^n)^3*B*x^n*n^3/x^(7/2)-9/2*(a+b*x^n)^3 *B*x^n*n^2/x^(7/2)+23/4*(a+b*x^n)^3*B*x^n*n/x^(7/2)-15/8*(a+b*x^n)^3*(A+B* x^n)/x^(7/2))+32/(384*n^4+400*n^3+140*n^2+20*n+1)*x^5*(105/16*(a+b*x^n)^3* (A+B*x^n)/x^(9/2)+108*(a+b*x^n)*B*(x^n)^3*n^4/x^(9/2)*b^2+42*(a+b*x^n)^2*B *(x^n)^2*n^4/x^(9/2)*b-48*b^3*(x^n)^3*n^3/x^(9/2)*(A+B*x^n)-144*(a+b*x^n)* B*(x^n)^3*n^3/x^(9/2)*b^2-144*(a+b*x^n)*(A+B*x^n)/x^(9/2)*b^2*(x^n)^2*n^3- 8*(a+b*x^n)^3*B*x^n*n^3/x^(9/2)+129*(a+b*x^n)^2*B*(x^n)^2*n^2/x^(9/2)*b...
Leaf count of result is larger than twice the leaf count of optimal. 330 vs. \(2 (107) = 214\).
Time = 0.09 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.82 \[ \int \frac {\left (a+b x^n\right )^3 \left (A+B x^n\right )}{\sqrt {x}} \, dx=\frac {2 \, {\left ({\left (48 \, B b^{3} n^{3} + 44 \, B b^{3} n^{2} + 12 \, B b^{3} n + B b^{3}\right )} \sqrt {x} x^{4 \, n} + {\left (3 \, B a b^{2} + A b^{3} + 64 \, {\left (3 \, B a b^{2} + A b^{3}\right )} n^{3} + 56 \, {\left (3 \, B a b^{2} + A b^{3}\right )} n^{2} + 14 \, {\left (3 \, B a b^{2} + A b^{3}\right )} n\right )} \sqrt {x} x^{3 \, n} + 3 \, {\left (B a^{2} b + A a b^{2} + 96 \, {\left (B a^{2} b + A a b^{2}\right )} n^{3} + 76 \, {\left (B a^{2} b + A a b^{2}\right )} n^{2} + 16 \, {\left (B a^{2} b + A a b^{2}\right )} n\right )} \sqrt {x} x^{2 \, n} + {\left (B a^{3} + 3 \, A a^{2} b + 192 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} n^{3} + 104 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} n^{2} + 18 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} n\right )} \sqrt {x} x^{n} + {\left (384 \, A a^{3} n^{4} + 400 \, A a^{3} n^{3} + 140 \, A a^{3} n^{2} + 20 \, A a^{3} n + A a^{3}\right )} \sqrt {x}\right )}}{384 \, n^{4} + 400 \, n^{3} + 140 \, n^{2} + 20 \, n + 1} \] Input:
integrate((a+b*x^n)^3*(A+B*x^n)/x^(1/2),x, algorithm="fricas")
Output:
2*((48*B*b^3*n^3 + 44*B*b^3*n^2 + 12*B*b^3*n + B*b^3)*sqrt(x)*x^(4*n) + (3 *B*a*b^2 + A*b^3 + 64*(3*B*a*b^2 + A*b^3)*n^3 + 56*(3*B*a*b^2 + A*b^3)*n^2 + 14*(3*B*a*b^2 + A*b^3)*n)*sqrt(x)*x^(3*n) + 3*(B*a^2*b + A*a*b^2 + 96*( B*a^2*b + A*a*b^2)*n^3 + 76*(B*a^2*b + A*a*b^2)*n^2 + 16*(B*a^2*b + A*a*b^ 2)*n)*sqrt(x)*x^(2*n) + (B*a^3 + 3*A*a^2*b + 192*(B*a^3 + 3*A*a^2*b)*n^3 + 104*(B*a^3 + 3*A*a^2*b)*n^2 + 18*(B*a^3 + 3*A*a^2*b)*n)*sqrt(x)*x^n + (38 4*A*a^3*n^4 + 400*A*a^3*n^3 + 140*A*a^3*n^2 + 20*A*a^3*n + A*a^3)*sqrt(x)) /(384*n^4 + 400*n^3 + 140*n^2 + 20*n + 1)
Time = 3.61 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.84 \[ \int \frac {\left (a+b x^n\right )^3 \left (A+B x^n\right )}{\sqrt {x}} \, dx=2 A a^{3} \sqrt {x} - 6 A a^{2} b \left (\begin {cases} \frac {\sqrt {x} x^{n}}{- 2 n - 1} & \text {for}\: n \neq - \frac {1}{2} \\\sqrt {x} x^{n} \log {\left (\frac {1}{\sqrt {x}} \right )} & \text {otherwise} \end {cases}\right ) - 6 A a b^{2} \left (\begin {cases} \frac {\sqrt {x} x^{2 n}}{- 4 n - 1} & \text {for}\: n \neq - \frac {1}{4} \\\sqrt {x} x^{2 n} \log {\left (\frac {1}{\sqrt {x}} \right )} & \text {otherwise} \end {cases}\right ) - 2 A b^{3} \left (\begin {cases} \frac {\sqrt {x} x^{3 n}}{- 6 n - 1} & \text {for}\: n \neq - \frac {1}{6} \\\sqrt {x} x^{3 n} \log {\left (\frac {1}{\sqrt {x}} \right )} & \text {otherwise} \end {cases}\right ) - 2 B a^{3} \left (\begin {cases} \frac {\sqrt {x} x^{n}}{- 2 n - 1} & \text {for}\: n \neq - \frac {1}{2} \\\sqrt {x} x^{n} \log {\left (\frac {1}{\sqrt {x}} \right )} & \text {otherwise} \end {cases}\right ) - 6 B a^{2} b \left (\begin {cases} \frac {\sqrt {x} x^{2 n}}{- 4 n - 1} & \text {for}\: n \neq - \frac {1}{4} \\\sqrt {x} x^{2 n} \log {\left (\frac {1}{\sqrt {x}} \right )} & \text {otherwise} \end {cases}\right ) - 6 B a b^{2} \left (\begin {cases} \frac {\sqrt {x} x^{3 n}}{- 6 n - 1} & \text {for}\: n \neq - \frac {1}{6} \\\sqrt {x} x^{3 n} \log {\left (\frac {1}{\sqrt {x}} \right )} & \text {otherwise} \end {cases}\right ) - 2 B b^{3} \left (\begin {cases} \frac {\sqrt {x} x^{4 n}}{- 8 n - 1} & \text {for}\: n \neq - \frac {1}{8} \\\sqrt {x} x^{4 n} \log {\left (\frac {1}{\sqrt {x}} \right )} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((a+b*x**n)**3*(A+B*x**n)/x**(1/2),x)
Output:
2*A*a**3*sqrt(x) - 6*A*a**2*b*Piecewise((sqrt(x)*x**n/(-2*n - 1), Ne(n, -1 /2)), (sqrt(x)*x**n*log(1/sqrt(x)), True)) - 6*A*a*b**2*Piecewise((sqrt(x) *x**(2*n)/(-4*n - 1), Ne(n, -1/4)), (sqrt(x)*x**(2*n)*log(1/sqrt(x)), True )) - 2*A*b**3*Piecewise((sqrt(x)*x**(3*n)/(-6*n - 1), Ne(n, -1/6)), (sqrt( x)*x**(3*n)*log(1/sqrt(x)), True)) - 2*B*a**3*Piecewise((sqrt(x)*x**n/(-2* n - 1), Ne(n, -1/2)), (sqrt(x)*x**n*log(1/sqrt(x)), True)) - 6*B*a**2*b*Pi ecewise((sqrt(x)*x**(2*n)/(-4*n - 1), Ne(n, -1/4)), (sqrt(x)*x**(2*n)*log( 1/sqrt(x)), True)) - 6*B*a*b**2*Piecewise((sqrt(x)*x**(3*n)/(-6*n - 1), Ne (n, -1/6)), (sqrt(x)*x**(3*n)*log(1/sqrt(x)), True)) - 2*B*b**3*Piecewise( (sqrt(x)*x**(4*n)/(-8*n - 1), Ne(n, -1/8)), (sqrt(x)*x**(4*n)*log(1/sqrt(x )), True))
Time = 0.03 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b x^n\right )^3 \left (A+B x^n\right )}{\sqrt {x}} \, dx=2 \, A a^{3} \sqrt {x} + \frac {2 \, B b^{3} x^{4 \, n + \frac {1}{2}}}{8 \, n + 1} + \frac {6 \, B a b^{2} x^{3 \, n + \frac {1}{2}}}{6 \, n + 1} + \frac {2 \, A b^{3} x^{3 \, n + \frac {1}{2}}}{6 \, n + 1} + \frac {6 \, B a^{2} b x^{2 \, n + \frac {1}{2}}}{4 \, n + 1} + \frac {6 \, A a b^{2} x^{2 \, n + \frac {1}{2}}}{4 \, n + 1} + \frac {2 \, B a^{3} x^{n + \frac {1}{2}}}{2 \, n + 1} + \frac {6 \, A a^{2} b x^{n + \frac {1}{2}}}{2 \, n + 1} \] Input:
integrate((a+b*x^n)^3*(A+B*x^n)/x^(1/2),x, algorithm="maxima")
Output:
2*A*a^3*sqrt(x) + 2*B*b^3*x^(4*n + 1/2)/(8*n + 1) + 6*B*a*b^2*x^(3*n + 1/2 )/(6*n + 1) + 2*A*b^3*x^(3*n + 1/2)/(6*n + 1) + 6*B*a^2*b*x^(2*n + 1/2)/(4 *n + 1) + 6*A*a*b^2*x^(2*n + 1/2)/(4*n + 1) + 2*B*a^3*x^(n + 1/2)/(2*n + 1 ) + 6*A*a^2*b*x^(n + 1/2)/(2*n + 1)
Time = 0.13 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b x^n\right )^3 \left (A+B x^n\right )}{\sqrt {x}} \, dx=2 \, A a^{3} \sqrt {x} + \frac {2 \, B b^{3} x^{4 \, n + \frac {1}{2}}}{8 \, n + 1} + \frac {6 \, B a b^{2} x^{3 \, n + \frac {1}{2}}}{6 \, n + 1} + \frac {2 \, A b^{3} x^{3 \, n + \frac {1}{2}}}{6 \, n + 1} + \frac {6 \, B a^{2} b x^{2 \, n + \frac {1}{2}}}{4 \, n + 1} + \frac {6 \, A a b^{2} x^{2 \, n + \frac {1}{2}}}{4 \, n + 1} + \frac {2 \, B a^{3} x^{n + \frac {1}{2}}}{2 \, n + 1} + \frac {6 \, A a^{2} b x^{n + \frac {1}{2}}}{2 \, n + 1} \] Input:
integrate((a+b*x^n)^3*(A+B*x^n)/x^(1/2),x, algorithm="giac")
Output:
2*A*a^3*sqrt(x) + 2*B*b^3*x^(4*n + 1/2)/(8*n + 1) + 6*B*a*b^2*x^(3*n + 1/2 )/(6*n + 1) + 2*A*b^3*x^(3*n + 1/2)/(6*n + 1) + 6*B*a^2*b*x^(2*n + 1/2)/(4 *n + 1) + 6*A*a*b^2*x^(2*n + 1/2)/(4*n + 1) + 2*B*a^3*x^(n + 1/2)/(2*n + 1 ) + 6*A*a^2*b*x^(n + 1/2)/(2*n + 1)
Time = 4.33 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^n\right )^3 \left (A+B x^n\right )}{\sqrt {x}} \, dx=2\,A\,a^3\,\sqrt {x}+\frac {x^n\,\sqrt {x}\,\left (2\,B\,a^3+6\,A\,b\,a^2\right )}{2\,n+1}+\frac {x^{3\,n}\,\sqrt {x}\,\left (2\,A\,b^3+6\,B\,a\,b^2\right )}{6\,n+1}+\frac {2\,B\,b^3\,x^{4\,n}\,\sqrt {x}}{8\,n+1}+\frac {6\,a\,b\,x^{2\,n}\,\sqrt {x}\,\left (A\,b+B\,a\right )}{4\,n+1} \] Input:
int(((A + B*x^n)*(a + b*x^n)^3)/x^(1/2),x)
Output:
2*A*a^3*x^(1/2) + (x^n*x^(1/2)*(2*B*a^3 + 6*A*a^2*b))/(2*n + 1) + (x^(3*n) *x^(1/2)*(2*A*b^3 + 6*B*a*b^2))/(6*n + 1) + (2*B*b^3*x^(4*n)*x^(1/2))/(8*n + 1) + (6*a*b*x^(2*n)*x^(1/2)*(A*b + B*a))/(4*n + 1)
Time = 0.20 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.21 \[ \int \frac {\left (a+b x^n\right )^3 \left (A+B x^n\right )}{\sqrt {x}} \, dx=\frac {2 \sqrt {x}\, \left (48 x^{4 n} b^{4} n^{3}+44 x^{4 n} b^{4} n^{2}+12 x^{4 n} b^{4} n +x^{4 n} b^{4}+256 x^{3 n} a \,b^{3} n^{3}+224 x^{3 n} a \,b^{3} n^{2}+56 x^{3 n} a \,b^{3} n +4 x^{3 n} a \,b^{3}+576 x^{2 n} a^{2} b^{2} n^{3}+456 x^{2 n} a^{2} b^{2} n^{2}+96 x^{2 n} a^{2} b^{2} n +6 x^{2 n} a^{2} b^{2}+768 x^{n} a^{3} b \,n^{3}+416 x^{n} a^{3} b \,n^{2}+72 x^{n} a^{3} b n +4 x^{n} a^{3} b +384 a^{4} n^{4}+400 a^{4} n^{3}+140 a^{4} n^{2}+20 a^{4} n +a^{4}\right )}{384 n^{4}+400 n^{3}+140 n^{2}+20 n +1} \] Input:
int((a+b*x^n)^3*(A+B*x^n)/x^(1/2),x)
Output:
(2*sqrt(x)*(48*x**(4*n)*b**4*n**3 + 44*x**(4*n)*b**4*n**2 + 12*x**(4*n)*b* *4*n + x**(4*n)*b**4 + 256*x**(3*n)*a*b**3*n**3 + 224*x**(3*n)*a*b**3*n**2 + 56*x**(3*n)*a*b**3*n + 4*x**(3*n)*a*b**3 + 576*x**(2*n)*a**2*b**2*n**3 + 456*x**(2*n)*a**2*b**2*n**2 + 96*x**(2*n)*a**2*b**2*n + 6*x**(2*n)*a**2* b**2 + 768*x**n*a**3*b*n**3 + 416*x**n*a**3*b*n**2 + 72*x**n*a**3*b*n + 4* x**n*a**3*b + 384*a**4*n**4 + 400*a**4*n**3 + 140*a**4*n**2 + 20*a**4*n + a**4))/(384*n**4 + 400*n**3 + 140*n**2 + 20*n + 1)