Integrand size = 22, antiderivative size = 117 \[ \int \frac {\left (a+b x^n\right )^3 \left (A+B x^n\right )}{x^{3/2}} \, dx=-\frac {2 a^3 A}{\sqrt {x}}-\frac {2 a^2 (3 A b+a B) x^{-\frac {1}{2}+n}}{1-2 n}-\frac {6 a b (A b+a B) x^{-\frac {1}{2}+2 n}}{1-4 n}-\frac {2 b^2 (A b+3 a B) x^{-\frac {1}{2}+3 n}}{1-6 n}-\frac {2 b^3 B x^{-\frac {1}{2}+4 n}}{1-8 n} \] 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 = 0.42 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^n\right )^3 \left (A+B x^n\right )}{x^{3/2}} \, dx=2 \left (-\frac {a^3 A}{\sqrt {x}}+\frac {a^2 (3 A b+a B) x^{-\frac {1}{2}+n}}{-1+2 n}+\frac {3 a b (A b+a B) x^{-\frac {1}{2}+2 n}}{-1+4 n}+\frac {b^2 (A b+3 a B) x^{-\frac {1}{2}+3 n}}{-1+6 n}+\frac {b^3 B x^{-\frac {1}{2}+4 n}}{-1+8 n}\right ) \] Input:
Integrate[((a + b*x^n)^3*(A + B*x^n))/x^(3/2),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.
\(\displaystyle \int \frac {\left (a+b x^n\right )^3 \left (A+B x^n\right )}{x^{3/2}} \, dx\) |
\(\Big \downarrow \) 950 |
\(\displaystyle \int \left (\frac {a^3 A}{x^{3/2}}+a^2 x^{n-\frac {3}{2}} (a B+3 A b)+b^2 x^{3 n-\frac {3}{2}} (3 a B+A b)+3 a b x^{2 n-\frac {3}{2}} (a B+A b)+b^3 B x^{4 n-\frac {3}{2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 a^3 A}{\sqrt {x}}-\frac {2 a^2 x^{n-\frac {1}{2}} (a B+3 A b)}{1-2 n}-\frac {2 b^2 x^{3 n-\frac {1}{2}} (3 a B+A b)}{1-6 n}-\frac {6 a b x^{2 n-\frac {1}{2}} (a B+A b)}{1-4 n}-\frac {2 b^3 B x^{4 n-\frac {1}{2}}}{1-8 n}\) |
Input:
Int[((a + b*x^n)^3*(A + B*x^n))/x^(3/2),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. \(1238\) vs. \(2(107)=214\).
Time = 0.53 (sec) , antiderivative size = 1239, normalized size of antiderivative = 10.59
Input:
int((a+b*x^n)^3*(A+B*x^n)/x^(3/2),x,method=_RETURNVERBOSE)
Output:
-2*(192*n^3-704*n^2+558*n-121)/x^(1/2)/(-1+4*n)/(48*n^2-14*n+1)*(a+b*x^n)^ 3*(A+B*x^n)+40*x^2*(40*n^3-126*n^2+116*n-33)/(96*n^3-76*n^2+16*n-1)/(-1+4* n)*(3*(a+b*x^n)^2*(A+B*x^n)/x^(5/2)*b*x^n*n+(a+b*x^n)^3*B*x^n*n/x^(5/2)-3/ 2*(a+b*x^n)^3*(A+B*x^n)/x^(5/2))-80*x^3*(14*n^2-32*n+17)/(96*n^3-76*n^2+16 *n-1)/(-1+4*n)*(6*(a+b*x^n)*(A+B*x^n)/x^(7/2)*b^2*(x^n)^2*n^2+6*(a+b*x^n)^ 2*B*(x^n)^2*n^2/x^(7/2)*b-12*(a+b*x^n)^2*(A+B*x^n)/x^(7/2)*b*x^n*n+3*(a+b* x^n)^2*(A+B*x^n)/x^(7/2)*b*x^n*n^2+(a+b*x^n)^3*B*x^n*n^2/x^(7/2)-4*(a+b*x^ n)^3*B*x^n*n/x^(7/2)+15/4*(a+b*x^n)^3*(A+B*x^n)/x^(7/2))+80*x^4*(-5+4*n)/( 384*n^4-400*n^3+140*n^2-20*n+1)*(6*b^3*(x^n)^3*n^3/x^(9/2)*(A+B*x^n)+18*(a +b*x^n)*B*(x^n)^3*n^3/x^(9/2)*b^2-45*(a+b*x^n)*(A+B*x^n)/x^(9/2)*b^2*(x^n) ^2*n^2+18*(a+b*x^n)*(A+B*x^n)/x^(9/2)*b^2*(x^n)^2*n^3+18*(a+b*x^n)^2*B*(x^ n)^2*n^3/x^(9/2)*b-45*(a+b*x^n)^2*B*(x^n)^2*n^2/x^(9/2)*b+213/4*(a+b*x^n)^ 2*(A+B*x^n)/x^(9/2)*b*x^n*n-45/2*(a+b*x^n)^2*(A+B*x^n)/x^(9/2)*b*x^n*n^2+3 *(a+b*x^n)^2*(A+B*x^n)/x^(9/2)*b*x^n*n^3+(a+b*x^n)^3*B*x^n*n^3/x^(9/2)-15/ 2*(a+b*x^n)^3*B*x^n*n^2/x^(9/2)+71/4*(a+b*x^n)^3*B*x^n*n/x^(9/2)-105/8*(a+ b*x^n)^3*(A+B*x^n)/x^(9/2))-32/(384*n^4-400*n^3+140*n^2-20*n+1)*x^5*(42*(a +b*x^n)*(A+B*x^n)/x^(11/2)*b^2*(x^n)^2*n^4-216*(a+b*x^n)^2*B*(x^n)^2*n^3/x ^(11/2)*b+309*(a+b*x^n)*(A+B*x^n)/x^(11/2)*b^2*(x^n)^2*n^2-279*(a+b*x^n)^2 *(A+B*x^n)/x^(11/2)*b*x^n*n+108*(a+b*x^n)*B*(x^n)^3*n^4/x^(11/2)*b^2+945/1 6*(a+b*x^n)^3*(A+B*x^n)/x^(11/2)-12*(a+b*x^n)^3*B*x^n*n^3/x^(11/2)+36*b...
Leaf count of result is larger than twice the leaf count of optimal. 337 vs. \(2 (107) = 214\).
Time = 0.10 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.88 \[ \int \frac {\left (a+b x^n\right )^3 \left (A+B x^n\right )}{x^{3/2}} \, dx=\frac {2 \, {\left ({\left (48 \, B b^{3} n^{3} - 44 \, B b^{3} n^{2} + 12 \, B b^{3} n - B b^{3}\right )} \sqrt {x} x^{4 \, n} - {\left (3 \, B a b^{2} + A b^{3} - 64 \, {\left (3 \, B a b^{2} + A b^{3}\right )} n^{3} + 56 \, {\left (3 \, B a b^{2} + A b^{3}\right )} n^{2} - 14 \, {\left (3 \, B a b^{2} + A b^{3}\right )} n\right )} \sqrt {x} x^{3 \, n} - 3 \, {\left (B a^{2} b + A a b^{2} - 96 \, {\left (B a^{2} b + A a b^{2}\right )} n^{3} + 76 \, {\left (B a^{2} b + A a b^{2}\right )} n^{2} - 16 \, {\left (B a^{2} b + A a b^{2}\right )} n\right )} \sqrt {x} x^{2 \, n} - {\left (B a^{3} + 3 \, A a^{2} b - 192 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} n^{3} + 104 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} n^{2} - 18 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} n\right )} \sqrt {x} x^{n} - {\left (384 \, A a^{3} n^{4} - 400 \, A a^{3} n^{3} + 140 \, A a^{3} n^{2} - 20 \, A a^{3} n + A a^{3}\right )} \sqrt {x}\right )}}{{\left (384 \, n^{4} - 400 \, n^{3} + 140 \, n^{2} - 20 \, n + 1\right )} x} \] Input:
integrate((a+b*x^n)^3*(A+B*x^n)/x^(3/2),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)*x)
Time = 1.71 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.35 \[ \int \frac {\left (a+b x^n\right )^3 \left (A+B x^n\right )}{x^{3/2}} \, dx=- \frac {2 A a^{3}}{\sqrt {x}} + 3 A a^{2} b \left (\begin {cases} \frac {x^{n}}{\sqrt {x} \left (n - \frac {1}{2}\right )} & \text {for}\: n \neq \frac {1}{2} \\\frac {x^{n} \log {\left (x \right )}}{\sqrt {x}} & \text {otherwise} \end {cases}\right ) + 3 A a b^{2} \left (\begin {cases} \frac {x^{2 n}}{\sqrt {x} \left (2 n - \frac {1}{2}\right )} & \text {for}\: n \neq \frac {1}{4} \\\frac {x^{2 n} \log {\left (x \right )}}{\sqrt {x}} & \text {otherwise} \end {cases}\right ) + A b^{3} \left (\begin {cases} \frac {x^{3 n}}{\sqrt {x} \left (3 n - \frac {1}{2}\right )} & \text {for}\: n \neq \frac {1}{6} \\\frac {x^{3 n} \log {\left (x \right )}}{\sqrt {x}} & \text {otherwise} \end {cases}\right ) + B a^{3} \left (\begin {cases} \frac {x^{n}}{\sqrt {x} \left (n - \frac {1}{2}\right )} & \text {for}\: n \neq \frac {1}{2} \\\frac {x^{n} \log {\left (x \right )}}{\sqrt {x}} & \text {otherwise} \end {cases}\right ) + 3 B a^{2} b \left (\begin {cases} \frac {x^{2 n}}{\sqrt {x} \left (2 n - \frac {1}{2}\right )} & \text {for}\: n \neq \frac {1}{4} \\\frac {x^{2 n} \log {\left (x \right )}}{\sqrt {x}} & \text {otherwise} \end {cases}\right ) + 3 B a b^{2} \left (\begin {cases} \frac {x^{3 n}}{\sqrt {x} \left (3 n - \frac {1}{2}\right )} & \text {for}\: n \neq \frac {1}{6} \\\frac {x^{3 n} \log {\left (x \right )}}{\sqrt {x}} & \text {otherwise} \end {cases}\right ) + B b^{3} \left (\begin {cases} \frac {x^{4 n}}{\sqrt {x} \left (4 n - \frac {1}{2}\right )} & \text {for}\: n \neq \frac {1}{8} \\\frac {x^{4 n} \log {\left (x \right )}}{\sqrt {x}} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((a+b*x**n)**3*(A+B*x**n)/x**(3/2),x)
Output:
-2*A*a**3/sqrt(x) + 3*A*a**2*b*Piecewise((x**n/(sqrt(x)*(n - 1/2)), Ne(n, 1/2)), (x**n*log(x)/sqrt(x), True)) + 3*A*a*b**2*Piecewise((x**(2*n)/(sqrt (x)*(2*n - 1/2)), Ne(n, 1/4)), (x**(2*n)*log(x)/sqrt(x), True)) + A*b**3*P iecewise((x**(3*n)/(sqrt(x)*(3*n - 1/2)), Ne(n, 1/6)), (x**(3*n)*log(x)/sq rt(x), True)) + B*a**3*Piecewise((x**n/(sqrt(x)*(n - 1/2)), Ne(n, 1/2)), ( x**n*log(x)/sqrt(x), True)) + 3*B*a**2*b*Piecewise((x**(2*n)/(sqrt(x)*(2*n - 1/2)), Ne(n, 1/4)), (x**(2*n)*log(x)/sqrt(x), True)) + 3*B*a*b**2*Piece wise((x**(3*n)/(sqrt(x)*(3*n - 1/2)), Ne(n, 1/6)), (x**(3*n)*log(x)/sqrt(x ), True)) + B*b**3*Piecewise((x**(4*n)/(sqrt(x)*(4*n - 1/2)), Ne(n, 1/8)), (x**(4*n)*log(x)/sqrt(x), True))
Exception generated. \[ \int \frac {\left (a+b x^n\right )^3 \left (A+B x^n\right )}{x^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+b*x^n)^3*(A+B*x^n)/x^(3/2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(n-3/2>0)', see `assume?` for mor e details)
\[ \int \frac {\left (a+b x^n\right )^3 \left (A+B x^n\right )}{x^{3/2}} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{3}}{x^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*x^n)^3*(A+B*x^n)/x^(3/2),x, algorithm="giac")
Output:
integrate((B*x^n + A)*(b*x^n + a)^3/x^(3/2), x)
Time = 4.67 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a+b x^n\right )^3 \left (A+B x^n\right )}{x^{3/2}} \, dx=\frac {B\,a^3\,x^{n-\frac {1}{2}}}{n-\frac {1}{2}}-\frac {2\,A\,a^3}{\sqrt {x}}+\frac {2\,A\,b^3\,x^{3\,n-\frac {1}{2}}}{6\,n-1}+\frac {2\,B\,b^3\,x^{4\,n-\frac {1}{2}}}{8\,n-1}+\frac {6\,A\,a\,b^2\,x^{2\,n-\frac {1}{2}}}{4\,n-1}+\frac {6\,B\,a^2\,b\,x^{2\,n-\frac {1}{2}}}{4\,n-1}+\frac {6\,B\,a\,b^2\,x^{3\,n-\frac {1}{2}}}{6\,n-1}+\frac {3\,A\,a^2\,b\,x^{n-\frac {1}{2}}}{n-\frac {1}{2}} \] Input:
int(((A + B*x^n)*(a + b*x^n)^3)/x^(3/2),x)
Output:
(B*a^3*x^(n - 1/2))/(n - 1/2) - (2*A*a^3)/x^(1/2) + (2*A*b^3*x^(3*n - 1/2) )/(6*n - 1) + (2*B*b^3*x^(4*n - 1/2))/(8*n - 1) + (6*A*a*b^2*x^(2*n - 1/2) )/(4*n - 1) + (6*B*a^2*b*x^(2*n - 1/2))/(4*n - 1) + (6*B*a*b^2*x^(3*n - 1/ 2))/(6*n - 1) + (3*A*a^2*b*x^(n - 1/2))/(n - 1/2)
Time = 0.23 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.26 \[ \int \frac {\left (a+b x^n\right )^3 \left (A+B x^n\right )}{x^{3/2}} \, dx=\frac {96 x^{4 n} b^{4} n^{3}-88 x^{4 n} b^{4} n^{2}+24 x^{4 n} b^{4} n -2 x^{4 n} b^{4}+512 x^{3 n} a \,b^{3} n^{3}-448 x^{3 n} a \,b^{3} n^{2}+112 x^{3 n} a \,b^{3} n -8 x^{3 n} a \,b^{3}+1152 x^{2 n} a^{2} b^{2} n^{3}-912 x^{2 n} a^{2} b^{2} n^{2}+192 x^{2 n} a^{2} b^{2} n -12 x^{2 n} a^{2} b^{2}+1536 x^{n} a^{3} b \,n^{3}-832 x^{n} a^{3} b \,n^{2}+144 x^{n} a^{3} b n -8 x^{n} a^{3} b -768 a^{4} n^{4}+800 a^{4} n^{3}-280 a^{4} n^{2}+40 a^{4} n -2 a^{4}}{\sqrt {x}\, \left (384 n^{4}-400 n^{3}+140 n^{2}-20 n +1\right )} \] Input:
int((a+b*x^n)^3*(A+B*x^n)/x^(3/2),x)
Output:
(2*(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 + 7 68*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))/( sqrt(x)*(384*n**4 - 400*n**3 + 140*n**2 - 20*n + 1))