Integrand size = 13, antiderivative size = 100 \[ \int \left (a+b \sqrt {x}\right )^p x \, dx=-\frac {2 a^3 \left (a+b \sqrt {x}\right )^{1+p}}{b^4 (1+p)}+\frac {6 a^2 \left (a+b \sqrt {x}\right )^{2+p}}{b^4 (2+p)}-\frac {6 a \left (a+b \sqrt {x}\right )^{3+p}}{b^4 (3+p)}+\frac {2 \left (a+b \sqrt {x}\right )^{4+p}}{b^4 (4+p)} \] Output:
-2*a^3*(a+b*x^(1/2))^(p+1)/b^4/(p+1)+6*a^2*(a+b*x^(1/2))^(2+p)/b^4/(2+p)-6 *a*(a+b*x^(1/2))^(3+p)/b^4/(3+p)+2*(a+b*x^(1/2))^(4+p)/b^4/(4+p)
Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.95 \[ \int \left (a+b \sqrt {x}\right )^p x \, dx=\frac {2 \left (a+b \sqrt {x}\right )^{1+p} \left (-6 a^3+6 a^2 b (1+p) \sqrt {x}-3 a b^2 \left (2+3 p+p^2\right ) x+b^3 \left (6+11 p+6 p^2+p^3\right ) x^{3/2}\right )}{b^4 (1+p) (2+p) (3+p) (4+p)} \] Input:
Integrate[(a + b*Sqrt[x])^p*x,x]
Output:
(2*(a + b*Sqrt[x])^(1 + p)*(-6*a^3 + 6*a^2*b*(1 + p)*Sqrt[x] - 3*a*b^2*(2 + 3*p + p^2)*x + b^3*(6 + 11*p + 6*p^2 + p^3)*x^(3/2)))/(b^4*(1 + p)*(2 + p)*(3 + p)*(4 + p))
Time = 0.39 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {798, 53, 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.
\(\displaystyle \int x \left (a+b \sqrt {x}\right )^p \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \left (a+b \sqrt {x}\right )^p x^{3/2}d\sqrt {x}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle 2 \int \left (-\frac {a^3 \left (a+b \sqrt {x}\right )^p}{b^3}+\frac {3 a^2 \left (a+b \sqrt {x}\right )^{p+1}}{b^3}-\frac {3 a \left (a+b \sqrt {x}\right )^{p+2}}{b^3}+\frac {\left (a+b \sqrt {x}\right )^{p+3}}{b^3}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {a^3 \left (a+b \sqrt {x}\right )^{p+1}}{b^4 (p+1)}+\frac {3 a^2 \left (a+b \sqrt {x}\right )^{p+2}}{b^4 (p+2)}-\frac {3 a \left (a+b \sqrt {x}\right )^{p+3}}{b^4 (p+3)}+\frac {\left (a+b \sqrt {x}\right )^{p+4}}{b^4 (p+4)}\right )\) |
Input:
Int[(a + b*Sqrt[x])^p*x,x]
Output:
2*(-((a^3*(a + b*Sqrt[x])^(1 + p))/(b^4*(1 + p))) + (3*a^2*(a + b*Sqrt[x]) ^(2 + p))/(b^4*(2 + p)) - (3*a*(a + b*Sqrt[x])^(3 + p))/(b^4*(3 + p)) + (a + b*Sqrt[x])^(4 + p)/(b^4*(4 + p)))
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
\[\int \left (a +b \sqrt {x}\right )^{p} x d x\]
Input:
int((a+b*x^(1/2))^p*x,x)
Output:
int((a+b*x^(1/2))^p*x,x)
Time = 0.11 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.48 \[ \int \left (a+b \sqrt {x}\right )^p x \, dx=-\frac {2 \, {\left (6 \, a^{4} - {\left (b^{4} p^{3} + 6 \, b^{4} p^{2} + 11 \, b^{4} p + 6 \, b^{4}\right )} x^{2} + 3 \, {\left (a^{2} b^{2} p^{2} + a^{2} b^{2} p\right )} x - {\left (6 \, a^{3} b p + {\left (a b^{3} p^{3} + 3 \, a b^{3} p^{2} + 2 \, a b^{3} p\right )} x\right )} \sqrt {x}\right )} {\left (b \sqrt {x} + a\right )}^{p}}{b^{4} p^{4} + 10 \, b^{4} p^{3} + 35 \, b^{4} p^{2} + 50 \, b^{4} p + 24 \, b^{4}} \] Input:
integrate((a+b*x^(1/2))^p*x,x, algorithm="fricas")
Output:
-2*(6*a^4 - (b^4*p^3 + 6*b^4*p^2 + 11*b^4*p + 6*b^4)*x^2 + 3*(a^2*b^2*p^2 + a^2*b^2*p)*x - (6*a^3*b*p + (a*b^3*p^3 + 3*a*b^3*p^2 + 2*a*b^3*p)*x)*sqr t(x))*(b*sqrt(x) + a)^p/(b^4*p^4 + 10*b^4*p^3 + 35*b^4*p^2 + 50*b^4*p + 24 *b^4)
Leaf count of result is larger than twice the leaf count of optimal. 14683 vs. \(2 (88) = 176\).
Time = 36.50 (sec) , antiderivative size = 14683, normalized size of antiderivative = 146.83 \[ \int \left (a+b \sqrt {x}\right )^p x \, dx=\text {Too large to display} \] Input:
integrate((a+b*x**(1/2))**p*x,x)
Output:
-12*a**6*a**(p + 4)*x**8*(1 + b*sqrt(x)/a)**(p + 4)/(a**6*b**4*p**4*x**8 + 10*a**6*b**4*p**3*x**8 + 35*a**6*b**4*p**2*x**8 + 50*a**6*b**4*p*x**8 + 2 4*a**6*b**4*x**8 + 6*a**5*b**5*p**4*x**(17/2) + 60*a**5*b**5*p**3*x**(17/2 ) + 210*a**5*b**5*p**2*x**(17/2) + 300*a**5*b**5*p*x**(17/2) + 144*a**5*b* *5*x**(17/2) + 15*a**4*b**6*p**4*x**9 + 150*a**4*b**6*p**3*x**9 + 525*a**4 *b**6*p**2*x**9 + 750*a**4*b**6*p*x**9 + 360*a**4*b**6*x**9 + 20*a**3*b**7 *p**4*x**(19/2) + 200*a**3*b**7*p**3*x**(19/2) + 700*a**3*b**7*p**2*x**(19 /2) + 1000*a**3*b**7*p*x**(19/2) + 480*a**3*b**7*x**(19/2) + 15*a**2*b**8* p**4*x**10 + 150*a**2*b**8*p**3*x**10 + 525*a**2*b**8*p**2*x**10 + 750*a** 2*b**8*p*x**10 + 360*a**2*b**8*x**10 + 6*a*b**9*p**4*x**(21/2) + 60*a*b**9 *p**3*x**(21/2) + 210*a*b**9*p**2*x**(21/2) + 300*a*b**9*p*x**(21/2) + 144 *a*b**9*x**(21/2) + b**10*p**4*x**11 + 10*b**10*p**3*x**11 + 35*b**10*p**2 *x**11 + 50*b**10*p*x**11 + 24*b**10*x**11) + 12*a**6*a**(p + 4)*x**8/(a** 6*b**4*p**4*x**8 + 10*a**6*b**4*p**3*x**8 + 35*a**6*b**4*p**2*x**8 + 50*a* *6*b**4*p*x**8 + 24*a**6*b**4*x**8 + 6*a**5*b**5*p**4*x**(17/2) + 60*a**5* b**5*p**3*x**(17/2) + 210*a**5*b**5*p**2*x**(17/2) + 300*a**5*b**5*p*x**(1 7/2) + 144*a**5*b**5*x**(17/2) + 15*a**4*b**6*p**4*x**9 + 150*a**4*b**6*p* *3*x**9 + 525*a**4*b**6*p**2*x**9 + 750*a**4*b**6*p*x**9 + 360*a**4*b**6*x **9 + 20*a**3*b**7*p**4*x**(19/2) + 200*a**3*b**7*p**3*x**(19/2) + 700*a** 3*b**7*p**2*x**(19/2) + 1000*a**3*b**7*p*x**(19/2) + 480*a**3*b**7*x**(...
Time = 0.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.04 \[ \int \left (a+b \sqrt {x}\right )^p x \, dx=\frac {2 \, {\left ({\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} b^{4} x^{2} + {\left (p^{3} + 3 \, p^{2} + 2 \, p\right )} a b^{3} x^{\frac {3}{2}} - 3 \, {\left (p^{2} + p\right )} a^{2} b^{2} x + 6 \, a^{3} b p \sqrt {x} - 6 \, a^{4}\right )} {\left (b \sqrt {x} + a\right )}^{p}}{{\left (p^{4} + 10 \, p^{3} + 35 \, p^{2} + 50 \, p + 24\right )} b^{4}} \] Input:
integrate((a+b*x^(1/2))^p*x,x, algorithm="maxima")
Output:
2*((p^3 + 6*p^2 + 11*p + 6)*b^4*x^2 + (p^3 + 3*p^2 + 2*p)*a*b^3*x^(3/2) - 3*(p^2 + p)*a^2*b^2*x + 6*a^3*b*p*sqrt(x) - 6*a^4)*(b*sqrt(x) + a)^p/((p^4 + 10*p^3 + 35*p^2 + 50*p + 24)*b^4)
Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (92) = 184\).
Time = 0.13 (sec) , antiderivative size = 410, normalized size of antiderivative = 4.10 \[ \int \left (a+b \sqrt {x}\right )^p x \, dx=\frac {2 \, {\left ({\left (b \sqrt {x} + a\right )}^{4} {\left (b \sqrt {x} + a\right )}^{p} p^{3} - 3 \, {\left (b \sqrt {x} + a\right )}^{3} {\left (b \sqrt {x} + a\right )}^{p} a p^{3} + 3 \, {\left (b \sqrt {x} + a\right )}^{2} {\left (b \sqrt {x} + a\right )}^{p} a^{2} p^{3} - {\left (b \sqrt {x} + a\right )} {\left (b \sqrt {x} + a\right )}^{p} a^{3} p^{3} + 6 \, {\left (b \sqrt {x} + a\right )}^{4} {\left (b \sqrt {x} + a\right )}^{p} p^{2} - 21 \, {\left (b \sqrt {x} + a\right )}^{3} {\left (b \sqrt {x} + a\right )}^{p} a p^{2} + 24 \, {\left (b \sqrt {x} + a\right )}^{2} {\left (b \sqrt {x} + a\right )}^{p} a^{2} p^{2} - 9 \, {\left (b \sqrt {x} + a\right )} {\left (b \sqrt {x} + a\right )}^{p} a^{3} p^{2} + 11 \, {\left (b \sqrt {x} + a\right )}^{4} {\left (b \sqrt {x} + a\right )}^{p} p - 42 \, {\left (b \sqrt {x} + a\right )}^{3} {\left (b \sqrt {x} + a\right )}^{p} a p + 57 \, {\left (b \sqrt {x} + a\right )}^{2} {\left (b \sqrt {x} + a\right )}^{p} a^{2} p - 26 \, {\left (b \sqrt {x} + a\right )} {\left (b \sqrt {x} + a\right )}^{p} a^{3} p + 6 \, {\left (b \sqrt {x} + a\right )}^{4} {\left (b \sqrt {x} + a\right )}^{p} - 24 \, {\left (b \sqrt {x} + a\right )}^{3} {\left (b \sqrt {x} + a\right )}^{p} a + 36 \, {\left (b \sqrt {x} + a\right )}^{2} {\left (b \sqrt {x} + a\right )}^{p} a^{2} - 24 \, {\left (b \sqrt {x} + a\right )} {\left (b \sqrt {x} + a\right )}^{p} a^{3}\right )}}{{\left (b^{3} p^{4} + 10 \, b^{3} p^{3} + 35 \, b^{3} p^{2} + 50 \, b^{3} p + 24 \, b^{3}\right )} b} \] Input:
integrate((a+b*x^(1/2))^p*x,x, algorithm="giac")
Output:
2*((b*sqrt(x) + a)^4*(b*sqrt(x) + a)^p*p^3 - 3*(b*sqrt(x) + a)^3*(b*sqrt(x ) + a)^p*a*p^3 + 3*(b*sqrt(x) + a)^2*(b*sqrt(x) + a)^p*a^2*p^3 - (b*sqrt(x ) + a)*(b*sqrt(x) + a)^p*a^3*p^3 + 6*(b*sqrt(x) + a)^4*(b*sqrt(x) + a)^p*p ^2 - 21*(b*sqrt(x) + a)^3*(b*sqrt(x) + a)^p*a*p^2 + 24*(b*sqrt(x) + a)^2*( b*sqrt(x) + a)^p*a^2*p^2 - 9*(b*sqrt(x) + a)*(b*sqrt(x) + a)^p*a^3*p^2 + 1 1*(b*sqrt(x) + a)^4*(b*sqrt(x) + a)^p*p - 42*(b*sqrt(x) + a)^3*(b*sqrt(x) + a)^p*a*p + 57*(b*sqrt(x) + a)^2*(b*sqrt(x) + a)^p*a^2*p - 26*(b*sqrt(x) + a)*(b*sqrt(x) + a)^p*a^3*p + 6*(b*sqrt(x) + a)^4*(b*sqrt(x) + a)^p - 24* (b*sqrt(x) + a)^3*(b*sqrt(x) + a)^p*a + 36*(b*sqrt(x) + a)^2*(b*sqrt(x) + a)^p*a^2 - 24*(b*sqrt(x) + a)*(b*sqrt(x) + a)^p*a^3)/((b^3*p^4 + 10*b^3*p^ 3 + 35*b^3*p^2 + 50*b^3*p + 24*b^3)*b)
Time = 0.93 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.80 \[ \int \left (a+b \sqrt {x}\right )^p x \, dx={\left (a+b\,\sqrt {x}\right )}^p\,\left (\frac {2\,x^2\,\left (p^3+6\,p^2+11\,p+6\right )}{p^4+10\,p^3+35\,p^2+50\,p+24}-\frac {12\,a^4}{b^4\,\left (p^4+10\,p^3+35\,p^2+50\,p+24\right )}+\frac {12\,a^3\,p\,\sqrt {x}}{b^3\,\left (p^4+10\,p^3+35\,p^2+50\,p+24\right )}-\frac {6\,a^2\,p\,x\,\left (p+1\right )}{b^2\,\left (p^4+10\,p^3+35\,p^2+50\,p+24\right )}+\frac {2\,a\,p\,x^{3/2}\,\left (p^2+3\,p+2\right )}{b\,\left (p^4+10\,p^3+35\,p^2+50\,p+24\right )}\right ) \] Input:
int(x*(a + b*x^(1/2))^p,x)
Output:
(a + b*x^(1/2))^p*((2*x^2*(11*p + 6*p^2 + p^3 + 6))/(50*p + 35*p^2 + 10*p^ 3 + p^4 + 24) - (12*a^4)/(b^4*(50*p + 35*p^2 + 10*p^3 + p^4 + 24)) + (12*a ^3*p*x^(1/2))/(b^3*(50*p + 35*p^2 + 10*p^3 + p^4 + 24)) - (6*a^2*p*x*(p + 1))/(b^2*(50*p + 35*p^2 + 10*p^3 + p^4 + 24)) + (2*a*p*x^(3/2)*(3*p + p^2 + 2))/(b*(50*p + 35*p^2 + 10*p^3 + p^4 + 24)))
Time = 0.22 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.41 \[ \int \left (a+b \sqrt {x}\right )^p x \, dx=\frac {2 \left (\sqrt {x}\, b +a \right )^{p} \left (6 \sqrt {x}\, a^{3} b p +\sqrt {x}\, a \,b^{3} p^{3} x +3 \sqrt {x}\, a \,b^{3} p^{2} x +2 \sqrt {x}\, a \,b^{3} p x -6 a^{4}-3 a^{2} b^{2} p^{2} x -3 a^{2} b^{2} p x +b^{4} p^{3} x^{2}+6 b^{4} p^{2} x^{2}+11 b^{4} p \,x^{2}+6 b^{4} x^{2}\right )}{b^{4} \left (p^{4}+10 p^{3}+35 p^{2}+50 p +24\right )} \] Input:
int((a+b*x^(1/2))^p*x,x)
Output:
(2*(sqrt(x)*b + a)**p*(6*sqrt(x)*a**3*b*p + sqrt(x)*a*b**3*p**3*x + 3*sqrt (x)*a*b**3*p**2*x + 2*sqrt(x)*a*b**3*p*x - 6*a**4 - 3*a**2*b**2*p**2*x - 3 *a**2*b**2*p*x + b**4*p**3*x**2 + 6*b**4*p**2*x**2 + 11*b**4*p*x**2 + 6*b* *4*x**2))/(b**4*(p**4 + 10*p**3 + 35*p**2 + 50*p + 24))