Integrand size = 15, antiderivative size = 55 \[ \int x^m \left (a+b x^n\right )^{3/2} \, dx=\frac {x^{1+m} \left (a+b x^n\right )^{5/2} \operatorname {Hypergeometric2F1}\left (1,\frac {5}{2}+\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a (1+m)} \] Output:
x^(1+m)*(a+b*x^n)^(5/2)*hypergeom([1, 5/2+(1+m)/n],[(1+m+n)/n],-b*x^n/a)/a /(1+m)
Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.20 \[ \int x^m \left (a+b x^n\right )^{3/2} \, dx=\frac {a x^{1+m} \sqrt {a+b x^n} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1+m}{n},1+\frac {1+m}{n},-\frac {b x^n}{a}\right )}{(1+m) \sqrt {1+\frac {b x^n}{a}}} \] Input:
Integrate[x^m*(a + b*x^n)^(3/2),x]
Output:
(a*x^(1 + m)*Sqrt[a + b*x^n]*Hypergeometric2F1[-3/2, (1 + m)/n, 1 + (1 + m )/n, -((b*x^n)/a)])/((1 + m)*Sqrt[1 + (b*x^n)/a])
Time = 0.30 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {889, 888}
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.
\(\displaystyle \int x^m \left (a+b x^n\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 889 |
\(\displaystyle \frac {a \sqrt {a+b x^n} \int x^m \left (\frac {b x^n}{a}+1\right )^{3/2}dx}{\sqrt {\frac {b x^n}{a}+1}}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {a x^{m+1} \sqrt {a+b x^n} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {m+1}{n},\frac {m+n+1}{n},-\frac {b x^n}{a}\right )}{(m+1) \sqrt {\frac {b x^n}{a}+1}}\) |
Input:
Int[x^m*(a + b*x^n)^(3/2),x]
Output:
(a*x^(1 + m)*Sqrt[a + b*x^n]*Hypergeometric2F1[-3/2, (1 + m)/n, (1 + m + n )/n, -((b*x^n)/a)])/((1 + m)*Sqrt[1 + (b*x^n)/a])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I ntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) Int[(c*x) ^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0 ] && !(ILtQ[p, 0] || GtQ[a, 0])
\[\int x^{m} \left (a +b \,x^{n}\right )^{\frac {3}{2}}d x\]
Input:
int(x^m*(a+b*x^n)^(3/2),x)
Output:
int(x^m*(a+b*x^n)^(3/2),x)
Exception generated. \[ \int x^m \left (a+b x^n\right )^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^m*(a+b*x^n)^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
Result contains complex when optimal does not.
Time = 9.00 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.33 \[ \int x^m \left (a+b x^n\right )^{3/2} \, dx=\frac {a^{\frac {m}{n} + \frac {1}{n}} a^{- \frac {m}{n} + \frac {3}{2} - \frac {1}{n}} x^{m + 1} \Gamma \left (\frac {m}{n} + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {m}{n} + \frac {1}{n} \\ \frac {m}{n} + 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )} \] Input:
integrate(x**m*(a+b*x**n)**(3/2),x)
Output:
a**(m/n + 1/n)*a**(-m/n + 3/2 - 1/n)*x**(m + 1)*gamma(m/n + 1/n)*hyper((-3 /2, m/n + 1/n), (m/n + 1 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(m/n + 1 + 1/n))
\[ \int x^m \left (a+b x^n\right )^{3/2} \, dx=\int { {\left (b x^{n} + a\right )}^{\frac {3}{2}} x^{m} \,d x } \] Input:
integrate(x^m*(a+b*x^n)^(3/2),x, algorithm="maxima")
Output:
integrate((b*x^n + a)^(3/2)*x^m, x)
\[ \int x^m \left (a+b x^n\right )^{3/2} \, dx=\int { {\left (b x^{n} + a\right )}^{\frac {3}{2}} x^{m} \,d x } \] Input:
integrate(x^m*(a+b*x^n)^(3/2),x, algorithm="giac")
Output:
integrate((b*x^n + a)^(3/2)*x^m, x)
Timed out. \[ \int x^m \left (a+b x^n\right )^{3/2} \, dx=\int x^m\,{\left (a+b\,x^n\right )}^{3/2} \,d x \] Input:
int(x^m*(a + b*x^n)^(3/2),x)
Output:
int(x^m*(a + b*x^n)^(3/2), x)
\[ \int x^m \left (a+b x^n\right )^{3/2} \, dx=\frac {4 x^{m +n} \sqrt {x^{n} b +a}\, b m x +2 x^{m +n} \sqrt {x^{n} b +a}\, b n x +4 x^{m +n} \sqrt {x^{n} b +a}\, b x +4 x^{m} \sqrt {x^{n} b +a}\, a m x +8 x^{m} \sqrt {x^{n} b +a}\, a n x +4 x^{m} \sqrt {x^{n} b +a}\, a x +12 \left (\int \frac {x^{m} \sqrt {x^{n} b +a}}{4 x^{n} b \,m^{2}+8 x^{n} b m n +8 x^{n} b m +3 x^{n} b \,n^{2}+8 x^{n} b n +4 x^{n} b +4 a \,m^{2}+8 a m n +8 a m +3 a \,n^{2}+8 a n +4 a}d x \right ) a^{2} m^{2} n^{2}+24 \left (\int \frac {x^{m} \sqrt {x^{n} b +a}}{4 x^{n} b \,m^{2}+8 x^{n} b m n +8 x^{n} b m +3 x^{n} b \,n^{2}+8 x^{n} b n +4 x^{n} b +4 a \,m^{2}+8 a m n +8 a m +3 a \,n^{2}+8 a n +4 a}d x \right ) a^{2} m \,n^{3}+24 \left (\int \frac {x^{m} \sqrt {x^{n} b +a}}{4 x^{n} b \,m^{2}+8 x^{n} b m n +8 x^{n} b m +3 x^{n} b \,n^{2}+8 x^{n} b n +4 x^{n} b +4 a \,m^{2}+8 a m n +8 a m +3 a \,n^{2}+8 a n +4 a}d x \right ) a^{2} m \,n^{2}+9 \left (\int \frac {x^{m} \sqrt {x^{n} b +a}}{4 x^{n} b \,m^{2}+8 x^{n} b m n +8 x^{n} b m +3 x^{n} b \,n^{2}+8 x^{n} b n +4 x^{n} b +4 a \,m^{2}+8 a m n +8 a m +3 a \,n^{2}+8 a n +4 a}d x \right ) a^{2} n^{4}+24 \left (\int \frac {x^{m} \sqrt {x^{n} b +a}}{4 x^{n} b \,m^{2}+8 x^{n} b m n +8 x^{n} b m +3 x^{n} b \,n^{2}+8 x^{n} b n +4 x^{n} b +4 a \,m^{2}+8 a m n +8 a m +3 a \,n^{2}+8 a n +4 a}d x \right ) a^{2} n^{3}+12 \left (\int \frac {x^{m} \sqrt {x^{n} b +a}}{4 x^{n} b \,m^{2}+8 x^{n} b m n +8 x^{n} b m +3 x^{n} b \,n^{2}+8 x^{n} b n +4 x^{n} b +4 a \,m^{2}+8 a m n +8 a m +3 a \,n^{2}+8 a n +4 a}d x \right ) a^{2} n^{2}}{4 m^{2}+8 m n +3 n^{2}+8 m +8 n +4} \] Input:
int(x^m*(a+b*x^n)^(3/2),x)
Output:
(4*x**(m + n)*sqrt(x**n*b + a)*b*m*x + 2*x**(m + n)*sqrt(x**n*b + a)*b*n*x + 4*x**(m + n)*sqrt(x**n*b + a)*b*x + 4*x**m*sqrt(x**n*b + a)*a*m*x + 8*x **m*sqrt(x**n*b + a)*a*n*x + 4*x**m*sqrt(x**n*b + a)*a*x + 12*int((x**m*sq rt(x**n*b + a))/(4*x**n*b*m**2 + 8*x**n*b*m*n + 8*x**n*b*m + 3*x**n*b*n**2 + 8*x**n*b*n + 4*x**n*b + 4*a*m**2 + 8*a*m*n + 8*a*m + 3*a*n**2 + 8*a*n + 4*a),x)*a**2*m**2*n**2 + 24*int((x**m*sqrt(x**n*b + a))/(4*x**n*b*m**2 + 8*x**n*b*m*n + 8*x**n*b*m + 3*x**n*b*n**2 + 8*x**n*b*n + 4*x**n*b + 4*a*m* *2 + 8*a*m*n + 8*a*m + 3*a*n**2 + 8*a*n + 4*a),x)*a**2*m*n**3 + 24*int((x* *m*sqrt(x**n*b + a))/(4*x**n*b*m**2 + 8*x**n*b*m*n + 8*x**n*b*m + 3*x**n*b *n**2 + 8*x**n*b*n + 4*x**n*b + 4*a*m**2 + 8*a*m*n + 8*a*m + 3*a*n**2 + 8* a*n + 4*a),x)*a**2*m*n**2 + 9*int((x**m*sqrt(x**n*b + a))/(4*x**n*b*m**2 + 8*x**n*b*m*n + 8*x**n*b*m + 3*x**n*b*n**2 + 8*x**n*b*n + 4*x**n*b + 4*a*m **2 + 8*a*m*n + 8*a*m + 3*a*n**2 + 8*a*n + 4*a),x)*a**2*n**4 + 24*int((x** m*sqrt(x**n*b + a))/(4*x**n*b*m**2 + 8*x**n*b*m*n + 8*x**n*b*m + 3*x**n*b* n**2 + 8*x**n*b*n + 4*x**n*b + 4*a*m**2 + 8*a*m*n + 8*a*m + 3*a*n**2 + 8*a *n + 4*a),x)*a**2*n**3 + 12*int((x**m*sqrt(x**n*b + a))/(4*x**n*b*m**2 + 8 *x**n*b*m*n + 8*x**n*b*m + 3*x**n*b*n**2 + 8*x**n*b*n + 4*x**n*b + 4*a*m** 2 + 8*a*m*n + 8*a*m + 3*a*n**2 + 8*a*n + 4*a),x)*a**2*n**2)/(4*m**2 + 8*m* n + 8*m + 3*n**2 + 8*n + 4)