Integrand size = 11, antiderivative size = 50 \[ \int \left (a+b x^{5/2}\right )^p \, dx=x \left (a+b x^{5/2}\right )^p \left (1+\frac {b x^{5/2}}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2}{5},-p,\frac {7}{5},-\frac {b x^{5/2}}{a}\right ) \] Output:
x*(a+b*x^(5/2))^p*hypergeom([2/5, -p],[7/5],-b*x^(5/2)/a)/((1+b*x^(5/2)/a) ^p)
Time = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^{5/2}\right )^p \, dx=x \left (a+b x^{5/2}\right )^p \left (1+\frac {b x^{5/2}}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2}{5},-p,\frac {7}{5},-\frac {b x^{5/2}}{a}\right ) \] Input:
Integrate[(a + b*x^(5/2))^p,x]
Output:
(x*(a + b*x^(5/2))^p*Hypergeometric2F1[2/5, -p, 7/5, -((b*x^(5/2))/a)])/(1 + (b*x^(5/2))/a)^p
Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {774, 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.
\(\displaystyle \int \left (a+b x^{5/2}\right )^p \, dx\) |
\(\Big \downarrow \) 774 |
\(\displaystyle 2 \int \sqrt {x} \left (b x^{5/2}+a\right )^pd\sqrt {x}\) |
\(\Big \downarrow \) 889 |
\(\displaystyle 2 \left (a+b x^{5/2}\right )^p \left (\frac {b x^{5/2}}{a}+1\right )^{-p} \int \sqrt {x} \left (\frac {b x^{5/2}}{a}+1\right )^pd\sqrt {x}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle x \left (a+b x^{5/2}\right )^p \left (\frac {b x^{5/2}}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2}{5},-p,\frac {7}{5},-\frac {b x^{5/2}}{a}\right )\) |
Input:
Int[(a + b*x^(5/2))^p,x]
Output:
(x*(a + b*x^(5/2))^p*Hypergeometric2F1[2/5, -p, 7/5, -((b*x^(5/2))/a)])/(1 + (b*x^(5/2))/a)^p
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; Fre eQ[{a, b, p}, x] && FractionQ[n]
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])
\[\int \left (a +b \,x^{\frac {5}{2}}\right )^{p}d x\]
Input:
int((a+b*x^(5/2))^p,x)
Output:
int((a+b*x^(5/2))^p,x)
\[ \int \left (a+b x^{5/2}\right )^p \, dx=\int { {\left (b x^{\frac {5}{2}} + a\right )}^{p} \,d x } \] Input:
integrate((a+b*x^(5/2))^p,x, algorithm="fricas")
Output:
integral((b*x^(5/2) + a)^p, x)
Timed out. \[ \int \left (a+b x^{5/2}\right )^p \, dx=\text {Timed out} \] Input:
integrate((a+b*x**(5/2))**p,x)
Output:
Timed out
\[ \int \left (a+b x^{5/2}\right )^p \, dx=\int { {\left (b x^{\frac {5}{2}} + a\right )}^{p} \,d x } \] Input:
integrate((a+b*x^(5/2))^p,x, algorithm="maxima")
Output:
integrate((b*x^(5/2) + a)^p, x)
\[ \int \left (a+b x^{5/2}\right )^p \, dx=\int { {\left (b x^{\frac {5}{2}} + a\right )}^{p} \,d x } \] Input:
integrate((a+b*x^(5/2))^p,x, algorithm="giac")
Output:
integrate((b*x^(5/2) + a)^p, x)
Time = 0.23 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.82 \[ \int \left (a+b x^{5/2}\right )^p \, dx=\frac {x\,{\left (a+b\,x^{5/2}\right )}^p\,{{}}_2{\mathrm {F}}_1\left (\frac {2}{5},-p;\ \frac {7}{5};\ -\frac {b\,x^{5/2}}{a}\right )}{{\left (\frac {b\,x^{5/2}}{a}+1\right )}^p} \] Input:
int((a + b*x^(5/2))^p,x)
Output:
(x*(a + b*x^(5/2))^p*hypergeom([2/5, -p], 7/5, -(b*x^(5/2))/a))/((b*x^(5/2 ))/a + 1)^p
\[ \int \left (a+b x^{5/2}\right )^p \, dx=\frac {2 \left (\sqrt {x}\, b \,x^{2}+a \right )^{p} x +25 \left (\int \frac {\left (\sqrt {x}\, b \,x^{2}+a \right )^{p}}{-5 b^{2} p \,x^{5}-2 b^{2} x^{5}+5 a^{2} p +2 a^{2}}d x \right ) a^{2} p^{2}+10 \left (\int \frac {\left (\sqrt {x}\, b \,x^{2}+a \right )^{p}}{-5 b^{2} p \,x^{5}-2 b^{2} x^{5}+5 a^{2} p +2 a^{2}}d x \right ) a^{2} p -25 \left (\int \frac {\sqrt {x}\, \left (\sqrt {x}\, b \,x^{2}+a \right )^{p} x^{2}}{-5 b^{2} p \,x^{5}-2 b^{2} x^{5}+5 a^{2} p +2 a^{2}}d x \right ) a b \,p^{2}-10 \left (\int \frac {\sqrt {x}\, \left (\sqrt {x}\, b \,x^{2}+a \right )^{p} x^{2}}{-5 b^{2} p \,x^{5}-2 b^{2} x^{5}+5 a^{2} p +2 a^{2}}d x \right ) a b p}{5 p +2} \] Input:
int((a+b*x^(5/2))^p,x)
Output:
(2*(sqrt(x)*b*x**2 + a)**p*x + 25*int((sqrt(x)*b*x**2 + a)**p/(5*a**2*p + 2*a**2 - 5*b**2*p*x**5 - 2*b**2*x**5),x)*a**2*p**2 + 10*int((sqrt(x)*b*x** 2 + a)**p/(5*a**2*p + 2*a**2 - 5*b**2*p*x**5 - 2*b**2*x**5),x)*a**2*p - 25 *int((sqrt(x)*(sqrt(x)*b*x**2 + a)**p*x**2)/(5*a**2*p + 2*a**2 - 5*b**2*p* x**5 - 2*b**2*x**5),x)*a*b*p**2 - 10*int((sqrt(x)*(sqrt(x)*b*x**2 + a)**p* x**2)/(5*a**2*p + 2*a**2 - 5*b**2*p*x**5 - 2*b**2*x**5),x)*a*b*p)/(5*p + 2 )