Integrand size = 19, antiderivative size = 93 \[ \int \left (a+b x^n\right )^{5/2} \left (A+B x^n\right ) \, dx=\frac {2 B x \left (a+b x^n\right )^{7/2}}{b (2+7 n)}+\frac {a^2 \left (A-\frac {2 a B}{2 b+7 b n}\right ) x \sqrt {a+b x^n} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{\sqrt {1+\frac {b x^n}{a}}} \] Output:
2*B*x*(a+b*x^n)^(7/2)/b/(2+7*n)+a^2*(A-2*a*B/(7*b*n+2*b))*x*(a+b*x^n)^(1/2 )*hypergeom([-5/2, 1/n],[1+1/n],-b*x^n/a)/(1+b*x^n/a)^(1/2)
Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.97 \[ \int \left (a+b x^n\right )^{5/2} \left (A+B x^n\right ) \, dx=\frac {x \sqrt {a+b x^n} \left (B \left (a+b x^n\right )^3-\frac {a^2 \left (a B-\frac {1}{2} A b (2+7 n)\right ) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{\sqrt {1+\frac {b x^n}{a}}}\right )}{b+\frac {7 b n}{2}} \] Input:
Integrate[(a + b*x^n)^(5/2)*(A + B*x^n),x]
Output:
(x*Sqrt[a + b*x^n]*(B*(a + b*x^n)^3 - (a^2*(a*B - (A*b*(2 + 7*n))/2)*Hyper geometric2F1[-5/2, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/Sqrt[1 + (b*x^n)/a]) )/(b + (7*b*n)/2)
Time = 0.38 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {913, 779, 778}
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.
| \(\displaystyle \int \left (a+b x^n\right )^{5/2} \left (A+B x^n\right ) \, dx\) |
| \(\Big \downarrow \) 913 |
| \(\displaystyle \left (A-\frac {2 a B}{7 b n+2 b}\right ) \int \left (b x^n+a\right )^{5/2}dx+\frac {2 B x \left (a+b x^n\right )^{7/2}}{b (7 n+2)}\) |
| \(\Big \downarrow \) 779 |
| \(\displaystyle \frac {a^2 \sqrt {a+b x^n} \left (A-\frac {2 a B}{7 b n+2 b}\right ) \int \left (\frac {b x^n}{a}+1\right )^{5/2}dx}{\sqrt {\frac {b x^n}{a}+1}}+\frac {2 B x \left (a+b x^n\right )^{7/2}}{b (7 n+2)}\) |
| \(\Big \downarrow \) 778 |
| \(\displaystyle \frac {a^2 x \sqrt {a+b x^n} \left (A-\frac {2 a B}{7 b n+2 b}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{\sqrt {\frac {b x^n}{a}+1}}+\frac {2 B x \left (a+b x^n\right )^{7/2}}{b (7 n+2)}\) |
Input:
Int[(a + b*x^n)^(5/2)*(A + B*x^n),x]
Output:
(2*B*x*(a + b*x^n)^(7/2))/(b*(2 + 7*n)) + (a^2*(A - (2*a*B)/(2*b + 7*b*n)) *x*Sqrt[a + b*x^n]*Hypergeometric2F1[-5/2, n^(-1), 1 + n^(-1), -((b*x^n)/a )])/Sqrt[1 + (b*x^n)/a]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] && !IGtQ[p , 0] && !IntegerQ[1/n] && !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || GtQ[a, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x ^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) Int[(1 + b*(x^n/a))^p, x], x ] /; FreeQ[{a, b, n, p}, x] && !IGtQ[p, 0] && !IntegerQ[1/n] && !ILtQ[Si mplify[1/n + p], 0] && !(IntegerQ[p] || GtQ[a, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(p + 1) + 1))), x] - Simp[(a*d - b*c*(n*( p + 1) + 1))/(b*(n*(p + 1) + 1)) Int[(a + b*x^n)^p, x], x] /; FreeQ[{a, b , c, d, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
\[\int \left (a +b \,x^{n}\right )^{\frac {5}{2}} \left (A +B \,x^{n}\right )d x\]
Input:
int((a+b*x^n)^(5/2)*(A+B*x^n),x)
Output:
int((a+b*x^n)^(5/2)*(A+B*x^n),x)
Exception generated. \[ \int \left (a+b x^n\right )^{5/2} \left (A+B x^n\right ) \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b*x^n)^(5/2)*(A+B*x^n),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 = 8.45 (sec) , antiderivative size = 394, normalized size of antiderivative = 4.24 \[ \int \left (a+b x^n\right )^{5/2} \left (A+B x^n\right ) \, dx=\frac {A a^{2} a^{\frac {1}{n}} a^{\frac {1}{2} - \frac {1}{n}} x \Gamma \left (\frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{n} \\ 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac {1}{n}\right )} + \frac {2 A a a^{- \frac {1}{2} - \frac {1}{n}} a^{1 + \frac {1}{n}} b x^{n + 1} \Gamma \left (1 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 1 + \frac {1}{n} \\ 2 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 + \frac {1}{n}\right )} + \frac {A a^{- \frac {3}{2} - \frac {1}{n}} a^{2 + \frac {1}{n}} b^{2} x^{2 n + 1} \Gamma \left (2 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 2 + \frac {1}{n} \\ 3 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (3 + \frac {1}{n}\right )} + \frac {B a^{2} a^{- \frac {1}{2} - \frac {1}{n}} a^{1 + \frac {1}{n}} x^{n + 1} \Gamma \left (1 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 1 + \frac {1}{n} \\ 2 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 + \frac {1}{n}\right )} + \frac {2 B a a^{- \frac {3}{2} - \frac {1}{n}} a^{2 + \frac {1}{n}} b x^{2 n + 1} \Gamma \left (2 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 2 + \frac {1}{n} \\ 3 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (3 + \frac {1}{n}\right )} + \frac {B a^{- \frac {5}{2} - \frac {1}{n}} a^{3 + \frac {1}{n}} b^{2} x^{3 n + 1} \Gamma \left (3 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 3 + \frac {1}{n} \\ 4 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (4 + \frac {1}{n}\right )} \] Input:
integrate((a+b*x**n)**(5/2)*(A+B*x**n),x)
Output:
A*a**2*a**(1/n)*a**(1/2 - 1/n)*x*gamma(1/n)*hyper((-1/2, 1/n), (1 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(1 + 1/n)) + 2*A*a*a**(-1/2 - 1/n)*a**( 1 + 1/n)*b*x**(n + 1)*gamma(1 + 1/n)*hyper((-1/2, 1 + 1/n), (2 + 1/n,), b* x**n*exp_polar(I*pi)/a)/(n*gamma(2 + 1/n)) + A*a**(-3/2 - 1/n)*a**(2 + 1/n )*b**2*x**(2*n + 1)*gamma(2 + 1/n)*hyper((-1/2, 2 + 1/n), (3 + 1/n,), b*x* *n*exp_polar(I*pi)/a)/(n*gamma(3 + 1/n)) + B*a**2*a**(-1/2 - 1/n)*a**(1 + 1/n)*x**(n + 1)*gamma(1 + 1/n)*hyper((-1/2, 1 + 1/n), (2 + 1/n,), b*x**n*e xp_polar(I*pi)/a)/(n*gamma(2 + 1/n)) + 2*B*a*a**(-3/2 - 1/n)*a**(2 + 1/n)* b*x**(2*n + 1)*gamma(2 + 1/n)*hyper((-1/2, 2 + 1/n), (3 + 1/n,), b*x**n*ex p_polar(I*pi)/a)/(n*gamma(3 + 1/n)) + B*a**(-5/2 - 1/n)*a**(3 + 1/n)*b**2* x**(3*n + 1)*gamma(3 + 1/n)*hyper((-1/2, 3 + 1/n), (4 + 1/n,), b*x**n*exp_ polar(I*pi)/a)/(n*gamma(4 + 1/n))
\[ \int \left (a+b x^n\right )^{5/2} \left (A+B x^n\right ) \, dx=\int { {\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+b*x^n)^(5/2)*(A+B*x^n),x, algorithm="maxima")
Output:
integrate((B*x^n + A)*(b*x^n + a)^(5/2), x)
\[ \int \left (a+b x^n\right )^{5/2} \left (A+B x^n\right ) \, dx=\int { {\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+b*x^n)^(5/2)*(A+B*x^n),x, algorithm="giac")
Output:
integrate((B*x^n + A)*(b*x^n + a)^(5/2), x)
Timed out. \[ \int \left (a+b x^n\right )^{5/2} \left (A+B x^n\right ) \, dx=\int \left (A+B\,x^n\right )\,{\left (a+b\,x^n\right )}^{5/2} \,d x \] Input:
int((A + B*x^n)*(a + b*x^n)^(5/2),x)
Output:
int((A + B*x^n)*(a + b*x^n)^(5/2), x)
\[ \int \left (a+b x^n\right )^{5/2} \left (A+B x^n\right ) \, dx=\frac {30 x^{3 n} \sqrt {x^{n} b +a}\, b^{3} n^{3} x +92 x^{3 n} \sqrt {x^{n} b +a}\, b^{3} n^{2} x +72 x^{3 n} \sqrt {x^{n} b +a}\, b^{3} n x +16 x^{3 n} \sqrt {x^{n} b +a}\, b^{3} x +132 x^{2 n} \sqrt {x^{n} b +a}\, a \,b^{2} n^{3} x +388 x^{2 n} \sqrt {x^{n} b +a}\, a \,b^{2} n^{2} x +272 x^{2 n} \sqrt {x^{n} b +a}\, a \,b^{2} n x +48 x^{2 n} \sqrt {x^{n} b +a}\, a \,b^{2} x +244 x^{n} \sqrt {x^{n} b +a}\, a^{2} b \,n^{3} x +640 x^{n} \sqrt {x^{n} b +a}\, a^{2} b \,n^{2} x +328 x^{n} \sqrt {x^{n} b +a}\, a^{2} b n x +48 x^{n} \sqrt {x^{n} b +a}\, a^{2} b x +352 \sqrt {x^{n} b +a}\, a^{3} n^{3} x +344 \sqrt {x^{n} b +a}\, a^{3} n^{2} x +128 \sqrt {x^{n} b +a}\, a^{3} n x +16 \sqrt {x^{n} b +a}\, a^{3} x +11025 \left (\int \frac {\sqrt {x^{n} b +a}}{105 x^{n} b \,n^{4}+352 x^{n} b \,n^{3}+344 x^{n} b \,n^{2}+128 x^{n} b n +16 x^{n} b +105 a \,n^{4}+352 a \,n^{3}+344 a \,n^{2}+128 a n +16 a}d x \right ) a^{4} n^{8}+36960 \left (\int \frac {\sqrt {x^{n} b +a}}{105 x^{n} b \,n^{4}+352 x^{n} b \,n^{3}+344 x^{n} b \,n^{2}+128 x^{n} b n +16 x^{n} b +105 a \,n^{4}+352 a \,n^{3}+344 a \,n^{2}+128 a n +16 a}d x \right ) a^{4} n^{7}+36120 \left (\int \frac {\sqrt {x^{n} b +a}}{105 x^{n} b \,n^{4}+352 x^{n} b \,n^{3}+344 x^{n} b \,n^{2}+128 x^{n} b n +16 x^{n} b +105 a \,n^{4}+352 a \,n^{3}+344 a \,n^{2}+128 a n +16 a}d x \right ) a^{4} n^{6}+13440 \left (\int \frac {\sqrt {x^{n} b +a}}{105 x^{n} b \,n^{4}+352 x^{n} b \,n^{3}+344 x^{n} b \,n^{2}+128 x^{n} b n +16 x^{n} b +105 a \,n^{4}+352 a \,n^{3}+344 a \,n^{2}+128 a n +16 a}d x \right ) a^{4} n^{5}+1680 \left (\int \frac {\sqrt {x^{n} b +a}}{105 x^{n} b \,n^{4}+352 x^{n} b \,n^{3}+344 x^{n} b \,n^{2}+128 x^{n} b n +16 x^{n} b +105 a \,n^{4}+352 a \,n^{3}+344 a \,n^{2}+128 a n +16 a}d x \right ) a^{4} n^{4}}{105 n^{4}+352 n^{3}+344 n^{2}+128 n +16} \] Input:
int((a+b*x^n)^(5/2)*(A+B*x^n),x)
Output:
(30*x**(3*n)*sqrt(x**n*b + a)*b**3*n**3*x + 92*x**(3*n)*sqrt(x**n*b + a)*b **3*n**2*x + 72*x**(3*n)*sqrt(x**n*b + a)*b**3*n*x + 16*x**(3*n)*sqrt(x**n *b + a)*b**3*x + 132*x**(2*n)*sqrt(x**n*b + a)*a*b**2*n**3*x + 388*x**(2*n )*sqrt(x**n*b + a)*a*b**2*n**2*x + 272*x**(2*n)*sqrt(x**n*b + a)*a*b**2*n* x + 48*x**(2*n)*sqrt(x**n*b + a)*a*b**2*x + 244*x**n*sqrt(x**n*b + a)*a**2 *b*n**3*x + 640*x**n*sqrt(x**n*b + a)*a**2*b*n**2*x + 328*x**n*sqrt(x**n*b + a)*a**2*b*n*x + 48*x**n*sqrt(x**n*b + a)*a**2*b*x + 352*sqrt(x**n*b + a )*a**3*n**3*x + 344*sqrt(x**n*b + a)*a**3*n**2*x + 128*sqrt(x**n*b + a)*a* *3*n*x + 16*sqrt(x**n*b + a)*a**3*x + 11025*int(sqrt(x**n*b + a)/(105*x**n *b*n**4 + 352*x**n*b*n**3 + 344*x**n*b*n**2 + 128*x**n*b*n + 16*x**n*b + 1 05*a*n**4 + 352*a*n**3 + 344*a*n**2 + 128*a*n + 16*a),x)*a**4*n**8 + 36960 *int(sqrt(x**n*b + a)/(105*x**n*b*n**4 + 352*x**n*b*n**3 + 344*x**n*b*n**2 + 128*x**n*b*n + 16*x**n*b + 105*a*n**4 + 352*a*n**3 + 344*a*n**2 + 128*a *n + 16*a),x)*a**4*n**7 + 36120*int(sqrt(x**n*b + a)/(105*x**n*b*n**4 + 35 2*x**n*b*n**3 + 344*x**n*b*n**2 + 128*x**n*b*n + 16*x**n*b + 105*a*n**4 + 352*a*n**3 + 344*a*n**2 + 128*a*n + 16*a),x)*a**4*n**6 + 13440*int(sqrt(x* *n*b + a)/(105*x**n*b*n**4 + 352*x**n*b*n**3 + 344*x**n*b*n**2 + 128*x**n* b*n + 16*x**n*b + 105*a*n**4 + 352*a*n**3 + 344*a*n**2 + 128*a*n + 16*a),x )*a**4*n**5 + 1680*int(sqrt(x**n*b + a)/(105*x**n*b*n**4 + 352*x**n*b*n**3 + 344*x**n*b*n**2 + 128*x**n*b*n + 16*x**n*b + 105*a*n**4 + 352*a*n**3...