Integrand size = 19, antiderivative size = 90 \[ \int \sqrt {a+b x^n} \left (c+d x^n\right ) \, dx=\frac {2 d x \left (a+b x^n\right )^{3/2}}{b (2+3 n)}+\frac {\left (c-\frac {2 a d}{2 b+3 b n}\right ) x \sqrt {a+b x^n} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{\sqrt {1+\frac {b x^n}{a}}} \] Output:
2*d*x*(a+b*x^n)^(3/2)/b/(2+3*n)+(c-2*a*d/(3*b*n+2*b))*x*(a+b*x^n)^(1/2)*hy pergeom([-1/2, 1/n],[1+1/n],-b*x^n/a)/(1+b*x^n/a)^(1/2)
Time = 0.11 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.08 \[ \int \sqrt {a+b x^n} \left (c+d x^n\right ) \, dx=\frac {x \sqrt {a+b x^n} \left (2 d \left (a+b x^n\right ) \sqrt {1+\frac {b x^n}{a}}+(-2 a d+b c (2+3 n)) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )\right )}{b (2+3 n) \sqrt {1+\frac {b x^n}{a}}} \] Input:
Integrate[Sqrt[a + b*x^n]*(c + d*x^n),x]
Output:
(x*Sqrt[a + b*x^n]*(2*d*(a + b*x^n)*Sqrt[1 + (b*x^n)/a] + (-2*a*d + b*c*(2 + 3*n))*Hypergeometric2F1[-1/2, n^(-1), 1 + n^(-1), -((b*x^n)/a)]))/(b*(2 + 3*n)*Sqrt[1 + (b*x^n)/a])
Time = 0.38 (sec) , antiderivative size = 90, 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.
| \(\displaystyle \int \sqrt {a+b x^n} \left (c+d x^n\right ) \, dx\) |
| \(\Big \downarrow \) 913 |
| \(\displaystyle \left (c-\frac {2 a d}{3 b n+2 b}\right ) \int \sqrt {b x^n+a}dx+\frac {2 d x \left (a+b x^n\right )^{3/2}}{b (3 n+2)}\) |
| \(\Big \downarrow \) 779 |
| \(\displaystyle \frac {\sqrt {a+b x^n} \left (c-\frac {2 a d}{3 b n+2 b}\right ) \int \sqrt {\frac {b x^n}{a}+1}dx}{\sqrt {\frac {b x^n}{a}+1}}+\frac {2 d x \left (a+b x^n\right )^{3/2}}{b (3 n+2)}\) |
| \(\Big \downarrow \) 778 |
| \(\displaystyle \frac {x \sqrt {a+b x^n} \left (c-\frac {2 a d}{3 b n+2 b}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{\sqrt {\frac {b x^n}{a}+1}}+\frac {2 d x \left (a+b x^n\right )^{3/2}}{b (3 n+2)}\) |
Input:
Int[Sqrt[a + b*x^n]*(c + d*x^n),x]
Output:
(2*d*x*(a + b*x^n)^(3/2))/(b*(2 + 3*n)) + ((c - (2*a*d)/(2*b + 3*b*n))*x*S qrt[a + b*x^n]*Hypergeometric2F1[-1/2, 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]
\[\int \sqrt {a +b \,x^{n}}\, \left (c +d \,x^{n}\right )d x\]
Input:
int((a+b*x^n)^(1/2)*(c+d*x^n),x)
Output:
int((a+b*x^n)^(1/2)*(c+d*x^n),x)
Exception generated. \[ \int \sqrt {a+b x^n} \left (c+d x^n\right ) \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b*x^n)^(1/2)*(c+d*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 = 1.40 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.27 \[ \int \sqrt {a+b x^n} \left (c+d x^n\right ) \, dx=\frac {a^{\frac {1}{n}} a^{\frac {1}{2} - \frac {1}{n}} c x \Gamma \left (\frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{n} \\ 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac {1}{n}\right )} + \frac {a^{- \frac {1}{2} - \frac {1}{n}} a^{1 + \frac {1}{n}} d x^{n + 1} \Gamma \left (1 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 1 + \frac {1}{n} \\ 2 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 + \frac {1}{n}\right )} \] Input:
integrate((a+b*x**n)**(1/2)*(c+d*x**n),x)
Output:
a**(1/n)*a**(1/2 - 1/n)*c*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)) + a**(-1/2 - 1/n)*a**(1 + 1/n)*d* x**(n + 1)*gamma(1 + 1/n)*hyper((-1/2, 1 + 1/n), (2 + 1/n,), b*x**n*exp_po lar(I*pi)/a)/(n*gamma(2 + 1/n))
\[ \int \sqrt {a+b x^n} \left (c+d x^n\right ) \, dx=\int { \sqrt {b x^{n} + a} {\left (d x^{n} + c\right )} \,d x } \] Input:
integrate((a+b*x^n)^(1/2)*(c+d*x^n),x, algorithm="maxima")
Output:
integrate(sqrt(b*x^n + a)*(d*x^n + c), x)
\[ \int \sqrt {a+b x^n} \left (c+d x^n\right ) \, dx=\int { \sqrt {b x^{n} + a} {\left (d x^{n} + c\right )} \,d x } \] Input:
integrate((a+b*x^n)^(1/2)*(c+d*x^n),x, algorithm="giac")
Output:
integrate(sqrt(b*x^n + a)*(d*x^n + c), x)
Timed out. \[ \int \sqrt {a+b x^n} \left (c+d x^n\right ) \, dx=\int \sqrt {a+b\,x^n}\,\left (c+d\,x^n\right ) \,d x \] Input:
int((a + b*x^n)^(1/2)*(c + d*x^n),x)
Output:
int((a + b*x^n)^(1/2)*(c + d*x^n), x)
\[ \int \sqrt {a+b x^n} \left (c+d x^n\right ) \, dx=\frac {2 x^{n} \sqrt {x^{n} b +a}\, b d n x +4 x^{n} \sqrt {x^{n} b +a}\, b d x +2 \sqrt {x^{n} b +a}\, a d n x +6 \sqrt {x^{n} b +a}\, b c n x +4 \sqrt {x^{n} b +a}\, b c x -6 \left (\int \frac {\sqrt {x^{n} b +a}}{3 x^{n} b \,n^{2}+8 x^{n} b n +4 x^{n} b +3 a \,n^{2}+8 a n +4 a}d x \right ) a^{2} d \,n^{3}-16 \left (\int \frac {\sqrt {x^{n} b +a}}{3 x^{n} b \,n^{2}+8 x^{n} b n +4 x^{n} b +3 a \,n^{2}+8 a n +4 a}d x \right ) a^{2} d \,n^{2}-8 \left (\int \frac {\sqrt {x^{n} b +a}}{3 x^{n} b \,n^{2}+8 x^{n} b n +4 x^{n} b +3 a \,n^{2}+8 a n +4 a}d x \right ) a^{2} d n +9 \left (\int \frac {\sqrt {x^{n} b +a}}{3 x^{n} b \,n^{2}+8 x^{n} b n +4 x^{n} b +3 a \,n^{2}+8 a n +4 a}d x \right ) a b c \,n^{4}+30 \left (\int \frac {\sqrt {x^{n} b +a}}{3 x^{n} b \,n^{2}+8 x^{n} b n +4 x^{n} b +3 a \,n^{2}+8 a n +4 a}d x \right ) a b c \,n^{3}+28 \left (\int \frac {\sqrt {x^{n} b +a}}{3 x^{n} b \,n^{2}+8 x^{n} b n +4 x^{n} b +3 a \,n^{2}+8 a n +4 a}d x \right ) a b c \,n^{2}+8 \left (\int \frac {\sqrt {x^{n} b +a}}{3 x^{n} b \,n^{2}+8 x^{n} b n +4 x^{n} b +3 a \,n^{2}+8 a n +4 a}d x \right ) a b c n}{b \left (3 n^{2}+8 n +4\right )} \] Input:
int((a+b*x^n)^(1/2)*(c+d*x^n),x)
Output:
(2*x**n*sqrt(x**n*b + a)*b*d*n*x + 4*x**n*sqrt(x**n*b + a)*b*d*x + 2*sqrt( x**n*b + a)*a*d*n*x + 6*sqrt(x**n*b + a)*b*c*n*x + 4*sqrt(x**n*b + a)*b*c* x - 6*int(sqrt(x**n*b + a)/(3*x**n*b*n**2 + 8*x**n*b*n + 4*x**n*b + 3*a*n* *2 + 8*a*n + 4*a),x)*a**2*d*n**3 - 16*int(sqrt(x**n*b + a)/(3*x**n*b*n**2 + 8*x**n*b*n + 4*x**n*b + 3*a*n**2 + 8*a*n + 4*a),x)*a**2*d*n**2 - 8*int(s qrt(x**n*b + a)/(3*x**n*b*n**2 + 8*x**n*b*n + 4*x**n*b + 3*a*n**2 + 8*a*n + 4*a),x)*a**2*d*n + 9*int(sqrt(x**n*b + a)/(3*x**n*b*n**2 + 8*x**n*b*n + 4*x**n*b + 3*a*n**2 + 8*a*n + 4*a),x)*a*b*c*n**4 + 30*int(sqrt(x**n*b + a) /(3*x**n*b*n**2 + 8*x**n*b*n + 4*x**n*b + 3*a*n**2 + 8*a*n + 4*a),x)*a*b*c *n**3 + 28*int(sqrt(x**n*b + a)/(3*x**n*b*n**2 + 8*x**n*b*n + 4*x**n*b + 3 *a*n**2 + 8*a*n + 4*a),x)*a*b*c*n**2 + 8*int(sqrt(x**n*b + a)/(3*x**n*b*n* *2 + 8*x**n*b*n + 4*x**n*b + 3*a*n**2 + 8*a*n + 4*a),x)*a*b*c*n)/(b*(3*n** 2 + 8*n + 4))