Integrand size = 19, antiderivative size = 93 \[ \int \frac {c+d x^n}{\left (a+b x^n\right )^{3/2}} \, dx=\frac {2 d x}{b (2-n) \sqrt {a+b x^n}}+\frac {\left (\frac {c}{a}-\frac {2 d}{b (2-n)}\right ) x \sqrt {1+\frac {b x^n}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{\sqrt {a+b x^n}} \] Output:
2*d*x/b/(2-n)/(a+b*x^n)^(1/2)+(c/a-2*d/b/(2-n))*x*(1+b*x^n/a)^(1/2)*hyperg eom([3/2, 1/n],[1+1/n],-b*x^n/a)/(a+b*x^n)^(1/2)
Time = 0.12 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.83 \[ \int \frac {c+d x^n}{\left (a+b x^n\right )^{3/2}} \, dx=\frac {-2 a d x+(2 a d+b c (-2+n)) x \sqrt {1+\frac {b x^n}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a b (-2+n) \sqrt {a+b x^n}} \] Input:
Integrate[(c + d*x^n)/(a + b*x^n)^(3/2),x]
Output:
(-2*a*d*x + (2*a*d + b*c*(-2 + n))*x*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1 [3/2, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/(a*b*(-2 + n)*Sqrt[a + b*x^n])
Time = 0.39 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {910, 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.
| \(\displaystyle \int \frac {c+d x^n}{\left (a+b x^n\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 910 |
| \(\displaystyle \frac {(2 a d-b c (2-n)) \int \frac {1}{\sqrt {b x^n+a}}dx}{a b n}+\frac {2 x (b c-a d)}{a b n \sqrt {a+b x^n}}\) |
| \(\Big \downarrow \) 779 |
| \(\displaystyle \frac {\sqrt {\frac {b x^n}{a}+1} (2 a d-b c (2-n)) \int \frac {1}{\sqrt {\frac {b x^n}{a}+1}}dx}{a b n \sqrt {a+b x^n}}+\frac {2 x (b c-a d)}{a b n \sqrt {a+b x^n}}\) |
| \(\Big \downarrow \) 778 |
| \(\displaystyle \frac {x \sqrt {\frac {b x^n}{a}+1} (2 a d-b c (2-n)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a b n \sqrt {a+b x^n}}+\frac {2 x (b c-a d)}{a b n \sqrt {a+b x^n}}\) |
Input:
Int[(c + d*x^n)/(a + b*x^n)^(3/2),x]
Output:
(2*(b*c - a*d)*x)/(a*b*n*Sqrt[a + b*x^n]) + ((2*a*d - b*c*(2 - n))*x*Sqrt[ 1 + (b*x^n)/a]*Hypergeometric2F1[1/2, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/( a*b*n*Sqrt[a + b*x^n])
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[(-(b*c - a*d))*x*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] - Simp[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)) Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/ n + p, 0])
\[\int \frac {c +d \,x^{n}}{\left (a +b \,x^{n}\right )^{\frac {3}{2}}}d x\]
Input:
int((c+d*x^n)/(a+b*x^n)^(3/2),x)
Output:
int((c+d*x^n)/(a+b*x^n)^(3/2),x)
Exception generated. \[ \int \frac {c+d x^n}{\left (a+b x^n\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c+d*x^n)/(a+b*x^n)^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Result contains complex when optimal does not.
Time = 2.37 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.20 \[ \int \frac {c+d x^n}{\left (a+b x^n\right )^{3/2}} \, dx=\frac {a^{\frac {1}{n}} a^{- \frac {3}{2} - \frac {1}{n}} c x \Gamma \left (\frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {1}{n} \\ 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac {1}{n}\right )} + \frac {a^{- \frac {5}{2} - \frac {1}{n}} a^{1 + \frac {1}{n}} d x^{n + 1} \Gamma \left (1 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, 1 + \frac {1}{n} \\ 2 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 + \frac {1}{n}\right )} \] Input:
integrate((c+d*x**n)/(a+b*x**n)**(3/2),x)
Output:
a**(1/n)*a**(-3/2 - 1/n)*c*x*gamma(1/n)*hyper((3/2, 1/n), (1 + 1/n,), b*x* *n*exp_polar(I*pi)/a)/(n*gamma(1 + 1/n)) + a**(-5/2 - 1/n)*a**(1 + 1/n)*d* x**(n + 1)*gamma(1 + 1/n)*hyper((3/2, 1 + 1/n), (2 + 1/n,), b*x**n*exp_pol ar(I*pi)/a)/(n*gamma(2 + 1/n))
\[ \int \frac {c+d x^n}{\left (a+b x^n\right )^{3/2}} \, dx=\int { \frac {d x^{n} + c}{{\left (b x^{n} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((c+d*x^n)/(a+b*x^n)^(3/2),x, algorithm="maxima")
Output:
integrate((d*x^n + c)/(b*x^n + a)^(3/2), x)
\[ \int \frac {c+d x^n}{\left (a+b x^n\right )^{3/2}} \, dx=\int { \frac {d x^{n} + c}{{\left (b x^{n} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((c+d*x^n)/(a+b*x^n)^(3/2),x, algorithm="giac")
Output:
integrate((d*x^n + c)/(b*x^n + a)^(3/2), x)
Timed out. \[ \int \frac {c+d x^n}{\left (a+b x^n\right )^{3/2}} \, dx=\int \frac {c+d\,x^n}{{\left (a+b\,x^n\right )}^{3/2}} \,d x \] Input:
int((c + d*x^n)/(a + b*x^n)^(3/2),x)
Output:
int((c + d*x^n)/(a + b*x^n)^(3/2), x)
\[ \int \frac {c+d x^n}{\left (a+b x^n\right )^{3/2}} \, dx=\frac {-2 \sqrt {x^{n} b +a}\, d x +2 x^{n} \left (\int \frac {\sqrt {x^{n} b +a}}{x^{2 n} b^{2} n -2 x^{2 n} b^{2}+2 x^{n} a b n -4 x^{n} a b +a^{2} n -2 a^{2}}d x \right ) a b d n -4 x^{n} \left (\int \frac {\sqrt {x^{n} b +a}}{x^{2 n} b^{2} n -2 x^{2 n} b^{2}+2 x^{n} a b n -4 x^{n} a b +a^{2} n -2 a^{2}}d x \right ) a b d +x^{n} \left (\int \frac {\sqrt {x^{n} b +a}}{x^{2 n} b^{2} n -2 x^{2 n} b^{2}+2 x^{n} a b n -4 x^{n} a b +a^{2} n -2 a^{2}}d x \right ) b^{2} c \,n^{2}-4 x^{n} \left (\int \frac {\sqrt {x^{n} b +a}}{x^{2 n} b^{2} n -2 x^{2 n} b^{2}+2 x^{n} a b n -4 x^{n} a b +a^{2} n -2 a^{2}}d x \right ) b^{2} c n +4 x^{n} \left (\int \frac {\sqrt {x^{n} b +a}}{x^{2 n} b^{2} n -2 x^{2 n} b^{2}+2 x^{n} a b n -4 x^{n} a b +a^{2} n -2 a^{2}}d x \right ) b^{2} c +2 \left (\int \frac {\sqrt {x^{n} b +a}}{x^{2 n} b^{2} n -2 x^{2 n} b^{2}+2 x^{n} a b n -4 x^{n} a b +a^{2} n -2 a^{2}}d x \right ) a^{2} d n -4 \left (\int \frac {\sqrt {x^{n} b +a}}{x^{2 n} b^{2} n -2 x^{2 n} b^{2}+2 x^{n} a b n -4 x^{n} a b +a^{2} n -2 a^{2}}d x \right ) a^{2} d +\left (\int \frac {\sqrt {x^{n} b +a}}{x^{2 n} b^{2} n -2 x^{2 n} b^{2}+2 x^{n} a b n -4 x^{n} a b +a^{2} n -2 a^{2}}d x \right ) a b c \,n^{2}-4 \left (\int \frac {\sqrt {x^{n} b +a}}{x^{2 n} b^{2} n -2 x^{2 n} b^{2}+2 x^{n} a b n -4 x^{n} a b +a^{2} n -2 a^{2}}d x \right ) a b c n +4 \left (\int \frac {\sqrt {x^{n} b +a}}{x^{2 n} b^{2} n -2 x^{2 n} b^{2}+2 x^{n} a b n -4 x^{n} a b +a^{2} n -2 a^{2}}d x \right ) a b c}{b \left (x^{n} b n -2 x^{n} b +a n -2 a \right )} \] Input:
int((c+d*x^n)/(a+b*x^n)^(3/2),x)
Output:
( - 2*sqrt(x**n*b + a)*d*x + 2*x**n*int(sqrt(x**n*b + a)/(x**(2*n)*b**2*n - 2*x**(2*n)*b**2 + 2*x**n*a*b*n - 4*x**n*a*b + a**2*n - 2*a**2),x)*a*b*d* n - 4*x**n*int(sqrt(x**n*b + a)/(x**(2*n)*b**2*n - 2*x**(2*n)*b**2 + 2*x** n*a*b*n - 4*x**n*a*b + a**2*n - 2*a**2),x)*a*b*d + x**n*int(sqrt(x**n*b + a)/(x**(2*n)*b**2*n - 2*x**(2*n)*b**2 + 2*x**n*a*b*n - 4*x**n*a*b + a**2*n - 2*a**2),x)*b**2*c*n**2 - 4*x**n*int(sqrt(x**n*b + a)/(x**(2*n)*b**2*n - 2*x**(2*n)*b**2 + 2*x**n*a*b*n - 4*x**n*a*b + a**2*n - 2*a**2),x)*b**2*c* n + 4*x**n*int(sqrt(x**n*b + a)/(x**(2*n)*b**2*n - 2*x**(2*n)*b**2 + 2*x** n*a*b*n - 4*x**n*a*b + a**2*n - 2*a**2),x)*b**2*c + 2*int(sqrt(x**n*b + a) /(x**(2*n)*b**2*n - 2*x**(2*n)*b**2 + 2*x**n*a*b*n - 4*x**n*a*b + a**2*n - 2*a**2),x)*a**2*d*n - 4*int(sqrt(x**n*b + a)/(x**(2*n)*b**2*n - 2*x**(2*n )*b**2 + 2*x**n*a*b*n - 4*x**n*a*b + a**2*n - 2*a**2),x)*a**2*d + int(sqrt (x**n*b + a)/(x**(2*n)*b**2*n - 2*x**(2*n)*b**2 + 2*x**n*a*b*n - 4*x**n*a* b + a**2*n - 2*a**2),x)*a*b*c*n**2 - 4*int(sqrt(x**n*b + a)/(x**(2*n)*b**2 *n - 2*x**(2*n)*b**2 + 2*x**n*a*b*n - 4*x**n*a*b + a**2*n - 2*a**2),x)*a*b *c*n + 4*int(sqrt(x**n*b + a)/(x**(2*n)*b**2*n - 2*x**(2*n)*b**2 + 2*x**n* a*b*n - 4*x**n*a*b + a**2*n - 2*a**2),x)*a*b*c)/(b*(x**n*b*n - 2*x**n*b + a*n - 2*a))