3.6.84 \(\int \frac {(c x)^m (d+e x^n+f x^{2 n}+g x^{3 n})}{\sqrt {a+b x^n}} \, dx\) [584]

3.6.84.1 Optimal result

Integrand size = 38, antiderivative size = 305 \[ \int \frac {(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{\sqrt {a+b x^n}} \, dx=\frac {d (c x)^{1+m} \sqrt {1+\frac {b x^n}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{c (1+m) \sqrt {a+b x^n}}+\frac {e x^{1+n} (c x)^m \sqrt {1+\frac {b x^n}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m+n}{n},\frac {1+m+2 n}{n},-\frac {b x^n}{a}\right )}{(1+m+n) \sqrt {a+b x^n}}+\frac {f x^{1+2 n} (c x)^m \sqrt {1+\frac {b x^n}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m+2 n}{n},\frac {1+m+3 n}{n},-\frac {b x^n}{a}\right )}{(1+m+2 n) \sqrt {a+b x^n}}+\frac {g x^{1+3 n} (c x)^m \sqrt {1+\frac {b x^n}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m+3 n}{n},\frac {1+m+4 n}{n},-\frac {b x^n}{a}\right )}{(1+m+3 n) \sqrt {a+b x^n}} \]

output

d*(c*x)^(1+m)*hypergeom([1/2, (1+m)/n],[(1+m+n)/n],-b*x^n/a)*(1+b*x^n/a)^( 
1/2)/c/(1+m)/(a+b*x^n)^(1/2)+e*x^(1+n)*(c*x)^m*hypergeom([1/2, (1+m+n)/n], 
[(1+m+2*n)/n],-b*x^n/a)*(1+b*x^n/a)^(1/2)/(1+m+n)/(a+b*x^n)^(1/2)+f*x^(1+2 
*n)*(c*x)^m*hypergeom([1/2, (1+m+2*n)/n],[(1+m+3*n)/n],-b*x^n/a)*(1+b*x^n/ 
a)^(1/2)/(1+m+2*n)/(a+b*x^n)^(1/2)+g*x^(1+3*n)*(c*x)^m*hypergeom([1/2, (1+ 
m+3*n)/n],[(1+m+4*n)/n],-b*x^n/a)*(1+b*x^n/a)^(1/2)/(1+m+3*n)/(a+b*x^n)^(1 
/2)
 

3.6.84.2 Mathematica [A] (verified)

Time = 1.12 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.68 \[ \int \frac {(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{\sqrt {a+b x^n}} \, dx=\frac {x (c x)^m \sqrt {1+\frac {b x^n}{a}} \left (\frac {d \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{1+m}+x^n \left (\frac {e \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m+n}{n},\frac {1+m+2 n}{n},-\frac {b x^n}{a}\right )}{1+m+n}+x^n \left (\frac {f \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m+2 n}{n},\frac {1+m+3 n}{n},-\frac {b x^n}{a}\right )}{1+m+2 n}+\frac {g x^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m+3 n}{n},\frac {1+m+4 n}{n},-\frac {b x^n}{a}\right )}{1+m+3 n}\right )\right )\right )}{\sqrt {a+b x^n}} \]

input

Integrate[((c*x)^m*(d + e*x^n + f*x^(2*n) + g*x^(3*n)))/Sqrt[a + b*x^n],x]
 

output

(x*(c*x)^m*Sqrt[1 + (b*x^n)/a]*((d*Hypergeometric2F1[1/2, (1 + m)/n, (1 + 
m + n)/n, -((b*x^n)/a)])/(1 + m) + x^n*((e*Hypergeometric2F1[1/2, (1 + m + 
 n)/n, (1 + m + 2*n)/n, -((b*x^n)/a)])/(1 + m + n) + x^n*((f*Hypergeometri 
c2F1[1/2, (1 + m + 2*n)/n, (1 + m + 3*n)/n, -((b*x^n)/a)])/(1 + m + 2*n) + 
 (g*x^n*Hypergeometric2F1[1/2, (1 + m + 3*n)/n, (1 + m + 4*n)/n, -((b*x^n) 
/a)])/(1 + m + 3*n)))))/Sqrt[a + b*x^n]
 

3.6.84.3 Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2383, 2009}

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.

\(\displaystyle \int \frac {(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{\sqrt {a+b x^n}} \, dx\)

\(\Big \downarrow \) 2383

\(\displaystyle \int \left (\frac {d (c x)^m}{\sqrt {a+b x^n}}+\frac {e x^n (c x)^m}{\sqrt {a+b x^n}}+\frac {f x^{2 n} (c x)^m}{\sqrt {a+b x^n}}+\frac {g x^{3 n} (c x)^m}{\sqrt {a+b x^n}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d (c x)^{m+1} \sqrt {\frac {b x^n}{a}+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{n},\frac {m+n+1}{n},-\frac {b x^n}{a}\right )}{c (m+1) \sqrt {a+b x^n}}+\frac {e x^{n+1} (c x)^m \sqrt {\frac {b x^n}{a}+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+n+1}{n},\frac {m+2 n+1}{n},-\frac {b x^n}{a}\right )}{(m+n+1) \sqrt {a+b x^n}}+\frac {f x^{2 n+1} (c x)^m \sqrt {\frac {b x^n}{a}+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2 n+1}{n},\frac {m+3 n+1}{n},-\frac {b x^n}{a}\right )}{(m+2 n+1) \sqrt {a+b x^n}}+\frac {g x^{3 n+1} (c x)^m \sqrt {\frac {b x^n}{a}+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3 n+1}{n},\frac {m+4 n+1}{n},-\frac {b x^n}{a}\right )}{(m+3 n+1) \sqrt {a+b x^n}}\)

input

Int[((c*x)^m*(d + e*x^n + f*x^(2*n) + g*x^(3*n)))/Sqrt[a + b*x^n],x]
 

output

(d*(c*x)^(1 + m)*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2, (1 + m)/n, (1 
+ m + n)/n, -((b*x^n)/a)])/(c*(1 + m)*Sqrt[a + b*x^n]) + (e*x^(1 + n)*(c*x 
)^m*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2, (1 + m + n)/n, (1 + m + 2*n 
)/n, -((b*x^n)/a)])/((1 + m + n)*Sqrt[a + b*x^n]) + (f*x^(1 + 2*n)*(c*x)^m 
*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2, (1 + m + 2*n)/n, (1 + m + 3*n) 
/n, -((b*x^n)/a)])/((1 + m + 2*n)*Sqrt[a + b*x^n]) + (g*x^(1 + 3*n)*(c*x)^ 
m*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2, (1 + m + 3*n)/n, (1 + m + 4*n 
)/n, -((b*x^n)/a)])/((1 + m + 3*n)*Sqrt[a + b*x^n])
 

3.6.84.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2383

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> I 
nt[ExpandIntegrand[(c*x)^m*Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n 
, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !IGtQ[m, 0]
 

3.6.84.4 Maple [F]

\[\int \frac {\left (c x \right )^{m} \left (d +e \,x^{n}+f \,x^{2 n}+g \,x^{3 n}\right )}{\sqrt {a +b \,x^{n}}}d x\]

input

int((c*x)^m*(d+e*x^n+f*x^(2*n)+g*x^(3*n))/(a+b*x^n)^(1/2),x)
 

output

int((c*x)^m*(d+e*x^n+f*x^(2*n)+g*x^(3*n))/(a+b*x^n)^(1/2),x)
 

3.6.84.5 Fricas [F(-2)]

Exception generated. \[ \int \frac {(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{\sqrt {a+b x^n}} \, dx=\text {Exception raised: TypeError} \]

input

integrate((c*x)^m*(d+e*x^n+f*x^(2*n)+g*x^(3*n))/(a+b*x^n)^(1/2),x, algorit 
hm="fricas")
 

output

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

3.6.84.6 Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 20.81 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.12 \[ \int \frac {(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{\sqrt {a+b x^n}} \, dx=\frac {a^{\frac {m}{n} + \frac {1}{n}} a^{- \frac {m}{n} - \frac {1}{2} - \frac {1}{n}} c^{m} d x^{m + 1} \Gamma \left (\frac {m}{n} + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{n} + \frac {1}{n} \\ \frac {m}{n} + 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )} + \frac {a^{- \frac {m}{n} - \frac {7}{2} - \frac {1}{n}} a^{\frac {m}{n} + 3 + \frac {1}{n}} c^{m} g x^{m + 3 n + 1} \Gamma \left (\frac {m}{n} + 3 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{n} + 3 + \frac {1}{n} \\ \frac {m}{n} + 4 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 4 + \frac {1}{n}\right )} + \frac {a^{- \frac {m}{n} - \frac {5}{2} - \frac {1}{n}} a^{\frac {m}{n} + 2 + \frac {1}{n}} c^{m} f x^{m + 2 n + 1} \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{n} + 2 + \frac {1}{n} \\ \frac {m}{n} + 3 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 3 + \frac {1}{n}\right )} + \frac {a^{- \frac {m}{n} - \frac {3}{2} - \frac {1}{n}} a^{\frac {m}{n} + 1 + \frac {1}{n}} c^{m} e x^{m + n + 1} \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{n} + 1 + \frac {1}{n} \\ \frac {m}{n} + 2 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} \]

input

integrate((c*x)**m*(d+e*x**n+f*x**(2*n)+g*x**(3*n))/(a+b*x**n)**(1/2),x)
 

output

a**(m/n + 1/n)*a**(-m/n - 1/2 - 1/n)*c**m*d*x**(m + 1)*gamma(m/n + 1/n)*hy 
per((1/2, m/n + 1/n), (m/n + 1 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma 
(m/n + 1 + 1/n)) + a**(-m/n - 7/2 - 1/n)*a**(m/n + 3 + 1/n)*c**m*g*x**(m + 
 3*n + 1)*gamma(m/n + 3 + 1/n)*hyper((1/2, m/n + 3 + 1/n), (m/n + 4 + 1/n, 
), b*x**n*exp_polar(I*pi)/a)/(n*gamma(m/n + 4 + 1/n)) + a**(-m/n - 5/2 - 1 
/n)*a**(m/n + 2 + 1/n)*c**m*f*x**(m + 2*n + 1)*gamma(m/n + 2 + 1/n)*hyper( 
(1/2, m/n + 2 + 1/n), (m/n + 3 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma 
(m/n + 3 + 1/n)) + a**(-m/n - 3/2 - 1/n)*a**(m/n + 1 + 1/n)*c**m*e*x**(m + 
 n + 1)*gamma(m/n + 1 + 1/n)*hyper((1/2, m/n + 1 + 1/n), (m/n + 2 + 1/n,), 
 b*x**n*exp_polar(I*pi)/a)/(n*gamma(m/n + 2 + 1/n))
 

3.6.84.7 Maxima [F]

\[ \int \frac {(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{\sqrt {a+b x^n}} \, dx=\int { \frac {{\left (g x^{3 \, n} + f x^{2 \, n} + e x^{n} + d\right )} \left (c x\right )^{m}}{\sqrt {b x^{n} + a}} \,d x } \]

input

integrate((c*x)^m*(d+e*x^n+f*x^(2*n)+g*x^(3*n))/(a+b*x^n)^(1/2),x, algorit 
hm="maxima")
 

output

integrate((g*x^(3*n) + f*x^(2*n) + e*x^n + d)*(c*x)^m/sqrt(b*x^n + a), x)
 

3.6.84.8 Giac [F]

\[ \int \frac {(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{\sqrt {a+b x^n}} \, dx=\int { \frac {{\left (g x^{3 \, n} + f x^{2 \, n} + e x^{n} + d\right )} \left (c x\right )^{m}}{\sqrt {b x^{n} + a}} \,d x } \]

input

integrate((c*x)^m*(d+e*x^n+f*x^(2*n)+g*x^(3*n))/(a+b*x^n)^(1/2),x, algorit 
hm="giac")
 

output

integrate((g*x^(3*n) + f*x^(2*n) + e*x^n + d)*(c*x)^m/sqrt(b*x^n + a), x)
 

3.6.84.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{\sqrt {a+b x^n}} \, dx=\int \frac {{\left (c\,x\right )}^m\,\left (d+e\,x^n+f\,x^{2\,n}+g\,x^{3\,n}\right )}{\sqrt {a+b\,x^n}} \,d x \]

input

int(((c*x)^m*(d + e*x^n + f*x^(2*n) + g*x^(3*n)))/(a + b*x^n)^(1/2),x)
 

output

int(((c*x)^m*(d + e*x^n + f*x^(2*n) + g*x^(3*n)))/(a + b*x^n)^(1/2), x)