\(\int (d x)^m (c (a+b x^n))^{3/2} \, dx\) [67]

Optimal result

Integrand size = 19, antiderivative size = 74 \[ \int (d x)^m \left (c \left (a+b x^n\right )\right )^{3/2} \, dx=\frac {a c (d x)^{1+m} \sqrt {a c+b c x^n} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{d (1+m) \sqrt {1+\frac {b x^n}{a}}} \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.93 \[ \int (d x)^m \left (c \left (a+b x^n\right )\right )^{3/2} \, dx=\frac {a c x (d x)^m \sqrt {c \left (a+b x^n\right )} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{(1+m) \sqrt {1+\frac {b x^n}{a}}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 74, 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 = {2073, 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.

\(\displaystyle \int (d x)^m \left (c \left (a+b x^n\right )\right )^{3/2} \, dx\)

\(\Big \downarrow \) 2073

\(\displaystyle \int (d x)^m \left (a c+b c x^n\right )^{3/2}dx\)

\(\Big \downarrow \) 889

\(\displaystyle \frac {a c \sqrt {a c+b c x^n} \int (d x)^m \left (\frac {b x^n}{a}+1\right )^{3/2}dx}{\sqrt {\frac {b x^n}{a}+1}}\)

\(\Big \downarrow \) 888

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 888

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])
 

rule 889

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])
 

rule 2073

Int[(u_)^(p_.)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x)^m*ExpandToSum[u, 
x]^p, x] /; FreeQ[{c, m, p}, x] && BinomialQ[u, x] &&  !BinomialMatchQ[u, x 
]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate((d*x)^m*(c*(a+b*x^n))^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int (d x)^m \left (c \left (a+b x^n\right )\right )^{3/2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int (d x)^m \left (c \left (a+b x^n\right )\right )^{3/2} \, dx=\int { \left ({\left (b x^{n} + a\right )} c\right )^{\frac {3}{2}} \left (d x\right )^{m} \,d x } \] Input:

integrate((d*x)^m*(c*(a+b*x^n))^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int (d x)^m \left (c \left (a+b x^n\right )\right )^{3/2} \, dx=\int { \left ({\left (b x^{n} + a\right )} c\right )^{\frac {3}{2}} \left (d x\right )^{m} \,d x } \] Input:

integrate((d*x)^m*(c*(a+b*x^n))^(3/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int (d x)^m \left (c \left (a+b x^n\right )\right )^{3/2} \, dx=\int {\left (d\,x\right )}^m\,{\left (c\,\left (a+b\,x^n\right )\right )}^{3/2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int (d x)^m \left (c \left (a+b x^n\right )\right )^{3/2} \, dx=\frac {d^{m} \sqrt {c}\, c \left (4 x^{m +n} \sqrt {x^{n} b +a}\, b m x +2 x^{m +n} \sqrt {x^{n} b +a}\, b n x +4 x^{m +n} \sqrt {x^{n} b +a}\, b x +4 x^{m} \sqrt {x^{n} b +a}\, a m x +8 x^{m} \sqrt {x^{n} b +a}\, a n x +4 x^{m} \sqrt {x^{n} b +a}\, a x +12 \left (\int \frac {x^{m} \sqrt {x^{n} b +a}}{4 x^{n} b \,m^{2}+8 x^{n} b m n +8 x^{n} b m +3 x^{n} b \,n^{2}+8 x^{n} b n +4 x^{n} b +4 a \,m^{2}+8 a m n +8 a m +3 a \,n^{2}+8 a n +4 a}d x \right ) a^{2} m^{2} n^{2}+24 \left (\int \frac {x^{m} \sqrt {x^{n} b +a}}{4 x^{n} b \,m^{2}+8 x^{n} b m n +8 x^{n} b m +3 x^{n} b \,n^{2}+8 x^{n} b n +4 x^{n} b +4 a \,m^{2}+8 a m n +8 a m +3 a \,n^{2}+8 a n +4 a}d x \right ) a^{2} m \,n^{3}+24 \left (\int \frac {x^{m} \sqrt {x^{n} b +a}}{4 x^{n} b \,m^{2}+8 x^{n} b m n +8 x^{n} b m +3 x^{n} b \,n^{2}+8 x^{n} b n +4 x^{n} b +4 a \,m^{2}+8 a m n +8 a m +3 a \,n^{2}+8 a n +4 a}d x \right ) a^{2} m \,n^{2}+9 \left (\int \frac {x^{m} \sqrt {x^{n} b +a}}{4 x^{n} b \,m^{2}+8 x^{n} b m n +8 x^{n} b m +3 x^{n} b \,n^{2}+8 x^{n} b n +4 x^{n} b +4 a \,m^{2}+8 a m n +8 a m +3 a \,n^{2}+8 a n +4 a}d x \right ) a^{2} n^{4}+24 \left (\int \frac {x^{m} \sqrt {x^{n} b +a}}{4 x^{n} b \,m^{2}+8 x^{n} b m n +8 x^{n} b m +3 x^{n} b \,n^{2}+8 x^{n} b n +4 x^{n} b +4 a \,m^{2}+8 a m n +8 a m +3 a \,n^{2}+8 a n +4 a}d x \right ) a^{2} n^{3}+12 \left (\int \frac {x^{m} \sqrt {x^{n} b +a}}{4 x^{n} b \,m^{2}+8 x^{n} b m n +8 x^{n} b m +3 x^{n} b \,n^{2}+8 x^{n} b n +4 x^{n} b +4 a \,m^{2}+8 a m n +8 a m +3 a \,n^{2}+8 a n +4 a}d x \right ) a^{2} n^{2}\right )}{4 m^{2}+8 m n +3 n^{2}+8 m +8 n +4} \] Input:

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

Output:

(d**m*sqrt(c)*c*(4*x**(m + n)*sqrt(x**n*b + a)*b*m*x + 2*x**(m + n)*sqrt(x 
**n*b + a)*b*n*x + 4*x**(m + n)*sqrt(x**n*b + a)*b*x + 4*x**m*sqrt(x**n*b 
+ a)*a*m*x + 8*x**m*sqrt(x**n*b + a)*a*n*x + 4*x**m*sqrt(x**n*b + a)*a*x + 
 12*int((x**m*sqrt(x**n*b + a))/(4*x**n*b*m**2 + 8*x**n*b*m*n + 8*x**n*b*m 
 + 3*x**n*b*n**2 + 8*x**n*b*n + 4*x**n*b + 4*a*m**2 + 8*a*m*n + 8*a*m + 3* 
a*n**2 + 8*a*n + 4*a),x)*a**2*m**2*n**2 + 24*int((x**m*sqrt(x**n*b + a))/( 
4*x**n*b*m**2 + 8*x**n*b*m*n + 8*x**n*b*m + 3*x**n*b*n**2 + 8*x**n*b*n + 4 
*x**n*b + 4*a*m**2 + 8*a*m*n + 8*a*m + 3*a*n**2 + 8*a*n + 4*a),x)*a**2*m*n 
**3 + 24*int((x**m*sqrt(x**n*b + a))/(4*x**n*b*m**2 + 8*x**n*b*m*n + 8*x** 
n*b*m + 3*x**n*b*n**2 + 8*x**n*b*n + 4*x**n*b + 4*a*m**2 + 8*a*m*n + 8*a*m 
 + 3*a*n**2 + 8*a*n + 4*a),x)*a**2*m*n**2 + 9*int((x**m*sqrt(x**n*b + a))/ 
(4*x**n*b*m**2 + 8*x**n*b*m*n + 8*x**n*b*m + 3*x**n*b*n**2 + 8*x**n*b*n + 
4*x**n*b + 4*a*m**2 + 8*a*m*n + 8*a*m + 3*a*n**2 + 8*a*n + 4*a),x)*a**2*n* 
*4 + 24*int((x**m*sqrt(x**n*b + a))/(4*x**n*b*m**2 + 8*x**n*b*m*n + 8*x**n 
*b*m + 3*x**n*b*n**2 + 8*x**n*b*n + 4*x**n*b + 4*a*m**2 + 8*a*m*n + 8*a*m 
+ 3*a*n**2 + 8*a*n + 4*a),x)*a**2*n**3 + 12*int((x**m*sqrt(x**n*b + a))/(4 
*x**n*b*m**2 + 8*x**n*b*m*n + 8*x**n*b*m + 3*x**n*b*n**2 + 8*x**n*b*n + 4* 
x**n*b + 4*a*m**2 + 8*a*m*n + 8*a*m + 3*a*n**2 + 8*a*n + 4*a),x)*a**2*n**2 
))/(4*m**2 + 8*m*n + 8*m + 3*n**2 + 8*n + 4)