\(\int (e x)^m (c+d x) \sqrt {a x^n+b x^{1+n}} \, dx\) [372]

Optimal result

Integrand size = 28, antiderivative size = 141 \[ \int (e x)^m (c+d x) \sqrt {a x^n+b x^{1+n}} \, dx=\frac {2 d x^{1-n} (e x)^m \left (a x^n+b x^{1+n}\right )^{3/2}}{b (5+2 m+n)}+\frac {2 \left (b c-\frac {a d (2+2 m+n)}{5+2 m+n}\right ) x^{-n} \left (-\frac {b x}{a}\right )^{-m-\frac {n}{2}} (e x)^m \left (a x^n+b x^{1+n}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-m-\frac {n}{2},\frac {5}{2},1+\frac {b x}{a}\right )}{3 b^2} \] Output:

2*d*x^(1-n)*(e*x)^m*(a*x^n+b*x^(1+n))^(3/2)/b/(5+2*m+n)+2/3*(b*c-a*d*(2+2* 
m+n)/(5+2*m+n))*(-b*x/a)^(-m-1/2*n)*(e*x)^m*(a*x^n+b*x^(1+n))^(3/2)*hyperg 
eom([3/2, -m-1/2*n],[5/2],1+b*x/a)/b^2/(x^n)
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.86 \[ \int (e x)^m (c+d x) \sqrt {a x^n+b x^{1+n}} \, dx=\frac {2 \left (-\frac {b x}{a}\right )^{-m-\frac {n}{2}} (e x)^m (a+b x) \sqrt {x^n (a+b x)} \left (3 b d x \left (-\frac {b x}{a}\right )^{m+\frac {n}{2}}+(-a d (2+2 m+n)+b c (5+2 m+n)) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-m-\frac {n}{2},\frac {5}{2},1+\frac {b x}{a}\right )\right )}{3 b^2 (5+2 m+n)} \] Input:

Integrate[(e*x)^m*(c + d*x)*Sqrt[a*x^n + b*x^(1 + n)],x]
 

Output:

(2*(-((b*x)/a))^(-m - n/2)*(e*x)^m*(a + b*x)*Sqrt[x^n*(a + b*x)]*(3*b*d*x* 
(-((b*x)/a))^(m + n/2) + (-(a*d*(2 + 2*m + n)) + b*c*(5 + 2*m + n))*Hyperg 
eometric2F1[3/2, -m - n/2, 5/2, 1 + (b*x)/a]))/(3*b^2*(5 + 2*m + n))
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.18, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1948, 90, 77, 75}

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.

\(\displaystyle \int (c+d x) (e x)^m \sqrt {a x^n+b x^{n+1}} \, dx\)

\(\Big \downarrow \) 1948

\(\displaystyle \frac {(e x)^m x^{-m-\frac {n}{2}} \sqrt {a x^n+b x^{n+1}} \int x^{m+\frac {n}{2}} \sqrt {a+b x} (c+d x)dx}{\sqrt {a+b x}}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {(e x)^m x^{-m-\frac {n}{2}} \sqrt {a x^n+b x^{n+1}} \left (\left (c-\frac {a d (2 m+n+2)}{b (2 m+n+5)}\right ) \int x^{m+\frac {n}{2}} \sqrt {a+b x}dx+\frac {2 d (a+b x)^{3/2} x^{m+\frac {n}{2}+1}}{b (2 m+n+5)}\right )}{\sqrt {a+b x}}\)

\(\Big \downarrow \) 77

\(\displaystyle \frac {(e x)^m x^{-m-\frac {n}{2}} \sqrt {a x^n+b x^{n+1}} \left (x^{m+\frac {n}{2}} \left (-\frac {b x}{a}\right )^{-m-\frac {n}{2}} \left (c-\frac {a d (2 m+n+2)}{b (2 m+n+5)}\right ) \int \left (-\frac {b x}{a}\right )^{m+\frac {n}{2}} \sqrt {a+b x}dx+\frac {2 d (a+b x)^{3/2} x^{m+\frac {n}{2}+1}}{b (2 m+n+5)}\right )}{\sqrt {a+b x}}\)

\(\Big \downarrow \) 75

\(\displaystyle \frac {(e x)^m x^{-m-\frac {n}{2}} \sqrt {a x^n+b x^{n+1}} \left (\frac {2 (a+b x)^{3/2} x^{m+\frac {n}{2}} \left (-\frac {b x}{a}\right )^{-m-\frac {n}{2}} \left (c-\frac {a d (2 m+n+2)}{b (2 m+n+5)}\right ) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-m-\frac {n}{2},\frac {5}{2},\frac {b x}{a}+1\right )}{3 b}+\frac {2 d (a+b x)^{3/2} x^{m+\frac {n}{2}+1}}{b (2 m+n+5)}\right )}{\sqrt {a+b x}}\)

Input:

Int[(e*x)^m*(c + d*x)*Sqrt[a*x^n + b*x^(1 + n)],x]
 

Output:

(x^(-m - n/2)*(e*x)^m*Sqrt[a*x^n + b*x^(1 + n)]*((2*d*x^(1 + m + n/2)*(a + 
 b*x)^(3/2))/(b*(5 + 2*m + n)) + (2*(c - (a*d*(2 + 2*m + n))/(b*(5 + 2*m + 
 n)))*x^(m + n/2)*(-((b*x)/a))^(-m - n/2)*(a + b*x)^(3/2)*Hypergeometric2F 
1[3/2, -m - n/2, 5/2, 1 + (b*x)/a])/(3*b)))/Sqrt[a + b*x]
 

Defintions of rubi rules used

rule 75

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x 
)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + 
 d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (IntegerQ[m] 
 || GtQ[-d/(b*c), 0])
 

rule 77

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((-b)*(c/ 
d))^IntPart[m]*((b*x)^FracPart[m]/((-d)*(x/c))^FracPart[m])   Int[((-d)*(x/ 
c))^m*(c + d*x)^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && 
 !IntegerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-d/(b*c), 0]
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 1948

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + 
(d_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Simp[e^IntPart[m]*(e*x)^FracPart[m]*( 
(a*x^j + b*x^(j + n))^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a + b*x 
^n)^FracPart[p]))   Int[x^(m + j*p)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; 
FreeQ[{a, b, c, d, e, j, m, n, p, q}, x] && EqQ[jn, j + n] &&  !IntegerQ[p] 
 && NeQ[b*c - a*d, 0] &&  !(EqQ[n, 1] && EqQ[j, 1])
 

Maple [F]

\[\int \left (e x \right )^{m} \left (d x +c \right ) \sqrt {a \,x^{n}+b \,x^{1+n}}d x\]

Input:

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

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int (e x)^m (c+d x) \sqrt {a x^n+b x^{1+n}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((e*x)^m*(d*x+c)*(a*x^n+b*x^(1+n))^(1/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int (e x)^m (c+d x) \sqrt {a x^n+b x^{1+n}} \, dx=\int \left (e x\right )^{m} \left (c + d x\right ) \sqrt {a x^{n} + b x^{n + 1}}\, dx \] Input:

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

Output:

Integral((e*x)**m*(c + d*x)*sqrt(a*x**n + b*x**(n + 1)), x)
 

Maxima [F]

\[ \int (e x)^m (c+d x) \sqrt {a x^n+b x^{1+n}} \, dx=\int { \sqrt {b x^{n + 1} + a x^{n}} {\left (d x + c\right )} \left (e x\right )^{m} \,d x } \] Input:

integrate((e*x)^m*(d*x+c)*(a*x^n+b*x^(1+n))^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(b*x^(n + 1) + a*x^n)*(d*x + c)*(e*x)^m, x)
 

Giac [F]

\[ \int (e x)^m (c+d x) \sqrt {a x^n+b x^{1+n}} \, dx=\int { \sqrt {b x^{n + 1} + a x^{n}} {\left (d x + c\right )} \left (e x\right )^{m} \,d x } \] Input:

integrate((e*x)^m*(d*x+c)*(a*x^n+b*x^(1+n))^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(b*x^(n + 1) + a*x^n)*(d*x + c)*(e*x)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^m (c+d x) \sqrt {a x^n+b x^{1+n}} \, dx=\int {\left (e\,x\right )}^m\,\sqrt {a\,x^n+b\,x^{n+1}}\,\left (c+d\,x\right ) \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int (e x)^m (c+d x) \sqrt {a x^n+b x^{1+n}} \, dx=e^{m} \left (\left (\int x^{m +\frac {n}{2}} \sqrt {b x +a}\, x d x \right ) d +\left (\int x^{m +\frac {n}{2}} \sqrt {b x +a}d x \right ) c \right ) \] Input:

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

Output:

e**m*(int(x**((2*m + n)/2)*sqrt(a + b*x)*x,x)*d + int(x**((2*m + n)/2)*sqr 
t(a + b*x),x)*c)