\(\int \frac {(e x)^m (c+d x)}{(a x^n+b x^{1+n})^{5/2}} \, dx\) [375]

Optimal result

Integrand size = 28, antiderivative size = 146 \[ \int \frac {(e x)^m (c+d x)}{\left (a x^n+b x^{1+n}\right )^{5/2}} \, dx=-\frac {2 d x^{1-n} (e x)^m}{b (1-2 m+5 n) \left (a x^n+b x^{1+n}\right )^{3/2}}-\frac {2 \left (b c+\frac {a d (2+2 m-5 n)}{1-2 m+5 n}\right ) x^{-n} \left (-\frac {b x}{a}\right )^{-m+\frac {5 n}{2}} (e x)^m \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-m+\frac {5 n}{2},-\frac {1}{2},1+\frac {b x}{a}\right )}{3 b^2 \left (a x^n+b x^{1+n}\right )^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.79 \[ \int \frac {(e x)^m (c+d x)}{\left (a x^n+b x^{1+n}\right )^{5/2}} \, dx=\frac {2 (e x)^m (a+b x) \left (b (b c-a d) x+(b c (-1+2 m-5 n)+a d (-2-2 m+5 n)) \left (-\frac {b x}{a}\right )^{-m+\frac {5 n}{2}} (a+b x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-m+\frac {5 n}{2},\frac {1}{2},1+\frac {b x}{a}\right )\right )}{3 a b^2 \left (x^n (a+b x)\right )^{5/2}} \] Input:

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

Output:

(2*(e*x)^m*(a + b*x)*(b*(b*c - a*d)*x + (b*c*(-1 + 2*m - 5*n) + a*d*(-2 - 
2*m + 5*n))*(-((b*x)/a))^(-m + (5*n)/2)*(a + b*x)*Hypergeometric2F1[-1/2, 
-m + (5*n)/2, 1/2, 1 + (b*x)/a]))/(3*a*b^2*(x^n*(a + b*x))^(5/2))
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 173, 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, 87, 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.

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

\(\Big \downarrow \) 1948

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 77

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

\(\Big \downarrow \) 75

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

Input:

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

Output:

(x^((-2*m + n)/2)*(e*x)^m*Sqrt[a + b*x]*((2*(b*c - a*d)*x^(1 + m - (5*n)/2 
))/(3*a*b*(a + b*x)^(3/2)) - (2*(a*d*(2 + 2*m - 5*n) + b*c*(1 - 2*m + 5*n) 
)*x^(m - (5*n)/2)*(-((b*x)/a))^(-m + (5*n)/2)*Hypergeometric2F1[-1/2, -m + 
 (5*n)/2, 1/2, 1 + (b*x)/a])/(3*a*b^2*Sqrt[a + b*x])))/Sqrt[a*x^n + b*x^(1 
 + n)]
 

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 87

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

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]

\[\int \frac {\left (e x \right )^{m} \left (d x +c \right )}{\left (a \,x^{n}+b \,x^{1+n}\right )^{\frac {5}{2}}}d x\]

Input:

int((e*x)^m*(d*x+c)/(a*x^n+b*x^(1+n))^(5/2),x)
 

Output:

int((e*x)^m*(d*x+c)/(a*x^n+b*x^(1+n))^(5/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {(e x)^m (c+d x)}{\left (a x^n+b x^{1+n}\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((e*x)^m*(d*x+c)/(a*x^n+b*x^(1+n))^(5/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {(e x)^m (c+d x)}{\left (a x^n+b x^{1+n}\right )^{5/2}} \, dx=\int \frac {\left (e x\right )^{m} \left (c + d x\right )}{\left (a x^{n} + b x^{n + 1}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((e*x)**m*(d*x+c)/(a*x**n+b*x**(1+n))**(5/2),x)
 

Output:

Integral((e*x)**m*(c + d*x)/(a*x**n + b*x**(n + 1))**(5/2), x)
 

Maxima [F]

\[ \int \frac {(e x)^m (c+d x)}{\left (a x^n+b x^{1+n}\right )^{5/2}} \, dx=\int { \frac {{\left (d x + c\right )} \left (e x\right )^{m}}{{\left (b x^{n + 1} + a x^{n}\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((e*x)^m*(d*x+c)/(a*x^n+b*x^(1+n))^(5/2),x, algorithm="maxima")
 

Output:

integrate((d*x + c)*(e*x)^m/(b*x^(n + 1) + a*x^n)^(5/2), x)
 

Giac [F]

\[ \int \frac {(e x)^m (c+d x)}{\left (a x^n+b x^{1+n}\right )^{5/2}} \, dx=\int { \frac {{\left (d x + c\right )} \left (e x\right )^{m}}{{\left (b x^{n + 1} + a x^{n}\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((e*x)^m*(d*x+c)/(a*x^n+b*x^(1+n))^(5/2),x, algorithm="giac")
 

Output:

integrate((d*x + c)*(e*x)^m/(b*x^(n + 1) + a*x^n)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^m (c+d x)}{\left (a x^n+b x^{1+n}\right )^{5/2}} \, dx=\int \frac {{\left (e\,x\right )}^m\,\left (c+d\,x\right )}{{\left (a\,x^n+b\,x^{n+1}\right )}^{5/2}} \,d x \] Input:

int(((e*x)^m*(c + d*x))/(a*x^n + b*x^(n + 1))^(5/2),x)
 

Output:

int(((e*x)^m*(c + d*x))/(a*x^n + b*x^(n + 1))^(5/2), x)
 

Reduce [F]

\[ \int \frac {(e x)^m (c+d x)}{\left (a x^n+b x^{1+n}\right )^{5/2}} \, dx=e^{m} \left (\left (\int \frac {x^{m}}{x^{\frac {5 n}{2}} \sqrt {b x +a}\, a^{2}+2 x^{\frac {5 n}{2}} \sqrt {b x +a}\, a b x +x^{\frac {5 n}{2}} \sqrt {b x +a}\, b^{2} x^{2}}d x \right ) c +\left (\int \frac {x^{m} x}{x^{\frac {5 n}{2}} \sqrt {b x +a}\, a^{2}+2 x^{\frac {5 n}{2}} \sqrt {b x +a}\, a b x +x^{\frac {5 n}{2}} \sqrt {b x +a}\, b^{2} x^{2}}d x \right ) d \right ) \] Input:

int((e*x)^m*(d*x+c)/(a*x^n+b*x^(1+n))^(5/2),x)
 

Output:

e**m*(int(x**m/(x**((5*n)/2)*sqrt(a + b*x)*a**2 + 2*x**((5*n)/2)*sqrt(a + 
b*x)*a*b*x + x**((5*n)/2)*sqrt(a + b*x)*b**2*x**2),x)*c + int((x**m*x)/(x* 
*((5*n)/2)*sqrt(a + b*x)*a**2 + 2*x**((5*n)/2)*sqrt(a + b*x)*a*b*x + x**(( 
5*n)/2)*sqrt(a + b*x)*b**2*x**2),x)*d)