\(\int \frac {(d x)^m (a+b x)^p}{(c x^2)^{3/2}} \, dx\) [467]

Optimal result

Integrand size = 22, antiderivative size = 65 \[ \int \frac {(d x)^m (a+b x)^p}{\left (c x^2\right )^{3/2}} \, dx=-\frac {d (d x)^{-1+m} (a+b x)^p \left (1+\frac {b x}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-2+m,-p,-1+m,-\frac {b x}{a}\right )}{c (2-m) \sqrt {c x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.88 \[ \int \frac {(d x)^m (a+b x)^p}{\left (c x^2\right )^{3/2}} \, dx=\frac {x (d x)^m (a+b x)^p \left (1+\frac {b x}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-2+m,-p,-1+m,-\frac {b x}{a}\right )}{(-2+m) \left (c x^2\right )^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {30, 76, 74}

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.

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

\(\Big \downarrow \) 30

\(\displaystyle \frac {d^3 x \int (d x)^{m-3} (a+b x)^pdx}{c \sqrt {c x^2}}\)

\(\Big \downarrow \) 76

\(\displaystyle \frac {d^3 x (a+b x)^p \left (\frac {b x}{a}+1\right )^{-p} \int (d x)^{m-3} \left (\frac {b x}{a}+1\right )^pdx}{c \sqrt {c x^2}}\)

\(\Big \downarrow \) 74

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 30

Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I 
ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) 
Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & 
&  !IntegerQ[p]
 

rule 74

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

rule 76

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(c*x^2)*(b*x + a)^p*(d*x)^m/(c^2*x^4), x)
 

Sympy [F]

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

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

Output:

Integral((d*x)**m*(a + b*x)**p/(c*x**2)**(3/2), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(d**m*sqrt(c)*(x**m*(a + b*x)**p + int((x**m*(a + b*x)**p)/(a*m*x**3 + a*p 
*x**3 - 2*a*x**3 + b*m*x**4 + b*p*x**4 - 2*b*x**4),x)*a*m*p*x**2 + int((x* 
*m*(a + b*x)**p)/(a*m*x**3 + a*p*x**3 - 2*a*x**3 + b*m*x**4 + b*p*x**4 - 2 
*b*x**4),x)*a*p**2*x**2 - 2*int((x**m*(a + b*x)**p)/(a*m*x**3 + a*p*x**3 - 
 2*a*x**3 + b*m*x**4 + b*p*x**4 - 2*b*x**4),x)*a*p*x**2))/(c**2*x**2*(m + 
p - 2))