\(\int (d x)^m (c x^2)^{3/2} (a+b x)^p \, dx\) [464]

Optimal result

Integrand size = 22, antiderivative size = 62 \[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x)^p \, dx=\frac {c (d x)^{3+m} \sqrt {c x^2} (a+b x)^p \left (1+\frac {b x}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (4+m,-p,5+m,-\frac {b x}{a}\right )}{d^3 (4+m)} \] Output:

c*(d*x)^(3+m)*(c*x^2)^(1/2)*(b*x+a)^p*hypergeom([-p, 4+m],[5+m],-b*x/a)/d^ 
3/(4+m)/((1+b*x/a)^p)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.92 \[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x)^p \, dx=\frac {x (d x)^m \left (c x^2\right )^{3/2} (a+b x)^p \left (1+\frac {b x}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (4+m,-p,5+m,-\frac {b x}{a}\right )}{4+m} \] Input:

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

Output:

(x*(d*x)^m*(c*x^2)^(3/2)*(a + b*x)^p*Hypergeometric2F1[4 + m, -p, 5 + m, - 
((b*x)/a)])/((4 + m)*(1 + (b*x)/a)^p)
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 65, 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.

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

\(\Big \downarrow \) 30

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

\(\Big \downarrow \) 76

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

\(\Big \downarrow \) 74

\(\displaystyle \frac {c \sqrt {c x^2} (d x)^{m+4} (a+b x)^p \left (\frac {b x}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (m+4,-p,m+5,-\frac {b x}{a}\right )}{d^4 (m+4) x}\)

Input:

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

Output:

(c*(d*x)^(4 + m)*Sqrt[c*x^2]*(a + b*x)^p*Hypergeometric2F1[4 + m, -p, 5 + 
m, -((b*x)/a)])/(d^4*(4 + m)*x*(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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(d**m*sqrt(c)*c*( - x**m*(a + b*x)**p*a**4*m**3*p - 6*x**m*(a + b*x)**p*a* 
*4*m**2*p - 11*x**m*(a + b*x)**p*a**4*m*p - 6*x**m*(a + b*x)**p*a**4*p + x 
**m*(a + b*x)**p*a**3*b*m**3*p*x + x**m*(a + b*x)**p*a**3*b*m**2*p**2*x + 
5*x**m*(a + b*x)**p*a**3*b*m**2*p*x + 5*x**m*(a + b*x)**p*a**3*b*m*p**2*x 
+ 6*x**m*(a + b*x)**p*a**3*b*m*p*x + 6*x**m*(a + b*x)**p*a**3*b*p**2*x - x 
**m*(a + b*x)**p*a**2*b**2*m**3*p*x**2 - 2*x**m*(a + b*x)**p*a**2*b**2*m** 
2*p**2*x**2 - 4*x**m*(a + b*x)**p*a**2*b**2*m**2*p*x**2 - x**m*(a + b*x)** 
p*a**2*b**2*m*p**3*x**2 - 7*x**m*(a + b*x)**p*a**2*b**2*m*p**2*x**2 - 3*x* 
*m*(a + b*x)**p*a**2*b**2*m*p*x**2 - 3*x**m*(a + b*x)**p*a**2*b**2*p**3*x* 
*2 - 3*x**m*(a + b*x)**p*a**2*b**2*p**2*x**2 + x**m*(a + b*x)**p*a*b**3*m* 
*3*p*x**3 + 3*x**m*(a + b*x)**p*a*b**3*m**2*p**2*x**3 + 3*x**m*(a + b*x)** 
p*a*b**3*m**2*p*x**3 + 3*x**m*(a + b*x)**p*a*b**3*m*p**3*x**3 + 6*x**m*(a 
+ b*x)**p*a*b**3*m*p**2*x**3 + 2*x**m*(a + b*x)**p*a*b**3*m*p*x**3 + x**m* 
(a + b*x)**p*a*b**3*p**4*x**3 + 3*x**m*(a + b*x)**p*a*b**3*p**3*x**3 + 2*x 
**m*(a + b*x)**p*a*b**3*p**2*x**3 + x**m*(a + b*x)**p*b**4*m**4*x**4 + 4*x 
**m*(a + b*x)**p*b**4*m**3*p*x**4 + 6*x**m*(a + b*x)**p*b**4*m**3*x**4 + 6 
*x**m*(a + b*x)**p*b**4*m**2*p**2*x**4 + 18*x**m*(a + b*x)**p*b**4*m**2*p* 
x**4 + 11*x**m*(a + b*x)**p*b**4*m**2*x**4 + 4*x**m*(a + b*x)**p*b**4*m*p* 
*3*x**4 + 18*x**m*(a + b*x)**p*b**4*m*p**2*x**4 + 22*x**m*(a + b*x)**p*b** 
4*m*p*x**4 + 6*x**m*(a + b*x)**p*b**4*m*x**4 + x**m*(a + b*x)**p*b**4*p...