\(\int \frac {(b x+c x^2)^p}{(d+e x)^{3/2}} \, dx\) [257]

Optimal result

Integrand size = 21, antiderivative size = 81 \[ \int \frac {\left (b x+c x^2\right )^p}{(d+e x)^{3/2}} \, dx=\frac {x \left (1+\frac {c x}{b}\right )^{-p} \sqrt {1+\frac {e x}{d}} \left (b x+c x^2\right )^p \operatorname {AppellF1}\left (1+p,-p,\frac {3}{2},2+p,-\frac {c x}{b},-\frac {e x}{d}\right )}{d (1+p) \sqrt {d+e x}} \] Output:

x*(1+e*x/d)^(1/2)*(c*x^2+b*x)^p*AppellF1(p+1,-p,3/2,2+p,-c*x/b,-e*x/d)/d/( 
p+1)/((1+c*x/b)^p)/(e*x+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.96 \[ \int \frac {\left (b x+c x^2\right )^p}{(d+e x)^{3/2}} \, dx=\frac {x \left (\frac {b+c x}{b}\right )^{-p} (x (b+c x))^p \left (\frac {d+e x}{d}\right )^{3/2} \operatorname {AppellF1}\left (1+p,-p,\frac {3}{2},2+p,-\frac {c x}{b},-\frac {e x}{d}\right )}{(1+p) (d+e x)^{3/2}} \] Input:

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

Output:

(x*(x*(b + c*x))^p*((d + e*x)/d)^(3/2)*AppellF1[1 + p, -p, 3/2, 2 + p, -(( 
c*x)/b), -((e*x)/d)])/((1 + p)*((b + c*x)/b)^p*(d + e*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.22, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1179, 150}

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.

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

\(\Big \downarrow \) 1179

\(\displaystyle \frac {\left (b x+c x^2\right )^p \left (-\frac {e x}{d}\right )^{-p} \left (1-\frac {c (d+e x)}{c d-b e}\right )^{-p} \int \frac {\left (1-\frac {d+e x}{d}\right )^p \left (1-\frac {c (d+e x)}{c d-b e}\right )^p}{(d+e x)^{3/2}}d(d+e x)}{e}\)

\(\Big \downarrow \) 150

\(\displaystyle -\frac {2 \left (b x+c x^2\right )^p \left (-\frac {e x}{d}\right )^{-p} \left (1-\frac {c (d+e x)}{c d-b e}\right )^{-p} \operatorname {AppellF1}\left (-\frac {1}{2},-p,-p,\frac {1}{2},\frac {d+e x}{d},\frac {c (d+e x)}{c d-b e}\right )}{e \sqrt {d+e x}}\)

Input:

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

Output:

(-2*(b*x + c*x^2)^p*AppellF1[-1/2, -p, -p, 1/2, (d + e*x)/d, (c*(d + e*x)) 
/(c*d - b*e)])/(e*(-((e*x)/d))^p*Sqrt[d + e*x]*(1 - (c*(d + e*x))/(c*d - b 
*e))^p)
 

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 1179

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(a + b*x + c*x^2)^p/(e*(1 - ( 
d + e*x)/(d - e*((b - q)/(2*c))))^p*(1 - (d + e*x)/(d - e*((b + q)/(2*c)))) 
^p)   Subst[Int[x^m*Simp[1 - x/(d - e*((b - q)/(2*c))), x]^p*Simp[1 - x/(d 
- e*((b + q)/(2*c))), x]^p, x], x, d + e*x], x]] /; FreeQ[{a, b, c, d, e, m 
, p}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(e*x + d)*(c*x^2 + b*x)^p/(e^2*x^2 + 2*d*e*x + d^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*(sqrt(d + e*x)*(b*x + c*x**2)**p*b - 2*int((sqrt(d + e*x)*(b*x + c*x**2 
)**p*x)/(2*b**2*d**2*e*p - b**2*d**2*e + 4*b**2*d*e**2*p*x - 2*b**2*d*e**2 
*x + 2*b**2*e**3*p*x**2 - b**2*e**3*x**2 + 4*b*c*d**3*p + 10*b*c*d**2*e*p* 
x - b*c*d**2*e*x + 8*b*c*d*e**2*p*x**2 - 2*b*c*d*e**2*x**2 + 2*b*c*e**3*p* 
x**3 - b*c*e**3*x**3 + 4*c**2*d**3*p*x + 8*c**2*d**2*e*p*x**2 + 4*c**2*d*e 
**2*p*x**3),x)*b**2*c*d*e**2*p**2 + int((sqrt(d + e*x)*(b*x + c*x**2)**p*x 
)/(2*b**2*d**2*e*p - b**2*d**2*e + 4*b**2*d*e**2*p*x - 2*b**2*d*e**2*x + 2 
*b**2*e**3*p*x**2 - b**2*e**3*x**2 + 4*b*c*d**3*p + 10*b*c*d**2*e*p*x - b* 
c*d**2*e*x + 8*b*c*d*e**2*p*x**2 - 2*b*c*d*e**2*x**2 + 2*b*c*e**3*p*x**3 - 
 b*c*e**3*x**3 + 4*c**2*d**3*p*x + 8*c**2*d**2*e*p*x**2 + 4*c**2*d*e**2*p* 
x**3),x)*b**2*c*d*e**2*p - 2*int((sqrt(d + e*x)*(b*x + c*x**2)**p*x)/(2*b* 
*2*d**2*e*p - b**2*d**2*e + 4*b**2*d*e**2*p*x - 2*b**2*d*e**2*x + 2*b**2*e 
**3*p*x**2 - b**2*e**3*x**2 + 4*b*c*d**3*p + 10*b*c*d**2*e*p*x - b*c*d**2* 
e*x + 8*b*c*d*e**2*p*x**2 - 2*b*c*d*e**2*x**2 + 2*b*c*e**3*p*x**3 - b*c*e* 
*3*x**3 + 4*c**2*d**3*p*x + 8*c**2*d**2*e*p*x**2 + 4*c**2*d*e**2*p*x**3),x 
)*b**2*c*e**3*p**2*x + int((sqrt(d + e*x)*(b*x + c*x**2)**p*x)/(2*b**2*d** 
2*e*p - b**2*d**2*e + 4*b**2*d*e**2*p*x - 2*b**2*d*e**2*x + 2*b**2*e**3*p* 
x**2 - b**2*e**3*x**2 + 4*b*c*d**3*p + 10*b*c*d**2*e*p*x - b*c*d**2*e*x + 
8*b*c*d*e**2*p*x**2 - 2*b*c*d*e**2*x**2 + 2*b*c*e**3*p*x**3 - b*c*e**3*x** 
3 + 4*c**2*d**3*p*x + 8*c**2*d**2*e*p*x**2 + 4*c**2*d*e**2*p*x**3),x)*b...