\(\int \frac {(c+d x)^n (a+b x^2)}{x^{9/2}} \, dx\) [234]

Optimal result

Integrand size = 20, antiderivative size = 122 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )}{x^{9/2}} \, dx=-\frac {2 a (c+d x)^{1+n}}{7 c x^{7/2}}-\frac {2 b (c+d x)^{1+n}}{d (3-2 n) x^{5/2}}+\frac {2 \left (\frac {a d (5-2 n)}{c}+\frac {35 b c}{3 d-2 d n}\right ) (c+d x)^n \left (\frac {c+d x}{c}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-n,-\frac {3}{2},-\frac {d x}{c}\right )}{35 x^{5/2}} \] Output:

-2/7*a*(d*x+c)^(1+n)/c/x^(7/2)-2*b*(d*x+c)^(1+n)/d/(3-2*n)/x^(5/2)+2/35*(a 
*d*(5-2*n)/c+35*b*c/(-2*d*n+3*d))*(d*x+c)^n*hypergeom([-5/2, -n],[-3/2],-d 
*x/c)/x^(5/2)/(((d*x+c)/c)^n)
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.90 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )}{x^{9/2}} \, dx=-\frac {2 (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \left (b c^2 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},-2-n,-\frac {5}{2},-\frac {d x}{c}\right )-2 b c^2 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},-1-n,-\frac {5}{2},-\frac {d x}{c}\right )+\left (b c^2+a d^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},-n,-\frac {5}{2},-\frac {d x}{c}\right )\right )}{7 d^2 x^{7/2}} \] Input:

Integrate[((c + d*x)^n*(a + b*x^2))/x^(9/2),x]
 

Output:

(-2*(c + d*x)^n*(b*c^2*Hypergeometric2F1[-7/2, -2 - n, -5/2, -((d*x)/c)] - 
 2*b*c^2*Hypergeometric2F1[-7/2, -1 - n, -5/2, -((d*x)/c)] + (b*c^2 + a*d^ 
2)*Hypergeometric2F1[-7/2, -n, -5/2, -((d*x)/c)]))/(7*d^2*x^(7/2)*(1 + (d* 
x)/c)^n)
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {520, 27, 87, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right ) (c+d x)^n}{x^{9/2}} \, dx\)

\(\Big \downarrow \) 520

\(\displaystyle -\frac {2 \int \frac {(a d (5-2 n)-7 b c x) (c+d x)^n}{2 x^{7/2}}dx}{7 c}-\frac {2 a (c+d x)^{n+1}}{7 c x^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {(a d (5-2 n)-7 b c x) (c+d x)^n}{x^{7/2}}dx}{7 c}-\frac {2 a (c+d x)^{n+1}}{7 c x^{7/2}}\)

\(\Big \downarrow \) 87

\(\displaystyle -\frac {-\frac {\left (a d^2 (3-2 n) (5-2 n)+35 b c^2\right ) \int \frac {(c+d x)^n}{x^{5/2}}dx}{5 c}-\frac {2 a d (5-2 n) (c+d x)^{n+1}}{5 c x^{5/2}}}{7 c}-\frac {2 a (c+d x)^{n+1}}{7 c x^{7/2}}\)

\(\Big \downarrow \) 76

\(\displaystyle -\frac {-\frac {(c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \left (a d^2 (3-2 n) (5-2 n)+35 b c^2\right ) \int \frac {\left (\frac {d x}{c}+1\right )^n}{x^{5/2}}dx}{5 c}-\frac {2 a d (5-2 n) (c+d x)^{n+1}}{5 c x^{5/2}}}{7 c}-\frac {2 a (c+d x)^{n+1}}{7 c x^{7/2}}\)

\(\Big \downarrow \) 74

\(\displaystyle -\frac {\frac {2 (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \left (a d^2 (3-2 n) (5-2 n)+35 b c^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-n,-\frac {1}{2},-\frac {d x}{c}\right )}{15 c x^{3/2}}-\frac {2 a d (5-2 n) (c+d x)^{n+1}}{5 c x^{5/2}}}{7 c}-\frac {2 a (c+d x)^{n+1}}{7 c x^{7/2}}\)

Input:

Int[((c + d*x)^n*(a + b*x^2))/x^(9/2),x]
 

Output:

(-2*a*(c + d*x)^(1 + n))/(7*c*x^(7/2)) - ((-2*a*d*(5 - 2*n)*(c + d*x)^(1 + 
 n))/(5*c*x^(5/2)) + (2*(35*b*c^2 + a*d^2*(3 - 2*n)*(5 - 2*n))*(c + d*x)^n 
*Hypergeometric2F1[-3/2, -n, -1/2, -((d*x)/c)])/(15*c*x^(3/2)*(1 + (d*x)/c 
)^n))/(7*c)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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])
 

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 520

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), 
 x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2)^p, e*x, x], R = Pol 
ynomialRemainder[(a + b*x^2)^p, e*x, x]}, Simp[R*(e*x)^(m + 1)*((c + d*x)^( 
n + 1)/((m + 1)*(e*c))), x] + Simp[1/((m + 1)*(e*c))   Int[(e*x)^(m + 1)*(c 
 + d*x)^n*ExpandToSum[(m + 1)*(e*c)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr 
eeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[m, -1] &&  !IntegerQ[n]
 

Maple [F]

\[\int \frac {\left (d x +c \right )^{n} \left (b \,x^{2}+a \right )}{x^{\frac {9}{2}}}d x\]

Input:

int((d*x+c)^n*(b*x^2+a)/x^(9/2),x)
 

Output:

int((d*x+c)^n*(b*x^2+a)/x^(9/2),x)
 

Fricas [F]

\[ \int \frac {(c+d x)^n \left (a+b x^2\right )}{x^{9/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )} {\left (d x + c\right )}^{n}}{x^{\frac {9}{2}}} \,d x } \] Input:

integrate((d*x+c)^n*(b*x^2+a)/x^(9/2),x, algorithm="fricas")
 

Output:

integral((b*x^2 + a)*(d*x + c)^n/x^(9/2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^n \left (a+b x^2\right )}{x^{9/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**n*(b*x**2+a)/x**(9/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(c+d x)^n \left (a+b x^2\right )}{x^{9/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )} {\left (d x + c\right )}^{n}}{x^{\frac {9}{2}}} \,d x } \] Input:

integrate((d*x+c)^n*(b*x^2+a)/x^(9/2),x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)*(d*x + c)^n/x^(9/2), x)
 

Giac [F]

\[ \int \frac {(c+d x)^n \left (a+b x^2\right )}{x^{9/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )} {\left (d x + c\right )}^{n}}{x^{\frac {9}{2}}} \,d x } \] Input:

integrate((d*x+c)^n*(b*x^2+a)/x^(9/2),x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)*(d*x + c)^n/x^(9/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^n \left (a+b x^2\right )}{x^{9/2}} \, dx=\int \frac {\left (b\,x^2+a\right )\,{\left (c+d\,x\right )}^n}{x^{9/2}} \,d x \] Input:

int(((a + b*x^2)*(c + d*x)^n)/x^(9/2),x)
 

Output:

int(((a + b*x^2)*(c + d*x)^n)/x^(9/2), x)
 

Reduce [F]

\[ \int \frac {(c+d x)^n \left (a+b x^2\right )}{x^{9/2}} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^n*(b*x^2+a)/x^(9/2),x)
 

Output:

(2*(4*(c + d*x)**n*a*d**2*n**2 - 16*(c + d*x)**n*a*d**2*n + 15*(c + d*x)** 
n*a*d**2 + 10*(c + d*x)**n*b*c**2*n + 4*(c + d*x)**n*b*c*d*n**2*x - 14*(c 
+ d*x)**n*b*c*d*n*x + 4*(c + d*x)**n*b*d**2*n**2*x**2 - 24*(c + d*x)**n*b* 
d**2*n*x**2 + 35*(c + d*x)**n*b*d**2*x**2 + 32*sqrt(x)*int((sqrt(x)*(c + d 
*x)**n)/(8*c*n**3*x**5 - 60*c*n**2*x**5 + 142*c*n*x**5 - 105*c*x**5 + 8*d* 
n**3*x**6 - 60*d*n**2*x**6 + 142*d*n*x**6 - 105*d*x**6),x)*a*c*d**2*n**6*x 
**3 - 368*sqrt(x)*int((sqrt(x)*(c + d*x)**n)/(8*c*n**3*x**5 - 60*c*n**2*x* 
*5 + 142*c*n*x**5 - 105*c*x**5 + 8*d*n**3*x**6 - 60*d*n**2*x**6 + 142*d*n* 
x**6 - 105*d*x**6),x)*a*c*d**2*n**5*x**3 + 1648*sqrt(x)*int((sqrt(x)*(c + 
d*x)**n)/(8*c*n**3*x**5 - 60*c*n**2*x**5 + 142*c*n*x**5 - 105*c*x**5 + 8*d 
*n**3*x**6 - 60*d*n**2*x**6 + 142*d*n*x**6 - 105*d*x**6),x)*a*c*d**2*n**4* 
x**3 - 3592*sqrt(x)*int((sqrt(x)*(c + d*x)**n)/(8*c*n**3*x**5 - 60*c*n**2* 
x**5 + 142*c*n*x**5 - 105*c*x**5 + 8*d*n**3*x**6 - 60*d*n**2*x**6 + 142*d* 
n*x**6 - 105*d*x**6),x)*a*c*d**2*n**3*x**3 + 3810*sqrt(x)*int((sqrt(x)*(c 
+ d*x)**n)/(8*c*n**3*x**5 - 60*c*n**2*x**5 + 142*c*n*x**5 - 105*c*x**5 + 8 
*d*n**3*x**6 - 60*d*n**2*x**6 + 142*d*n*x**6 - 105*d*x**6),x)*a*c*d**2*n** 
2*x**3 - 1575*sqrt(x)*int((sqrt(x)*(c + d*x)**n)/(8*c*n**3*x**5 - 60*c*n** 
2*x**5 + 142*c*n*x**5 - 105*c*x**5 + 8*d*n**3*x**6 - 60*d*n**2*x**6 + 142* 
d*n*x**6 - 105*d*x**6),x)*a*c*d**2*n*x**3 + 280*sqrt(x)*int((sqrt(x)*(c + 
d*x)**n)/(8*c*n**3*x**5 - 60*c*n**2*x**5 + 142*c*n*x**5 - 105*c*x**5 + ...