\(\int \frac {(c+d x)^n (a+b x^2)^2}{x^{11/2}} \, dx\) [242]

Optimal result

Integrand size = 22, antiderivative size = 216 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^2}{x^{11/2}} \, dx=-\frac {2 a^2 (c+d x)^{1+n}}{9 c x^{9/2}}+\frac {2 a^2 d (7-2 n) (c+d x)^{1+n}}{63 c^2 x^{7/2}}+\frac {6 b^2 c (c+d x)^{1+n}}{d^2 \left (3-8 n+4 n^2\right ) x^{5/2}}-\frac {2 b^2 (c+d x)^{1+n}}{d (1-2 n) x^{3/2}}-\frac {2 \left (126 a b c^2+a^2 d^2 \left (35-24 n+4 n^2\right )+\frac {945 b^2 c^4}{d^2 \left (3-8 n+4 n^2\right )}\right ) (c+d x)^n \left (\frac {c+d x}{c}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-n,-\frac {3}{2},-\frac {d x}{c}\right )}{315 c^2 x^{5/2}} \] Output:

-2/9*a^2*(d*x+c)^(1+n)/c/x^(9/2)+2/63*a^2*d*(7-2*n)*(d*x+c)^(1+n)/c^2/x^(7 
/2)+6*b^2*c*(d*x+c)^(1+n)/d^2/(4*n^2-8*n+3)/x^(5/2)-2*b^2*(d*x+c)^(1+n)/d/ 
(1-2*n)/x^(3/2)-2/315*(126*a*b*c^2+a^2*d^2*(4*n^2-24*n+35)+945*b^2*c^4/d^2 
/(4*n^2-8*n+3))*(d*x+c)^n*hypergeom([-5/2, -n],[-3/2],-d*x/c)/c^2/x^(5/2)/ 
(((d*x+c)/c)^n)
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.26 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^2}{x^{11/2}} \, dx=-\frac {2 (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \left (b^2 c^4 \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},-4-n,-\frac {7}{2},-\frac {d x}{c}\right )-4 b^2 c^4 \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},-3-n,-\frac {7}{2},-\frac {d x}{c}\right )+6 b^2 c^4 \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},-2-n,-\frac {7}{2},-\frac {d x}{c}\right )+2 a b c^2 d^2 \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},-2-n,-\frac {7}{2},-\frac {d x}{c}\right )-4 b^2 c^4 \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},-1-n,-\frac {7}{2},-\frac {d x}{c}\right )-4 a b c^2 d^2 \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},-1-n,-\frac {7}{2},-\frac {d x}{c}\right )+b^2 c^4 \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},-n,-\frac {7}{2},-\frac {d x}{c}\right )+2 a b c^2 d^2 \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},-n,-\frac {7}{2},-\frac {d x}{c}\right )+a^2 d^4 \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},-n,-\frac {7}{2},-\frac {d x}{c}\right )\right )}{9 d^4 x^{9/2}} \] Input:

Integrate[((c + d*x)^n*(a + b*x^2)^2)/x^(11/2),x]
 

Output:

(-2*(c + d*x)^n*(b^2*c^4*Hypergeometric2F1[-9/2, -4 - n, -7/2, -((d*x)/c)] 
 - 4*b^2*c^4*Hypergeometric2F1[-9/2, -3 - n, -7/2, -((d*x)/c)] + 6*b^2*c^4 
*Hypergeometric2F1[-9/2, -2 - n, -7/2, -((d*x)/c)] + 2*a*b*c^2*d^2*Hyperge 
ometric2F1[-9/2, -2 - n, -7/2, -((d*x)/c)] - 4*b^2*c^4*Hypergeometric2F1[- 
9/2, -1 - n, -7/2, -((d*x)/c)] - 4*a*b*c^2*d^2*Hypergeometric2F1[-9/2, -1 
- n, -7/2, -((d*x)/c)] + b^2*c^4*Hypergeometric2F1[-9/2, -n, -7/2, -((d*x) 
/c)] + 2*a*b*c^2*d^2*Hypergeometric2F1[-9/2, -n, -7/2, -((d*x)/c)] + a^2*d 
^4*Hypergeometric2F1[-9/2, -n, -7/2, -((d*x)/c)]))/(9*d^4*x^(9/2)*(1 + (d* 
x)/c)^n)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.24, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {520, 27, 2124, 27, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^2 (c+d x)^n}{x^{11/2}} \, dx\)

\(\Big \downarrow \) 520

\(\displaystyle -\frac {2 \int \frac {(c+d x)^n \left (-9 b^2 c x^3-18 a b c x+a^2 d (7-2 n)\right )}{2 x^{9/2}}dx}{9 c}-\frac {2 a^2 (c+d x)^{n+1}}{9 c x^{9/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {(c+d x)^n \left (-9 b^2 c x^3-18 a b c x+a^2 d (7-2 n)\right )}{x^{9/2}}dx}{9 c}-\frac {2 a^2 (c+d x)^{n+1}}{9 c x^{9/2}}\)

\(\Big \downarrow \) 2124

\(\displaystyle -\frac {-\frac {2 \int \frac {(c+d x)^n \left (63 b^2 c^2 x^2+a \left (126 b c^2+a d^2 \left (4 n^2-24 n+35\right )\right )\right )}{2 x^{7/2}}dx}{7 c}-\frac {2 a^2 d (7-2 n) (c+d x)^{n+1}}{7 c x^{7/2}}}{9 c}-\frac {2 a^2 (c+d x)^{n+1}}{9 c x^{9/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {(c+d x)^n \left (63 b^2 c^2 x^2+a \left (126 b c^2+a d^2 \left (4 n^2-24 n+35\right )\right )\right )}{x^{7/2}}dx}{7 c}-\frac {2 a^2 d (7-2 n) (c+d x)^{n+1}}{7 c x^{7/2}}}{9 c}-\frac {2 a^2 (c+d x)^{n+1}}{9 c x^{9/2}}\)

\(\Big \downarrow \) 520

\(\displaystyle -\frac {-\frac {-\frac {2 \int \frac {\left (a d (3-2 n) \left (126 b c^2+a d^2 \left (4 n^2-24 n+35\right )\right )-315 b^2 c^3 x\right ) (c+d x)^n}{2 x^{5/2}}dx}{5 c}-\frac {2 a (c+d x)^{n+1} \left (a d^2 \left (4 n^2-24 n+35\right )+126 b c^2\right )}{5 c x^{5/2}}}{7 c}-\frac {2 a^2 d (7-2 n) (c+d x)^{n+1}}{7 c x^{7/2}}}{9 c}-\frac {2 a^2 (c+d x)^{n+1}}{9 c x^{9/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {-\frac {\int \frac {\left (a d (3-2 n) \left (126 b c^2+a d^2 \left (4 n^2-24 n+35\right )\right )-315 b^2 c^3 x\right ) (c+d x)^n}{x^{5/2}}dx}{5 c}-\frac {2 a (c+d x)^{n+1} \left (a d^2 \left (4 n^2-24 n+35\right )+126 b c^2\right )}{5 c x^{5/2}}}{7 c}-\frac {2 a^2 d (7-2 n) (c+d x)^{n+1}}{7 c x^{7/2}}}{9 c}-\frac {2 a^2 (c+d x)^{n+1}}{9 c x^{9/2}}\)

\(\Big \downarrow \) 87

\(\displaystyle -\frac {-\frac {-\frac {-\frac {\left (a d^2 (1-2 n) (3-2 n) \left (a d^2 \left (4 n^2-24 n+35\right )+126 b c^2\right )+945 b^2 c^4\right ) \int \frac {(c+d x)^n}{x^{3/2}}dx}{3 c}-\frac {2 a d (3-2 n) (c+d x)^{n+1} \left (a d^2 \left (4 n^2-24 n+35\right )+126 b c^2\right )}{3 c x^{3/2}}}{5 c}-\frac {2 a (c+d x)^{n+1} \left (a d^2 \left (4 n^2-24 n+35\right )+126 b c^2\right )}{5 c x^{5/2}}}{7 c}-\frac {2 a^2 d (7-2 n) (c+d x)^{n+1}}{7 c x^{7/2}}}{9 c}-\frac {2 a^2 (c+d x)^{n+1}}{9 c x^{9/2}}\)

\(\Big \downarrow \) 76

\(\displaystyle -\frac {-\frac {-\frac {-\frac {(c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \left (a d^2 (1-2 n) (3-2 n) \left (a d^2 \left (4 n^2-24 n+35\right )+126 b c^2\right )+945 b^2 c^4\right ) \int \frac {\left (\frac {d x}{c}+1\right )^n}{x^{3/2}}dx}{3 c}-\frac {2 a d (3-2 n) (c+d x)^{n+1} \left (a d^2 \left (4 n^2-24 n+35\right )+126 b c^2\right )}{3 c x^{3/2}}}{5 c}-\frac {2 a (c+d x)^{n+1} \left (a d^2 \left (4 n^2-24 n+35\right )+126 b c^2\right )}{5 c x^{5/2}}}{7 c}-\frac {2 a^2 d (7-2 n) (c+d x)^{n+1}}{7 c x^{7/2}}}{9 c}-\frac {2 a^2 (c+d x)^{n+1}}{9 c x^{9/2}}\)

\(\Big \downarrow \) 74

\(\displaystyle -\frac {-\frac {2 a^2 d (7-2 n) (c+d x)^{n+1}}{7 c x^{7/2}}-\frac {-\frac {\frac {2 (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \left (a d^2 (1-2 n) (3-2 n) \left (a d^2 \left (4 n^2-24 n+35\right )+126 b c^2\right )+945 b^2 c^4\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-n,\frac {1}{2},-\frac {d x}{c}\right )}{3 c \sqrt {x}}-\frac {2 a d (3-2 n) (c+d x)^{n+1} \left (a d^2 \left (4 n^2-24 n+35\right )+126 b c^2\right )}{3 c x^{3/2}}}{5 c}-\frac {2 a (c+d x)^{n+1} \left (a d^2 \left (4 n^2-24 n+35\right )+126 b c^2\right )}{5 c x^{5/2}}}{7 c}}{9 c}-\frac {2 a^2 (c+d x)^{n+1}}{9 c x^{9/2}}\)

Input:

Int[((c + d*x)^n*(a + b*x^2)^2)/x^(11/2),x]
 

Output:

(-2*a^2*(c + d*x)^(1 + n))/(9*c*x^(9/2)) - ((-2*a^2*d*(7 - 2*n)*(c + d*x)^ 
(1 + n))/(7*c*x^(7/2)) - ((-2*a*(126*b*c^2 + a*d^2*(35 - 24*n + 4*n^2))*(c 
 + d*x)^(1 + n))/(5*c*x^(5/2)) - ((-2*a*d*(3 - 2*n)*(126*b*c^2 + a*d^2*(35 
 - 24*n + 4*n^2))*(c + d*x)^(1 + n))/(3*c*x^(3/2)) + (2*(945*b^2*c^4 + a*d 
^2*(1 - 2*n)*(3 - 2*n)*(126*b*c^2 + a*d^2*(35 - 24*n + 4*n^2)))*(c + d*x)^ 
n*Hypergeometric2F1[-1/2, -n, 1/2, -((d*x)/c)])/(3*c*Sqrt[x]*(1 + (d*x)/c) 
^n))/(5*c))/(7*c))/(9*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]
 

rule 2124

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px 
, a + b*x, x]}, Simp[R*(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((m + 1)*(b*c - 
 a*d))), x] + Simp[1/((m + 1)*(b*c - a*d))   Int[(a + b*x)^(m + 1)*(c + d*x 
)^n*ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr 
eeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && LtQ[m, -1] && (IntegerQ[m] ||  ! 
ILtQ[n, -1])
 

Maple [F]

\[\int \frac {\left (d x +c \right )^{n} \left (b \,x^{2}+a \right )^{2}}{x^{\frac {11}{2}}}d x\]

Input:

int((d*x+c)^n*(b*x^2+a)^2/x^(11/2),x)
 

Output:

int((d*x+c)^n*(b*x^2+a)^2/x^(11/2),x)
 

Fricas [F]

\[ \int \frac {(c+d x)^n \left (a+b x^2\right )^2}{x^{11/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} {\left (d x + c\right )}^{n}}{x^{\frac {11}{2}}} \,d x } \] Input:

integrate((d*x+c)^n*(b*x^2+a)^2/x^(11/2),x, algorithm="fricas")
 

Output:

integral((b^2*x^4 + 2*a*b*x^2 + a^2)*(d*x + c)^n/x^(11/2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^2}{x^{11/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**n*(b*x**2+a)**2/x**(11/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(c+d x)^n \left (a+b x^2\right )^2}{x^{11/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} {\left (d x + c\right )}^{n}}{x^{\frac {11}{2}}} \,d x } \] Input:

integrate((d*x+c)^n*(b*x^2+a)^2/x^(11/2),x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^2*(d*x + c)^n/x^(11/2), x)
 

Giac [F]

\[ \int \frac {(c+d x)^n \left (a+b x^2\right )^2}{x^{11/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} {\left (d x + c\right )}^{n}}{x^{\frac {11}{2}}} \,d x } \] Input:

integrate((d*x+c)^n*(b*x^2+a)^2/x^(11/2),x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^2*(d*x + c)^n/x^(11/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^2}{x^{11/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2\,{\left (c+d\,x\right )}^n}{x^{11/2}} \,d x \] Input:

int(((a + b*x^2)^2*(c + d*x)^n)/x^(11/2),x)
 

Output:

int(((a + b*x^2)^2*(c + d*x)^n)/x^(11/2), x)
 

Reduce [F]

\[ \int \frac {(c+d x)^n \left (a+b x^2\right )^2}{x^{11/2}} \, dx=\text {too large to display} \] Input:

int((d*x+c)^n*(b*x^2+a)^2/x^(11/2),x)
                                                                                    
                                                                                    
 

Output:

(2*(16*(c + d*x)**n*a**2*d**4*n**4 - 128*(c + d*x)**n*a**2*d**4*n**3 + 344 
*(c + d*x)**n*a**2*d**4*n**2 - 352*(c + d*x)**n*a**2*d**4*n + 105*(c + d*x 
)**n*a**2*d**4 + 112*(c + d*x)**n*a*b*c**2*d**2*n**3 - 224*(c + d*x)**n*a* 
b*c**2*d**2*n**2 + 84*(c + d*x)**n*a*b*c**2*d**2*n + 32*(c + d*x)**n*a*b*c 
*d**3*n**4*x - 208*(c + d*x)**n*a*b*c*d**3*n**3*x + 312*(c + d*x)**n*a*b*c 
*d**3*n**2*x - 108*(c + d*x)**n*a*b*c*d**3*n*x + 32*(c + d*x)**n*a*b*d**4* 
n**4*x**2 - 320*(c + d*x)**n*a*b*d**4*n**3*x**2 + 1040*(c + d*x)**n*a*b*d* 
*4*n**2*x**2 - 1200*(c + d*x)**n*a*b*d**4*n*x**2 + 378*(c + d*x)**n*a*b*d* 
*4*x**2 + 210*(c + d*x)**n*b**2*c**4*n + 60*(c + d*x)**n*b**2*c**3*d*n**2* 
x - 270*(c + d*x)**n*b**2*c**3*d*n*x + 24*(c + d*x)**n*b**2*c**2*d**2*n**3 
*x**2 - 192*(c + d*x)**n*b**2*c**2*d**2*n**2*x**2 + 378*(c + d*x)**n*b**2* 
c**2*d**2*n*x**2 + 16*(c + d*x)**n*b**2*c*d**3*n**4*x**3 - 168*(c + d*x)** 
n*b**2*c*d**3*n**3*x**3 + 572*(c + d*x)**n*b**2*c*d**3*n**2*x**3 - 630*(c 
+ d*x)**n*b**2*c*d**3*n*x**3 + 16*(c + d*x)**n*b**2*d**4*n**4*x**4 - 192*( 
c + d*x)**n*b**2*d**4*n**3*x**4 + 824*(c + d*x)**n*b**2*d**4*n**2*x**4 - 1 
488*(c + d*x)**n*b**2*d**4*n*x**4 + 945*(c + d*x)**n*b**2*d**4*x**4 + 512* 
sqrt(x)*int((sqrt(x)*(c + d*x)**n)/(32*c*n**5*x**6 - 400*c*n**4*x**6 + 184 
0*c*n**3*x**6 - 3800*c*n**2*x**6 + 3378*c*n*x**6 - 945*c*x**6 + 32*d*n**5* 
x**7 - 400*d*n**4*x**7 + 1840*d*n**3*x**7 - 3800*d*n**2*x**7 + 3378*d*n*x* 
*7 - 945*d*x**7),x)*a**2*c*d**4*n**10*x**4 - 10496*sqrt(x)*int((sqrt(x)...