\(\int \frac {(c+d x)^n (a+b x^2)^3}{x^7} \, dx\) [212]

Optimal result

Integrand size = 20, antiderivative size = 354 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^3}{x^7} \, dx=-\frac {a^3 (c+d x)^{1+n}}{6 c x^6}+\frac {a^3 d (5-n) (c+d x)^{1+n}}{30 c^2 x^5}-\frac {3 b^2 c \left (2 b c^2-3 a d^2 (1-n) n\right ) (c+d x)^{1+n}}{d^4 (1-n) (2-n) (3-n) n x^4}+\frac {b^2 \left (2 b c^2-3 a d^2 (1-n) n\right ) (c+d x)^{1+n}}{d^3 (1-n) (2-n) n x^3}-\frac {b^3 c (c+d x)^{1+n}}{d^2 (1-n) n x^2}+\frac {b^3 (c+d x)^{1+n}}{d n x}+\frac {\left (360 b^2 c^4 \left (2 b c^2-3 a d^2 (1-n) n\right )-a^2 d^4 (3-n) n \left (2-3 n+n^2\right ) \left (90 b c^2+a d^2 \left (20-9 n+n^2\right )\right )\right ) (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (5,1+n,2+n,1+\frac {d x}{c}\right )}{30 c^7 (1-n) (2-n) (3-n) n (1+n)} \] Output:

-1/6*a^3*(d*x+c)^(1+n)/c/x^6+1/30*a^3*d*(5-n)*(d*x+c)^(1+n)/c^2/x^5-3*b^2* 
c*(2*b*c^2-3*a*d^2*(1-n)*n)*(d*x+c)^(1+n)/d^4/(1-n)/(2-n)/(3-n)/n/x^4+b^2* 
(2*b*c^2-3*a*d^2*(1-n)*n)*(d*x+c)^(1+n)/d^3/(1-n)/(2-n)/n/x^3-b^3*c*(d*x+c 
)^(1+n)/d^2/(1-n)/n/x^2+b^3*(d*x+c)^(1+n)/d/n/x+1/30*(360*b^2*c^4*(2*b*c^2 
-3*a*d^2*(1-n)*n)-a^2*d^4*(3-n)*n*(n^2-3*n+2)*(90*b*c^2+a*d^2*(n^2-9*n+20) 
))*(d*x+c)^(1+n)*hypergeom([5, 1+n],[2+n],1+d*x/c)/c^7/(1-n)/(2-n)/(3-n)/n 
/(1+n)
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.34 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^3}{x^7} \, dx=-\frac {(c+d x)^{1+n} \left (b^3 c^6 \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {d x}{c}\right )+a d^2 \left (3 b^2 c^4 \operatorname {Hypergeometric2F1}\left (3,1+n,2+n,1+\frac {d x}{c}\right )+a d^2 \left (3 b c^2 \operatorname {Hypergeometric2F1}\left (5,1+n,2+n,1+\frac {d x}{c}\right )+a d^2 \operatorname {Hypergeometric2F1}\left (7,1+n,2+n,1+\frac {d x}{c}\right )\right )\right )\right )}{c^7 (1+n)} \] Input:

Integrate[((c + d*x)^n*(a + b*x^2)^3)/x^7,x]
 

Output:

-(((c + d*x)^(1 + n)*(b^3*c^6*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (d*x) 
/c] + a*d^2*(3*b^2*c^4*Hypergeometric2F1[3, 1 + n, 2 + n, 1 + (d*x)/c] + a 
*d^2*(3*b*c^2*Hypergeometric2F1[5, 1 + n, 2 + n, 1 + (d*x)/c] + a*d^2*Hype 
rgeometric2F1[7, 1 + n, 2 + n, 1 + (d*x)/c]))))/(c^7*(1 + n)))
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.23, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {520, 2124, 2124, 2124, 520, 87, 75}

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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^3 (c+d x)^n}{x^7} \, dx\)

\(\Big \downarrow \) 520

\(\displaystyle -\frac {\int \frac {(c+d x)^n \left (-6 b^3 c x^5-18 a b^2 c x^3-18 a^2 b c x+a^3 d (5-n)\right )}{x^6}dx}{6 c}-\frac {a^3 (c+d x)^{n+1}}{6 c x^6}\)

\(\Big \downarrow \) 2124

\(\displaystyle -\frac {-\frac {\int \frac {(c+d x)^n \left (30 b^3 c^2 x^4+90 a b^2 c^2 x^2+a^2 \left (90 b c^2+a d^2 \left (n^2-9 n+20\right )\right )\right )}{x^5}dx}{5 c}-\frac {a^3 d (5-n) (c+d x)^{n+1}}{5 c x^5}}{6 c}-\frac {a^3 (c+d x)^{n+1}}{6 c x^6}\)

\(\Big \downarrow \) 2124

\(\displaystyle -\frac {-\frac {-\frac {\int \frac {(c+d x)^n \left (-120 b^3 x^3 c^3-360 a b^2 x c^3+a^2 d (3-n) \left (90 b c^2+a d^2 \left (n^2-9 n+20\right )\right )\right )}{x^4}dx}{4 c}-\frac {a^2 (c+d x)^{n+1} \left (a d^2 \left (n^2-9 n+20\right )+90 b c^2\right )}{4 c x^4}}{5 c}-\frac {a^3 d (5-n) (c+d x)^{n+1}}{5 c x^5}}{6 c}-\frac {a^3 (c+d x)^{n+1}}{6 c x^6}\)

\(\Big \downarrow \) 2124

\(\displaystyle -\frac {-\frac {-\frac {-\frac {\int \frac {(c+d x)^n \left (360 b^3 x^2 c^4+a \left (1080 b^2 c^4+90 a b d^2 \left (n^2-5 n+6\right ) c^2+a^2 d^4 \left (n^4-14 n^3+71 n^2-154 n+120\right )\right )\right )}{x^3}dx}{3 c}-\frac {a^2 d (3-n) (c+d x)^{n+1} \left (a d^2 \left (n^2-9 n+20\right )+90 b c^2\right )}{3 c x^3}}{4 c}-\frac {a^2 (c+d x)^{n+1} \left (a d^2 \left (n^2-9 n+20\right )+90 b c^2\right )}{4 c x^4}}{5 c}-\frac {a^3 d (5-n) (c+d x)^{n+1}}{5 c x^5}}{6 c}-\frac {a^3 (c+d x)^{n+1}}{6 c x^6}\)

\(\Big \downarrow \) 520

\(\displaystyle -\frac {-\frac {-\frac {-\frac {-\frac {\int \frac {\left (a d (1-n) \left (1080 b^2 c^4+90 a b d^2 \left (n^2-5 n+6\right ) c^2+a^2 d^4 \left (n^4-14 n^3+71 n^2-154 n+120\right )\right )-720 b^3 c^5 x\right ) (c+d x)^n}{x^2}dx}{2 c}-\frac {a (c+d x)^{n+1} \left (a^2 d^4 \left (n^4-14 n^3+71 n^2-154 n+120\right )+90 a b c^2 d^2 \left (n^2-5 n+6\right )+1080 b^2 c^4\right )}{2 c x^2}}{3 c}-\frac {a^2 d (3-n) (c+d x)^{n+1} \left (a d^2 \left (n^2-9 n+20\right )+90 b c^2\right )}{3 c x^3}}{4 c}-\frac {a^2 (c+d x)^{n+1} \left (a d^2 \left (n^2-9 n+20\right )+90 b c^2\right )}{4 c x^4}}{5 c}-\frac {a^3 d (5-n) (c+d x)^{n+1}}{5 c x^5}}{6 c}-\frac {a^3 (c+d x)^{n+1}}{6 c x^6}\)

\(\Big \downarrow \) 87

\(\displaystyle -\frac {-\frac {-\frac {-\frac {-\frac {-\frac {\left (720 b^3 c^6-a d^2 (1-n) n \left (a^2 d^4 \left (n^4-14 n^3+71 n^2-154 n+120\right )+90 a b c^2 d^2 \left (n^2-5 n+6\right )+1080 b^2 c^4\right )\right ) \int \frac {(c+d x)^n}{x}dx}{c}-\frac {a d (1-n) (c+d x)^{n+1} \left (a^2 d^4 \left (n^4-14 n^3+71 n^2-154 n+120\right )+90 a b c^2 d^2 \left (n^2-5 n+6\right )+1080 b^2 c^4\right )}{c x}}{2 c}-\frac {a (c+d x)^{n+1} \left (a^2 d^4 \left (n^4-14 n^3+71 n^2-154 n+120\right )+90 a b c^2 d^2 \left (n^2-5 n+6\right )+1080 b^2 c^4\right )}{2 c x^2}}{3 c}-\frac {a^2 d (3-n) (c+d x)^{n+1} \left (a d^2 \left (n^2-9 n+20\right )+90 b c^2\right )}{3 c x^3}}{4 c}-\frac {a^2 (c+d x)^{n+1} \left (a d^2 \left (n^2-9 n+20\right )+90 b c^2\right )}{4 c x^4}}{5 c}-\frac {a^3 d (5-n) (c+d x)^{n+1}}{5 c x^5}}{6 c}-\frac {a^3 (c+d x)^{n+1}}{6 c x^6}\)

\(\Big \downarrow \) 75

\(\displaystyle -\frac {a^3 (c+d x)^{n+1}}{6 c x^6}-\frac {-\frac {a^3 d (5-n) (c+d x)^{n+1}}{5 c x^5}-\frac {-\frac {-\frac {-\frac {a (c+d x)^{n+1} \left (a^2 d^4 \left (n^4-14 n^3+71 n^2-154 n+120\right )+90 a b c^2 d^2 \left (n^2-5 n+6\right )+1080 b^2 c^4\right )}{2 c x^2}-\frac {\frac {(c+d x)^{n+1} \left (720 b^3 c^6-a d^2 (1-n) n \left (a^2 d^4 \left (n^4-14 n^3+71 n^2-154 n+120\right )+90 a b c^2 d^2 \left (n^2-5 n+6\right )+1080 b^2 c^4\right )\right ) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {d x}{c}+1\right )}{c^2 (n+1)}-\frac {a d (1-n) (c+d x)^{n+1} \left (a^2 d^4 \left (n^4-14 n^3+71 n^2-154 n+120\right )+90 a b c^2 d^2 \left (n^2-5 n+6\right )+1080 b^2 c^4\right )}{c x}}{2 c}}{3 c}-\frac {a^2 d (3-n) (c+d x)^{n+1} \left (a d^2 \left (n^2-9 n+20\right )+90 b c^2\right )}{3 c x^3}}{4 c}-\frac {a^2 (c+d x)^{n+1} \left (a d^2 \left (n^2-9 n+20\right )+90 b c^2\right )}{4 c x^4}}{5 c}}{6 c}\)

Input:

Int[((c + d*x)^n*(a + b*x^2)^3)/x^7,x]
 

Output:

-1/6*(a^3*(c + d*x)^(1 + n))/(c*x^6) - (-1/5*(a^3*d*(5 - n)*(c + d*x)^(1 + 
 n))/(c*x^5) - (-1/4*(a^2*(90*b*c^2 + a*d^2*(20 - 9*n + n^2))*(c + d*x)^(1 
 + n))/(c*x^4) - (-1/3*(a^2*d*(3 - n)*(90*b*c^2 + a*d^2*(20 - 9*n + n^2))* 
(c + d*x)^(1 + n))/(c*x^3) - (-1/2*(a*(1080*b^2*c^4 + 90*a*b*c^2*d^2*(6 - 
5*n + n^2) + a^2*d^4*(120 - 154*n + 71*n^2 - 14*n^3 + n^4))*(c + d*x)^(1 + 
 n))/(c*x^2) - (-((a*d*(1 - n)*(1080*b^2*c^4 + 90*a*b*c^2*d^2*(6 - 5*n + n 
^2) + a^2*d^4*(120 - 154*n + 71*n^2 - 14*n^3 + n^4))*(c + d*x)^(1 + n))/(c 
*x)) + ((720*b^3*c^6 - a*d^2*(1 - n)*n*(1080*b^2*c^4 + 90*a*b*c^2*d^2*(6 - 
 5*n + n^2) + a^2*d^4*(120 - 154*n + 71*n^2 - 14*n^3 + n^4)))*(c + d*x)^(1 
 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (d*x)/c])/(c^2*(1 + n)))/(2*c 
))/(3*c))/(4*c))/(5*c))/(6*c)
 

Defintions of rubi rules used

rule 75

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

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]

\[\int \frac {\left (d x +c \right )^{n} \left (b \,x^{2}+a \right )^{3}}{x^{7}}d x\]

Input:

int((d*x+c)^n*(b*x^2+a)^3/x^7,x)
 

Output:

int((d*x+c)^n*(b*x^2+a)^3/x^7,x)
 

Fricas [F]

\[ \int \frac {(c+d x)^n \left (a+b x^2\right )^3}{x^7} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{3} {\left (d x + c\right )}^{n}}{x^{7}} \,d x } \] Input:

integrate((d*x+c)^n*(b*x^2+a)^3/x^7,x, algorithm="fricas")
 

Output:

integral((b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3)*(d*x + c)^n/x^7, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^3}{x^7} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**n*(b*x**2+a)**3/x**7,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(c+d x)^n \left (a+b x^2\right )^3}{x^7} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{3} {\left (d x + c\right )}^{n}}{x^{7}} \,d x } \] Input:

integrate((d*x+c)^n*(b*x^2+a)^3/x^7,x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^3*(d*x + c)^n/x^7, x)
 

Giac [F]

\[ \int \frac {(c+d x)^n \left (a+b x^2\right )^3}{x^7} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{3} {\left (d x + c\right )}^{n}}{x^{7}} \,d x } \] Input:

integrate((d*x+c)^n*(b*x^2+a)^3/x^7,x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^3*(d*x + c)^n/x^7, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^3}{x^7} \, dx=\int \frac {{\left (b\,x^2+a\right )}^3\,{\left (c+d\,x\right )}^n}{x^7} \,d x \] Input:

int(((a + b*x^2)^3*(c + d*x)^n)/x^7,x)
 

Output:

int(((a + b*x^2)^3*(c + d*x)^n)/x^7, x)
 

Reduce [F]

\[ \int \frac {(c+d x)^n \left (a+b x^2\right )^3}{x^7} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^n*(b*x^2+a)^3/x^7,x)
 

Output:

( - 120*(c + d*x)**n*a**3*c**5*n - 24*(c + d*x)**n*a**3*c**4*d*n**2*x - 6* 
(c + d*x)**n*a**3*c**3*d**2*n**3*x**2 + 30*(c + d*x)**n*a**3*c**3*d**2*n** 
2*x**2 - 2*(c + d*x)**n*a**3*c**2*d**3*n**4*x**3 + 18*(c + d*x)**n*a**3*c* 
*2*d**3*n**3*x**3 - 40*(c + d*x)**n*a**3*c**2*d**3*n**2*x**3 - (c + d*x)** 
n*a**3*c*d**4*n**5*x**4 + 12*(c + d*x)**n*a**3*c*d**4*n**4*x**4 - 47*(c + 
d*x)**n*a**3*c*d**4*n**3*x**4 + 60*(c + d*x)**n*a**3*c*d**4*n**2*x**4 - (c 
 + d*x)**n*a**3*d**5*n**6*x**5 + 14*(c + d*x)**n*a**3*d**5*n**5*x**5 - 71* 
(c + d*x)**n*a**3*d**5*n**4*x**5 + 154*(c + d*x)**n*a**3*d**5*n**3*x**5 - 
120*(c + d*x)**n*a**3*d**5*n**2*x**5 - 540*(c + d*x)**n*a**2*b*c**5*n*x**2 
 - 180*(c + d*x)**n*a**2*b*c**4*d*n**2*x**3 - 90*(c + d*x)**n*a**2*b*c**3* 
d**2*n**3*x**4 + 270*(c + d*x)**n*a**2*b*c**3*d**2*n**2*x**4 - 90*(c + d*x 
)**n*a**2*b*c**2*d**3*n**4*x**5 + 450*(c + d*x)**n*a**2*b*c**2*d**3*n**3*x 
**5 - 540*(c + d*x)**n*a**2*b*c**2*d**3*n**2*x**5 - 1080*(c + d*x)**n*a*b* 
*2*c**5*n*x**4 - 1080*(c + d*x)**n*a*b**2*c**4*d*n**2*x**5 + 720*(c + d*x) 
**n*b**3*c**5*x**6 + int((c + d*x)**n/(c*x + d*x**2),x)*a**3*d**6*n**7*x** 
6 - 15*int((c + d*x)**n/(c*x + d*x**2),x)*a**3*d**6*n**6*x**6 + 85*int((c 
+ d*x)**n/(c*x + d*x**2),x)*a**3*d**6*n**5*x**6 - 225*int((c + d*x)**n/(c* 
x + d*x**2),x)*a**3*d**6*n**4*x**6 + 274*int((c + d*x)**n/(c*x + d*x**2),x 
)*a**3*d**6*n**3*x**6 - 120*int((c + d*x)**n/(c*x + d*x**2),x)*a**3*d**6*n 
**2*x**6 + 90*int((c + d*x)**n/(c*x + d*x**2),x)*a**2*b*c**2*d**4*n**5*...