\(\int \frac {(c+d x)^n (a+b x^2)^3}{x^3} \, dx\) [208]

Optimal result

Integrand size = 20, antiderivative size = 221 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^3}{x^3} \, dx=-\frac {b^2 c \left (b c^2+3 a d^2\right ) (c+d x)^{1+n}}{d^4 (1+n)}-\frac {a^3 (c+d x)^{1+n}}{2 c x^2}+\frac {3 a^2 b (c+d x)^{1+n}}{d n x}+\frac {3 b^2 \left (b c^2+a d^2\right ) (c+d x)^{2+n}}{d^4 (2+n)}-\frac {3 b^3 c (c+d x)^{3+n}}{d^4 (3+n)}+\frac {b^3 (c+d x)^{4+n}}{d^4 (4+n)}+\frac {a^2 \left (6 b c^2-a d^2 (1-n) n\right ) (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {d x}{c}\right )}{2 c^3 n (1+n)} \] Output:

-b^2*c*(3*a*d^2+b*c^2)*(d*x+c)^(1+n)/d^4/(1+n)-1/2*a^3*(d*x+c)^(1+n)/c/x^2 
+3*a^2*b*(d*x+c)^(1+n)/d/n/x+3*b^2*(a*d^2+b*c^2)*(d*x+c)^(2+n)/d^4/(2+n)-3 
*b^3*c*(d*x+c)^(3+n)/d^4/(3+n)+b^3*(d*x+c)^(4+n)/d^4/(4+n)+1/2*a^2*(6*b*c^ 
2-a*d^2*(1-n)*n)*(d*x+c)^(1+n)*hypergeom([2, 1+n],[2+n],1+d*x/c)/c^3/n/(1+ 
n)
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.76 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^3}{x^3} \, dx=(c+d x)^{1+n} \left (-\frac {b^2 c \left (b c^2+3 a d^2\right )}{d^4 (1+n)}+\frac {3 b^2 \left (b c^2+a d^2\right ) (c+d x)}{d^4 (2+n)}-\frac {3 b^3 c (c+d x)^2}{d^4 (3+n)}+\frac {b^3 (c+d x)^3}{d^4 (4+n)}-\frac {3 a^2 b \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c+d x}{c}\right )}{c+c n}-\frac {a^3 d^2 \operatorname {Hypergeometric2F1}\left (3,1+n,2+n,1+\frac {d x}{c}\right )}{c^3 (1+n)}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {520, 2124, 2123, 2009}

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.

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

\(\Big \downarrow \) 520

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

\(\Big \downarrow \) 2124

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

\(\Big \downarrow \) 2123

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a^3 (c+d x)^{n+1}}{2 c x^2}-\frac {-\frac {a^3 d (1-n) (c+d x)^{n+1}}{c x}-\frac {-\frac {a^2 (c+d x)^{n+1} \left (6 b c^2-a d^2 (1-n) n\right ) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {d x}{c}+1\right )}{c (n+1)}+\frac {6 b^2 c^2 \left (a d^2+b c^2\right ) (c+d x)^{n+2}}{d^4 (n+2)}-\frac {2 b^2 c^3 \left (3 a d^2+b c^2\right ) (c+d x)^{n+1}}{d^4 (n+1)}-\frac {6 b^3 c^3 (c+d x)^{n+3}}{d^4 (n+3)}+\frac {2 b^3 c^2 (c+d x)^{n+4}}{d^4 (n+4)}}{c}}{2 c}\)

Input:

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

Output:

-1/2*(a^3*(c + d*x)^(1 + n))/(c*x^2) - (-((a^3*d*(1 - n)*(c + d*x)^(1 + n) 
)/(c*x)) - ((-2*b^2*c^3*(b*c^2 + 3*a*d^2)*(c + d*x)^(1 + n))/(d^4*(1 + n)) 
 + (6*b^2*c^2*(b*c^2 + a*d^2)*(c + d*x)^(2 + n))/(d^4*(2 + n)) - (6*b^3*c^ 
3*(c + d*x)^(3 + n))/(d^4*(3 + n)) + (2*b^3*c^2*(c + d*x)^(4 + n))/(d^4*(4 
 + n)) - (a^2*(6*b*c^2 - a*d^2*(1 - n)*n)*(c + d*x)^(1 + n)*Hypergeometric 
2F1[1, 1 + n, 2 + n, 1 + (d*x)/c])/(c*(1 + n)))/c)/(2*c)
 

Defintions of rubi rules used

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2123

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] 
:> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c 
, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1321 vs. \(2 (192) = 384\).

Time = 4.58 (sec) , antiderivative size = 2419, normalized size of antiderivative = 10.95 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^3}{x^3} \, dx=\text {Too large to display} \] Input:

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

Output:

a**3*c**2*d**(n + 3)*n**3*(c/d + x)**(n + 1)*lerchphi(1 + d*x/c, 1, n + 1) 
*gamma(n + 1)/(-2*c**5*gamma(n + 2) - 4*c**4*d*x*gamma(n + 2) + 2*c**3*d** 
2*(c/d + x)**2*gamma(n + 2)) - a**3*c**2*d**(n + 3)*n*(c/d + x)**(n + 1)*l 
erchphi(1 + d*x/c, 1, n + 1)*gamma(n + 1)/(-2*c**5*gamma(n + 2) - 4*c**4*d 
*x*gamma(n + 2) + 2*c**3*d**2*(c/d + x)**2*gamma(n + 2)) - a**3*c**2*d**(n 
 + 3)*n*(c/d + x)**(n + 1)*gamma(n + 1)/(-2*c**5*gamma(n + 2) - 4*c**4*d*x 
*gamma(n + 2) + 2*c**3*d**2*(c/d + x)**2*gamma(n + 2)) - a**3*c**2*d**(n + 
 3)*(c/d + x)**(n + 1)*gamma(n + 1)/(-2*c**5*gamma(n + 2) - 4*c**4*d*x*gam 
ma(n + 2) + 2*c**3*d**2*(c/d + x)**2*gamma(n + 2)) + 2*a**3*c*d*d**(n + 3) 
*n**3*x*(c/d + x)**(n + 1)*lerchphi(1 + d*x/c, 1, n + 1)*gamma(n + 1)/(-2* 
c**5*gamma(n + 2) - 4*c**4*d*x*gamma(n + 2) + 2*c**3*d**2*(c/d + x)**2*gam 
ma(n + 2)) - a**3*c*d*d**(n + 3)*n**2*x*(c/d + x)**(n + 1)*gamma(n + 1)/(- 
2*c**5*gamma(n + 2) - 4*c**4*d*x*gamma(n + 2) + 2*c**3*d**2*(c/d + x)**2*g 
amma(n + 2)) - 2*a**3*c*d*d**(n + 3)*n*x*(c/d + x)**(n + 1)*lerchphi(1 + d 
*x/c, 1, n + 1)*gamma(n + 1)/(-2*c**5*gamma(n + 2) - 4*c**4*d*x*gamma(n + 
2) + 2*c**3*d**2*(c/d + x)**2*gamma(n + 2)) + a**3*c*d*d**(n + 3)*x*(c/d + 
 x)**(n + 1)*gamma(n + 1)/(-2*c**5*gamma(n + 2) - 4*c**4*d*x*gamma(n + 2) 
+ 2*c**3*d**2*(c/d + x)**2*gamma(n + 2)) - a**3*d**2*d**(n + 3)*n**3*(c/d 
+ x)**2*(c/d + x)**(n + 1)*lerchphi(1 + d*x/c, 1, n + 1)*gamma(n + 1)/(-2* 
c**5*gamma(n + 2) - 4*c**4*d*x*gamma(n + 2) + 2*c**3*d**2*(c/d + x)**2*...
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - (c + d*x)**n*a**3*c*d**4*n**5 - 10*(c + d*x)**n*a**3*c*d**4*n**4 - 35* 
(c + d*x)**n*a**3*c*d**4*n**3 - 50*(c + d*x)**n*a**3*c*d**4*n**2 - 24*(c + 
 d*x)**n*a**3*c*d**4*n - (c + d*x)**n*a**3*d**5*n**6*x - 10*(c + d*x)**n*a 
**3*d**5*n**5*x - 35*(c + d*x)**n*a**3*d**5*n**4*x - 50*(c + d*x)**n*a**3* 
d**5*n**3*x - 24*(c + d*x)**n*a**3*d**5*n**2*x + 6*(c + d*x)**n*a**2*b*c*d 
**4*n**4*x**2 + 60*(c + d*x)**n*a**2*b*c*d**4*n**3*x**2 + 210*(c + d*x)**n 
*a**2*b*c*d**4*n**2*x**2 + 300*(c + d*x)**n*a**2*b*c*d**4*n*x**2 + 144*(c 
+ d*x)**n*a**2*b*c*d**4*x**2 - 6*(c + d*x)**n*a*b**2*c**3*d**2*n**3*x**2 - 
 42*(c + d*x)**n*a*b**2*c**3*d**2*n**2*x**2 - 72*(c + d*x)**n*a*b**2*c**3* 
d**2*n*x**2 + 6*(c + d*x)**n*a*b**2*c**2*d**3*n**4*x**3 + 42*(c + d*x)**n* 
a*b**2*c**2*d**3*n**3*x**3 + 72*(c + d*x)**n*a*b**2*c**2*d**3*n**2*x**3 + 
6*(c + d*x)**n*a*b**2*c*d**4*n**4*x**4 + 48*(c + d*x)**n*a*b**2*c*d**4*n** 
3*x**4 + 114*(c + d*x)**n*a*b**2*c*d**4*n**2*x**4 + 72*(c + d*x)**n*a*b**2 
*c*d**4*n*x**4 - 12*(c + d*x)**n*b**3*c**5*n*x**2 + 12*(c + d*x)**n*b**3*c 
**4*d*n**2*x**3 - 6*(c + d*x)**n*b**3*c**3*d**2*n**3*x**4 - 6*(c + d*x)**n 
*b**3*c**3*d**2*n**2*x**4 + 2*(c + d*x)**n*b**3*c**2*d**3*n**4*x**5 + 6*(c 
 + d*x)**n*b**3*c**2*d**3*n**3*x**5 + 4*(c + d*x)**n*b**3*c**2*d**3*n**2*x 
**5 + 2*(c + d*x)**n*b**3*c*d**4*n**4*x**6 + 12*(c + d*x)**n*b**3*c*d**4*n 
**3*x**6 + 22*(c + d*x)**n*b**3*c*d**4*n**2*x**6 + 12*(c + d*x)**n*b**3*c* 
d**4*n*x**6 + int((c + d*x)**n/(c*x + d*x**2),x)*a**3*d**6*n**7*x**2 + ...