Integrand size = 20, antiderivative size = 246 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^3}{x} \, dx=-\frac {b c \left (b^2 c^4+3 a b c^2 d^2+3 a^2 d^4\right ) (c+d x)^{1+n}}{d^6 (1+n)}+\frac {b \left (5 b^2 c^4+9 a b c^2 d^2+3 a^2 d^4\right ) (c+d x)^{2+n}}{d^6 (2+n)}-\frac {b^2 c \left (10 b c^2+9 a d^2\right ) (c+d x)^{3+n}}{d^6 (3+n)}+\frac {b^2 \left (10 b c^2+3 a d^2\right ) (c+d x)^{4+n}}{d^6 (4+n)}-\frac {5 b^3 c (c+d x)^{5+n}}{d^6 (5+n)}+\frac {b^3 (c+d x)^{6+n}}{d^6 (6+n)}-\frac {a^3 (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {d x}{c}\right )}{c (1+n)} \] Output:
-b*c*(3*a^2*d^4+3*a*b*c^2*d^2+b^2*c^4)*(d*x+c)^(1+n)/d^6/(1+n)+b*(3*a^2*d^ 4+9*a*b*c^2*d^2+5*b^2*c^4)*(d*x+c)^(2+n)/d^6/(2+n)-b^2*c*(9*a*d^2+10*b*c^2 )*(d*x+c)^(3+n)/d^6/(3+n)+b^2*(3*a*d^2+10*b*c^2)*(d*x+c)^(4+n)/d^6/(4+n)-5 *b^3*c*(d*x+c)^(5+n)/d^6/(5+n)+b^3*(d*x+c)^(6+n)/d^6/(6+n)-a^3*(d*x+c)^(1+ n)*hypergeom([1, 1+n],[2+n],1+d*x/c)/c/(1+n)
Time = 0.11 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.92 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^3}{x} \, dx=(c+d x)^{1+n} \left (-\frac {b c \left (b^2 c^4+3 a b c^2 d^2+3 a^2 d^4\right )}{d^6 (1+n)}+\frac {b \left (5 b^2 c^4+9 a b c^2 d^2+3 a^2 d^4\right ) (c+d x)}{d^6 (2+n)}-\frac {b^2 c \left (10 b c^2+9 a d^2\right ) (c+d x)^2}{d^6 (3+n)}+\frac {b^2 \left (10 b c^2+3 a d^2\right ) (c+d x)^3}{d^6 (4+n)}-\frac {5 b^3 c (c+d x)^4}{d^6 (5+n)}+\frac {b^3 (c+d x)^5}{d^6 (6+n)}-\frac {a^3 \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c+d x}{c}\right )}{c+c n}\right ) \] Input:
Integrate[((c + d*x)^n*(a + b*x^2)^3)/x,x]
Output:
(c + d*x)^(1 + n)*(-((b*c*(b^2*c^4 + 3*a*b*c^2*d^2 + 3*a^2*d^4))/(d^6*(1 + n))) + (b*(5*b^2*c^4 + 9*a*b*c^2*d^2 + 3*a^2*d^4)*(c + d*x))/(d^6*(2 + n) ) - (b^2*c*(10*b*c^2 + 9*a*d^2)*(c + d*x)^2)/(d^6*(3 + n)) + (b^2*(10*b*c^ 2 + 3*a*d^2)*(c + d*x)^3)/(d^6*(4 + n)) - (5*b^3*c*(c + d*x)^4)/(d^6*(5 + n)) + (b^3*(c + d*x)^5)/(d^6*(6 + n)) - (a^3*Hypergeometric2F1[1, 1 + n, 2 + n, (c + d*x)/c])/(c + c*n))
Time = 0.41 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {522, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^3 (c+d x)^n}{x} \, dx\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \int \left (\frac {a^3 (c+d x)^n}{x}-\frac {b c \left (3 a^2 d^4+3 a b c^2 d^2+b^2 c^4\right ) (c+d x)^n}{d^5}+\frac {b \left (3 a^2 d^4+9 a b c^2 d^2+5 b^2 c^4\right ) (c+d x)^{n+1}}{d^5}-\frac {b^2 c \left (9 a d^2+10 b c^2\right ) (c+d x)^{n+2}}{d^5}+\frac {b^2 \left (3 a d^2+10 b c^2\right ) (c+d x)^{n+3}}{d^5}-\frac {5 b^3 c (c+d x)^{n+4}}{d^5}+\frac {b^3 (c+d x)^{n+5}}{d^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^3 (c+d x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {d x}{c}+1\right )}{c (n+1)}-\frac {b c \left (3 a^2 d^4+3 a b c^2 d^2+b^2 c^4\right ) (c+d x)^{n+1}}{d^6 (n+1)}+\frac {b \left (3 a^2 d^4+9 a b c^2 d^2+5 b^2 c^4\right ) (c+d x)^{n+2}}{d^6 (n+2)}-\frac {b^2 c \left (9 a d^2+10 b c^2\right ) (c+d x)^{n+3}}{d^6 (n+3)}+\frac {b^2 \left (3 a d^2+10 b c^2\right ) (c+d x)^{n+4}}{d^6 (n+4)}-\frac {5 b^3 c (c+d x)^{n+5}}{d^6 (n+5)}+\frac {b^3 (c+d x)^{n+6}}{d^6 (n+6)}\) |
Input:
Int[((c + d*x)^n*(a + b*x^2)^3)/x,x]
Output:
-((b*c*(b^2*c^4 + 3*a*b*c^2*d^2 + 3*a^2*d^4)*(c + d*x)^(1 + n))/(d^6*(1 + n))) + (b*(5*b^2*c^4 + 9*a*b*c^2*d^2 + 3*a^2*d^4)*(c + d*x)^(2 + n))/(d^6* (2 + n)) - (b^2*c*(10*b*c^2 + 9*a*d^2)*(c + d*x)^(3 + n))/(d^6*(3 + n)) + (b^2*(10*b*c^2 + 3*a*d^2)*(c + d*x)^(4 + n))/(d^6*(4 + n)) - (5*b^3*c*(c + d*x)^(5 + n))/(d^6*(5 + n)) + (b^3*(c + d*x)^(6 + n))/(d^6*(6 + n)) - (a^ 3*(c + d*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (d*x)/c])/(c*(1 + n))
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
\[\int \frac {\left (d x +c \right )^{n} \left (b \,x^{2}+a \right )^{3}}{x}d x\]
Input:
int((d*x+c)^n*(b*x^2+a)^3/x,x)
Output:
int((d*x+c)^n*(b*x^2+a)^3/x,x)
\[ \int \frac {(c+d x)^n \left (a+b x^2\right )^3}{x} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{3} {\left (d x + c\right )}^{n}}{x} \,d x } \] Input:
integrate((d*x+c)^n*(b*x^2+a)^3/x,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, x)
Leaf count of result is larger than twice the leaf count of optimal. 4007 vs. \(2 (226) = 452\).
Time = 4.80 (sec) , antiderivative size = 5622, normalized size of antiderivative = 22.85 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^3}{x} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)**n*(b*x**2+a)**3/x,x)
Output:
-a**3*d**(n + 1)*n*(c/d + x)**(n + 1)*lerchphi(1 + d*x/c, 1, n + 1)*gamma( n + 1)/(c*gamma(n + 2)) - a**3*d**(n + 1)*(c/d + x)**(n + 1)*lerchphi(1 + d*x/c, 1, n + 1)*gamma(n + 1)/(c*gamma(n + 2)) + 3*a**2*b*Piecewise((c**n* x**2/2, Eq(d, 0)), (c*log(c/d + x)/(c*d**2 + d**3*x) + c/(c*d**2 + d**3*x) + d*x*log(c/d + x)/(c*d**2 + d**3*x), Eq(n, -2)), (-c*log(c/d + x)/d**2 + x/d, Eq(n, -1)), (-c**2*(c + d*x)**n/(d**2*n**2 + 3*d**2*n + 2*d**2) + c* d*n*x*(c + d*x)**n/(d**2*n**2 + 3*d**2*n + 2*d**2) + d**2*n*x**2*(c + d*x) **n/(d**2*n**2 + 3*d**2*n + 2*d**2) + d**2*x**2*(c + d*x)**n/(d**2*n**2 + 3*d**2*n + 2*d**2), True)) + 3*a*b**2*Piecewise((c**n*x**4/4, Eq(d, 0)), ( 6*c**3*log(c/d + x)/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d** 7*x**3) + 11*c**3/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7* x**3) + 18*c**2*d*x*log(c/d + x)/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6 *x**2 + 6*d**7*x**3) + 27*c**2*d*x/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d* *6*x**2 + 6*d**7*x**3) + 18*c*d**2*x**2*log(c/d + x)/(6*c**3*d**4 + 18*c** 2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) + 18*c*d**2*x**2/(6*c**3*d**4 + 1 8*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) + 6*d**3*x**3*log(c/d + x)/( 6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3), Eq(n, -4)), (-6*c**3*log(c/d + x)/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 9*c**3/(2 *c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 12*c**2*d*x*log(c/d + x)/(2*c**2* d**4 + 4*c*d**5*x + 2*d**6*x**2) - 12*c**2*d*x/(2*c**2*d**4 + 4*c*d**5*...
\[ \int \frac {(c+d x)^n \left (a+b x^2\right )^3}{x} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{3} {\left (d x + c\right )}^{n}}{x} \,d x } \] Input:
integrate((d*x+c)^n*(b*x^2+a)^3/x,x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^3*(d*x + c)^n/x, x)
\[ \int \frac {(c+d x)^n \left (a+b x^2\right )^3}{x} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{3} {\left (d x + c\right )}^{n}}{x} \,d x } \] Input:
integrate((d*x+c)^n*(b*x^2+a)^3/x,x, algorithm="giac")
Output:
integrate((b*x^2 + a)^3*(d*x + c)^n/x, x)
Timed out. \[ \int \frac {(c+d x)^n \left (a+b x^2\right )^3}{x} \, dx=\int \frac {{\left (b\,x^2+a\right )}^3\,{\left (c+d\,x\right )}^n}{x} \,d x \] Input:
int(((a + b*x^2)^3*(c + d*x)^n)/x,x)
Output:
int(((a + b*x^2)^3*(c + d*x)^n)/x, x)
\[ \int \frac {(c+d x)^n \left (a+b x^2\right )^3}{x} \, dx=\text {too large to display} \] Input:
int((d*x+c)^n*(b*x^2+a)^3/x,x)
Output:
((c + d*x)**n*a**3*d**6*n**6 + 21*(c + d*x)**n*a**3*d**6*n**5 + 175*(c + d *x)**n*a**3*d**6*n**4 + 735*(c + d*x)**n*a**3*d**6*n**3 + 1624*(c + d*x)** n*a**3*d**6*n**2 + 1764*(c + d*x)**n*a**3*d**6*n + 720*(c + d*x)**n*a**3*d **6 - 3*(c + d*x)**n*a**2*b*c**2*d**4*n**5 - 54*(c + d*x)**n*a**2*b*c**2*d **4*n**4 - 357*(c + d*x)**n*a**2*b*c**2*d**4*n**3 - 1026*(c + d*x)**n*a**2 *b*c**2*d**4*n**2 - 1080*(c + d*x)**n*a**2*b*c**2*d**4*n + 3*(c + d*x)**n* a**2*b*c*d**5*n**6*x + 54*(c + d*x)**n*a**2*b*c*d**5*n**5*x + 357*(c + d*x )**n*a**2*b*c*d**5*n**4*x + 1026*(c + d*x)**n*a**2*b*c*d**5*n**3*x + 1080* (c + d*x)**n*a**2*b*c*d**5*n**2*x + 3*(c + d*x)**n*a**2*b*d**6*n**6*x**2 + 57*(c + d*x)**n*a**2*b*d**6*n**5*x**2 + 411*(c + d*x)**n*a**2*b*d**6*n**4 *x**2 + 1383*(c + d*x)**n*a**2*b*d**6*n**3*x**2 + 2106*(c + d*x)**n*a**2*b *d**6*n**2*x**2 + 1080*(c + d*x)**n*a**2*b*d**6*n*x**2 - 18*(c + d*x)**n*a *b**2*c**4*d**2*n**3 - 198*(c + d*x)**n*a*b**2*c**4*d**2*n**2 - 540*(c + d *x)**n*a*b**2*c**4*d**2*n + 18*(c + d*x)**n*a*b**2*c**3*d**3*n**4*x + 198* (c + d*x)**n*a*b**2*c**3*d**3*n**3*x + 540*(c + d*x)**n*a*b**2*c**3*d**3*n **2*x - 9*(c + d*x)**n*a*b**2*c**2*d**4*n**5*x**2 - 108*(c + d*x)**n*a*b** 2*c**2*d**4*n**4*x**2 - 369*(c + d*x)**n*a*b**2*c**2*d**4*n**3*x**2 - 270* (c + d*x)**n*a*b**2*c**2*d**4*n**2*x**2 + 3*(c + d*x)**n*a*b**2*c*d**5*n** 6*x**3 + 42*(c + d*x)**n*a*b**2*c*d**5*n**5*x**3 + 195*(c + d*x)**n*a*b**2 *c*d**5*n**4*x**3 + 336*(c + d*x)**n*a*b**2*c*d**5*n**3*x**3 + 180*(c +...