Integrand size = 15, antiderivative size = 70 \[ \int (c+d x)^n \left (a+b x^2\right ) \, dx=\frac {\left (b c^2+a d^2\right ) (c+d x)^{1+n}}{d^3 (1+n)}-\frac {2 b c (c+d x)^{2+n}}{d^3 (2+n)}+\frac {b (c+d x)^{3+n}}{d^3 (3+n)} \] Output:
(a*d^2+b*c^2)*(d*x+c)^(1+n)/d^3/(1+n)-2*b*c*(d*x+c)^(2+n)/d^3/(2+n)+b*(d*x +c)^(3+n)/d^3/(3+n)
Time = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.84 \[ \int (c+d x)^n \left (a+b x^2\right ) \, dx=\frac {(c+d x)^{1+n} \left (\frac {b c^2+a d^2}{1+n}-\frac {2 b c (c+d x)}{2+n}+\frac {b (c+d x)^2}{3+n}\right )}{d^3} \] Input:
Integrate[(c + d*x)^n*(a + b*x^2),x]
Output:
((c + d*x)^(1 + n)*((b*c^2 + a*d^2)/(1 + n) - (2*b*c*(c + d*x))/(2 + n) + (b*(c + d*x)^2)/(3 + n)))/d^3
Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {476, 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.
\(\displaystyle \int \left (a+b x^2\right ) (c+d x)^n \, dx\) |
\(\Big \downarrow \) 476 |
\(\displaystyle \int \left (\frac {\left (a d^2+b c^2\right ) (c+d x)^n}{d^2}-\frac {2 b c (c+d x)^{n+1}}{d^2}+\frac {b (c+d x)^{n+2}}{d^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (a d^2+b c^2\right ) (c+d x)^{n+1}}{d^3 (n+1)}-\frac {2 b c (c+d x)^{n+2}}{d^3 (n+2)}+\frac {b (c+d x)^{n+3}}{d^3 (n+3)}\) |
Input:
Int[(c + d*x)^n*(a + b*x^2),x]
Output:
((b*c^2 + a*d^2)*(c + d*x)^(1 + n))/(d^3*(1 + n)) - (2*b*c*(c + d*x)^(2 + n))/(d^3*(2 + n)) + (b*(c + d*x)^(3 + n))/(d^3*(3 + n))
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
Time = 0.26 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.43
method | result | size |
gosper | \(\frac {\left (d x +c \right )^{1+n} \left (b \,d^{2} n^{2} x^{2}+3 b \,d^{2} n \,x^{2}+a \,d^{2} n^{2}-2 b c d n x +2 b \,x^{2} d^{2}+5 a \,d^{2} n -2 b c d x +6 a \,d^{2}+2 b \,c^{2}\right )}{d^{3} \left (n^{3}+6 n^{2}+11 n +6\right )}\) | \(100\) |
orering | \(\frac {\left (d x +c \right ) \left (b \,d^{2} n^{2} x^{2}+3 b \,d^{2} n \,x^{2}+a \,d^{2} n^{2}-2 b c d n x +2 b \,x^{2} d^{2}+5 a \,d^{2} n -2 b c d x +6 a \,d^{2}+2 b \,c^{2}\right ) \left (d x +c \right )^{n}}{d^{3} \left (n^{3}+6 n^{2}+11 n +6\right )}\) | \(103\) |
risch | \(\frac {\left (b \,d^{3} n^{2} x^{3}+b c \,d^{2} n^{2} x^{2}+3 b \,d^{3} n \,x^{3}+a \,d^{3} n^{2} x +b c \,d^{2} n \,x^{2}+2 b \,d^{3} x^{3}+a c \,d^{2} n^{2}+5 a \,d^{3} n x -2 b \,c^{2} d n x +5 a c \,d^{2} n +6 a x \,d^{3}+6 a \,d^{2} c +2 b \,c^{3}\right ) \left (d x +c \right )^{n}}{\left (2+n \right ) \left (3+n \right ) \left (1+n \right ) d^{3}}\) | \(143\) |
norman | \(\frac {b \,x^{3} {\mathrm e}^{n \ln \left (d x +c \right )}}{3+n}+\frac {c \left (a \,d^{2} n^{2}+5 a \,d^{2} n +6 a \,d^{2}+2 b \,c^{2}\right ) {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{3} \left (n^{3}+6 n^{2}+11 n +6\right )}+\frac {\left (a \,d^{2} n^{2}+5 a \,d^{2} n -2 b \,c^{2} n +6 a \,d^{2}\right ) x \,{\mathrm e}^{n \ln \left (d x +c \right )}}{d^{2} \left (n^{3}+6 n^{2}+11 n +6\right )}+\frac {b c n \,x^{2} {\mathrm e}^{n \ln \left (d x +c \right )}}{d \left (n^{2}+5 n +6\right )}\) | \(165\) |
parallelrisch | \(\frac {x^{3} \left (d x +c \right )^{n} b \,d^{3} n^{2}+3 x^{3} \left (d x +c \right )^{n} b \,d^{3} n +x^{2} \left (d x +c \right )^{n} b c \,d^{2} n^{2}+2 x^{3} \left (d x +c \right )^{n} b \,d^{3}+x^{2} \left (d x +c \right )^{n} b c \,d^{2} n +x \left (d x +c \right )^{n} a \,d^{3} n^{2}+5 x \left (d x +c \right )^{n} a \,d^{3} n -2 x \left (d x +c \right )^{n} b \,c^{2} d n +\left (d x +c \right )^{n} a c \,d^{2} n^{2}+6 x \left (d x +c \right )^{n} a \,d^{3}+5 \left (d x +c \right )^{n} a c \,d^{2} n +6 \left (d x +c \right )^{n} a c \,d^{2}+2 \left (d x +c \right )^{n} b \,c^{3}}{d^{3} \left (n^{3}+6 n^{2}+11 n +6\right )}\) | \(227\) |
Input:
int((d*x+c)^n*(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/d^3*(d*x+c)^(1+n)/(n^3+6*n^2+11*n+6)*(b*d^2*n^2*x^2+3*b*d^2*n*x^2+a*d^2* n^2-2*b*c*d*n*x+2*b*d^2*x^2+5*a*d^2*n-2*b*c*d*x+6*a*d^2+2*b*c^2)
Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (70) = 140\).
Time = 0.08 (sec) , antiderivative size = 149, normalized size of antiderivative = 2.13 \[ \int (c+d x)^n \left (a+b x^2\right ) \, dx=\frac {{\left (a c d^{2} n^{2} + 5 \, a c d^{2} n + 2 \, b c^{3} + 6 \, a c d^{2} + {\left (b d^{3} n^{2} + 3 \, b d^{3} n + 2 \, b d^{3}\right )} x^{3} + {\left (b c d^{2} n^{2} + b c d^{2} n\right )} x^{2} + {\left (a d^{3} n^{2} + 6 \, a d^{3} - {\left (2 \, b c^{2} d - 5 \, a d^{3}\right )} n\right )} x\right )} {\left (d x + c\right )}^{n}}{d^{3} n^{3} + 6 \, d^{3} n^{2} + 11 \, d^{3} n + 6 \, d^{3}} \] Input:
integrate((d*x+c)^n*(b*x^2+a),x, algorithm="fricas")
Output:
(a*c*d^2*n^2 + 5*a*c*d^2*n + 2*b*c^3 + 6*a*c*d^2 + (b*d^3*n^2 + 3*b*d^3*n + 2*b*d^3)*x^3 + (b*c*d^2*n^2 + b*c*d^2*n)*x^2 + (a*d^3*n^2 + 6*a*d^3 - (2 *b*c^2*d - 5*a*d^3)*n)*x)*(d*x + c)^n/(d^3*n^3 + 6*d^3*n^2 + 11*d^3*n + 6* d^3)
Leaf count of result is larger than twice the leaf count of optimal. 952 vs. \(2 (61) = 122\).
Time = 0.49 (sec) , antiderivative size = 952, normalized size of antiderivative = 13.60 \[ \int (c+d x)^n \left (a+b x^2\right ) \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)**n*(b*x**2+a),x)
Output:
Piecewise((c**n*(a*x + b*x**3/3), Eq(d, 0)), (-a*d**2/(2*c**2*d**3 + 4*c*d **4*x + 2*d**5*x**2) + 2*b*c**2*log(c/d + x)/(2*c**2*d**3 + 4*c*d**4*x + 2 *d**5*x**2) + 3*b*c**2/(2*c**2*d**3 + 4*c*d**4*x + 2*d**5*x**2) + 4*b*c*d* x*log(c/d + x)/(2*c**2*d**3 + 4*c*d**4*x + 2*d**5*x**2) + 4*b*c*d*x/(2*c** 2*d**3 + 4*c*d**4*x + 2*d**5*x**2) + 2*b*d**2*x**2*log(c/d + x)/(2*c**2*d* *3 + 4*c*d**4*x + 2*d**5*x**2), Eq(n, -3)), (-a*d**2/(c*d**3 + d**4*x) - 2 *b*c**2*log(c/d + x)/(c*d**3 + d**4*x) - 2*b*c**2/(c*d**3 + d**4*x) - 2*b* c*d*x*log(c/d + x)/(c*d**3 + d**4*x) + b*d**2*x**2/(c*d**3 + d**4*x), Eq(n , -2)), (a*log(c/d + x)/d + b*c**2*log(c/d + x)/d**3 - b*c*x/d**2 + b*x**2 /(2*d), Eq(n, -1)), (a*c*d**2*n**2*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + 5*a*c*d**2*n*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + 6*a*c*d**2*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + a*d**3*n**2*x*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + 5*a*d**3*n*x*(c + d*x)**n/(d**3*n**3 + 6*d**3*n** 2 + 11*d**3*n + 6*d**3) + 6*a*d**3*x*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + 2*b*c**3*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) - 2*b*c**2*d*n*x*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + b*c*d**2*n**2*x**2*(c + d*x)**n/(d**3*n**3 + 6*d* *3*n**2 + 11*d**3*n + 6*d**3) + b*c*d**2*n*x**2*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + b*d**3*n**2*x**3*(c + d*x)**n/(d**3...
Time = 0.04 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.27 \[ \int (c+d x)^n \left (a+b x^2\right ) \, dx=\frac {{\left (d x + c\right )}^{n + 1} a}{d {\left (n + 1\right )}} + \frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} d^{3} x^{3} + {\left (n^{2} + n\right )} c d^{2} x^{2} - 2 \, c^{2} d n x + 2 \, c^{3}\right )} {\left (d x + c\right )}^{n} b}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d^{3}} \] Input:
integrate((d*x+c)^n*(b*x^2+a),x, algorithm="maxima")
Output:
(d*x + c)^(n + 1)*a/(d*(n + 1)) + ((n^2 + 3*n + 2)*d^3*x^3 + (n^2 + n)*c*d ^2*x^2 - 2*c^2*d*n*x + 2*c^3)*(d*x + c)^n*b/((n^3 + 6*n^2 + 11*n + 6)*d^3)
Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (70) = 140\).
Time = 0.12 (sec) , antiderivative size = 237, normalized size of antiderivative = 3.39 \[ \int (c+d x)^n \left (a+b x^2\right ) \, dx=\frac {{\left (d x + c\right )}^{n} b d^{3} n^{2} x^{3} + {\left (d x + c\right )}^{n} b c d^{2} n^{2} x^{2} + 3 \, {\left (d x + c\right )}^{n} b d^{3} n x^{3} + {\left (d x + c\right )}^{n} a d^{3} n^{2} x + {\left (d x + c\right )}^{n} b c d^{2} n x^{2} + 2 \, {\left (d x + c\right )}^{n} b d^{3} x^{3} + {\left (d x + c\right )}^{n} a c d^{2} n^{2} - 2 \, {\left (d x + c\right )}^{n} b c^{2} d n x + 5 \, {\left (d x + c\right )}^{n} a d^{3} n x + 5 \, {\left (d x + c\right )}^{n} a c d^{2} n + 6 \, {\left (d x + c\right )}^{n} a d^{3} x + 2 \, {\left (d x + c\right )}^{n} b c^{3} + 6 \, {\left (d x + c\right )}^{n} a c d^{2}}{d^{3} n^{3} + 6 \, d^{3} n^{2} + 11 \, d^{3} n + 6 \, d^{3}} \] Input:
integrate((d*x+c)^n*(b*x^2+a),x, algorithm="giac")
Output:
((d*x + c)^n*b*d^3*n^2*x^3 + (d*x + c)^n*b*c*d^2*n^2*x^2 + 3*(d*x + c)^n*b *d^3*n*x^3 + (d*x + c)^n*a*d^3*n^2*x + (d*x + c)^n*b*c*d^2*n*x^2 + 2*(d*x + c)^n*b*d^3*x^3 + (d*x + c)^n*a*c*d^2*n^2 - 2*(d*x + c)^n*b*c^2*d*n*x + 5 *(d*x + c)^n*a*d^3*n*x + 5*(d*x + c)^n*a*c*d^2*n + 6*(d*x + c)^n*a*d^3*x + 2*(d*x + c)^n*b*c^3 + 6*(d*x + c)^n*a*c*d^2)/(d^3*n^3 + 6*d^3*n^2 + 11*d^ 3*n + 6*d^3)
Time = 9.12 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.33 \[ \int (c+d x)^n \left (a+b x^2\right ) \, dx={\left (c+d\,x\right )}^n\,\left (\frac {b\,x^3\,\left (n^2+3\,n+2\right )}{n^3+6\,n^2+11\,n+6}+\frac {x\,\left (-2\,b\,c^2\,d\,n+a\,d^3\,n^2+5\,a\,d^3\,n+6\,a\,d^3\right )}{d^3\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {c\,\left (2\,b\,c^2+a\,d^2\,n^2+5\,a\,d^2\,n+6\,a\,d^2\right )}{d^3\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {b\,c\,n\,x^2\,\left (n+1\right )}{d\,\left (n^3+6\,n^2+11\,n+6\right )}\right ) \] Input:
int((a + b*x^2)*(c + d*x)^n,x)
Output:
(c + d*x)^n*((b*x^3*(3*n + n^2 + 2))/(11*n + 6*n^2 + n^3 + 6) + (x*(6*a*d^ 3 + a*d^3*n^2 + 5*a*d^3*n - 2*b*c^2*d*n))/(d^3*(11*n + 6*n^2 + n^3 + 6)) + (c*(6*a*d^2 + 2*b*c^2 + a*d^2*n^2 + 5*a*d^2*n))/(d^3*(11*n + 6*n^2 + n^3 + 6)) + (b*c*n*x^2*(n + 1))/(d*(11*n + 6*n^2 + n^3 + 6)))
Time = 0.25 (sec) , antiderivative size = 142, normalized size of antiderivative = 2.03 \[ \int (c+d x)^n \left (a+b x^2\right ) \, dx=\frac {\left (d x +c \right )^{n} \left (b \,d^{3} n^{2} x^{3}+b c \,d^{2} n^{2} x^{2}+3 b \,d^{3} n \,x^{3}+a \,d^{3} n^{2} x +b c \,d^{2} n \,x^{2}+2 b \,d^{3} x^{3}+a c \,d^{2} n^{2}+5 a \,d^{3} n x -2 b \,c^{2} d n x +5 a c \,d^{2} n +6 a \,d^{3} x +6 a c \,d^{2}+2 b \,c^{3}\right )}{d^{3} \left (n^{3}+6 n^{2}+11 n +6\right )} \] Input:
int((d*x+c)^n*(b*x^2+a),x)
Output:
((c + d*x)**n*(a*c*d**2*n**2 + 5*a*c*d**2*n + 6*a*c*d**2 + a*d**3*n**2*x + 5*a*d**3*n*x + 6*a*d**3*x + 2*b*c**3 - 2*b*c**2*d*n*x + b*c*d**2*n**2*x** 2 + b*c*d**2*n*x**2 + b*d**3*n**2*x**3 + 3*b*d**3*n*x**3 + 2*b*d**3*x**3)) /(d**3*(n**3 + 6*n**2 + 11*n + 6))