Integrand size = 23, antiderivative size = 95 \[ \int (f x)^m \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {b d n (f x)^{1+m}}{f (1+m)^2}-\frac {b e n (f x)^{3+m}}{f^3 (3+m)^2}+\frac {d (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}+\frac {e (f x)^{3+m} \left (a+b \log \left (c x^n\right )\right )}{f^3 (3+m)} \]
-b*d*n*(f*x)^(1+m)/f/(1+m)^2-b*e*n*(f*x)^(3+m)/f^3/(3+m)^2+d*(f*x)^(1+m)*( a+b*ln(c*x^n))/f/(1+m)+e*(f*x)^(3+m)*(a+b*ln(c*x^n))/f^3/(3+m)
Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.72 \[ \int (f x)^m \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=x (f x)^m \left (-\frac {b d n}{(1+m)^2}-\frac {b e n x^2}{(3+m)^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{1+m}+\frac {e x^2 \left (a+b \log \left (c x^n\right )\right )}{3+m}\right ) \]
x*(f*x)^m*(-((b*d*n)/(1 + m)^2) - (b*e*n*x^2)/(3 + m)^2 + (d*(a + b*Log[c* x^n]))/(1 + m) + (e*x^2*(a + b*Log[c*x^n]))/(3 + m))
Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2792, 244, 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.
\(\displaystyle \int \left (d+e x^2\right ) (f x)^m \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 2792 |
\(\displaystyle -b n \int (f x)^m \left (\frac {e x^2}{m+3}+\frac {d}{m+1}\right )dx+\frac {d (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac {e (f x)^{m+3} \left (a+b \log \left (c x^n\right )\right )}{f^3 (m+3)}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle -b n \int \left (\frac {d (f x)^m}{m+1}+\frac {e (f x)^{m+2}}{f^2 (m+3)}\right )dx+\frac {d (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac {e (f x)^{m+3} \left (a+b \log \left (c x^n\right )\right )}{f^3 (m+3)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {d (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac {e (f x)^{m+3} \left (a+b \log \left (c x^n\right )\right )}{f^3 (m+3)}-b n \left (\frac {d (f x)^{m+1}}{f (m+1)^2}+\frac {e (f x)^{m+3}}{f^3 (m+3)^2}\right )\) |
-(b*n*((d*(f*x)^(1 + m))/(f*(1 + m)^2) + (e*(f*x)^(3 + m))/(f^3*(3 + m)^2) )) + (d*(f*x)^(1 + m)*(a + b*Log[c*x^n]))/(f*(1 + m)) + (e*(f*x)^(3 + m)*( a + b*Log[c*x^n]))/(f^3*(3 + m))
3.4.20.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x] }, Simp[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2] ) || InverseFunctionFreeQ[u, x]] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x ] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(342\) vs. \(2(95)=190\).
Time = 0.90 (sec) , antiderivative size = 343, normalized size of antiderivative = 3.61
method | result | size |
parallelrisch | \(-\frac {-3 x^{3} \left (f x \right )^{m} a e -9 x \left (f x \right )^{m} a d -x \left (f x \right )^{m} a d \,m^{3}-7 x \left (f x \right )^{m} a d \,m^{2}-15 x \left (f x \right )^{m} a d m +9 x \left (f x \right )^{m} b d n -9 x \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b d -x^{3} \left (f x \right )^{m} a e \,m^{3}-5 x^{3} \left (f x \right )^{m} a e \,m^{2}-7 x^{3} \left (f x \right )^{m} a e m +x^{3} \left (f x \right )^{m} b e n -3 x^{3} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b e -x \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b d \,m^{3}-7 x \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b d \,m^{2}+x \left (f x \right )^{m} b d \,m^{2} n -15 x \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b d m +6 x \left (f x \right )^{m} b d m n -x^{3} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b e \,m^{3}-5 x^{3} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b e \,m^{2}+x^{3} \left (f x \right )^{m} b e \,m^{2} n -7 x^{3} \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b e m +2 x^{3} \left (f x \right )^{m} b e m n}{\left (3+m \right )^{2} \left (1+m \right )^{2}}\) | \(343\) |
risch | \(\text {Expression too large to display}\) | \(1114\) |
-(-3*x^3*(f*x)^m*a*e-9*x*(f*x)^m*a*d-x*(f*x)^m*a*d*m^3-7*x*(f*x)^m*a*d*m^2 -15*x*(f*x)^m*a*d*m+9*x*(f*x)^m*b*d*n-9*x*(f*x)^m*ln(c*x^n)*b*d-x^3*(f*x)^ m*a*e*m^3-5*x^3*(f*x)^m*a*e*m^2-7*x^3*(f*x)^m*a*e*m+x^3*(f*x)^m*b*e*n-3*x^ 3*(f*x)^m*ln(c*x^n)*b*e-x*(f*x)^m*ln(c*x^n)*b*d*m^3-7*x*(f*x)^m*ln(c*x^n)* b*d*m^2+x*(f*x)^m*b*d*m^2*n-15*x*(f*x)^m*ln(c*x^n)*b*d*m+6*x*(f*x)^m*b*d*m *n-x^3*(f*x)^m*ln(c*x^n)*b*e*m^3-5*x^3*(f*x)^m*ln(c*x^n)*b*e*m^2+x^3*(f*x) ^m*b*e*m^2*n-7*x^3*(f*x)^m*ln(c*x^n)*b*e*m+2*x^3*(f*x)^m*b*e*m*n)/(3+m)^2/ (1+m)^2
Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (95) = 190\).
Time = 0.29 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.47 \[ \int (f x)^m \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {{\left ({\left (a e m^{3} + 5 \, a e m^{2} + 7 \, a e m + 3 \, a e - {\left (b e m^{2} + 2 \, b e m + b e\right )} n\right )} x^{3} + {\left (a d m^{3} + 7 \, a d m^{2} + 15 \, a d m + 9 \, a d - {\left (b d m^{2} + 6 \, b d m + 9 \, b d\right )} n\right )} x + {\left ({\left (b e m^{3} + 5 \, b e m^{2} + 7 \, b e m + 3 \, b e\right )} x^{3} + {\left (b d m^{3} + 7 \, b d m^{2} + 15 \, b d m + 9 \, b d\right )} x\right )} \log \left (c\right ) + {\left ({\left (b e m^{3} + 5 \, b e m^{2} + 7 \, b e m + 3 \, b e\right )} n x^{3} + {\left (b d m^{3} + 7 \, b d m^{2} + 15 \, b d m + 9 \, b d\right )} n x\right )} \log \left (x\right )\right )} e^{\left (m \log \left (f\right ) + m \log \left (x\right )\right )}}{m^{4} + 8 \, m^{3} + 22 \, m^{2} + 24 \, m + 9} \]
((a*e*m^3 + 5*a*e*m^2 + 7*a*e*m + 3*a*e - (b*e*m^2 + 2*b*e*m + b*e)*n)*x^3 + (a*d*m^3 + 7*a*d*m^2 + 15*a*d*m + 9*a*d - (b*d*m^2 + 6*b*d*m + 9*b*d)*n )*x + ((b*e*m^3 + 5*b*e*m^2 + 7*b*e*m + 3*b*e)*x^3 + (b*d*m^3 + 7*b*d*m^2 + 15*b*d*m + 9*b*d)*x)*log(c) + ((b*e*m^3 + 5*b*e*m^2 + 7*b*e*m + 3*b*e)*n *x^3 + (b*d*m^3 + 7*b*d*m^2 + 15*b*d*m + 9*b*d)*n*x)*log(x))*e^(m*log(f) + m*log(x))/(m^4 + 8*m^3 + 22*m^2 + 24*m + 9)
Leaf count of result is larger than twice the leaf count of optimal. 920 vs. \(2 (87) = 174\).
Time = 2.71 (sec) , antiderivative size = 920, normalized size of antiderivative = 9.68 \[ \int (f x)^m \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\begin {cases} \frac {- \frac {a d}{2 x^{2}} + a e \log {\left (x \right )} + b d \left (- \frac {n}{4 x^{2}} - \frac {\log {\left (c x^{n} \right )}}{2 x^{2}}\right ) - b e \left (\begin {cases} - \log {\left (c \right )} \log {\left (x \right )} & \text {for}\: n = 0 \\- \frac {\log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases}\right )}{f^{3}} & \text {for}\: m = -3 \\\frac {\frac {a d \log {\left (c x^{n} \right )}}{n} + \frac {a e x^{2}}{2} + \frac {b d \log {\left (c x^{n} \right )}^{2}}{2 n} - \frac {b e n x^{2}}{4} + \frac {b e x^{2} \log {\left (c x^{n} \right )}}{2}}{f} & \text {for}\: m = -1 \\\frac {a d m^{3} x \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {7 a d m^{2} x \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {15 a d m x \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {9 a d x \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {a e m^{3} x^{3} \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {5 a e m^{2} x^{3} \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {7 a e m x^{3} \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {3 a e x^{3} \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {b d m^{3} x \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} - \frac {b d m^{2} n x \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {7 b d m^{2} x \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} - \frac {6 b d m n x \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {15 b d m x \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} - \frac {9 b d n x \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {9 b d x \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {b e m^{3} x^{3} \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} - \frac {b e m^{2} n x^{3} \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {5 b e m^{2} x^{3} \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} - \frac {2 b e m n x^{3} \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {7 b e m x^{3} \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} - \frac {b e n x^{3} \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {3 b e x^{3} \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} & \text {otherwise} \end {cases} \]
Piecewise(((-a*d/(2*x**2) + a*e*log(x) + b*d*(-n/(4*x**2) - log(c*x**n)/(2 *x**2)) - b*e*Piecewise((-log(c)*log(x), Eq(n, 0)), (-log(c*x**n)**2/(2*n) , True)))/f**3, Eq(m, -3)), ((a*d*log(c*x**n)/n + a*e*x**2/2 + b*d*log(c*x **n)**2/(2*n) - b*e*n*x**2/4 + b*e*x**2*log(c*x**n)/2)/f, Eq(m, -1)), (a*d *m**3*x*(f*x)**m/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) + 7*a*d*m**2*x*(f*x) **m/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) + 15*a*d*m*x*(f*x)**m/(m**4 + 8*m **3 + 22*m**2 + 24*m + 9) + 9*a*d*x*(f*x)**m/(m**4 + 8*m**3 + 22*m**2 + 24 *m + 9) + a*e*m**3*x**3*(f*x)**m/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) + 5* a*e*m**2*x**3*(f*x)**m/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) + 7*a*e*m*x**3 *(f*x)**m/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) + 3*a*e*x**3*(f*x)**m/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) + b*d*m**3*x*(f*x)**m*log(c*x**n)/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) - b*d*m**2*n*x*(f*x)**m/(m**4 + 8*m**3 + 22*m **2 + 24*m + 9) + 7*b*d*m**2*x*(f*x)**m*log(c*x**n)/(m**4 + 8*m**3 + 22*m* *2 + 24*m + 9) - 6*b*d*m*n*x*(f*x)**m/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) + 15*b*d*m*x*(f*x)**m*log(c*x**n)/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) - 9*b*d*n*x*(f*x)**m/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) + 9*b*d*x*(f*x)**m *log(c*x**n)/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) + b*e*m**3*x**3*(f*x)**m *log(c*x**n)/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) - b*e*m**2*n*x**3*(f*x)* *m/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) + 5*b*e*m**2*x**3*(f*x)**m*log(c*x **n)/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) - 2*b*e*m*n*x**3*(f*x)**m/(m*...
Time = 0.19 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.25 \[ \int (f x)^m \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b e f^{m} x^{3} x^{m} \log \left (c x^{n}\right )}{m + 3} + \frac {a e f^{m} x^{3} x^{m}}{m + 3} - \frac {b e f^{m} n x^{3} x^{m}}{{\left (m + 3\right )}^{2}} - \frac {b d f^{m} n x x^{m}}{{\left (m + 1\right )}^{2}} + \frac {\left (f x\right )^{m + 1} b d \log \left (c x^{n}\right )}{f {\left (m + 1\right )}} + \frac {\left (f x\right )^{m + 1} a d}{f {\left (m + 1\right )}} \]
b*e*f^m*x^3*x^m*log(c*x^n)/(m + 3) + a*e*f^m*x^3*x^m/(m + 3) - b*e*f^m*n*x ^3*x^m/(m + 3)^2 - b*d*f^m*n*x*x^m/(m + 1)^2 + (f*x)^(m + 1)*b*d*log(c*x^n )/(f*(m + 1)) + (f*x)^(m + 1)*a*d/(f*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (95) = 190\).
Time = 0.34 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.46 \[ \int (f x)^m \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b e f^{2} f^{m} x^{3} x^{m} \log \left (c\right )}{f^{2} m + 3 \, f^{2}} + \frac {b e f^{m} m n x^{3} x^{m} \log \left (x\right )}{m^{2} + 6 \, m + 9} + \frac {a e f^{2} f^{m} x^{3} x^{m}}{f^{2} m + 3 \, f^{2}} + \frac {3 \, b e f^{m} n x^{3} x^{m} \log \left (x\right )}{m^{2} + 6 \, m + 9} - \frac {b e f^{m} n x^{3} x^{m}}{m^{2} + 6 \, m + 9} + \frac {b d f^{m} m n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} + \frac {b d f^{m} n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} - \frac {b d f^{m} n x x^{m}}{m^{2} + 2 \, m + 1} + \frac {\left (f x\right )^{m} b d x \log \left (c\right )}{m + 1} + \frac {\left (f x\right )^{m} a d x}{m + 1} \]
b*e*f^2*f^m*x^3*x^m*log(c)/(f^2*m + 3*f^2) + b*e*f^m*m*n*x^3*x^m*log(x)/(m ^2 + 6*m + 9) + a*e*f^2*f^m*x^3*x^m/(f^2*m + 3*f^2) + 3*b*e*f^m*n*x^3*x^m* log(x)/(m^2 + 6*m + 9) - b*e*f^m*n*x^3*x^m/(m^2 + 6*m + 9) + b*d*f^m*m*n*x *x^m*log(x)/(m^2 + 2*m + 1) + b*d*f^m*n*x*x^m*log(x)/(m^2 + 2*m + 1) - b*d *f^m*n*x*x^m/(m^2 + 2*m + 1) + (f*x)^m*b*d*x*log(c)/(m + 1) + (f*x)^m*a*d* x/(m + 1)
Timed out. \[ \int (f x)^m \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int {\left (f\,x\right )}^m\,\left (e\,x^2+d\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]