Integrand size = 18, antiderivative size = 99 \[ \int (d+e x) \log \left (c \left (a+b x^2\right )^p\right ) \, dx=-2 d p x-\frac {1}{2} e p x^2+\frac {2 \sqrt {a} d p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {\left (b d^2-a e^2\right ) p \log \left (a+b x^2\right )}{2 b e}+\frac {(d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right )}{2 e} \] Output:
-2*d*p*x-1/2*e*p*x^2+2*a^(1/2)*d*p*arctan(b^(1/2)*x/a^(1/2))/b^(1/2)-1/2*( -a*e^2+b*d^2)*p*ln(b*x^2+a)/b/e+1/2*(e*x+d)^2*ln(c*(b*x^2+a)^p)/e
Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.84 \[ \int (d+e x) \log \left (c \left (a+b x^2\right )^p\right ) \, dx=-2 d p x+\frac {2 \sqrt {a} d p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+d x \log \left (c \left (a+b x^2\right )^p\right )+\frac {1}{2} e \left (-p x^2+\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b}\right ) \] Input:
Integrate[(d + e*x)*Log[c*(a + b*x^2)^p],x]
Output:
-2*d*p*x + (2*Sqrt[a]*d*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b] + d*x*Log[c *(a + b*x^2)^p] + (e*(-(p*x^2) + ((a + b*x^2)*Log[c*(a + b*x^2)^p])/b))/2
Time = 0.48 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2913, 525, 523, 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.
\(\displaystyle \int (d+e x) \log \left (c \left (a+b x^2\right )^p\right ) \, dx\) |
\(\Big \downarrow \) 2913 |
\(\displaystyle \frac {(d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right )}{2 e}-\frac {b p \int \frac {x (d+e x)^2}{b x^2+a}dx}{e}\) |
\(\Big \downarrow \) 525 |
\(\displaystyle \frac {(d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right )}{2 e}-\frac {b p \left (\frac {\int \frac {x \left (b d^2+2 b e x d-a e^2\right )}{b x^2+a}dx}{b}+\frac {e^2 x^2}{2 b}\right )}{e}\) |
\(\Big \downarrow \) 523 |
\(\displaystyle \frac {(d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right )}{2 e}-\frac {b p \left (\frac {\int \left (2 d e-\frac {2 a d e-\left (b d^2-a e^2\right ) x}{b x^2+a}\right )dx}{b}+\frac {e^2 x^2}{2 b}\right )}{e}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right )}{2 e}-\frac {b p \left (\frac {-\frac {2 \sqrt {a} d e \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\left (b d^2-a e^2\right ) \log \left (a+b x^2\right )}{2 b}+2 d e x}{b}+\frac {e^2 x^2}{2 b}\right )}{e}\) |
Input:
Int[(d + e*x)*Log[c*(a + b*x^2)^p],x]
Output:
-((b*p*((e^2*x^2)/(2*b) + (2*d*e*x - (2*Sqrt[a]*d*e*ArcTan[(Sqrt[b]*x)/Sqr t[a]])/Sqrt[b] + ((b*d^2 - a*e^2)*Log[a + b*x^2])/(2*b))/b))/e) + ((d + e* x)^2*Log[c*(a + b*x^2)^p])/(2*e)
Int[((x_)^(m_.)*((c_) + (d_.)*(x_)))/((a_) + (b_.)*(x_)^2), x_Symbol] :> In t[ExpandIntegrand[x^m*((c + d*x)/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d} , x] && IntegerQ[m]
Int[((x_)^(m_.)*((c_) + (d_.)*(x_))^(n_))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[d^n*(x^(m + n - 1)/(b*(m + n - 1))), x] + Simp[1/b Int[x^m*(Expan dToSum[b*(c + d*x)^n - b*d^n*x^n - a*d^n*x^(n - 2), x]/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 1] && IGtQ[m, -2] && NeQ[m + n - 1, 0 ]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.) + (g_. )*(x_))^(r_.), x_Symbol] :> Simp[(f + g*x)^(r + 1)*((a + b*Log[c*(d + e*x^n )^p])/(g*(r + 1))), x] - Simp[b*e*n*(p/(g*(r + 1))) Int[x^(n - 1)*((f + g *x)^(r + 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, r}, x ] && (IGtQ[r, 0] || RationalQ[n]) && NeQ[r, -1]
Time = 0.82 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.79
method | result | size |
default | \(d x \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )-2 x d p +\frac {2 d p a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}+\frac {e \left (\ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) \left (b \,x^{2}+a \right )-p \left (b \,x^{2}+a \right )\right )}{2 b}\) | \(78\) |
parts | \(\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) e \,x^{2}}{2}+d x \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )-p b \left (\frac {\frac {1}{2} e \,x^{2}+2 d x}{b}-\frac {a \left (\frac {e \ln \left (b \,x^{2}+a \right )}{2 b}+\frac {2 d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{b}\right )\) | \(93\) |
risch | \(\left (\frac {1}{2} e \,x^{2}+d x \right ) \ln \left (\left (b \,x^{2}+a \right )^{p}\right )+\frac {i x \pi d \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}}{2}-\frac {i \pi e \,x^{2} {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}}{4}+\frac {i \operatorname {csgn}\left (i c \right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} x^{2} e \pi }{4}-\frac {i \pi e \,x^{2} \operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{4}-\frac {i x \pi d {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}}{2}+\frac {i {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) x^{2} e \pi }{4}-\frac {i x \pi d \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}+\frac {i x \pi d {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\frac {\ln \left (c \right ) e \,x^{2}}{2}-\frac {e p \,x^{2}}{2}+\frac {p \ln \left (-\sqrt {-a b}\, x +a \right ) d \sqrt {-a b}}{b}+\frac {p \ln \left (-\sqrt {-a b}\, x +a \right ) e a}{2 b}-\frac {p \ln \left (\sqrt {-a b}\, x +a \right ) d \sqrt {-a b}}{b}+\frac {p \ln \left (\sqrt {-a b}\, x +a \right ) e a}{2 b}+x \ln \left (c \right ) d -2 x d p\) | \(387\) |
Input:
int((e*x+d)*ln(c*(b*x^2+a)^p),x,method=_RETURNVERBOSE)
Output:
d*x*ln(c*(b*x^2+a)^p)-2*x*d*p+2*d*p*a/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))+ 1/2*e/b*(ln(c*(b*x^2+a)^p)*(b*x^2+a)-p*(b*x^2+a))
Time = 0.09 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.00 \[ \int (d+e x) \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\left [-\frac {b e p x^{2} - 2 \, b d p \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 4 \, b d p x - {\left (b e p x^{2} + 2 \, b d p x + a e p\right )} \log \left (b x^{2} + a\right ) - {\left (b e x^{2} + 2 \, b d x\right )} \log \left (c\right )}{2 \, b}, -\frac {b e p x^{2} - 4 \, b d p \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 4 \, b d p x - {\left (b e p x^{2} + 2 \, b d p x + a e p\right )} \log \left (b x^{2} + a\right ) - {\left (b e x^{2} + 2 \, b d x\right )} \log \left (c\right )}{2 \, b}\right ] \] Input:
integrate((e*x+d)*log(c*(b*x^2+a)^p),x, algorithm="fricas")
Output:
[-1/2*(b*e*p*x^2 - 2*b*d*p*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/( b*x^2 + a)) + 4*b*d*p*x - (b*e*p*x^2 + 2*b*d*p*x + a*e*p)*log(b*x^2 + a) - (b*e*x^2 + 2*b*d*x)*log(c))/b, -1/2*(b*e*p*x^2 - 4*b*d*p*sqrt(a/b)*arctan (b*x*sqrt(a/b)/a) + 4*b*d*p*x - (b*e*p*x^2 + 2*b*d*p*x + a*e*p)*log(b*x^2 + a) - (b*e*x^2 + 2*b*d*x)*log(c))/b]
Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (92) = 184\).
Time = 4.05 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.01 \[ \int (d+e x) \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\begin {cases} \left (d x + \frac {e x^{2}}{2}\right ) \log {\left (0^{p} c \right )} & \text {for}\: a = 0 \wedge b = 0 \\\left (d x + \frac {e x^{2}}{2}\right ) \log {\left (a^{p} c \right )} & \text {for}\: b = 0 \\- 2 d p x + d x \log {\left (c \left (b x^{2}\right )^{p} \right )} - \frac {e p x^{2}}{2} + \frac {e x^{2} \log {\left (c \left (b x^{2}\right )^{p} \right )}}{2} & \text {for}\: a = 0 \\\frac {2 a d p \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{b \sqrt {- \frac {a}{b}}} - \frac {a d \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{b \sqrt {- \frac {a}{b}}} + \frac {a e \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{2 b} - 2 d p x + d x \log {\left (c \left (a + b x^{2}\right )^{p} \right )} - \frac {e p x^{2}}{2} + \frac {e x^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{2} & \text {otherwise} \end {cases} \] Input:
integrate((e*x+d)*ln(c*(b*x**2+a)**p),x)
Output:
Piecewise(((d*x + e*x**2/2)*log(0**p*c), Eq(a, 0) & Eq(b, 0)), ((d*x + e*x **2/2)*log(a**p*c), Eq(b, 0)), (-2*d*p*x + d*x*log(c*(b*x**2)**p) - e*p*x* *2/2 + e*x**2*log(c*(b*x**2)**p)/2, Eq(a, 0)), (2*a*d*p*log(x - sqrt(-a/b) )/(b*sqrt(-a/b)) - a*d*log(c*(a + b*x**2)**p)/(b*sqrt(-a/b)) + a*e*log(c*( a + b*x**2)**p)/(2*b) - 2*d*p*x + d*x*log(c*(a + b*x**2)**p) - e*p*x**2/2 + e*x**2*log(c*(a + b*x**2)**p)/2, True))
Time = 0.10 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.81 \[ \int (d+e x) \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {1}{2} \, {\left (\frac {4 \, a d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {a e \log \left (b x^{2} + a\right )}{b^{2}} - \frac {e x^{2} + 4 \, d x}{b}\right )} b p + \frac {1}{2} \, {\left (e x^{2} + 2 \, d x\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) \] Input:
integrate((e*x+d)*log(c*(b*x^2+a)^p),x, algorithm="maxima")
Output:
1/2*(4*a*d*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b) + a*e*log(b*x^2 + a)/b^2 - (e*x^2 + 4*d*x)/b)*b*p + 1/2*(e*x^2 + 2*d*x)*log((b*x^2 + a)^p*c)
Time = 0.13 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.86 \[ \int (d+e x) \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {2 \, a d p \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}} - \frac {1}{2} \, {\left (e p - e \log \left (c\right )\right )} x^{2} + \frac {a e p \log \left (b x^{2} + a\right )}{2 \, b} - {\left (2 \, d p - d \log \left (c\right )\right )} x + \frac {1}{2} \, {\left (e p x^{2} + 2 \, d p x\right )} \log \left (b x^{2} + a\right ) \] Input:
integrate((e*x+d)*log(c*(b*x^2+a)^p),x, algorithm="giac")
Output:
2*a*d*p*arctan(b*x/sqrt(a*b))/sqrt(a*b) - 1/2*(e*p - e*log(c))*x^2 + 1/2*a *e*p*log(b*x^2 + a)/b - (2*d*p - d*log(c))*x + 1/2*(e*p*x^2 + 2*d*p*x)*log (b*x^2 + a)
Time = 26.66 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.82 \[ \int (d+e x) \log \left (c \left (a+b x^2\right )^p\right ) \, dx=d\,x\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )-\frac {e\,p\,x^2}{2}-2\,d\,p\,x+\frac {e\,x^2\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{2}+\frac {a\,e\,p\,\ln \left (b\,x^2+a\right )}{2\,b}+\frac {2\,\sqrt {a}\,d\,p\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{\sqrt {b}} \] Input:
int(log(c*(a + b*x^2)^p)*(d + e*x),x)
Output:
d*x*log(c*(a + b*x^2)^p) - (e*p*x^2)/2 - 2*d*p*x + (e*x^2*log(c*(a + b*x^2 )^p))/2 + (a*e*p*log(a + b*x^2))/(2*b) + (2*a^(1/2)*d*p*atan((b^(1/2)*x)/a ^(1/2)))/b^(1/2)
Time = 0.18 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.91 \[ \int (d+e x) \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {4 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) d p +\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) a e +2 \,\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) b d x +\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) b e \,x^{2}-4 b d p x -b e p \,x^{2}}{2 b} \] Input:
int((e*x+d)*log(c*(b*x^2+a)^p),x)
Output:
(4*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*d*p + log((a + b*x**2)**p *c)*a*e + 2*log((a + b*x**2)**p*c)*b*d*x + log((a + b*x**2)**p*c)*b*e*x**2 - 4*b*d*p*x - b*e*p*x**2)/(2*b)