3.3.10 \(\int (d+e x)^m \log (c (a+\frac {b}{x^2})^p) \, dx\) [210]

3.3.10.1 Optimal result

Integrand size = 20, antiderivative size = 257 \[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx=\frac {\sqrt {-a} p (d+e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{e \left (\sqrt {-a} d-\sqrt {b} e\right ) (1+m) (2+m)}+\frac {\sqrt {-a} p (d+e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{e \left (\sqrt {-a} d+\sqrt {b} e\right ) (1+m) (2+m)}-\frac {2 p (d+e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,1+\frac {e x}{d}\right )}{d e \left (2+3 m+m^2\right )}+\frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e (1+m)} \]

output

-2*p*(e*x+d)^(2+m)*hypergeom([1, 2+m],[3+m],1+e*x/d)/d/e/(m^2+3*m+2)+(e*x+ 
d)^(1+m)*ln(c*(a+b/x^2)^p)/e/(1+m)+p*(e*x+d)^(2+m)*hypergeom([1, 2+m],[3+m 
],(e*x+d)*(-a)^(1/2)/(d*(-a)^(1/2)-e*b^(1/2)))*(-a)^(1/2)/e/(1+m)/(2+m)/(d 
*(-a)^(1/2)-e*b^(1/2))+p*(e*x+d)^(2+m)*hypergeom([1, 2+m],[3+m],(e*x+d)*(- 
a)^(1/2)/(d*(-a)^(1/2)+e*b^(1/2)))*(-a)^(1/2)/e/(1+m)/(2+m)/(d*(-a)^(1/2)+ 
e*b^(1/2))
 

3.3.10.2 Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.82 \[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx=\frac {(d+e x)^{1+m} \left (\frac {p (d+e x) \left (d \left (a d-\sqrt {-a} \sqrt {b} e\right ) \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )+d \left (a d+\sqrt {-a} \sqrt {b} e\right ) \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )-2 \left (a d^2+b e^2\right ) \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,1+\frac {e x}{d}\right )\right )}{d \left (a d^2+b e^2\right ) (2+m)}+\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )\right )}{e (1+m)} \]

input

Integrate[(d + e*x)^m*Log[c*(a + b/x^2)^p],x]
 

output

((d + e*x)^(1 + m)*((p*(d + e*x)*(d*(a*d - Sqrt[-a]*Sqrt[b]*e)*Hypergeomet 
ric2F1[1, 2 + m, 3 + m, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d - Sqrt[b]*e)] + d 
*(a*d + Sqrt[-a]*Sqrt[b]*e)*Hypergeometric2F1[1, 2 + m, 3 + m, (Sqrt[-a]*( 
d + e*x))/(Sqrt[-a]*d + Sqrt[b]*e)] - 2*(a*d^2 + b*e^2)*Hypergeometric2F1[ 
1, 2 + m, 3 + m, 1 + (e*x)/d]))/(d*(a*d^2 + b*e^2)*(2 + m)) + Log[c*(a + b 
/x^2)^p]))/(e*(1 + m))
 

3.3.10.3 Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2913, 1894, 615, 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.

\(\displaystyle \int (d+e x)^m \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx\)

\(\Big \downarrow \) 2913

\(\displaystyle \frac {2 b p \int \frac {(d+e x)^{m+1}}{\left (a+\frac {b}{x^2}\right ) x^3}dx}{e (m+1)}+\frac {(d+e x)^{m+1} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e (m+1)}\)

\(\Big \downarrow \) 1894

\(\displaystyle \frac {2 b p \int \frac {(d+e x)^{m+1}}{x \left (a x^2+b\right )}dx}{e (m+1)}+\frac {(d+e x)^{m+1} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e (m+1)}\)

\(\Big \downarrow \) 615

\(\displaystyle \frac {2 b p \int \left (\frac {(d+e x)^{m+1}}{b x}-\frac {a x (d+e x)^{m+1}}{b \left (a x^2+b\right )}\right )dx}{e (m+1)}+\frac {(d+e x)^{m+1} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e (m+1)}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(d+e x)^{m+1} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e (m+1)}+\frac {2 b p \left (\frac {\sqrt {-a} (d+e x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{2 b (m+2) \left (\sqrt {-a} d-\sqrt {b} e\right )}+\frac {\sqrt {-a} (d+e x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{2 b (m+2) \left (\sqrt {-a} d+\sqrt {b} e\right )}-\frac {(d+e x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,\frac {e x}{d}+1\right )}{b d (m+2)}\right )}{e (m+1)}\)

input

Int[(d + e*x)^m*Log[c*(a + b/x^2)^p],x]
 

output

(2*b*p*((Sqrt[-a]*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (Sq 
rt[-a]*(d + e*x))/(Sqrt[-a]*d - Sqrt[b]*e)])/(2*b*(Sqrt[-a]*d - Sqrt[b]*e) 
*(2 + m)) + (Sqrt[-a]*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, 
 (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d + Sqrt[b]*e)])/(2*b*(Sqrt[-a]*d + Sqrt[b 
]*e)*(2 + m)) - ((d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, 1 + 
(e*x)/d])/(b*d*(2 + m))))/(e*(1 + m)) + ((d + e*x)^(1 + m)*Log[c*(a + b/x^ 
2)^p])/(e*(1 + m))
 

3.3.10.3.1 Defintions of rubi rules used

rule 615

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] && ILtQ[p, 0]
 

rule 1894

Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_.)) 
^(q_.), x_Symbol] :> Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + a*x^(2*n))^p, x] 
/; FreeQ[{a, c, d, e, m, n, q}, x] && EqQ[mn2, -2*n] && IntegerQ[p]
 

rule 2009

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

rule 2913

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]
 

3.3.10.4 Maple [F]

\[\int \left (e x +d \right )^{m} \ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )d x\]

input

int((e*x+d)^m*ln(c*(a+b/x^2)^p),x)
 

output

int((e*x+d)^m*ln(c*(a+b/x^2)^p),x)
 

3.3.10.5 Fricas [F]

\[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right ) \,d x } \]

input

integrate((e*x+d)^m*log(c*(a+b/x^2)^p),x, algorithm="fricas")
 

output

integral((e*x + d)^m*log(c*((a*x^2 + b)/x^2)^p), x)
 

3.3.10.6 Sympy [F(-1)]

Timed out. \[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx=\text {Timed out} \]

input

integrate((e*x+d)**m*ln(c*(a+b/x**2)**p),x)
 

output

Timed out
 

3.3.10.7 Maxima [F]

\[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right ) \,d x } \]

input

integrate((e*x+d)^m*log(c*(a+b/x^2)^p),x, algorithm="maxima")
 

output

(e*p*x + d*p)*(e*x + d)^m*log(a*x^2 + b)/(e*(m + 1)) - integrate((2*a*d*p* 
x - (e*(m + 1)*log(c) - 2*e*p)*a*x^2 - b*e*(m + 1)*log(c) + 2*(a*e*(m + 1) 
*x^2 + b*e*(m + 1))*log(x^p))*(e*x + d)^m/(a*e*(m + 1)*x^2 + b*e*(m + 1)), 
 x)
 

3.3.10.8 Giac [F]

\[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right ) \,d x } \]

input

integrate((e*x+d)^m*log(c*(a+b/x^2)^p),x, algorithm="giac")
 

output

integrate((e*x + d)^m*log((a + b/x^2)^p*c), x)
 

3.3.10.9 Mupad [F(-1)]

Timed out. \[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx=\int \ln \left (c\,{\left (a+\frac {b}{x^2}\right )}^p\right )\,{\left (d+e\,x\right )}^m \,d x \]

input

int(log(c*(a + b/x^2)^p)*(d + e*x)^m,x)
 

output

int(log(c*(a + b/x^2)^p)*(d + e*x)^m, x)