3.160 \(\int (f x)^m \log (c (d+e x^2)^p) \, dx\)

Optimal. Leaf size=81 \[ \frac {(f x)^{m+1} \log \left (c \left (d+e x^2\right )^p\right )}{f (m+1)}-\frac {2 e p (f x)^{m+3} \, _2F_1\left (1,\frac {m+3}{2};\frac {m+5}{2};-\frac {e x^2}{d}\right )}{d f^3 (m+1) (m+3)} \]

[Out]

-2*e*p*(f*x)^(3+m)*hypergeom([1, 3/2+1/2*m],[5/2+1/2*m],-e*x^2/d)/d/f^3/(1+m)/(3+m)+(f*x)^(1+m)*ln(c*(e*x^2+d)
^p)/f/(1+m)

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2455, 16, 364} \[ \frac {(f x)^{m+1} \log \left (c \left (d+e x^2\right )^p\right )}{f (m+1)}-\frac {2 e p (f x)^{m+3} \, _2F_1\left (1,\frac {m+3}{2};\frac {m+5}{2};-\frac {e x^2}{d}\right )}{d f^3 (m+1) (m+3)} \]

Antiderivative was successfully verified.

[In]

Int[(f*x)^m*Log[c*(d + e*x^2)^p],x]

[Out]

(-2*e*p*(f*x)^(3 + m)*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, -((e*x^2)/d)])/(d*f^3*(1 + m)*(3 + m)) + ((f*
x)^(1 + m)*Log[c*(d + e*x^2)^p])/(f*(1 + m))

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 2455

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[((f*x)^(m
+ 1)*(a + b*Log[c*(d + e*x^n)^p]))/(f*(m + 1)), x] - Dist[(b*e*n*p)/(f*(m + 1)), Int[(x^(n - 1)*(f*x)^(m + 1))
/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int (f x)^m \log \left (c \left (d+e x^2\right )^p\right ) \, dx &=\frac {(f x)^{1+m} \log \left (c \left (d+e x^2\right )^p\right )}{f (1+m)}-\frac {(2 e p) \int \frac {x (f x)^{1+m}}{d+e x^2} \, dx}{f (1+m)}\\ &=\frac {(f x)^{1+m} \log \left (c \left (d+e x^2\right )^p\right )}{f (1+m)}-\frac {(2 e p) \int \frac {(f x)^{2+m}}{d+e x^2} \, dx}{f^2 (1+m)}\\ &=-\frac {2 e p (f x)^{3+m} \, _2F_1\left (1,\frac {3+m}{2};\frac {5+m}{2};-\frac {e x^2}{d}\right )}{d f^3 (1+m) (3+m)}+\frac {(f x)^{1+m} \log \left (c \left (d+e x^2\right )^p\right )}{f (1+m)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 70, normalized size = 0.86 \[ \frac {x (f x)^m \left (d (m+3) \log \left (c \left (d+e x^2\right )^p\right )-2 e p x^2 \, _2F_1\left (1,\frac {m+3}{2};\frac {m+5}{2};-\frac {e x^2}{d}\right )\right )}{d (m+1) (m+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(f*x)^m*(-2*e*p*x^2*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, -((e*x^2)/d)] + d*(3 + m)*Log[c*(d + e*x^2)^
p]))/(d*(1 + m)*(3 + m))

________________________________________________________________________________________

fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (f x\right )^{m} \log \left ({\left (e x^{2} + d\right )}^{p} c\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((f*x)^m*log((e*x^2 + d)^p*c), x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (f x\right )^{m} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((f*x)^m*log((e*x^2 + d)^p*c), x)

________________________________________________________________________________________

maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^m*ln(c*(e*x^2+d)^p),x)

[Out]

int((f*x)^m*ln(c*(e*x^2+d)^p),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {f^{m} p x x^{m} \log \left (e x^{2} + d\right )}{m + 1} + \int \frac {{\left (d f^{m} {\left (m + 1\right )} \log \relax (c) + {\left (e f^{m} {\left (m + 1\right )} \log \relax (c) - 2 \, e f^{m} p\right )} x^{2}\right )} x^{m}}{e {\left (m + 1\right )} x^{2} + d {\left (m + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,{\left (f\,x\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(d + e*x^2)^p)*(f*x)^m,x)

[Out]

int(log(c*(d + e*x^2)^p)*(f*x)^m, x)

________________________________________________________________________________________

sympy [A]  time = 94.77, size = 359, normalized size = 4.43 \[ - 2 e p \left (\begin {cases} \frac {0^{m} \sqrt {- \frac {d}{e^{3}}} \log {\left (- e \sqrt {- \frac {d}{e^{3}}} + x \right )}}{2} - \frac {0^{m} \sqrt {- \frac {d}{e^{3}}} \log {\left (e \sqrt {- \frac {d}{e^{3}}} + x \right )}}{2} + \frac {0^{m} x}{e} & \text {for}\: \left (f = 0 \wedge m \neq -1\right ) \vee f = 0 \\\frac {f f^{m} m x^{3} x^{m} \Phi \left (\frac {e x^{2} e^{i \pi }}{d}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{4 d f m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 4 d f \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {3 f f^{m} x^{3} x^{m} \Phi \left (\frac {e x^{2} e^{i \pi }}{d}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{4 d f m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 4 d f \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} & \text {for}\: m > -\infty \wedge m < \infty \wedge m \neq -1 \\- \frac {\begin {cases} \log {\relax (d )} \log {\relax (x )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {for}\: \left |{x}\right | < 1 \\- \log {\relax (d )} \log {\left (\frac {1}{x} \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\relax (d )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\relax (d )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {otherwise} \end {cases}}{2 e f} + \frac {\log {\left (f x \right )} \log {\left (d + e x^{2} \right )}}{2 e f} & \text {otherwise} \end {cases}\right ) + \left (\begin {cases} 0^{m} x & \text {for}\: f = 0 \\\frac {\begin {cases} \frac {\left (f x\right )^{m + 1}}{m + 1} & \text {for}\: m \neq -1 \\\log {\left (f x \right )} & \text {otherwise} \end {cases}}{f} & \text {otherwise} \end {cases}\right ) \log {\left (c \left (d + e x^{2}\right )^{p} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)**m*ln(c*(e*x**2+d)**p),x)

[Out]

-2*e*p*Piecewise((0**m*sqrt(-d/e**3)*log(-e*sqrt(-d/e**3) + x)/2 - 0**m*sqrt(-d/e**3)*log(e*sqrt(-d/e**3) + x)
/2 + 0**m*x/e, Eq(f, 0) | (Eq(f, 0) & Ne(m, -1))), (f*f**m*m*x**3*x**m*lerchphi(e*x**2*exp_polar(I*pi)/d, 1, m
/2 + 3/2)*gamma(m/2 + 3/2)/(4*d*f*m*gamma(m/2 + 5/2) + 4*d*f*gamma(m/2 + 5/2)) + 3*f*f**m*x**3*x**m*lerchphi(e
*x**2*exp_polar(I*pi)/d, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(4*d*f*m*gamma(m/2 + 5/2) + 4*d*f*gamma(m/2 + 5/2)), (
m > -oo) & (m < oo) & Ne(m, -1)), (-Piecewise((log(d)*log(x) - polylog(2, e*x**2*exp_polar(I*pi)/d)/2, Abs(x)
< 1), (-log(d)*log(1/x) - polylog(2, e*x**2*exp_polar(I*pi)/d)/2, 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0,
0), ()), x)*log(d) + meijerg(((1, 1), ()), ((), (0, 0)), x)*log(d) - polylog(2, e*x**2*exp_polar(I*pi)/d)/2, T
rue))/(2*e*f) + log(f*x)*log(d + e*x**2)/(2*e*f), True)) + Piecewise((0**m*x, Eq(f, 0)), (Piecewise(((f*x)**(m
+ 1)/(m + 1), Ne(m, -1)), (log(f*x), True))/f, True))*log(c*(d + e*x**2)**p)

________________________________________________________________________________________