3.59 \(\int (f x)^m \log (c (d+\frac {e}{x^2})^p) \, dx\)

Optimal. Leaf size=82 \[ \frac {(f x)^{m+1} \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f (m+1)}-\frac {2 e f p (f x)^{m-1} \, _2F_1\left (1,\frac {1-m}{2};\frac {3-m}{2};-\frac {e}{d x^2}\right )}{d \left (1-m^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2455, 16, 339, 364} \[ \frac {(f x)^{m+1} \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f (m+1)}-\frac {2 e f p (f x)^{m-1} \, _2F_1\left (1,\frac {1-m}{2};\frac {3-m}{2};-\frac {e}{d x^2}\right )}{d \left (1-m^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(f*x)^m*Log[c*(d + e/x^2)^p],x]

[Out]

(-2*e*f*p*(f*x)^(-1 + m)*Hypergeometric2F1[1, (1 - m)/2, (3 - m)/2, -(e/(d*x^2))])/(d*(1 - m^2)) + ((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 339

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Dist[((c*x)^(m + 1)*(1/x)^(m + 1))/c, Subst
[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, m, p}, x] && ILtQ[n, 0] &&  !RationalQ[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+\frac {e}{x^2}\right )^p\right ) \, dx &=\frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f (1+m)}+\frac {(2 e p) \int \frac {(f x)^{1+m}}{\left (d+\frac {e}{x^2}\right ) x^3} \, dx}{f (1+m)}\\ &=\frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f (1+m)}+\frac {\left (2 e f^2 p\right ) \int \frac {(f x)^{-2+m}}{d+\frac {e}{x^2}} \, dx}{1+m}\\ &=\frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f (1+m)}-\frac {\left (2 e f p \left (\frac {1}{x}\right )^{-1+m} (f x)^{-1+m}\right ) \operatorname {Subst}\left (\int \frac {x^{-m}}{d+e x^2} \, dx,x,\frac {1}{x}\right )}{1+m}\\ &=-\frac {2 e f p (f x)^{-1+m} \, _2F_1\left (1,\frac {1-m}{2};\frac {3-m}{2};-\frac {e}{d x^2}\right )}{d \left (1-m^2\right )}+\frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f (1+m)}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 76, normalized size = 0.93 \[ \frac {(f x)^m \left (d (m-1) x^2 \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )+2 e p \, _2F_1\left (1,\frac {1}{2}-\frac {m}{2};\frac {3}{2}-\frac {m}{2};-\frac {e}{d x^2}\right )\right )}{d (m-1) (m+1) x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 1.36, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (f x\right )^{m} \log \left (c \left (\frac {d x^{2} + e}{x^{2}}\right )^{p}\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F(-1)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \ln \left (c \left (d +\frac {e}{x^{2}}\right )^{p}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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)

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sympy [A]  time = 77.64, size = 348, normalized size = 4.24 \[ 2 e p \left (\begin {cases} - \frac {0^{m} \sqrt {- \frac {1}{d e}} \log {\left (- e \sqrt {- \frac {1}{d e}} + x \right )}}{2} + \frac {0^{m} \sqrt {- \frac {1}{d e}} \log {\left (e \sqrt {- \frac {1}{d e}} + x \right )}}{2} & \text {for}\: \left (f = 0 \wedge m \neq -1\right ) \vee f = 0 \\\frac {f f^{m} m x^{m} \Phi \left (\frac {e e^{i \pi }}{d x^{2}}, 1, \frac {1}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {1}{2} - \frac {m}{2}\right )}{4 d f m x \Gamma \left (\frac {3}{2} - \frac {m}{2}\right ) + 4 d f x \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )} - \frac {f f^{m} x^{m} \Phi \left (\frac {e e^{i \pi }}{d x^{2}}, 1, \frac {1}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {1}{2} - \frac {m}{2}\right )}{4 d f m x \Gamma \left (\frac {3}{2} - \frac {m}{2}\right ) + 4 d f x \Gamma \left (\frac {3}{2} - \frac {m}{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 e^{i \pi }}{d x^{2}}\right )}{2} & \text {for}\: \left |{x}\right | < 1 \\- \log {\relax (d )} \log {\left (\frac {1}{x} \right )} + \frac {\operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x^{2}}\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 e^{i \pi }}{d x^{2}}\right )}{2} & \text {otherwise} \end {cases}}{2 e f} - \frac {\log {\left (f x \right )} \log {\left (d + \frac {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 + \frac {e}{x^{2}}\right )^{p} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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