\(\int (f x)^m \log ^2(c (d+e x^2)^p) \, dx\) [159]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int (f x)^m \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {(f x)^{1+m} \log ^2\left (c \left (d+e x^2\right )^p\right )}{f (1+m)}-\frac {4 e p \text {Int}\left (\frac {(f x)^{2+m} \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2},x\right )}{f^2 (1+m)} \]

[Out]

(f*x)^(1+m)*ln(c*(e*x^2+d)^p)^2/f/(1+m)-4*e*p*Unintegrable((f*x)^(2+m)*ln(c*(e*x^2+d)^p)/(e*x^2+d),x)/f^2/(1+m
)

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (f x)^m \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int (f x)^m \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx \]

[In]

Int[(f*x)^m*Log[c*(d + e*x^2)^p]^2,x]

[Out]

((f*x)^(1 + m)*Log[c*(d + e*x^2)^p]^2)/(f*(1 + m)) - (4*e*p*Defer[Int][((f*x)^(2 + m)*Log[c*(d + e*x^2)^p])/(d
+ e*x^2), x])/(f^2*(1 + m))

Rubi steps \begin{align*} \text {integral}& = \frac {(f x)^{1+m} \log ^2\left (c \left (d+e x^2\right )^p\right )}{f (1+m)}-\frac {(4 e p) \int \frac {(f x)^{2+m} \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{f^2 (1+m)} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(466\) vs. \(2(75)=150\).

Time = 0.53 (sec) , antiderivative size = 466, normalized size of antiderivative = 23.30 \[ \int (f x)^m \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {(f x)^m \left (4 p^2 x \left (\frac {2 e x^2 \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{2},\frac {5+m}{2},-\frac {e x^2}{d}\right )}{d (3+m)}-\log \left (d+e x^2\right )\right )+(1+m) p^2 x \log ^2\left (d+e x^2\right )+\frac {4 d (1+m) p^2 \left (\frac {e x^2}{d+e x^2}\right )^{\frac {1}{2}-\frac {m}{2}} \left (-2 \, _3F_2\left (\frac {1}{2}-\frac {m}{2},\frac {1}{2}-\frac {m}{2},\frac {1}{2}-\frac {m}{2};\frac {3}{2}-\frac {m}{2},\frac {3}{2}-\frac {m}{2};\frac {d}{d+e x^2}\right )+(-1+m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {m}{2},\frac {1}{2}-\frac {m}{2},\frac {3}{2}-\frac {m}{2},\frac {d}{d+e x^2}\right ) \log \left (d+e x^2\right )\right )}{e (-1+m)^2 x}+\frac {2 p \left (2 e x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{2},\frac {5+m}{2},-\frac {e x^2}{d}\right )-d (3+m) x \log \left (d+e x^2\right )\right ) \left (p \log \left (d+e x^2\right )-\log \left (c \left (d+e x^2\right )^p\right )\right )}{d (3+m)}-\frac {2 m p \left (-2 e x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{2},\frac {5+m}{2},-\frac {e x^2}{d}\right )+d (3+m) x \log \left (d+e x^2\right )\right ) \left (p \log \left (d+e x^2\right )-\log \left (c \left (d+e x^2\right )^p\right )\right )}{d (3+m)}+x \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^2+m x \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^2\right )}{(1+m)^2} \]

[In]

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

[Out]

((f*x)^m*(4*p^2*x*((2*e*x^2*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, -((e*x^2)/d)])/(d*(3 + m)) - Log[d + e*
x^2]) + (1 + m)*p^2*x*Log[d + e*x^2]^2 + (4*d*(1 + m)*p^2*((e*x^2)/(d + e*x^2))^(1/2 - m/2)*(-2*Hypergeometric
PFQ[{1/2 - m/2, 1/2 - m/2, 1/2 - m/2}, {3/2 - m/2, 3/2 - m/2}, d/(d + e*x^2)] + (-1 + m)*Hypergeometric2F1[1/2
- m/2, 1/2 - m/2, 3/2 - m/2, d/(d + e*x^2)]*Log[d + e*x^2]))/(e*(-1 + m)^2*x) + (2*p*(2*e*x^3*Hypergeometric2
F1[1, (3 + m)/2, (5 + m)/2, -((e*x^2)/d)] - d*(3 + m)*x*Log[d + e*x^2])*(p*Log[d + e*x^2] - Log[c*(d + e*x^2)^
p]))/(d*(3 + m)) - (2*m*p*(-2*e*x^3*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, -((e*x^2)/d)] + d*(3 + m)*x*Log
[d + e*x^2])*(p*Log[d + e*x^2] - Log[c*(d + e*x^2)^p]))/(d*(3 + m)) + x*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^
2)^p])^2 + m*x*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2))/(1 + m)^2

Maple [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (f x)^m \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 46.41 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int (f x)^m \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int \left (f x\right )^{m} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{2}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 126, normalized size of antiderivative = 6.30 \[ \int (f x)^m \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (f x)^m \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (f x)^m \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^2\,{\left (f\,x\right )}^m \,d x \]

[In]

int(log(c*(d + e*x^2)^p)^2*(f*x)^m,x)

[Out]

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