\(\int (f x)^m \log ^3(c (d+e x^2)^p) \, dx\) [158]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int (f x)^m \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {(f x)^{1+m} \log ^3\left (c \left (d+e x^2\right )^p\right )}{f (1+m)}-\frac {6 e p \text {Int}\left (\frac {(f x)^{2+m} \log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2},x\right )}{f^2 (1+m)} \]

[Out]

(f*x)^(1+m)*ln(c*(e*x^2+d)^p)^3/f/(1+m)-6*e*p*Unintegrable((f*x)^(2+m)*ln(c*(e*x^2+d)^p)^2/(e*x^2+d),x)/f^2/(1
+m)

Rubi [N/A]

Not integrable

Time = 0.07 (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 = {} \[ \int (f x)^m \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int (f x)^m \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx \]

[In]

Int[(f*x)^m*Log[c*(d + e*x^2)^p]^3,x]

[Out]

((f*x)^(1 + m)*Log[c*(d + e*x^2)^p]^3)/(f*(1 + m)) - (6*e*p*Defer[Int][((f*x)^(2 + m)*Log[c*(d + e*x^2)^p]^2)/
(d + e*x^2), x])/(f^2*(1 + m))

Rubi steps \begin{align*} \text {integral}& = \frac {(f x)^{1+m} \log ^3\left (c \left (d+e x^2\right )^p\right )}{f (1+m)}-\frac {(6 e p) \int \frac {(f x)^{2+m} \log ^2\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. \(994\) vs. \(2(77)=154\).

Time = 1.82 (sec) , antiderivative size = 994, normalized size of antiderivative = 49.70 \[ \int (f x)^m \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {(f x)^m \left ((1+m) p^3 x^2 \log ^3\left (d+e x^2\right )+\frac {6 p^3 \left (-\frac {e x^2}{d}\right )^{\frac {1-m}{2}} \left (-\left ((1+m) \left (d+e x^2\right ) \, _4F_3\left (1,1,1,\frac {1}{2}-\frac {m}{2};2,2,2;1+\frac {e x^2}{d}\right )\right )+(1+m) \left (d+e x^2\right ) \, _3F_2\left (1,1,\frac {1}{2}-\frac {m}{2};2,2;1+\frac {e x^2}{d}\right ) \log \left (d+e x^2\right )+d \left (-1+\left (-\frac {e x^2}{d}\right )^{\frac {1+m}{2}}\right ) \log ^2\left (d+e x^2\right )\right )}{e}+\frac {6 d (1+m) p^3 \left (\frac {e x^2}{d+e x^2}\right )^{\frac {1}{2}-\frac {m}{2}} \left (8 \, _4F_3\left (\frac {1}{2}-\frac {m}{2},\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 {3}{2}-\frac {m}{2};\frac {d}{d+e x^2}\right )+(-1+m) \log \left (d+e x^2\right ) \left (-4 \, _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 )\right )}{e (-1+m)^3}-\frac {3 p^2 \left (-\frac {e x^2}{d}\right )^{\frac {1-m}{2}} \left (-\left ((1+m) \left (d+e x^2\right ) \, _4F_3\left (1,1,1,\frac {1}{2}-\frac {m}{2};2,2,2;1+\frac {e x^2}{d}\right )\right )+(1+m) \left (d+e x^2\right ) \, _3F_2\left (1,1,\frac {1}{2}-\frac {m}{2};2,2;1+\frac {e x^2}{d}\right ) \log \left (d+e x^2\right )+d \left (-1+\left (-\frac {e x^2}{d}\right )^{\frac {1+m}{2}}\right ) \log ^2\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 )}{e}-\frac {3 m p^2 \left (-\frac {e x^2}{d}\right )^{\frac {1-m}{2}} \left (-\left ((1+m) \left (d+e x^2\right ) \, _4F_3\left (1,1,1,\frac {1}{2}-\frac {m}{2};2,2,2;1+\frac {e x^2}{d}\right )\right )+(1+m) \left (d+e x^2\right ) \, _3F_2\left (1,1,\frac {1}{2}-\frac {m}{2};2,2;1+\frac {e x^2}{d}\right ) \log \left (d+e x^2\right )+d \left (-1+\left (-\frac {e x^2}{d}\right )^{\frac {1+m}{2}}\right ) \log ^2\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 )}{e}+\frac {3 p x^2 \left (-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 ) \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^2}{d (3+m)}+\frac {3 m p x^2 \left (-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 ) \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^2}{d (3+m)}+x^2 \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^3+m x^2 \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^3\right )}{(1+m)^2 x} \]

[In]

Integrate[(f*x)^m*Log[c*(d + e*x^2)^p]^3,x]

[Out]

((f*x)^m*((1 + m)*p^3*x^2*Log[d + e*x^2]^3 + (6*p^3*(-((e*x^2)/d))^((1 - m)/2)*(-((1 + m)*(d + e*x^2)*Hypergeo
metricPFQ[{1, 1, 1, 1/2 - m/2}, {2, 2, 2}, 1 + (e*x^2)/d]) + (1 + m)*(d + e*x^2)*HypergeometricPFQ[{1, 1, 1/2
- m/2}, {2, 2}, 1 + (e*x^2)/d]*Log[d + e*x^2] + d*(-1 + (-((e*x^2)/d))^((1 + m)/2))*Log[d + e*x^2]^2))/e + (6*
d*(1 + m)*p^3*((e*x^2)/(d + e*x^2))^(1/2 - m/2)*(8*HypergeometricPFQ[{1/2 - m/2, 1/2 - m/2, 1/2 - m/2, 1/2 - m
/2}, {3/2 - m/2, 3/2 - m/2, 3/2 - m/2}, d/(d + e*x^2)] + (-1 + m)*Log[d + e*x^2]*(-4*HypergeometricPFQ[{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)^3) - (3*p^2*(-((e*x^2)/d))^((1 - m)/2)*(-((1 +
m)*(d + e*x^2)*HypergeometricPFQ[{1, 1, 1, 1/2 - m/2}, {2, 2, 2}, 1 + (e*x^2)/d]) + (1 + m)*(d + e*x^2)*Hyper
geometricPFQ[{1, 1, 1/2 - m/2}, {2, 2}, 1 + (e*x^2)/d]*Log[d + e*x^2] + d*(-1 + (-((e*x^2)/d))^((1 + m)/2))*Lo
g[d + e*x^2]^2)*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p]))/e - (3*m*p^2*(-((e*x^2)/d))^((1 - m)/2)*(-((1 +
m)*(d + e*x^2)*HypergeometricPFQ[{1, 1, 1, 1/2 - m/2}, {2, 2, 2}, 1 + (e*x^2)/d]) + (1 + m)*(d + e*x^2)*Hyperg
eometricPFQ[{1, 1, 1/2 - m/2}, {2, 2}, 1 + (e*x^2)/d]*Log[d + e*x^2] + d*(-1 + (-((e*x^2)/d))^((1 + m)/2))*Log
[d + e*x^2]^2)*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p]))/e + (3*p*x^2*(-2*e*x^2*Hypergeometric2F1[1, (3 +
m)/2, (5 + m)/2, -((e*x^2)/d)] + d*(3 + m)*Log[d + e*x^2])*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/(d*
(3 + m)) + (3*m*p*x^2*(-2*e*x^2*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, -((e*x^2)/d)] + d*(3 + m)*Log[d + e
*x^2])*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/(d*(3 + m)) + x^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x
^2)^p])^3 + m*x^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^3))/((1 + m)^2*x)

Maple [N/A]

Not integrable

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

\[\int \left (f x \right )^{m} {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}^{3}d x\]

[In]

int((f*x)^m*ln(c*(e*x^2+d)^p)^3,x)

[Out]

int((f*x)^m*ln(c*(e*x^2+d)^p)^3,x)

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (f x)^m \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3} \,d x } \]

[In]

integrate((f*x)^m*log(c*(e*x^2+d)^p)^3,x, algorithm="fricas")

[Out]

integral((f*x)^m*log((e*x^2 + d)^p*c)^3, x)

Sympy [N/A]

Not integrable

Time = 82.58 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int (f x)^m \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int \left (f x\right )^{m} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{3}\, dx \]

[In]

integrate((f*x)**m*ln(c*(e*x**2+d)**p)**3,x)

[Out]

Integral((f*x)**m*log(c*(d + e*x**2)**p)**3, x)

Maxima [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 171, normalized size of antiderivative = 8.55 \[ \int (f x)^m \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3} \,d x } \]

[In]

integrate((f*x)^m*log(c*(e*x^2+d)^p)^3,x, algorithm="maxima")

[Out]

f^m*x*x^m*log((e*x^2 + d)^p)^3/(m + 1) + integrate((3*(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)^2 + 3*(e*f^m*(m + 1)*x^2*log(c)^2 + d*f^m*(m + 1)*log(c)^2)*x^m*log((e*x^2 + d)
^p) + (e*f^m*(m + 1)*x^2*log(c)^3 + d*f^m*(m + 1)*log(c)^3)*x^m)/(e*(m + 1)*x^2 + d*(m + 1)), x)

Giac [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (f x)^m \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3} \,d x } \]

[In]

integrate((f*x)^m*log(c*(e*x^2+d)^p)^3,x, algorithm="giac")

[Out]

integrate((f*x)^m*log((e*x^2 + d)^p*c)^3, x)

Mupad [N/A]

Not integrable

Time = 1.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (f x)^m \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^3\,{\left (f\,x\right )}^m \,d x \]

[In]

int(log(c*(d + e*x^2)^p)^3*(f*x)^m,x)

[Out]

int(log(c*(d + e*x^2)^p)^3*(f*x)^m, x)