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

Optimal. Leaf size=77 \[ \frac {(f x)^{m+1} \log ^3\left (c \left (d+e x^2\right )^p\right )}{f (m+1)}-\frac {6 e p \text {Int}\left (\frac {(f x)^{m+2} \log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2},x\right )}{f^2 (m+1)} \]

[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)

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Rubi [A]  time = 0.12, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (f x)^m \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx \]

Verification is Not applicable to the result.

[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*} \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) \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*}

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Mathematica [A]  time = 2.20, size = 994, normalized size = 12.91 \[ \frac {(f x)^m \left (\frac {6 p^3 \left (d \left (\left (-\frac {e x^2}{d}\right )^{\frac {m+1}{2}}-1\right ) \log ^2\left (e x^2+d\right )+(m+1) \left (e x^2+d\right ) \, _3F_2\left (1,1,\frac {1}{2}-\frac {m}{2};2,2;\frac {e x^2}{d}+1\right ) \log \left (e x^2+d\right )-(m+1) \left (e x^2+d\right ) \, _4F_3\left (1,1,1,\frac {1}{2}-\frac {m}{2};2,2,2;\frac {e x^2}{d}+1\right )\right ) \left (-\frac {e x^2}{d}\right )^{\frac {1}{2}-\frac {m}{2}}}{e}-\frac {3 m p^2 \left (d \left (\left (-\frac {e x^2}{d}\right )^{\frac {m+1}{2}}-1\right ) \log ^2\left (e x^2+d\right )+(m+1) \left (e x^2+d\right ) \, _3F_2\left (1,1,\frac {1}{2}-\frac {m}{2};2,2;\frac {e x^2}{d}+1\right ) \log \left (e x^2+d\right )-(m+1) \left (e x^2+d\right ) \, _4F_3\left (1,1,1,\frac {1}{2}-\frac {m}{2};2,2,2;\frac {e x^2}{d}+1\right )\right ) \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right ) \left (-\frac {e x^2}{d}\right )^{\frac {1}{2}-\frac {m}{2}}}{e}-\frac {3 p^2 \left (d \left (\left (-\frac {e x^2}{d}\right )^{\frac {m+1}{2}}-1\right ) \log ^2\left (e x^2+d\right )+(m+1) \left (e x^2+d\right ) \, _3F_2\left (1,1,\frac {1}{2}-\frac {m}{2};2,2;\frac {e x^2}{d}+1\right ) \log \left (e x^2+d\right )-(m+1) \left (e x^2+d\right ) \, _4F_3\left (1,1,1,\frac {1}{2}-\frac {m}{2};2,2,2;\frac {e x^2}{d}+1\right )\right ) \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right ) \left (-\frac {e x^2}{d}\right )^{\frac {1}{2}-\frac {m}{2}}}{e}+(m+1) p^3 x^2 \log ^3\left (e x^2+d\right )+m x^2 \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right )^3+x^2 \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right )^3+\frac {3 m p x^2 \left (d (m+3) \log \left (e x^2+d\right )-2 e x^2 \, _2F_1\left (1,\frac {m+3}{2};\frac {m+5}{2};-\frac {e x^2}{d}\right )\right ) \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right )^2}{d (m+3)}+\frac {3 p x^2 \left (d (m+3) \log \left (e x^2+d\right )-2 e x^2 \, _2F_1\left (1,\frac {m+3}{2};\frac {m+5}{2};-\frac {e x^2}{d}\right )\right ) \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right )^2}{d (m+3)}+\frac {6 d (m+1) p^3 \left (\frac {e x^2}{e x^2+d}\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}{e x^2+d}\right )+(m-1) \log \left (e x^2+d\right ) \left ((m-1) \, _2F_1\left (\frac {1}{2}-\frac {m}{2},\frac {1}{2}-\frac {m}{2};\frac {3}{2}-\frac {m}{2};\frac {d}{e x^2+d}\right ) \log \left (e x^2+d\right )-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}{e x^2+d}\right )\right )\right )}{e (m-1)^3}\right )}{(m+1)^2 x} \]

Warning: Unable to verify antiderivative.

[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))^(1/2 - 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))^(1/2 - 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))^(1/2 - 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)

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fricas [A]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (f x\right )^{m} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (f x\right )^{m} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.87, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {f^{m} p^{3} x x^{m} \log \left (e x^{2} + d\right )^{3}}{m + 1} + \int \frac {3 \, {\left ({\left (m p^{2} + p^{2}\right )} d f^{m} \log \relax (c) - {\left (2 \, e f^{m} p^{3} - {\left (m p^{2} + p^{2}\right )} e f^{m} \log \relax (c)\right )} x^{2}\right )} x^{m} \log \left (e x^{2} + d\right )^{2} + 3 \, {\left ({\left (m p + p\right )} e f^{m} x^{2} \log \relax (c)^{2} + {\left (m p + p\right )} d f^{m} \log \relax (c)^{2}\right )} x^{m} \log \left (e x^{2} + d\right ) + {\left (e f^{m} {\left (m + 1\right )} x^{2} \log \relax (c)^{3} + d f^{m} {\left (m + 1\right )} \log \relax (c)^{3}\right )} x^{m}}{e {\left (m + 1\right )} x^{2} + d {\left (m + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^3\,{\left (f\,x\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(d + e*x^2)^p)^3*(f*x)^m,x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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