3.18 \(\int \frac {(a m x^m+b n q \log ^{-1+q}(c x^n)) (a x^m+b \log ^q(c x^n))}{x} \, dx\)

Optimal. Leaf size=22 \[ \frac {1}{2} \left (a x^m+b \log ^q\left (c x^n\right )\right )^2 \]

[Out]

1/2*(a*x^m+b*ln(c*x^n)^q)^2

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Rubi [A]  time = 0.11, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {2544} \[ \frac {1}{2} \left (a x^m+b \log ^q\left (c x^n\right )\right )^2 \]

Antiderivative was successfully verified.

[In]

Int[((a*m*x^m + b*n*q*Log[c*x^n]^(-1 + q))*(a*x^m + b*Log[c*x^n]^q))/x,x]

[Out]

(a*x^m + b*Log[c*x^n]^q)^2/2

Rule 2544

Int[((Log[(c_.)*(x_)^(n_.)]^(q_)*(b_.) + (a_.)*(x_)^(m_.))^(p_.)*(Log[(c_.)*(x_)^(n_.)]^(r_.)*(e_.) + (d_.)*(x
_)^(m_.)))/(x_), x_Symbol] :> Simp[(e*(a*x^m + b*Log[c*x^n]^q)^(p + 1))/(b*n*q*(p + 1)), x] /; FreeQ[{a, b, c,
d, e, m, n, p, q, r}, x] && EqQ[r, q - 1] && NeQ[p, -1] && EqQ[a*e*m - b*d*n*q, 0]

Rubi steps

\begin {align*} \int \frac {\left (a m x^m+b n q \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )}{x} \, dx &=\frac {1}{2} \left (a x^m+b \log ^q\left (c x^n\right )\right )^2\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 22, normalized size = 1.00 \[ \frac {1}{2} \left (a x^m+b \log ^q\left (c x^n\right )\right )^2 \]

Antiderivative was successfully verified.

[In]

Integrate[((a*m*x^m + b*n*q*Log[c*x^n]^(-1 + q))*(a*x^m + b*Log[c*x^n]^q))/x,x]

[Out]

(a*x^m + b*Log[c*x^n]^q)^2/2

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fricas [B]  time = 0.44, size = 42, normalized size = 1.91 \[ {\left (n \log \relax (x) + \log \relax (c)\right )}^{q} a b x^{m} + \frac {1}{2} \, {\left (n \log \relax (x) + \log \relax (c)\right )}^{2 \, q} b^{2} + \frac {1}{2} \, a^{2} x^{2 \, m} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*m*x^m+b*n*q*log(c*x^n)^(-1+q))*(a*x^m+b*log(c*x^n)^q)/x,x, algorithm="fricas")

[Out]

(n*log(x) + log(c))^q*a*b*x^m + 1/2*(n*log(x) + log(c))^(2*q)*b^2 + 1/2*a^2*x^(2*m)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b n q \log \left (c x^{n}\right )^{q - 1} + a m x^{m}\right )} {\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*m*x^m+b*n*q*log(c*x^n)^(-1+q))*(a*x^m+b*log(c*x^n)^q)/x,x, algorithm="giac")

[Out]

integrate((b*n*q*log(c*x^n)^(q - 1) + a*m*x^m)*(a*x^m + b*log(c*x^n)^q)/x, x)

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maple [C]  time = 0.67, size = 135, normalized size = 6.14 \[ a b \,x^{m} \left (-\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (\mathrm {csgn}\left (i x^{n}\right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}+\ln \relax (c )+\ln \left (x^{n}\right )\right )^{q}+\frac {a^{2} x^{2 m}}{2}+\frac {b^{2} \left (-\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (\mathrm {csgn}\left (i x^{n}\right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}+\ln \relax (c )+\ln \left (x^{n}\right )\right )^{2 q}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*n*q*ln(c*x^n)^(q-1)+a*m*x^m)*(a*x^m+b*ln(c*x^n)^q)/x,x)

[Out]

1/2*a^2*(x^m)^2+1/2*b^2*((-1/2*I*Pi*(csgn(I*c)-csgn(I*c*x^n))*(csgn(I*x^n)-csgn(I*c*x^n))*csgn(I*c*x^n)+ln(c)+
ln(x^n))^q)^2+a*b*x^m*(-1/2*I*Pi*(csgn(I*c)-csgn(I*c*x^n))*(csgn(I*x^n)-csgn(I*c*x^n))*csgn(I*c*x^n)+ln(c)+ln(
x^n))^q

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*m*x^m+b*n*q*log(c*x^n)^(-1+q))*(a*x^m+b*log(c*x^n)^q)/x,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

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mupad [B]  time = 0.68, size = 20, normalized size = 0.91 \[ \frac {{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )}^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*m*x^m + b*n*q*log(c*x^n)^(q - 1))*(a*x^m + b*log(c*x^n)^q))/x,x)

[Out]

(a*x^m + b*log(c*x^n)^q)^2/2

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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((a*m*x**m+b*n*q*ln(c*x**n)**(-1+q))*(a*x**m+b*ln(c*x**n)**q)/x,x)

[Out]

Timed out

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