3.89 \(\int (d x)^{5/2} (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=41 \[ \frac {2 (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 d}-\frac {4 b n (d x)^{7/2}}{49 d} \]

[Out]

-4/49*b*n*(d*x)^(7/2)/d+2/7*(d*x)^(7/2)*(a+b*ln(c*x^n))/d

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Rubi [A]  time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2304} \[ \frac {2 (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 d}-\frac {4 b n (d x)^{7/2}}{49 d} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^(5/2)*(a + b*Log[c*x^n]),x]

[Out]

(-4*b*n*(d*x)^(7/2))/(49*d) + (2*(d*x)^(7/2)*(a + b*Log[c*x^n]))/(7*d)

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin {align*} \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {4 b n (d x)^{7/2}}{49 d}+\frac {2 (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 d}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 29, normalized size = 0.71 \[ \frac {2}{49} x (d x)^{5/2} \left (7 a+7 b \log \left (c x^n\right )-2 b n\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(5/2)*(a + b*Log[c*x^n]),x]

[Out]

(2*x*(d*x)^(5/2)*(7*a - 2*b*n + 7*b*Log[c*x^n]))/49

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fricas [A]  time = 0.48, size = 50, normalized size = 1.22 \[ \frac {2}{49} \, {\left (7 \, b d^{2} n x^{3} \log \relax (x) + 7 \, b d^{2} x^{3} \log \relax (c) - {\left (2 \, b d^{2} n - 7 \, a d^{2}\right )} x^{3}\right )} \sqrt {d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

2/49*(7*b*d^2*n*x^3*log(x) + 7*b*d^2*x^3*log(c) - (2*b*d^2*n - 7*a*d^2)*x^3)*sqrt(d*x)

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giac [C]  time = 0.52, size = 117, normalized size = 2.85 \[ \left (\frac {1}{7} i + \frac {1}{7}\right ) \, \sqrt {2} b d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) \log \relax (x) - \left (\frac {1}{7} i - \frac {1}{7}\right ) \, \sqrt {2} b d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \log \relax (x) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) - \left (\frac {2}{49} i + \frac {2}{49}\right ) \, \sqrt {2} b d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) + \left (\frac {2}{49} i - \frac {2}{49}\right ) \, \sqrt {2} b d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) + \frac {2}{7} \, b d^{\frac {5}{2}} x^{\frac {7}{2}} \log \relax (c) + \frac {2}{7} \, a d^{\frac {5}{2}} x^{\frac {7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

(1/7*I + 1/7)*sqrt(2)*b*d^2*n*x^(7/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d))*log(x) - (1/7*I - 1/7)*sqrt(2)*b*d^2*n*x
^(7/2)*sqrt(abs(d))*log(x)*sin(1/4*pi*sgn(d)) - (2/49*I + 2/49)*sqrt(2)*b*d^2*n*x^(7/2)*sqrt(abs(d))*cos(1/4*p
i*sgn(d)) + (2/49*I - 2/49)*sqrt(2)*b*d^2*n*x^(7/2)*sqrt(abs(d))*sin(1/4*pi*sgn(d)) + 2/7*b*d^(5/2)*x^(7/2)*lo
g(c) + 2/7*a*d^(5/2)*x^(7/2)

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maple [C]  time = 0.15, size = 128, normalized size = 3.12 \[ \frac {2 b \,d^{3} x^{4} \ln \left (x^{n}\right )}{7 \sqrt {d x}}+\frac {\left (-7 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+7 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+7 i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-7 i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 b n +14 b \ln \relax (c )+14 a \right ) d^{3} x^{4}}{49 \sqrt {d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(5/2)*(b*ln(c*x^n)+a),x)

[Out]

2/7*d^3*b*x^4/(d*x)^(1/2)*ln(x^n)+1/49*d^3*(7*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-7*I*b*Pi*csgn(I*x^n)*csgn(I*c
*x^n)*csgn(I*c)-7*I*b*Pi*csgn(I*c*x^n)^3+7*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)+14*b*ln(c)-4*b*n+14*a)*x^4/(d*x)^(
1/2)

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maxima [A]  time = 0.53, size = 41, normalized size = 1.00 \[ -\frac {4 \, \left (d x\right )^{\frac {7}{2}} b n}{49 \, d} + \frac {2 \, \left (d x\right )^{\frac {7}{2}} b \log \left (c x^{n}\right )}{7 \, d} + \frac {2 \, \left (d x\right )^{\frac {7}{2}} a}{7 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

-4/49*(d*x)^(7/2)*b*n/d + 2/7*(d*x)^(7/2)*b*log(c*x^n)/d + 2/7*(d*x)^(7/2)*a/d

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (d\,x\right )}^{5/2}\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(5/2)*(a + b*log(c*x^n)),x)

[Out]

int((d*x)^(5/2)*(a + b*log(c*x^n)), 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((d*x)**(5/2)*(a+b*ln(c*x**n)),x)

[Out]

Timed out

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