3.878 \(\int \frac{x^{-1+3 n} \left (a+b x^n\right )}{c+d x^n} \, dx\)

Optimal. Leaf size=86 \[ -\frac{c^2 (b c-a d) \log \left (c+d x^n\right )}{d^4 n}+\frac{c x^n (b c-a d)}{d^3 n}-\frac{x^{2 n} (b c-a d)}{2 d^2 n}+\frac{b x^{3 n}}{3 d n} \]

[Out]

(c*(b*c - a*d)*x^n)/(d^3*n) - ((b*c - a*d)*x^(2*n))/(2*d^2*n) + (b*x^(3*n))/(3*d
*n) - (c^2*(b*c - a*d)*Log[c + d*x^n])/(d^4*n)

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Rubi [A]  time = 0.210023, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{c^2 (b c-a d) \log \left (c+d x^n\right )}{d^4 n}+\frac{c x^n (b c-a d)}{d^3 n}-\frac{x^{2 n} (b c-a d)}{2 d^2 n}+\frac{b x^{3 n}}{3 d n} \]

Antiderivative was successfully verified.

[In]  Int[(x^(-1 + 3*n)*(a + b*x^n))/(c + d*x^n),x]

[Out]

(c*(b*c - a*d)*x^n)/(d^3*n) - ((b*c - a*d)*x^(2*n))/(2*d^2*n) + (b*x^(3*n))/(3*d
*n) - (c^2*(b*c - a*d)*Log[c + d*x^n])/(d^4*n)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b x^{3 n}}{3 d n} + \frac{c^{2} \left (a d - b c\right ) \log{\left (c + d x^{n} \right )}}{d^{4} n} + \frac{\left (a d - b c\right ) \int ^{x^{n}} x\, dx}{d^{2} n} - \frac{\left (a d - b c\right ) \int ^{x^{n}} c\, dx}{d^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+3*n)*(a+b*x**n)/(c+d*x**n),x)

[Out]

b*x**(3*n)/(3*d*n) + c**2*(a*d - b*c)*log(c + d*x**n)/(d**4*n) + (a*d - b*c)*Int
egral(x, (x, x**n))/(d**2*n) - (a*d - b*c)*Integral(c, (x, x**n))/(d**3*n)

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Mathematica [A]  time = 0.101414, size = 76, normalized size = 0.88 \[ \frac{d x^n \left (3 a d \left (d x^n-2 c\right )+b \left (6 c^2-3 c d x^n+2 d^2 x^{2 n}\right )\right )+6 c^2 (a d-b c) \log \left (c+d x^n\right )}{6 d^4 n} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(-1 + 3*n)*(a + b*x^n))/(c + d*x^n),x]

[Out]

(d*x^n*(3*a*d*(-2*c + d*x^n) + b*(6*c^2 - 3*c*d*x^n + 2*d^2*x^(2*n))) + 6*c^2*(-
(b*c) + a*d)*Log[c + d*x^n])/(6*d^4*n)

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Maple [A]  time = 0.036, size = 125, normalized size = 1.5 \[{\frac{b \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{3\,dn}}+{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}a}{2\,dn}}-{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}bc}{2\,{d}^{2}n}}-{\frac{c{{\rm e}^{n\ln \left ( x \right ) }}a}{{d}^{2}n}}+{\frac{{c}^{2}{{\rm e}^{n\ln \left ( x \right ) }}b}{{d}^{3}n}}+{\frac{{c}^{2}\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) a}{{d}^{3}n}}-{\frac{{c}^{3}\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) b}{{d}^{4}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+3*n)*(a+b*x^n)/(c+d*x^n),x)

[Out]

1/3*b/d/n*exp(n*ln(x))^3+1/2/d/n*exp(n*ln(x))^2*a-1/2/d^2/n*exp(n*ln(x))^2*b*c-c
/d^2/n*exp(n*ln(x))*a+c^2/d^3/n*exp(n*ln(x))*b+c^2/d^3/n*ln(c+d*exp(n*ln(x)))*a-
c^3/d^4/n*ln(c+d*exp(n*ln(x)))*b

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Maxima [A]  time = 1.44172, size = 151, normalized size = 1.76 \[ -\frac{1}{6} \, b{\left (\frac{6 \, c^{3} \log \left (\frac{d x^{n} + c}{d}\right )}{d^{4} n} - \frac{2 \, d^{2} x^{3 \, n} - 3 \, c d x^{2 \, n} + 6 \, c^{2} x^{n}}{d^{3} n}\right )} + \frac{1}{2} \, a{\left (\frac{2 \, c^{2} \log \left (\frac{d x^{n} + c}{d}\right )}{d^{3} n} + \frac{d x^{2 \, n} - 2 \, c x^{n}}{d^{2} n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)*x^(3*n - 1)/(d*x^n + c),x, algorithm="maxima")

[Out]

-1/6*b*(6*c^3*log((d*x^n + c)/d)/(d^4*n) - (2*d^2*x^(3*n) - 3*c*d*x^(2*n) + 6*c^
2*x^n)/(d^3*n)) + 1/2*a*(2*c^2*log((d*x^n + c)/d)/(d^3*n) + (d*x^(2*n) - 2*c*x^n
)/(d^2*n))

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Fricas [A]  time = 0.232691, size = 111, normalized size = 1.29 \[ \frac{2 \, b d^{3} x^{3 \, n} - 3 \,{\left (b c d^{2} - a d^{3}\right )} x^{2 \, n} + 6 \,{\left (b c^{2} d - a c d^{2}\right )} x^{n} - 6 \,{\left (b c^{3} - a c^{2} d\right )} \log \left (d x^{n} + c\right )}{6 \, d^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)*x^(3*n - 1)/(d*x^n + c),x, algorithm="fricas")

[Out]

1/6*(2*b*d^3*x^(3*n) - 3*(b*c*d^2 - a*d^3)*x^(2*n) + 6*(b*c^2*d - a*c*d^2)*x^n -
6*(b*c^3 - a*c^2*d)*log(d*x^n + c))/(d^4*n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1+3*n)*(a+b*x**n)/(c+d*x**n),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{n} + a\right )} x^{3 \, n - 1}}{d x^{n} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)*x^(3*n - 1)/(d*x^n + c),x, algorithm="giac")

[Out]

integrate((b*x^n + a)*x^(3*n - 1)/(d*x^n + c), x)