3.871 \(\int \frac{x^{-1+2 n} \left (a+b x^n\right )^2}{c+d x^n} \, dx\)

Optimal. Leaf size=90 \[ -\frac{c (b c-a d)^2 \log \left (c+d x^n\right )}{d^4 n}+\frac{x^n (b c-a d)^2}{d^3 n}-\frac{b x^{2 n} (b c-2 a d)}{2 d^2 n}+\frac{b^2 x^{3 n}}{3 d n} \]

[Out]

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

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Rubi [A]  time = 0.232734, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{c (b c-a d)^2 \log \left (c+d x^n\right )}{d^4 n}+\frac{x^n (b c-a d)^2}{d^3 n}-\frac{b x^{2 n} (b c-2 a d)}{2 d^2 n}+\frac{b^2 x^{3 n}}{3 d n} \]

Antiderivative was successfully verified.

[In]  Int[(x^(-1 + 2*n)*(a + b*x^n)^2)/(c + d*x^n),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b^{2} x^{3 n}}{3 d n} + \frac{b \left (2 a d - b c\right ) \int ^{x^{n}} x\, dx}{d^{2} n} - \frac{c \left (a d - b c\right )^{2} \log{\left (c + d x^{n} \right )}}{d^{4} n} + \frac{\left (a d - b c\right )^{2} \int ^{x^{n}} \frac{1}{d^{3}}\, dx}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+2*n)*(a+b*x**n)**2/(c+d*x**n),x)

[Out]

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

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Mathematica [A]  time = 0.134348, size = 87, normalized size = 0.97 \[ \frac{d x^n \left (6 a^2 d^2+6 a b d \left (d x^n-2 c\right )+b^2 \left (6 c^2-3 c d x^n+2 d^2 x^{2 n}\right )\right )-6 c (b c-a d)^2 \log \left (c+d x^n\right )}{6 d^4 n} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(-1 + 2*n)*(a + b*x^n)^2)/(c + d*x^n),x]

[Out]

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

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Maple [A]  time = 0.038, size = 173, normalized size = 1.9 \[{\frac{{{\rm e}^{n\ln \left ( x \right ) }}{a}^{2}}{dn}}-2\,{\frac{c{{\rm e}^{n\ln \left ( x \right ) }}ab}{{d}^{2}n}}+{\frac{{b}^{2}{{\rm e}^{n\ln \left ( x \right ) }}{c}^{2}}{{d}^{3}n}}+{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{3\,dn}}+{\frac{b \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}a}{dn}}-{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}c}{2\,{d}^{2}n}}-{\frac{c\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ){a}^{2}}{{d}^{2}n}}+2\,{\frac{{c}^{2}\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) ab}{{d}^{3}n}}-{\frac{{c}^{3}\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ){b}^{2}}{{d}^{4}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+2*n)*(a+b*x^n)^2/(c+d*x^n),x)

[Out]

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

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Maxima [A]  time = 1.4019, size = 203, normalized size = 2.26 \[ a^{2}{\left (\frac{x^{n}}{d n} - \frac{c \log \left (\frac{d x^{n} + c}{d}\right )}{d^{2} n}\right )} - \frac{1}{6} \, b^{2}{\left (\frac{6 \, c^{3} \log \left (\frac{d x^{n} + c}{d}\right )}{d^{4} n} - \frac{2 \, d^{2} x^{3 \, n} - 3 \, c d x^{2 \, n} + 6 \, c^{2} x^{n}}{d^{3} n}\right )} + a b{\left (\frac{2 \, c^{2} \log \left (\frac{d x^{n} + c}{d}\right )}{d^{3} n} + \frac{d x^{2 \, n} - 2 \, c x^{n}}{d^{2} n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^2*x^(2*n - 1)/(d*x^n + c),x, algorithm="maxima")

[Out]

a^2*(x^n/(d*n) - c*log((d*x^n + c)/d)/(d^2*n)) - 1/6*b^2*(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)) + a*b*(2*c^2*l
og((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 = 0.233482, size = 146, normalized size = 1.62 \[ \frac{2 \, b^{2} d^{3} x^{3 \, n} - 3 \,{\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{2 \, n} + 6 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{n} - 6 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \log \left (d x^{n} + c\right )}{6 \, d^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^2*x^(2*n - 1)/(d*x^n + c),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 144.008, size = 202, normalized size = 2.24 \[ \begin{cases} \frac{\left (a + b\right )^{2} \log{\left (x \right )}}{c} & \text{for}\: d = 0 \wedge n = 0 \\\frac{\frac{a^{2} x^{2 n}}{2 n} + \frac{2 a b x^{3 n}}{3 n} + \frac{b^{2} x^{4 n}}{4 n}}{c} & \text{for}\: d = 0 \\\frac{\left (a + b\right )^{2} \log{\left (x \right )}}{c + d} & \text{for}\: n = 0 \\- \frac{a^{2} c \log{\left (\frac{c}{d} + x^{n} \right )}}{d^{2} n} + \frac{a^{2} x^{n}}{d n} + \frac{2 a b c^{2} \log{\left (\frac{c}{d} + x^{n} \right )}}{d^{3} n} - \frac{2 a b c x^{n}}{d^{2} n} + \frac{a b x^{2 n}}{d n} - \frac{b^{2} c^{3} \log{\left (\frac{c}{d} + x^{n} \right )}}{d^{4} n} + \frac{b^{2} c^{2} x^{n}}{d^{3} n} - \frac{b^{2} c x^{2 n}}{2 d^{2} n} + \frac{b^{2} x^{3 n}}{3 d n} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1+2*n)*(a+b*x**n)**2/(c+d*x**n),x)

[Out]

Piecewise(((a + b)**2*log(x)/c, Eq(d, 0) & Eq(n, 0)), ((a**2*x**(2*n)/(2*n) + 2*
a*b*x**(3*n)/(3*n) + b**2*x**(4*n)/(4*n))/c, Eq(d, 0)), ((a + b)**2*log(x)/(c +
d), Eq(n, 0)), (-a**2*c*log(c/d + x**n)/(d**2*n) + a**2*x**n/(d*n) + 2*a*b*c**2*
log(c/d + x**n)/(d**3*n) - 2*a*b*c*x**n/(d**2*n) + a*b*x**(2*n)/(d*n) - b**2*c**
3*log(c/d + x**n)/(d**4*n) + b**2*c**2*x**n/(d**3*n) - b**2*c*x**(2*n)/(2*d**2*n
) + b**2*x**(3*n)/(3*d*n), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{n} + a\right )}^{2} x^{2 \, n - 1}}{d x^{n} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^2*x^(2*n - 1)/(d*x^n + c),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^2*x^(2*n - 1)/(d*x^n + c), x)