3.870 \(\int \frac{x^{-1+2 n} \left (a+b x^n\right )^3}{c+d x^n} \, dx\)

Optimal. Leaf size=130 \[ \frac{b x^{2 n} \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{2 d^3 n}-\frac{b^2 x^{3 n} (b c-3 a d)}{3 d^2 n}+\frac{c (b c-a d)^3 \log \left (c+d x^n\right )}{d^5 n}-\frac{x^n (b c-a d)^3}{d^4 n}+\frac{b^3 x^{4 n}}{4 d n} \]

[Out]

-(((b*c - a*d)^3*x^n)/(d^4*n)) + (b*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2)*x^(2*n))/(
2*d^3*n) - (b^2*(b*c - 3*a*d)*x^(3*n))/(3*d^2*n) + (b^3*x^(4*n))/(4*d*n) + (c*(b
*c - a*d)^3*Log[c + d*x^n])/(d^5*n)

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Rubi [A]  time = 0.336232, antiderivative size = 130, 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 \[ \frac{b x^{2 n} \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{2 d^3 n}-\frac{b^2 x^{3 n} (b c-3 a d)}{3 d^2 n}+\frac{c (b c-a d)^3 \log \left (c+d x^n\right )}{d^5 n}-\frac{x^n (b c-a d)^3}{d^4 n}+\frac{b^3 x^{4 n}}{4 d n} \]

Antiderivative was successfully verified.

[In]  Int[(x^(-1 + 2*n)*(a + b*x^n)^3)/(c + d*x^n),x]

[Out]

-(((b*c - a*d)^3*x^n)/(d^4*n)) + (b*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2)*x^(2*n))/(
2*d^3*n) - (b^2*(b*c - 3*a*d)*x^(3*n))/(3*d^2*n) + (b^3*x^(4*n))/(4*d*n) + (c*(b
*c - a*d)^3*Log[c + d*x^n])/(d^5*n)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b^{3} x^{4 n}}{4 d n} + \frac{b^{2} x^{3 n} \left (3 a d - b c\right )}{3 d^{2} n} + \frac{b \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) \int ^{x^{n}} x\, dx}{d^{3} n} - \frac{c \left (a d - b c\right )^{3} \log{\left (c + d x^{n} \right )}}{d^{5} n} + \frac{\left (a d - b c\right )^{3} \int ^{x^{n}} \frac{1}{d^{4}}\, dx}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+2*n)*(a+b*x**n)**3/(c+d*x**n),x)

[Out]

b**3*x**(4*n)/(4*d*n) + b**2*x**(3*n)*(3*a*d - b*c)/(3*d**2*n) + b*(3*a**2*d**2
- 3*a*b*c*d + b**2*c**2)*Integral(x, (x, x**n))/(d**3*n) - c*(a*d - b*c)**3*log(
c + d*x**n)/(d**5*n) + (a*d - b*c)**3*Integral(d**(-4), (x, x**n))/n

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Mathematica [A]  time = 0.21763, size = 134, normalized size = 1.03 \[ \frac{d x^n \left (12 a^3 d^3+18 a^2 b d^2 \left (d x^n-2 c\right )+6 a b^2 d \left (6 c^2-3 c d x^n+2 d^2 x^{2 n}\right )+b^3 \left (-12 c^3+6 c^2 d x^n-4 c d^2 x^{2 n}+3 d^3 x^{3 n}\right )\right )+12 c (b c-a d)^3 \log \left (c+d x^n\right )}{12 d^5 n} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(-1 + 2*n)*(a + b*x^n)^3)/(c + d*x^n),x]

[Out]

(d*x^n*(12*a^3*d^3 + 18*a^2*b*d^2*(-2*c + d*x^n) + 6*a*b^2*d*(6*c^2 - 3*c*d*x^n
+ 2*d^2*x^(2*n)) + b^3*(-12*c^3 + 6*c^2*d*x^n - 4*c*d^2*x^(2*n) + 3*d^3*x^(3*n))
) + 12*c*(b*c - a*d)^3*Log[c + d*x^n])/(12*d^5*n)

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Maple [B]  time = 0.041, size = 284, normalized size = 2.2 \[{\frac{{{\rm e}^{n\ln \left ( x \right ) }}{a}^{3}}{dn}}-3\,{\frac{{{\rm e}^{n\ln \left ( x \right ) }}{a}^{2}cb}{{d}^{2}n}}+3\,{\frac{{{\rm e}^{n\ln \left ( x \right ) }}a{c}^{2}{b}^{2}}{{d}^{3}n}}-{\frac{{{\rm e}^{n\ln \left ( x \right ) }}{c}^{3}{b}^{3}}{{d}^{4}n}}+{\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{4\,dn}}+{\frac{3\,b \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}{a}^{2}}{2\,dn}}-{\frac{3\,{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}ca}{2\,{d}^{2}n}}+{\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}{c}^{2}}{2\,{d}^{3}n}}+{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}a}{dn}}-{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}{b}^{3}c}{3\,{d}^{2}n}}-{\frac{c\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ){a}^{3}}{{d}^{2}n}}+3\,{\frac{{c}^{2}\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ){a}^{2}b}{{d}^{3}n}}-3\,{\frac{{c}^{3}\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) a{b}^{2}}{{d}^{4}n}}+{\frac{{c}^{4}\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ){b}^{3}}{{d}^{5}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+2*n)*(a+b*x^n)^3/(c+d*x^n),x)

[Out]

1/d/n*exp(n*ln(x))*a^3-3/d^2/n*exp(n*ln(x))*a^2*c*b+3/d^3/n*exp(n*ln(x))*a*c^2*b
^2-1/d^4/n*exp(n*ln(x))*c^3*b^3+1/4*b^3/d/n*exp(n*ln(x))^4+3/2*b/d/n*exp(n*ln(x)
)^2*a^2-3/2*b^2/d^2/n*exp(n*ln(x))^2*c*a+1/2*b^3/d^3/n*exp(n*ln(x))^2*c^2+b^2/d/
n*exp(n*ln(x))^3*a-1/3*b^3/d^2/n*exp(n*ln(x))^3*c-c/d^2/n*ln(c+d*exp(n*ln(x)))*a
^3+3*c^2/d^3/n*ln(c+d*exp(n*ln(x)))*a^2*b-3*c^3/d^4/n*ln(c+d*exp(n*ln(x)))*a*b^2
+c^4/d^5/n*ln(c+d*exp(n*ln(x)))*b^3

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Maxima [A]  time = 1.42483, size = 312, normalized size = 2.4 \[ a^{3}{\left (\frac{x^{n}}{d n} - \frac{c \log \left (\frac{d x^{n} + c}{d}\right )}{d^{2} n}\right )} + \frac{1}{12} \, b^{3}{\left (\frac{12 \, c^{4} \log \left (\frac{d x^{n} + c}{d}\right )}{d^{5} n} + \frac{3 \, d^{3} x^{4 \, n} - 4 \, c d^{2} x^{3 \, n} + 6 \, c^{2} d x^{2 \, n} - 12 \, c^{3} x^{n}}{d^{4} n}\right )} - \frac{1}{2} \, a b^{2}{\left (\frac{6 \, c^{3} \log \left (\frac{d x^{n} + c}{d}\right )}{d^{4} n} - \frac{2 \, d^{2} x^{3 \, n} - 3 \, c d x^{2 \, n} + 6 \, c^{2} x^{n}}{d^{3} n}\right )} + \frac{3}{2} \, a^{2} b{\left (\frac{2 \, c^{2} \log \left (\frac{d x^{n} + c}{d}\right )}{d^{3} n} + \frac{d x^{2 \, n} - 2 \, c x^{n}}{d^{2} n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^3*x^(2*n - 1)/(d*x^n + c),x, algorithm="maxima")

[Out]

a^3*(x^n/(d*n) - c*log((d*x^n + c)/d)/(d^2*n)) + 1/12*b^3*(12*c^4*log((d*x^n + c
)/d)/(d^5*n) + (3*d^3*x^(4*n) - 4*c*d^2*x^(3*n) + 6*c^2*d*x^(2*n) - 12*c^3*x^n)/
(d^4*n)) - 1/2*a*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)) + 3/2*a^2*b*(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 = 0.236106, size = 239, normalized size = 1.84 \[ \frac{3 \, b^{3} d^{4} x^{4 \, n} - 4 \,{\left (b^{3} c d^{3} - 3 \, a b^{2} d^{4}\right )} x^{3 \, n} + 6 \,{\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3} + 3 \, a^{2} b d^{4}\right )} x^{2 \, n} - 12 \,{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x^{n} + 12 \,{\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} \log \left (d x^{n} + c\right )}{12 \, d^{5} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^3*x^(2*n - 1)/(d*x^n + c),x, algorithm="fricas")

[Out]

1/12*(3*b^3*d^4*x^(4*n) - 4*(b^3*c*d^3 - 3*a*b^2*d^4)*x^(3*n) + 6*(b^3*c^2*d^2 -
3*a*b^2*c*d^3 + 3*a^2*b*d^4)*x^(2*n) - 12*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*
b*c*d^3 - a^3*d^4)*x^n + 12*(b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d
^3)*log(d*x^n + c))/(d^5*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+2*n)*(a+b*x**n)**3/(c+d*x**n),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{n} + a\right )}^{3} x^{2 \, n - 1}}{d x^{n} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^3*x^(2*n - 1)/(d*x^n + c),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^3*x^(2*n - 1)/(d*x^n + c), x)