3.11.45 \(\int \frac {x^{-1+3 n} (a+b x^n)^3}{c+d x^n} \, dx\) [1045]

Optimal. Leaf size=158 \[ \frac {c (b c-a d)^3 x^n}{d^5 n}-\frac {(b c-a d)^3 x^{2 n}}{2 d^4 n}+\frac {b \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) x^{3 n}}{3 d^3 n}-\frac {b^2 (b c-3 a d) x^{4 n}}{4 d^2 n}+\frac {b^3 x^{5 n}}{5 d n}-\frac {c^2 (b c-a d)^3 \log \left (c+d x^n\right )}{d^6 n} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {457, 90} \begin {gather*} \frac {b x^{3 n} \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{3 d^3 n}-\frac {b^2 x^{4 n} (b c-3 a d)}{4 d^2 n}-\frac {c^2 (b c-a d)^3 \log \left (c+d x^n\right )}{d^6 n}+\frac {c x^n (b c-a d)^3}{d^5 n}-\frac {x^{2 n} (b c-a d)^3}{2 d^4 n}+\frac {b^3 x^{5 n}}{5 d n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^{-1+3 n} \left (a+b x^n\right )^3}{c+d x^n} \, dx &=\frac {\text {Subst}\left (\int \frac {x^2 (a+b x)^3}{c+d x} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (\frac {c (b c-a d)^3}{d^5}+\frac {(-b c+a d)^3 x}{d^4}+\frac {b \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) x^2}{d^3}-\frac {b^2 (b c-3 a d) x^3}{d^2}+\frac {b^3 x^4}{d}-\frac {c^2 (b c-a d)^3}{d^5 (c+d x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {c (b c-a d)^3 x^n}{d^5 n}-\frac {(b c-a d)^3 x^{2 n}}{2 d^4 n}+\frac {b \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) x^{3 n}}{3 d^3 n}-\frac {b^2 (b c-3 a d) x^{4 n}}{4 d^2 n}+\frac {b^3 x^{5 n}}{5 d n}-\frac {c^2 (b c-a d)^3 \log \left (c+d x^n\right )}{d^6 n}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 185, normalized size = 1.17 \begin {gather*} \frac {d x^n \left (30 a^3 d^3 \left (-2 c+d x^n\right )+30 a^2 b d^2 \left (6 c^2-3 c d x^n+2 d^2 x^{2 n}\right )+15 a b^2 d \left (-12 c^3+6 c^2 d x^n-4 c d^2 x^{2 n}+3 d^3 x^{3 n}\right )+b^3 \left (60 c^4-30 c^3 d x^n+20 c^2 d^2 x^{2 n}-15 c d^3 x^{3 n}+12 d^4 x^{4 n}\right )\right )-60 c^2 (b c-a d)^3 \log \left (c+d x^n\right )}{60 d^6 n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*x^n*(30*a^3*d^3*(-2*c + d*x^n) + 30*a^2*b*d^2*(6*c^2 - 3*c*d*x^n + 2*d^2*x^(2*n)) + 15*a*b^2*d*(-12*c^3 + 6
*c^2*d*x^n - 4*c*d^2*x^(2*n) + 3*d^3*x^(3*n)) + b^3*(60*c^4 - 30*c^3*d*x^n + 20*c^2*d^2*x^(2*n) - 15*c*d^3*x^(
3*n) + 12*d^4*x^(4*n))) - 60*c^2*(b*c - a*d)^3*Log[c + d*x^n])/(60*d^6*n)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(341\) vs. \(2(150)=300\).
time = 0.37, size = 342, normalized size = 2.16

method	result	size
risch	\(\frac {b^{3} x^{5 n}}{5 d n}+\frac {3 b^{2} x^{4 n} a}{4 d n}-\frac {b^{3} x^{4 n} c}{4 d^{2} n}+\frac {b \,x^{3 n} a^{2}}{d n}-\frac {b^{2} x^{3 n} a c}{d^{2} n}+\frac {b^{3} x^{3 n} c^{2}}{3 d^{3} n}+\frac {x^{2 n} a^{3}}{2 d n}-\frac {3 x^{2 n} a^{2} b c}{2 d^{2} n}+\frac {3 x^{2 n} a \,b^{2} c^{2}}{2 d^{3} n}-\frac {x^{2 n} b^{3} c^{3}}{2 d^{4} n}-\frac {c \,x^{n} a^{3}}{d^{2} n}+\frac {3 c^{2} x^{n} a^{2} b}{d^{3} n}-\frac {3 c^{3} x^{n} a \,b^{2}}{d^{4} n}+\frac {c^{4} x^{n} b^{3}}{d^{5} n}+\frac {c^{2} \ln \left (x^{n}+\frac {c}{d}\right ) a^{3}}{d^{3} n}-\frac {3 c^{3} \ln \left (x^{n}+\frac {c}{d}\right ) a^{2} b}{d^{4} n}+\frac {3 c^{4} \ln \left (x^{n}+\frac {c}{d}\right ) a \,b^{2}}{d^{5} n}-\frac {c^{5} \ln \left (x^{n}+\frac {c}{d}\right ) b^{3}}{d^{6} n}\)	\(342\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+3*n)*(a+b*x^n)^3/(c+d*x^n),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.31, size = 286, normalized size = 1.81 \begin {gather*} -\frac {1}{60} \, b^{3} {\left (\frac {60 \, c^{5} \log \left (\frac {d x^{n} + c}{d}\right )}{d^{6} n} - \frac {12 \, d^{4} x^{5 \, n} - 15 \, c d^{3} x^{4 \, n} + 20 \, c^{2} d^{2} x^{3 \, n} - 30 \, c^{3} d x^{2 \, n} + 60 \, c^{4} x^{n}}{d^{5} n}\right )} + \frac {1}{4} \, a b^{2} {\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^{2} 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^{3} {\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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*b^3*(60*c^5*log((d*x^n + c)/d)/(d^6*n) - (12*d^4*x^(5*n) - 15*c*d^3*x^(4*n) + 20*c^2*d^2*x^(3*n) - 30*c^
3*d*x^(2*n) + 60*c^4*x^n)/(d^5*n)) + 1/4*a*b^2*(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^2*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^3*(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 = 2.66, size = 230, normalized size = 1.46 \begin {gather*} \frac {12 \, b^{3} d^{5} x^{5 \, n} - 15 \, {\left (b^{3} c d^{4} - 3 \, a b^{2} d^{5}\right )} x^{4 \, n} + 20 \, {\left (b^{3} c^{2} d^{3} - 3 \, a b^{2} c d^{4} + 3 \, a^{2} b d^{5}\right )} x^{3 \, n} - 30 \, {\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} x^{2 \, n} + 60 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} x^{n} - 60 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} \log \left (d x^{n} + c\right )}{60 \, d^{6} n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*b^3*d^5*x^(5*n) - 15*(b^3*c*d^4 - 3*a*b^2*d^5)*x^(4*n) + 20*(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^
5)*x^(3*n) - 30*(b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*x^(2*n) + 60*(b^3*c^4*d - 3*a*b^2*c^
3*d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*x^n - 60*(b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3)*log(d*
x^n + c))/(d^6*n)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (138) = 276\).
time = 55.06, size = 401, normalized size = 2.54 \begin {gather*} \begin {cases} \frac {\left (a + b\right )^{3} \log {\left (x \right )}}{c} & \text {for}\: d = 0 \wedge n = 0 \\\frac {\left (a + b\right )^{3} \log {\left (x \right )}}{c + d} & \text {for}\: n = 0 \\\frac {\frac {a^{3} x^{3 n}}{3 n} + \frac {3 a^{2} b x^{4 n}}{4 n} + \frac {3 a b^{2} x^{5 n}}{5 n} + \frac {b^{3} x^{6 n}}{6 n}}{c} & \text {for}\: d = 0 \\\frac {a^{3} c^{2} \log {\left (\frac {c}{d} + x^{n} \right )}}{d^{3} n} - \frac {a^{3} c x^{n}}{d^{2} n} + \frac {a^{3} x^{2 n}}{2 d n} - \frac {3 a^{2} b c^{3} \log {\left (\frac {c}{d} + x^{n} \right )}}{d^{4} n} + \frac {3 a^{2} b c^{2} x^{n}}{d^{3} n} - \frac {3 a^{2} b c x^{2 n}}{2 d^{2} n} + \frac {a^{2} b x^{3 n}}{d n} + \frac {3 a b^{2} c^{4} \log {\left (\frac {c}{d} + x^{n} \right )}}{d^{5} n} - \frac {3 a b^{2} c^{3} x^{n}}{d^{4} n} + \frac {3 a b^{2} c^{2} x^{2 n}}{2 d^{3} n} - \frac {a b^{2} c x^{3 n}}{d^{2} n} + \frac {3 a b^{2} x^{4 n}}{4 d n} - \frac {b^{3} c^{5} \log {\left (\frac {c}{d} + x^{n} \right )}}{d^{6} n} + \frac {b^{3} c^{4} x^{n}}{d^{5} n} - \frac {b^{3} c^{3} x^{2 n}}{2 d^{4} n} + \frac {b^{3} c^{2} x^{3 n}}{3 d^{3} n} - \frac {b^{3} c x^{4 n}}{4 d^{2} n} + \frac {b^{3} x^{5 n}}{5 d n} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((a + b)**3*log(x)/c, Eq(d, 0) & Eq(n, 0)), ((a + b)**3*log(x)/(c + d), Eq(n, 0)), ((a**3*x**(3*n)/(
3*n) + 3*a**2*b*x**(4*n)/(4*n) + 3*a*b**2*x**(5*n)/(5*n) + b**3*x**(6*n)/(6*n))/c, Eq(d, 0)), (a**3*c**2*log(c
/d + x**n)/(d**3*n) - a**3*c*x**n/(d**2*n) + a**3*x**(2*n)/(2*d*n) - 3*a**2*b*c**3*log(c/d + x**n)/(d**4*n) +
3*a**2*b*c**2*x**n/(d**3*n) - 3*a**2*b*c*x**(2*n)/(2*d**2*n) + a**2*b*x**(3*n)/(d*n) + 3*a*b**2*c**4*log(c/d +
x**n)/(d**5*n) - 3*a*b**2*c**3*x**n/(d**4*n) + 3*a*b**2*c**2*x**(2*n)/(2*d**3*n) - a*b**2*c*x**(3*n)/(d**2*n)
+ 3*a*b**2*x**(4*n)/(4*d*n) - b**3*c**5*log(c/d + x**n)/(d**6*n) + b**3*c**4*x**n/(d**5*n) - b**3*c**3*x**(2*
n)/(2*d**4*n) + b**3*c**2*x**(3*n)/(3*d**3*n) - b**3*c*x**(4*n)/(4*d**2*n) + b**3*x**(5*n)/(5*d*n), True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{3\,n-1}\,{\left (a+b\,x^n\right )}^3}{c+d\,x^n} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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