3.2.26 \(\int x^m (a+b x^3) (A+B x^3) \, dx\)

Optimal. Leaf size=45 \[ \frac {x^{m+4} (a B+A b)}{m+4}+\frac {a A x^{m+1}}{m+1}+\frac {b B x^{m+7}}{m+7} \]

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Rubi [A]  time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {448} \begin {gather*} \frac {x^{m+4} (a B+A b)}{m+4}+\frac {a A x^{m+1}}{m+1}+\frac {b B x^{m+7}}{m+7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*x^3)*(A + B*x^3),x]

[Out]

(a*A*x^(1 + m))/(1 + m) + ((A*b + a*B)*x^(4 + m))/(4 + m) + (b*B*x^(7 + m))/(7 + m)

Rule 448

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

Rubi steps

\begin {align*} \int x^m \left (a+b x^3\right ) \left (A+B x^3\right ) \, dx &=\int \left (a A x^m+(A b+a B) x^{3+m}+b B x^{6+m}\right ) \, dx\\ &=\frac {a A x^{1+m}}{1+m}+\frac {(A b+a B) x^{4+m}}{4+m}+\frac {b B x^{7+m}}{7+m}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 42, normalized size = 0.93 \begin {gather*} x^{m+1} \left (\frac {x^3 (a B+A b)}{m+4}+\frac {a A}{m+1}+\frac {b B x^6}{m+7}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a + b*x^3)*(A + B*x^3),x]

[Out]

x^(1 + m)*((a*A)/(1 + m) + ((A*b + a*B)*x^3)/(4 + m) + (b*B*x^6)/(7 + m))

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IntegrateAlgebraic [F]  time = 0.03, size = 0, normalized size = 0.00 \begin {gather*} \int x^m \left (a+b x^3\right ) \left (A+B x^3\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^m*(a + b*x^3)*(A + B*x^3),x]

[Out]

Defer[IntegrateAlgebraic][x^m*(a + b*x^3)*(A + B*x^3), x]

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fricas [B]  time = 0.85, size = 92, normalized size = 2.04 \begin {gather*} \frac {{\left ({\left (B b m^{2} + 5 \, B b m + 4 \, B b\right )} x^{7} + {\left ({\left (B a + A b\right )} m^{2} + 7 \, B a + 7 \, A b + 8 \, {\left (B a + A b\right )} m\right )} x^{4} + {\left (A a m^{2} + 11 \, A a m + 28 \, A a\right )} x\right )} x^{m}}{m^{3} + 12 \, m^{2} + 39 \, m + 28} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^3+a)*(B*x^3+A),x, algorithm="fricas")

[Out]

((B*b*m^2 + 5*B*b*m + 4*B*b)*x^7 + ((B*a + A*b)*m^2 + 7*B*a + 7*A*b + 8*(B*a + A*b)*m)*x^4 + (A*a*m^2 + 11*A*a
*m + 28*A*a)*x)*x^m/(m^3 + 12*m^2 + 39*m + 28)

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giac [B]  time = 0.17, size = 143, normalized size = 3.18 \begin {gather*} \frac {B b m^{2} x^{7} x^{m} + 5 \, B b m x^{7} x^{m} + 4 \, B b x^{7} x^{m} + B a m^{2} x^{4} x^{m} + A b m^{2} x^{4} x^{m} + 8 \, B a m x^{4} x^{m} + 8 \, A b m x^{4} x^{m} + 7 \, B a x^{4} x^{m} + 7 \, A b x^{4} x^{m} + A a m^{2} x x^{m} + 11 \, A a m x x^{m} + 28 \, A a x x^{m}}{m^{3} + 12 \, m^{2} + 39 \, m + 28} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^3+a)*(B*x^3+A),x, algorithm="giac")

[Out]

(B*b*m^2*x^7*x^m + 5*B*b*m*x^7*x^m + 4*B*b*x^7*x^m + B*a*m^2*x^4*x^m + A*b*m^2*x^4*x^m + 8*B*a*m*x^4*x^m + 8*A
*b*m*x^4*x^m + 7*B*a*x^4*x^m + 7*A*b*x^4*x^m + A*a*m^2*x*x^m + 11*A*a*m*x*x^m + 28*A*a*x*x^m)/(m^3 + 12*m^2 +
39*m + 28)

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maple [B]  time = 0.04, size = 110, normalized size = 2.44 \begin {gather*} \frac {\left (B b \,m^{2} x^{6}+5 B b m \,x^{6}+4 B b \,x^{6}+A b \,m^{2} x^{3}+B a \,m^{2} x^{3}+8 A b m \,x^{3}+8 B a m \,x^{3}+7 A b \,x^{3}+7 B a \,x^{3}+A a \,m^{2}+11 A a m +28 A a \right ) x^{m +1}}{\left (m +7\right ) \left (m +4\right ) \left (m +1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(b*x^3+a)*(B*x^3+A),x)

[Out]

x^(m+1)*(B*b*m^2*x^6+5*B*b*m*x^6+4*B*b*x^6+A*b*m^2*x^3+B*a*m^2*x^3+8*A*b*m*x^3+8*B*a*m*x^3+7*A*b*x^3+7*B*a*x^3
+A*a*m^2+11*A*a*m+28*A*a)/(m+7)/(m+4)/(m+1)

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maxima [A]  time = 0.61, size = 53, normalized size = 1.18 \begin {gather*} \frac {B b x^{m + 7}}{m + 7} + \frac {B a x^{m + 4}}{m + 4} + \frac {A b x^{m + 4}}{m + 4} + \frac {A a x^{m + 1}}{m + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^3+a)*(B*x^3+A),x, algorithm="maxima")

[Out]

B*b*x^(m + 7)/(m + 7) + B*a*x^(m + 4)/(m + 4) + A*b*x^(m + 4)/(m + 4) + A*a*x^(m + 1)/(m + 1)

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mupad [B]  time = 2.65, size = 95, normalized size = 2.11 \begin {gather*} x^m\,\left (\frac {x^4\,\left (A\,b+B\,a\right )\,\left (m^2+8\,m+7\right )}{m^3+12\,m^2+39\,m+28}+\frac {B\,b\,x^7\,\left (m^2+5\,m+4\right )}{m^3+12\,m^2+39\,m+28}+\frac {A\,a\,x\,\left (m^2+11\,m+28\right )}{m^3+12\,m^2+39\,m+28}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(A + B*x^3)*(a + b*x^3),x)

[Out]

x^m*((x^4*(A*b + B*a)*(8*m + m^2 + 7))/(39*m + 12*m^2 + m^3 + 28) + (B*b*x^7*(5*m + m^2 + 4))/(39*m + 12*m^2 +
m^3 + 28) + (A*a*x*(11*m + m^2 + 28))/(39*m + 12*m^2 + m^3 + 28))

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sympy [A]  time = 2.89, size = 410, normalized size = 9.11 \begin {gather*} \begin {cases} - \frac {A a}{6 x^{6}} - \frac {A b}{3 x^{3}} - \frac {B a}{3 x^{3}} + B b \log {\relax (x )} & \text {for}\: m = -7 \\- \frac {A a}{3 x^{3}} + A b \log {\relax (x )} + B a \log {\relax (x )} + \frac {B b x^{3}}{3} & \text {for}\: m = -4 \\A a \log {\relax (x )} + \frac {A b x^{3}}{3} + \frac {B a x^{3}}{3} + \frac {B b x^{6}}{6} & \text {for}\: m = -1 \\\frac {A a m^{2} x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {11 A a m x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {28 A a x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {A b m^{2} x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {8 A b m x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {7 A b x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {B a m^{2} x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {8 B a m x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {7 B a x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {B b m^{2} x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {5 B b m x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {4 B b x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(b*x**3+a)*(B*x**3+A),x)

[Out]

Piecewise((-A*a/(6*x**6) - A*b/(3*x**3) - B*a/(3*x**3) + B*b*log(x), Eq(m, -7)), (-A*a/(3*x**3) + A*b*log(x) +
B*a*log(x) + B*b*x**3/3, Eq(m, -4)), (A*a*log(x) + A*b*x**3/3 + B*a*x**3/3 + B*b*x**6/6, Eq(m, -1)), (A*a*m**
2*x*x**m/(m**3 + 12*m**2 + 39*m + 28) + 11*A*a*m*x*x**m/(m**3 + 12*m**2 + 39*m + 28) + 28*A*a*x*x**m/(m**3 + 1
2*m**2 + 39*m + 28) + A*b*m**2*x**4*x**m/(m**3 + 12*m**2 + 39*m + 28) + 8*A*b*m*x**4*x**m/(m**3 + 12*m**2 + 39
*m + 28) + 7*A*b*x**4*x**m/(m**3 + 12*m**2 + 39*m + 28) + B*a*m**2*x**4*x**m/(m**3 + 12*m**2 + 39*m + 28) + 8*
B*a*m*x**4*x**m/(m**3 + 12*m**2 + 39*m + 28) + 7*B*a*x**4*x**m/(m**3 + 12*m**2 + 39*m + 28) + B*b*m**2*x**7*x*
*m/(m**3 + 12*m**2 + 39*m + 28) + 5*B*b*m*x**7*x**m/(m**3 + 12*m**2 + 39*m + 28) + 4*B*b*x**7*x**m/(m**3 + 12*
m**2 + 39*m + 28), True))

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