3.2.25 \(\int x^m (a+b x^3)^2 (A+B x^3) \, dx\)

Optimal. Leaf size=71 \[ \frac {a^2 A x^{m+1}}{m+1}+\frac {a x^{m+4} (a B+2 A b)}{m+4}+\frac {b x^{m+7} (2 a B+A b)}{m+7}+\frac {b^2 B x^{m+10}}{m+10} \]

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Rubi [A]  time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {448} \begin {gather*} \frac {a^2 A x^{m+1}}{m+1}+\frac {a x^{m+4} (a B+2 A b)}{m+4}+\frac {b x^{m+7} (2 a B+A b)}{m+7}+\frac {b^2 B x^{m+10}}{m+10} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*x^3)^2*(A + B*x^3),x]

[Out]

(a^2*A*x^(1 + m))/(1 + m) + (a*(2*A*b + a*B)*x^(4 + m))/(4 + m) + (b*(A*b + 2*a*B)*x^(7 + m))/(7 + m) + (b^2*B
*x^(10 + m))/(10 + 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 )^2 \left (A+B x^3\right ) \, dx &=\int \left (a^2 A x^m+a (2 A b+a B) x^{3+m}+b (A b+2 a B) x^{6+m}+b^2 B x^{9+m}\right ) \, dx\\ &=\frac {a^2 A x^{1+m}}{1+m}+\frac {a (2 A b+a B) x^{4+m}}{4+m}+\frac {b (A b+2 a B) x^{7+m}}{7+m}+\frac {b^2 B x^{10+m}}{10+m}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 66, normalized size = 0.93 \begin {gather*} x^{m+1} \left (\frac {a^2 A}{m+1}+\frac {b x^6 (2 a B+A b)}{m+7}+\frac {a x^3 (a B+2 A b)}{m+4}+\frac {b^2 B x^9}{m+10}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a + b*x^3)^2*(A + B*x^3),x]

[Out]

x^(1 + m)*((a^2*A)/(1 + m) + (a*(2*A*b + a*B)*x^3)/(4 + m) + (b*(A*b + 2*a*B)*x^6)/(7 + m) + (b^2*B*x^9)/(10 +
m))

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IntegrateAlgebraic [F]  time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int x^m \left (a+b x^3\right )^2 \left (A+B x^3\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^m*(a + b*x^3)^2*(A + B*x^3),x]

[Out]

Defer[IntegrateAlgebraic][x^m*(a + b*x^3)^2*(A + B*x^3), x]

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fricas [B]  time = 0.89, size = 215, normalized size = 3.03 \begin {gather*} \frac {{\left ({\left (B b^{2} m^{3} + 12 \, B b^{2} m^{2} + 39 \, B b^{2} m + 28 \, B b^{2}\right )} x^{10} + {\left ({\left (2 \, B a b + A b^{2}\right )} m^{3} + 80 \, B a b + 40 \, A b^{2} + 15 \, {\left (2 \, B a b + A b^{2}\right )} m^{2} + 54 \, {\left (2 \, B a b + A b^{2}\right )} m\right )} x^{7} + {\left ({\left (B a^{2} + 2 \, A a b\right )} m^{3} + 70 \, B a^{2} + 140 \, A a b + 18 \, {\left (B a^{2} + 2 \, A a b\right )} m^{2} + 87 \, {\left (B a^{2} + 2 \, A a b\right )} m\right )} x^{4} + {\left (A a^{2} m^{3} + 21 \, A a^{2} m^{2} + 138 \, A a^{2} m + 280 \, A a^{2}\right )} x\right )} x^{m}}{m^{4} + 22 \, m^{3} + 159 \, m^{2} + 418 \, m + 280} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^3+a)^2*(B*x^3+A),x, algorithm="fricas")

[Out]

((B*b^2*m^3 + 12*B*b^2*m^2 + 39*B*b^2*m + 28*B*b^2)*x^10 + ((2*B*a*b + A*b^2)*m^3 + 80*B*a*b + 40*A*b^2 + 15*(
2*B*a*b + A*b^2)*m^2 + 54*(2*B*a*b + A*b^2)*m)*x^7 + ((B*a^2 + 2*A*a*b)*m^3 + 70*B*a^2 + 140*A*a*b + 18*(B*a^2
+ 2*A*a*b)*m^2 + 87*(B*a^2 + 2*A*a*b)*m)*x^4 + (A*a^2*m^3 + 21*A*a^2*m^2 + 138*A*a^2*m + 280*A*a^2)*x)*x^m/(m
^4 + 22*m^3 + 159*m^2 + 418*m + 280)

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giac [B]  time = 0.20, size = 332, normalized size = 4.68 \begin {gather*} \frac {B b^{2} m^{3} x^{10} x^{m} + 12 \, B b^{2} m^{2} x^{10} x^{m} + 39 \, B b^{2} m x^{10} x^{m} + 2 \, B a b m^{3} x^{7} x^{m} + A b^{2} m^{3} x^{7} x^{m} + 28 \, B b^{2} x^{10} x^{m} + 30 \, B a b m^{2} x^{7} x^{m} + 15 \, A b^{2} m^{2} x^{7} x^{m} + 108 \, B a b m x^{7} x^{m} + 54 \, A b^{2} m x^{7} x^{m} + B a^{2} m^{3} x^{4} x^{m} + 2 \, A a b m^{3} x^{4} x^{m} + 80 \, B a b x^{7} x^{m} + 40 \, A b^{2} x^{7} x^{m} + 18 \, B a^{2} m^{2} x^{4} x^{m} + 36 \, A a b m^{2} x^{4} x^{m} + 87 \, B a^{2} m x^{4} x^{m} + 174 \, A a b m x^{4} x^{m} + A a^{2} m^{3} x x^{m} + 70 \, B a^{2} x^{4} x^{m} + 140 \, A a b x^{4} x^{m} + 21 \, A a^{2} m^{2} x x^{m} + 138 \, A a^{2} m x x^{m} + 280 \, A a^{2} x x^{m}}{m^{4} + 22 \, m^{3} + 159 \, m^{2} + 418 \, m + 280} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^3+a)^2*(B*x^3+A),x, algorithm="giac")

[Out]

(B*b^2*m^3*x^10*x^m + 12*B*b^2*m^2*x^10*x^m + 39*B*b^2*m*x^10*x^m + 2*B*a*b*m^3*x^7*x^m + A*b^2*m^3*x^7*x^m +
28*B*b^2*x^10*x^m + 30*B*a*b*m^2*x^7*x^m + 15*A*b^2*m^2*x^7*x^m + 108*B*a*b*m*x^7*x^m + 54*A*b^2*m*x^7*x^m + B
*a^2*m^3*x^4*x^m + 2*A*a*b*m^3*x^4*x^m + 80*B*a*b*x^7*x^m + 40*A*b^2*x^7*x^m + 18*B*a^2*m^2*x^4*x^m + 36*A*a*b
*m^2*x^4*x^m + 87*B*a^2*m*x^4*x^m + 174*A*a*b*m*x^4*x^m + A*a^2*m^3*x*x^m + 70*B*a^2*x^4*x^m + 140*A*a*b*x^4*x
^m + 21*A*a^2*m^2*x*x^m + 138*A*a^2*m*x*x^m + 280*A*a^2*x*x^m)/(m^4 + 22*m^3 + 159*m^2 + 418*m + 280)

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maple [B]  time = 0.05, size = 262, normalized size = 3.69 \begin {gather*} \frac {\left (B \,b^{2} m^{3} x^{9}+12 B \,b^{2} m^{2} x^{9}+39 B \,b^{2} m \,x^{9}+A \,b^{2} m^{3} x^{6}+2 B a b \,m^{3} x^{6}+28 b^{2} B \,x^{9}+15 A \,b^{2} m^{2} x^{6}+30 B a b \,m^{2} x^{6}+54 A \,b^{2} m \,x^{6}+108 B a b m \,x^{6}+2 A a b \,m^{3} x^{3}+40 A \,b^{2} x^{6}+B \,a^{2} m^{3} x^{3}+80 B a b \,x^{6}+36 A a b \,m^{2} x^{3}+18 B \,a^{2} m^{2} x^{3}+174 A a b m \,x^{3}+87 B \,a^{2} m \,x^{3}+A \,a^{2} m^{3}+140 A a b \,x^{3}+70 B \,a^{2} x^{3}+21 A \,a^{2} m^{2}+138 A \,a^{2} m +280 a^{2} A \right ) x^{m +1}}{\left (m +10\right ) \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)^2*(B*x^3+A),x)

[Out]

x^(m+1)*(B*b^2*m^3*x^9+12*B*b^2*m^2*x^9+39*B*b^2*m*x^9+A*b^2*m^3*x^6+2*B*a*b*m^3*x^6+28*B*b^2*x^9+15*A*b^2*m^2
*x^6+30*B*a*b*m^2*x^6+54*A*b^2*m*x^6+108*B*a*b*m*x^6+2*A*a*b*m^3*x^3+40*A*b^2*x^6+B*a^2*m^3*x^3+80*B*a*b*x^6+3
6*A*a*b*m^2*x^3+18*B*a^2*m^2*x^3+174*A*a*b*m*x^3+87*B*a^2*m*x^3+A*a^2*m^3+140*A*a*b*x^3+70*B*a^2*x^3+21*A*a^2*
m^2+138*A*a^2*m+280*A*a^2)/(m+10)/(m+7)/(m+4)/(m+1)

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maxima [A]  time = 0.49, size = 91, normalized size = 1.28 \begin {gather*} \frac {B b^{2} x^{m + 10}}{m + 10} + \frac {2 \, B a b x^{m + 7}}{m + 7} + \frac {A b^{2} x^{m + 7}}{m + 7} + \frac {B a^{2} x^{m + 4}}{m + 4} + \frac {2 \, A a b x^{m + 4}}{m + 4} + \frac {A a^{2} 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)^2*(B*x^3+A),x, algorithm="maxima")

[Out]

B*b^2*x^(m + 10)/(m + 10) + 2*B*a*b*x^(m + 7)/(m + 7) + A*b^2*x^(m + 7)/(m + 7) + B*a^2*x^(m + 4)/(m + 4) + 2*
A*a*b*x^(m + 4)/(m + 4) + A*a^2*x^(m + 1)/(m + 1)

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mupad [B]  time = 2.72, size = 177, normalized size = 2.49 \begin {gather*} x^m\,\left (\frac {B\,b^2\,x^{10}\,\left (m^3+12\,m^2+39\,m+28\right )}{m^4+22\,m^3+159\,m^2+418\,m+280}+\frac {A\,a^2\,x\,\left (m^3+21\,m^2+138\,m+280\right )}{m^4+22\,m^3+159\,m^2+418\,m+280}+\frac {a\,x^4\,\left (2\,A\,b+B\,a\right )\,\left (m^3+18\,m^2+87\,m+70\right )}{m^4+22\,m^3+159\,m^2+418\,m+280}+\frac {b\,x^7\,\left (A\,b+2\,B\,a\right )\,\left (m^3+15\,m^2+54\,m+40\right )}{m^4+22\,m^3+159\,m^2+418\,m+280}\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)^2,x)

[Out]

x^m*((B*b^2*x^10*(39*m + 12*m^2 + m^3 + 28))/(418*m + 159*m^2 + 22*m^3 + m^4 + 280) + (A*a^2*x*(138*m + 21*m^2
+ m^3 + 280))/(418*m + 159*m^2 + 22*m^3 + m^4 + 280) + (a*x^4*(2*A*b + B*a)*(87*m + 18*m^2 + m^3 + 70))/(418*
m + 159*m^2 + 22*m^3 + m^4 + 280) + (b*x^7*(A*b + 2*B*a)*(54*m + 15*m^2 + m^3 + 40))/(418*m + 159*m^2 + 22*m^3
+ m^4 + 280))

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sympy [A]  time = 6.31, size = 1057, normalized size = 14.89 \begin {gather*} \begin {cases} - \frac {A a^{2}}{9 x^{9}} - \frac {A a b}{3 x^{6}} - \frac {A b^{2}}{3 x^{3}} - \frac {B a^{2}}{6 x^{6}} - \frac {2 B a b}{3 x^{3}} + B b^{2} \log {\relax (x )} & \text {for}\: m = -10 \\- \frac {A a^{2}}{6 x^{6}} - \frac {2 A a b}{3 x^{3}} + A b^{2} \log {\relax (x )} - \frac {B a^{2}}{3 x^{3}} + 2 B a b \log {\relax (x )} + \frac {B b^{2} x^{3}}{3} & \text {for}\: m = -7 \\- \frac {A a^{2}}{3 x^{3}} + 2 A a b \log {\relax (x )} + \frac {A b^{2} x^{3}}{3} + B a^{2} \log {\relax (x )} + \frac {2 B a b x^{3}}{3} + \frac {B b^{2} x^{6}}{6} & \text {for}\: m = -4 \\A a^{2} \log {\relax (x )} + \frac {2 A a b x^{3}}{3} + \frac {A b^{2} x^{6}}{6} + \frac {B a^{2} x^{3}}{3} + \frac {B a b x^{6}}{3} + \frac {B b^{2} x^{9}}{9} & \text {for}\: m = -1 \\\frac {A a^{2} m^{3} x x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {21 A a^{2} m^{2} x x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {138 A a^{2} m x x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {280 A a^{2} x x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {2 A a b m^{3} x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {36 A a b m^{2} x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {174 A a b m x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {140 A a b x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {A b^{2} m^{3} x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {15 A b^{2} m^{2} x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {54 A b^{2} m x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {40 A b^{2} x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {B a^{2} m^{3} x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {18 B a^{2} m^{2} x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {87 B a^{2} m x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {70 B a^{2} x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {2 B a b m^{3} x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {30 B a b m^{2} x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {108 B a b m x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {80 B a b x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {B b^{2} m^{3} x^{10} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {12 B b^{2} m^{2} x^{10} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {39 B b^{2} m x^{10} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {28 B b^{2} x^{10} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(b*x**3+a)**2*(B*x**3+A),x)

[Out]

Piecewise((-A*a**2/(9*x**9) - A*a*b/(3*x**6) - A*b**2/(3*x**3) - B*a**2/(6*x**6) - 2*B*a*b/(3*x**3) + B*b**2*l
og(x), Eq(m, -10)), (-A*a**2/(6*x**6) - 2*A*a*b/(3*x**3) + A*b**2*log(x) - B*a**2/(3*x**3) + 2*B*a*b*log(x) +
B*b**2*x**3/3, Eq(m, -7)), (-A*a**2/(3*x**3) + 2*A*a*b*log(x) + A*b**2*x**3/3 + B*a**2*log(x) + 2*B*a*b*x**3/3
+ B*b**2*x**6/6, Eq(m, -4)), (A*a**2*log(x) + 2*A*a*b*x**3/3 + A*b**2*x**6/6 + B*a**2*x**3/3 + B*a*b*x**6/3 +
B*b**2*x**9/9, Eq(m, -1)), (A*a**2*m**3*x*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 21*A*a**2*m**2*x*x
**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 138*A*a**2*m*x*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280)
+ 280*A*a**2*x*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 2*A*a*b*m**3*x**4*x**m/(m**4 + 22*m**3 + 159*m
**2 + 418*m + 280) + 36*A*a*b*m**2*x**4*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 174*A*a*b*m*x**4*x**m
/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 140*A*a*b*x**4*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + A
*b**2*m**3*x**7*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 15*A*b**2*m**2*x**7*x**m/(m**4 + 22*m**3 + 15
9*m**2 + 418*m + 280) + 54*A*b**2*m*x**7*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 40*A*b**2*x**7*x**m/
(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + B*a**2*m**3*x**4*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) +
18*B*a**2*m**2*x**4*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 87*B*a**2*m*x**4*x**m/(m**4 + 22*m**3 + 1
59*m**2 + 418*m + 280) + 70*B*a**2*x**4*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 2*B*a*b*m**3*x**7*x**
m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 30*B*a*b*m**2*x**7*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280
) + 108*B*a*b*m*x**7*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 80*B*a*b*x**7*x**m/(m**4 + 22*m**3 + 159
*m**2 + 418*m + 280) + B*b**2*m**3*x**10*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 12*B*b**2*m**2*x**10
*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 39*B*b**2*m*x**10*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m +
280) + 28*B*b**2*x**10*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280), True))

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