3.2.75 \(\int x^8 \sqrt {a+b x^3} (A+B x^3) \, dx\)

Optimal. Leaf size=103 \[ \frac {2 a^2 \left (a+b x^3\right )^{3/2} (A b-a B)}{9 b^4}+\frac {2 \left (a+b x^3\right )^{7/2} (A b-3 a B)}{21 b^4}-\frac {2 a \left (a+b x^3\right )^{5/2} (2 A b-3 a B)}{15 b^4}+\frac {2 B \left (a+b x^3\right )^{9/2}}{27 b^4} \]

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Rubi [A]  time = 0.08, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {446, 77} \begin {gather*} \frac {2 a^2 \left (a+b x^3\right )^{3/2} (A b-a B)}{9 b^4}+\frac {2 \left (a+b x^3\right )^{7/2} (A b-3 a B)}{21 b^4}-\frac {2 a \left (a+b x^3\right )^{5/2} (2 A b-3 a B)}{15 b^4}+\frac {2 B \left (a+b x^3\right )^{9/2}}{27 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^8*Sqrt[a + b*x^3]*(A + B*x^3),x]

[Out]

(2*a^2*(A*b - a*B)*(a + b*x^3)^(3/2))/(9*b^4) - (2*a*(2*A*b - 3*a*B)*(a + b*x^3)^(5/2))/(15*b^4) + (2*(A*b - 3
*a*B)*(a + b*x^3)^(7/2))/(21*b^4) + (2*B*(a + b*x^3)^(9/2))/(27*b^4)

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 446

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 x^8 \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int x^2 \sqrt {a+b x} (A+B x) \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {a^2 (-A b+a B) \sqrt {a+b x}}{b^3}+\frac {a (-2 A b+3 a B) (a+b x)^{3/2}}{b^3}+\frac {(A b-3 a B) (a+b x)^{5/2}}{b^3}+\frac {B (a+b x)^{7/2}}{b^3}\right ) \, dx,x,x^3\right )\\ &=\frac {2 a^2 (A b-a B) \left (a+b x^3\right )^{3/2}}{9 b^4}-\frac {2 a (2 A b-3 a B) \left (a+b x^3\right )^{5/2}}{15 b^4}+\frac {2 (A b-3 a B) \left (a+b x^3\right )^{7/2}}{21 b^4}+\frac {2 B \left (a+b x^3\right )^{9/2}}{27 b^4}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 75, normalized size = 0.73 \begin {gather*} \frac {2 \left (a+b x^3\right )^{3/2} \left (-16 a^3 B+24 a^2 b \left (A+B x^3\right )-6 a b^2 x^3 \left (6 A+5 B x^3\right )+5 b^3 x^6 \left (9 A+7 B x^3\right )\right )}{945 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^8*Sqrt[a + b*x^3]*(A + B*x^3),x]

[Out]

(2*(a + b*x^3)^(3/2)*(-16*a^3*B + 24*a^2*b*(A + B*x^3) - 6*a*b^2*x^3*(6*A + 5*B*x^3) + 5*b^3*x^6*(9*A + 7*B*x^
3)))/(945*b^4)

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IntegrateAlgebraic [A]  time = 0.05, size = 80, normalized size = 0.78 \begin {gather*} -\frac {2 \left (a+b x^3\right )^{3/2} \left (16 a^3 B-24 a^2 A b-24 a^2 b B x^3+36 a A b^2 x^3+30 a b^2 B x^6-45 A b^3 x^6-35 b^3 B x^9\right )}{945 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^8*Sqrt[a + b*x^3]*(A + B*x^3),x]

[Out]

(-2*(a + b*x^3)^(3/2)*(-24*a^2*A*b + 16*a^3*B + 36*a*A*b^2*x^3 - 24*a^2*b*B*x^3 - 45*A*b^3*x^6 + 30*a*b^2*B*x^
6 - 35*b^3*B*x^9))/(945*b^4)

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fricas [A]  time = 0.79, size = 99, normalized size = 0.96 \begin {gather*} \frac {2 \, {\left (35 \, B b^{4} x^{12} + 5 \, {\left (B a b^{3} + 9 \, A b^{4}\right )} x^{9} - 3 \, {\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{6} - 16 \, B a^{4} + 24 \, A a^{3} b + 4 \, {\left (2 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{945 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(B*x^3+A)*(b*x^3+a)^(1/2),x, algorithm="fricas")

[Out]

2/945*(35*B*b^4*x^12 + 5*(B*a*b^3 + 9*A*b^4)*x^9 - 3*(2*B*a^2*b^2 - 3*A*a*b^3)*x^6 - 16*B*a^4 + 24*A*a^3*b + 4
*(2*B*a^3*b - 3*A*a^2*b^2)*x^3)*sqrt(b*x^3 + a)/b^4

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giac [A]  time = 0.16, size = 104, normalized size = 1.01 \begin {gather*} \frac {2 \, {\left (35 \, {\left (b x^{3} + a\right )}^{\frac {9}{2}} B - 135 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} B a + 189 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} B a^{2} - 105 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} B a^{3} + 45 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} A b - 126 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} A a b + 105 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} A a^{2} b\right )}}{945 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(B*x^3+A)*(b*x^3+a)^(1/2),x, algorithm="giac")

[Out]

2/945*(35*(b*x^3 + a)^(9/2)*B - 135*(b*x^3 + a)^(7/2)*B*a + 189*(b*x^3 + a)^(5/2)*B*a^2 - 105*(b*x^3 + a)^(3/2
)*B*a^3 + 45*(b*x^3 + a)^(7/2)*A*b - 126*(b*x^3 + a)^(5/2)*A*a*b + 105*(b*x^3 + a)^(3/2)*A*a^2*b)/b^4

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maple [A]  time = 0.05, size = 77, normalized size = 0.75 \begin {gather*} \frac {2 \left (b \,x^{3}+a \right )^{\frac {3}{2}} \left (35 B \,x^{9} b^{3}+45 A \,b^{3} x^{6}-30 B a \,b^{2} x^{6}-36 A a \,b^{2} x^{3}+24 B \,a^{2} b \,x^{3}+24 A \,a^{2} b -16 B \,a^{3}\right )}{945 b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(B*x^3+A)*(b*x^3+a)^(1/2),x)

[Out]

2/945*(b*x^3+a)^(3/2)*(35*B*b^3*x^9+45*A*b^3*x^6-30*B*a*b^2*x^6-36*A*a*b^2*x^3+24*B*a^2*b*x^3+24*A*a^2*b-16*B*
a^3)/b^4

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maxima [A]  time = 0.62, size = 118, normalized size = 1.15 \begin {gather*} \frac {2}{945} \, B {\left (\frac {35 \, {\left (b x^{3} + a\right )}^{\frac {9}{2}}}{b^{4}} - \frac {135 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} a}{b^{4}} + \frac {189 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a^{2}}{b^{4}} - \frac {105 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{3}}{b^{4}}\right )} + \frac {2}{315} \, A {\left (\frac {15 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}}}{b^{3}} - \frac {42 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a}{b^{3}} + \frac {35 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2}}{b^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(B*x^3+A)*(b*x^3+a)^(1/2),x, algorithm="maxima")

[Out]

2/945*B*(35*(b*x^3 + a)^(9/2)/b^4 - 135*(b*x^3 + a)^(7/2)*a/b^4 + 189*(b*x^3 + a)^(5/2)*a^2/b^4 - 105*(b*x^3 +
a)^(3/2)*a^3/b^4) + 2/315*A*(15*(b*x^3 + a)^(7/2)/b^3 - 42*(b*x^3 + a)^(5/2)*a/b^3 + 35*(b*x^3 + a)^(3/2)*a^2
/b^3)

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mupad [B]  time = 2.72, size = 154, normalized size = 1.50 \begin {gather*} \frac {2\,B\,x^{12}\,\sqrt {b\,x^3+a}}{27}+\frac {x^9\,\sqrt {b\,x^3+a}\,\left (2\,A\,b+\frac {2\,B\,a}{9}\right )}{21\,b}+\frac {8\,a^2\,\left (2\,A\,a-\frac {6\,a\,\left (2\,A\,b+\frac {2\,B\,a}{9}\right )}{7\,b}\right )\,\sqrt {b\,x^3+a}}{45\,b^3}+\frac {x^6\,\left (2\,A\,a-\frac {6\,a\,\left (2\,A\,b+\frac {2\,B\,a}{9}\right )}{7\,b}\right )\,\sqrt {b\,x^3+a}}{15\,b}-\frac {4\,a\,x^3\,\left (2\,A\,a-\frac {6\,a\,\left (2\,A\,b+\frac {2\,B\,a}{9}\right )}{7\,b}\right )\,\sqrt {b\,x^3+a}}{45\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(A + B*x^3)*(a + b*x^3)^(1/2),x)

[Out]

(2*B*x^12*(a + b*x^3)^(1/2))/27 + (x^9*(a + b*x^3)^(1/2)*(2*A*b + (2*B*a)/9))/(21*b) + (8*a^2*(2*A*a - (6*a*(2
*A*b + (2*B*a)/9))/(7*b))*(a + b*x^3)^(1/2))/(45*b^3) + (x^6*(2*A*a - (6*a*(2*A*b + (2*B*a)/9))/(7*b))*(a + b*
x^3)^(1/2))/(15*b) - (4*a*x^3*(2*A*a - (6*a*(2*A*b + (2*B*a)/9))/(7*b))*(a + b*x^3)^(1/2))/(45*b^2)

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sympy [A]  time = 4.17, size = 219, normalized size = 2.13 \begin {gather*} \begin {cases} \frac {16 A a^{3} \sqrt {a + b x^{3}}}{315 b^{3}} - \frac {8 A a^{2} x^{3} \sqrt {a + b x^{3}}}{315 b^{2}} + \frac {2 A a x^{6} \sqrt {a + b x^{3}}}{105 b} + \frac {2 A x^{9} \sqrt {a + b x^{3}}}{21} - \frac {32 B a^{4} \sqrt {a + b x^{3}}}{945 b^{4}} + \frac {16 B a^{3} x^{3} \sqrt {a + b x^{3}}}{945 b^{3}} - \frac {4 B a^{2} x^{6} \sqrt {a + b x^{3}}}{315 b^{2}} + \frac {2 B a x^{9} \sqrt {a + b x^{3}}}{189 b} + \frac {2 B x^{12} \sqrt {a + b x^{3}}}{27} & \text {for}\: b \neq 0 \\\sqrt {a} \left (\frac {A x^{9}}{9} + \frac {B x^{12}}{12}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(B*x**3+A)*(b*x**3+a)**(1/2),x)

[Out]

Piecewise((16*A*a**3*sqrt(a + b*x**3)/(315*b**3) - 8*A*a**2*x**3*sqrt(a + b*x**3)/(315*b**2) + 2*A*a*x**6*sqrt
(a + b*x**3)/(105*b) + 2*A*x**9*sqrt(a + b*x**3)/21 - 32*B*a**4*sqrt(a + b*x**3)/(945*b**4) + 16*B*a**3*x**3*s
qrt(a + b*x**3)/(945*b**3) - 4*B*a**2*x**6*sqrt(a + b*x**3)/(315*b**2) + 2*B*a*x**9*sqrt(a + b*x**3)/(189*b) +
2*B*x**12*sqrt(a + b*x**3)/27, Ne(b, 0)), (sqrt(a)*(A*x**9/9 + B*x**12/12), True))

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