3.196 \(\int x^8 (a+b x^3)^{3/2} (A+B x^3) \, dx\)

Optimal. Leaf size=103 \[ \frac{2 a^2 \left (a+b x^3\right )^{5/2} (A b-a B)}{15 b^4}+\frac{2 \left (a+b x^3\right )^{9/2} (A b-3 a B)}{27 b^4}-\frac{2 a \left (a+b x^3\right )^{7/2} (2 A b-3 a B)}{21 b^4}+\frac{2 B \left (a+b x^3\right )^{11/2}}{33 b^4} \]

[Out]

(2*a^2*(A*b - a*B)*(a + b*x^3)^(5/2))/(15*b^4) - (2*a*(2*A*b - 3*a*B)*(a + b*x^3)^(7/2))/(21*b^4) + (2*(A*b -
3*a*B)*(a + b*x^3)^(9/2))/(27*b^4) + (2*B*(a + b*x^3)^(11/2))/(33*b^4)

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Rubi [A]  time = 0.0789919, antiderivative size = 103, normalized size of antiderivative = 1., 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} \[ \frac{2 a^2 \left (a+b x^3\right )^{5/2} (A b-a B)}{15 b^4}+\frac{2 \left (a+b x^3\right )^{9/2} (A b-3 a B)}{27 b^4}-\frac{2 a \left (a+b x^3\right )^{7/2} (2 A b-3 a B)}{21 b^4}+\frac{2 B \left (a+b x^3\right )^{11/2}}{33 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^8*(a + b*x^3)^(3/2)*(A + B*x^3),x]

[Out]

(2*a^2*(A*b - a*B)*(a + b*x^3)^(5/2))/(15*b^4) - (2*a*(2*A*b - 3*a*B)*(a + b*x^3)^(7/2))/(21*b^4) + (2*(A*b -
3*a*B)*(a + b*x^3)^(9/2))/(27*b^4) + (2*B*(a + b*x^3)^(11/2))/(33*b^4)

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]]

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]))))

Rubi steps

\begin{align*} \int x^8 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^2 (a+b x)^{3/2} (A+B x) \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a^2 (-A b+a B) (a+b x)^{3/2}}{b^3}+\frac{a (-2 A b+3 a B) (a+b x)^{5/2}}{b^3}+\frac{(A b-3 a B) (a+b x)^{7/2}}{b^3}+\frac{B (a+b x)^{9/2}}{b^3}\right ) \, dx,x,x^3\right )\\ &=\frac{2 a^2 (A b-a B) \left (a+b x^3\right )^{5/2}}{15 b^4}-\frac{2 a (2 A b-3 a B) \left (a+b x^3\right )^{7/2}}{21 b^4}+\frac{2 (A b-3 a B) \left (a+b x^3\right )^{9/2}}{27 b^4}+\frac{2 B \left (a+b x^3\right )^{11/2}}{33 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0598628, size = 78, normalized size = 0.76 \[ \frac{2 \left (a+b x^3\right )^{5/2} \left (8 a^2 b \left (11 A+15 B x^3\right )-48 a^3 B-10 a b^2 x^3 \left (22 A+21 B x^3\right )+35 b^3 x^6 \left (11 A+9 B x^3\right )\right )}{10395 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8*(a + b*x^3)^(3/2)*(A + B*x^3),x]

[Out]

(2*(a + b*x^3)^(5/2)*(-48*a^3*B + 35*b^3*x^6*(11*A + 9*B*x^3) + 8*a^2*b*(11*A + 15*B*x^3) - 10*a*b^2*x^3*(22*A
+ 21*B*x^3)))/(10395*b^4)

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Maple [A]  time = 0.007, size = 77, normalized size = 0.8 \begin{align*}{\frac{630\,B{x}^{9}{b}^{3}+770\,A{b}^{3}{x}^{6}-420\,Ba{b}^{2}{x}^{6}-440\,Aa{b}^{2}{x}^{3}+240\,B{a}^{2}b{x}^{3}+176\,A{a}^{2}b-96\,B{a}^{3}}{10395\,{b}^{4}} \left ( b{x}^{3}+a \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/10395*(b*x^3+a)^(5/2)*(315*B*b^3*x^9+385*A*b^3*x^6-210*B*a*b^2*x^6-220*A*a*b^2*x^3+120*B*a^2*b*x^3+88*A*a^2*
b-48*B*a^3)/b^4

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Maxima [A]  time = 0.949675, size = 159, normalized size = 1.54 \begin{align*} \frac{2}{945} \,{\left (\frac{35 \,{\left (b x^{3} + a\right )}^{\frac{9}{2}}}{b^{3}} - \frac{90 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} a}{b^{3}} + \frac{63 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a^{2}}{b^{3}}\right )} A + \frac{2}{3465} \,{\left (\frac{105 \,{\left (b x^{3} + a\right )}^{\frac{11}{2}}}{b^{4}} - \frac{385 \,{\left (b x^{3} + a\right )}^{\frac{9}{2}} a}{b^{4}} + \frac{495 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} a^{2}}{b^{4}} - \frac{231 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a^{3}}{b^{4}}\right )} B \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/945*(35*(b*x^3 + a)^(9/2)/b^3 - 90*(b*x^3 + a)^(7/2)*a/b^3 + 63*(b*x^3 + a)^(5/2)*a^2/b^3)*A + 2/3465*(105*(
b*x^3 + a)^(11/2)/b^4 - 385*(b*x^3 + a)^(9/2)*a/b^4 + 495*(b*x^3 + a)^(7/2)*a^2/b^4 - 231*(b*x^3 + a)^(5/2)*a^
3/b^4)*B

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Fricas [A]  time = 1.6951, size = 286, normalized size = 2.78 \begin{align*} \frac{2 \,{\left (315 \, B b^{5} x^{15} + 35 \,{\left (12 \, B a b^{4} + 11 \, A b^{5}\right )} x^{12} + 5 \,{\left (3 \, B a^{2} b^{3} + 110 \, A a b^{4}\right )} x^{9} - 3 \,{\left (6 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{6} - 48 \, B a^{5} + 88 \, A a^{4} b + 4 \,{\left (6 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x^{3}\right )} \sqrt{b x^{3} + a}}{10395 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/10395*(315*B*b^5*x^15 + 35*(12*B*a*b^4 + 11*A*b^5)*x^12 + 5*(3*B*a^2*b^3 + 110*A*a*b^4)*x^9 - 3*(6*B*a^3*b^2
- 11*A*a^2*b^3)*x^6 - 48*B*a^5 + 88*A*a^4*b + 4*(6*B*a^4*b - 11*A*a^3*b^2)*x^3)*sqrt(b*x^3 + a)/b^4

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Sympy [A]  time = 8.34159, size = 267, normalized size = 2.59 \begin{align*} \begin{cases} \frac{16 A a^{4} \sqrt{a + b x^{3}}}{945 b^{3}} - \frac{8 A a^{3} x^{3} \sqrt{a + b x^{3}}}{945 b^{2}} + \frac{2 A a^{2} x^{6} \sqrt{a + b x^{3}}}{315 b} + \frac{20 A a x^{9} \sqrt{a + b x^{3}}}{189} + \frac{2 A b x^{12} \sqrt{a + b x^{3}}}{27} - \frac{32 B a^{5} \sqrt{a + b x^{3}}}{3465 b^{4}} + \frac{16 B a^{4} x^{3} \sqrt{a + b x^{3}}}{3465 b^{3}} - \frac{4 B a^{3} x^{6} \sqrt{a + b x^{3}}}{1155 b^{2}} + \frac{2 B a^{2} x^{9} \sqrt{a + b x^{3}}}{693 b} + \frac{8 B a x^{12} \sqrt{a + b x^{3}}}{99} + \frac{2 B b x^{15} \sqrt{a + b x^{3}}}{33} & \text{for}\: b \neq 0 \\a^{\frac{3}{2}} \left (\frac{A x^{9}}{9} + \frac{B x^{12}}{12}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((16*A*a**4*sqrt(a + b*x**3)/(945*b**3) - 8*A*a**3*x**3*sqrt(a + b*x**3)/(945*b**2) + 2*A*a**2*x**6*s
qrt(a + b*x**3)/(315*b) + 20*A*a*x**9*sqrt(a + b*x**3)/189 + 2*A*b*x**12*sqrt(a + b*x**3)/27 - 32*B*a**5*sqrt(
a + b*x**3)/(3465*b**4) + 16*B*a**4*x**3*sqrt(a + b*x**3)/(3465*b**3) - 4*B*a**3*x**6*sqrt(a + b*x**3)/(1155*b
**2) + 2*B*a**2*x**9*sqrt(a + b*x**3)/(693*b) + 8*B*a*x**12*sqrt(a + b*x**3)/99 + 2*B*b*x**15*sqrt(a + b*x**3)
/33, Ne(b, 0)), (a**(3/2)*(A*x**9/9 + B*x**12/12), True))

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Giac [B]  time = 1.1598, size = 323, normalized size = 3.14 \begin{align*} \frac{2 \,{\left (\frac{33 \,{\left (15 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{2}\right )} A a}{b^{2}} + \frac{11 \,{\left (35 \,{\left (b x^{3} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{3}\right )} B a}{b^{3}} + \frac{11 \,{\left (35 \,{\left (b x^{3} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{3}\right )} A}{b^{2}} + \frac{{\left (315 \,{\left (b x^{3} + a\right )}^{\frac{11}{2}} - 1540 \,{\left (b x^{3} + a\right )}^{\frac{9}{2}} a + 2970 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a^{3} + 1155 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{4}\right )} B}{b^{3}}\right )}}{10395 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/10395*(33*(15*(b*x^3 + a)^(7/2) - 42*(b*x^3 + a)^(5/2)*a + 35*(b*x^3 + a)^(3/2)*a^2)*A*a/b^2 + 11*(35*(b*x^3
+ a)^(9/2) - 135*(b*x^3 + a)^(7/2)*a + 189*(b*x^3 + a)^(5/2)*a^2 - 105*(b*x^3 + a)^(3/2)*a^3)*B*a/b^3 + 11*(3
5*(b*x^3 + a)^(9/2) - 135*(b*x^3 + a)^(7/2)*a + 189*(b*x^3 + a)^(5/2)*a^2 - 105*(b*x^3 + a)^(3/2)*a^3)*A/b^2 +
(315*(b*x^3 + a)^(11/2) - 1540*(b*x^3 + a)^(9/2)*a + 2970*(b*x^3 + a)^(7/2)*a^2 - 2772*(b*x^3 + a)^(5/2)*a^3
+ 1155*(b*x^3 + a)^(3/2)*a^4)*B/b^3)/b