3.6.3 \(\int x^5 (a+b x^2)^{3/2} (A+B x^2) \, dx\)

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

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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 {a^2 \left (a+b x^2\right )^{5/2} (A b-a B)}{5 b^4}+\frac {\left (a+b x^2\right )^{9/2} (A b-3 a B)}{9 b^4}-\frac {a \left (a+b x^2\right )^{7/2} (2 A b-3 a B)}{7 b^4}+\frac {B \left (a+b x^2\right )^{11/2}}{11 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x^2)^(3/2)*(A + B*x^2),x]

[Out]

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

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Mathematica [A]  time = 0.06, size = 78, normalized size = 0.76 \begin {gather*} \frac {\left (a+b x^2\right )^{5/2} \left (-48 a^3 B+8 a^2 b \left (11 A+15 B x^2\right )-10 a b^2 x^2 \left (22 A+21 B x^2\right )+35 b^3 x^4 \left (11 A+9 B x^2\right )\right )}{3465 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x^2)^(3/2)*(A + B*x^2),x]

[Out]

((a + b*x^2)^(5/2)*(-48*a^3*B + 35*b^3*x^4*(11*A + 9*B*x^2) + 8*a^2*b*(11*A + 15*B*x^2) - 10*a*b^2*x^2*(22*A +
21*B*x^2)))/(3465*b^4)

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IntegrateAlgebraic [A]  time = 0.06, size = 80, normalized size = 0.78 \begin {gather*} \frac {\left (a+b x^2\right )^{5/2} \left (-48 a^3 B+88 a^2 A b+120 a^2 b B x^2-220 a A b^2 x^2-210 a b^2 B x^4+385 A b^3 x^4+315 b^3 B x^6\right )}{3465 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^5*(a + b*x^2)^(3/2)*(A + B*x^2),x]

[Out]

((a + b*x^2)^(5/2)*(88*a^2*A*b - 48*a^3*B - 220*a*A*b^2*x^2 + 120*a^2*b*B*x^2 + 385*A*b^3*x^4 - 210*a*b^2*B*x^
4 + 315*b^3*B*x^6))/(3465*b^4)

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fricas [A]  time = 0.85, size = 124, normalized size = 1.20 \begin {gather*} \frac {{\left (315 \, B b^{5} x^{10} + 35 \, {\left (12 \, B a b^{4} + 11 \, A b^{5}\right )} x^{8} + 5 \, {\left (3 \, B a^{2} b^{3} + 110 \, A a b^{4}\right )} x^{6} - 48 \, B a^{5} + 88 \, A a^{4} b - 3 \, {\left (6 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{4} + 4 \, {\left (6 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{3465 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^2+a)^(3/2)*(B*x^2+A),x, algorithm="fricas")

[Out]

1/3465*(315*B*b^5*x^10 + 35*(12*B*a*b^4 + 11*A*b^5)*x^8 + 5*(3*B*a^2*b^3 + 110*A*a*b^4)*x^6 - 48*B*a^5 + 88*A*
a^4*b - 3*(6*B*a^3*b^2 - 11*A*a^2*b^3)*x^4 + 4*(6*B*a^4*b - 11*A*a^3*b^2)*x^2)*sqrt(b*x^2 + a)/b^4

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giac [A]  time = 0.43, size = 104, normalized size = 1.01 \begin {gather*} \frac {315 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} B - 1155 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} B a + 1485 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a^{2} - 693 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a^{3} + 385 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} A b - 990 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A a b + 693 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A a^{2} b}{3465 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^2+a)^(3/2)*(B*x^2+A),x, algorithm="giac")

[Out]

1/3465*(315*(b*x^2 + a)^(11/2)*B - 1155*(b*x^2 + a)^(9/2)*B*a + 1485*(b*x^2 + a)^(7/2)*B*a^2 - 693*(b*x^2 + a)
^(5/2)*B*a^3 + 385*(b*x^2 + a)^(9/2)*A*b - 990*(b*x^2 + a)^(7/2)*A*a*b + 693*(b*x^2 + a)^(5/2)*A*a^2*b)/b^4

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maple [A]  time = 0.01, size = 77, normalized size = 0.75 \begin {gather*} \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (315 B \,x^{6} b^{3}+385 A \,b^{3} x^{4}-210 B a \,b^{2} x^{4}-220 A a \,b^{2} x^{2}+120 B \,a^{2} b \,x^{2}+88 A \,a^{2} b -48 B \,a^{3}\right )}{3465 b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(b*x^2+a)^(3/2)*(B*x^2+A),x)

[Out]

1/3465*(b*x^2+a)^(5/2)*(315*B*b^3*x^6+385*A*b^3*x^4-210*B*a*b^2*x^4-220*A*a*b^2*x^2+120*B*a^2*b*x^2+88*A*a^2*b
-48*B*a^3)/b^4

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maxima [A]  time = 1.05, size = 132, normalized size = 1.28 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B x^{6}}{11 \, b} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a x^{4}}{33 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A x^{4}}{9 \, b} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a^{2} x^{2}}{231 \, b^{3}} - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A a x^{2}}{63 \, b^{2}} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a^{3}}{1155 \, b^{4}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A a^{2}}{315 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^2+a)^(3/2)*(B*x^2+A),x, algorithm="maxima")

[Out]

1/11*(b*x^2 + a)^(5/2)*B*x^6/b - 2/33*(b*x^2 + a)^(5/2)*B*a*x^4/b^2 + 1/9*(b*x^2 + a)^(5/2)*A*x^4/b + 8/231*(b
*x^2 + a)^(5/2)*B*a^2*x^2/b^3 - 4/63*(b*x^2 + a)^(5/2)*A*a*x^2/b^2 - 16/1155*(b*x^2 + a)^(5/2)*B*a^3/b^4 + 8/3
15*(b*x^2 + a)^(5/2)*A*a^2/b^3

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mupad [B]  time = 0.74, size = 117, normalized size = 1.14 \begin {gather*} \sqrt {b\,x^2+a}\,\left (\frac {x^8\,\left (385\,A\,b^5+420\,B\,a\,b^4\right )}{3465\,b^4}-\frac {48\,B\,a^5-88\,A\,a^4\,b}{3465\,b^4}+\frac {B\,b\,x^{10}}{11}+\frac {a^2\,x^4\,\left (11\,A\,b-6\,B\,a\right )}{1155\,b^2}-\frac {4\,a^3\,x^2\,\left (11\,A\,b-6\,B\,a\right )}{3465\,b^3}+\frac {a\,x^6\,\left (110\,A\,b+3\,B\,a\right )}{693\,b}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(A + B*x^2)*(a + b*x^2)^(3/2),x)

[Out]

(a + b*x^2)^(1/2)*((x^8*(385*A*b^5 + 420*B*a*b^4))/(3465*b^4) - (48*B*a^5 - 88*A*a^4*b)/(3465*b^4) + (B*b*x^10
)/11 + (a^2*x^4*(11*A*b - 6*B*a))/(1155*b^2) - (4*a^3*x^2*(11*A*b - 6*B*a))/(3465*b^3) + (a*x^6*(110*A*b + 3*B
*a))/(693*b))

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sympy [A]  time = 8.36, size = 260, normalized size = 2.52 \begin {gather*} \begin {cases} \frac {8 A a^{4} \sqrt {a + b x^{2}}}{315 b^{3}} - \frac {4 A a^{3} x^{2} \sqrt {a + b x^{2}}}{315 b^{2}} + \frac {A a^{2} x^{4} \sqrt {a + b x^{2}}}{105 b} + \frac {10 A a x^{6} \sqrt {a + b x^{2}}}{63} + \frac {A b x^{8} \sqrt {a + b x^{2}}}{9} - \frac {16 B a^{5} \sqrt {a + b x^{2}}}{1155 b^{4}} + \frac {8 B a^{4} x^{2} \sqrt {a + b x^{2}}}{1155 b^{3}} - \frac {2 B a^{3} x^{4} \sqrt {a + b x^{2}}}{385 b^{2}} + \frac {B a^{2} x^{6} \sqrt {a + b x^{2}}}{231 b} + \frac {4 B a x^{8} \sqrt {a + b x^{2}}}{33} + \frac {B b x^{10} \sqrt {a + b x^{2}}}{11} & \text {for}\: b \neq 0 \\a^{\frac {3}{2}} \left (\frac {A x^{6}}{6} + \frac {B x^{8}}{8}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(b*x**2+a)**(3/2)*(B*x**2+A),x)

[Out]

Piecewise((8*A*a**4*sqrt(a + b*x**2)/(315*b**3) - 4*A*a**3*x**2*sqrt(a + b*x**2)/(315*b**2) + A*a**2*x**4*sqrt
(a + b*x**2)/(105*b) + 10*A*a*x**6*sqrt(a + b*x**2)/63 + A*b*x**8*sqrt(a + b*x**2)/9 - 16*B*a**5*sqrt(a + b*x*
*2)/(1155*b**4) + 8*B*a**4*x**2*sqrt(a + b*x**2)/(1155*b**3) - 2*B*a**3*x**4*sqrt(a + b*x**2)/(385*b**2) + B*a
**2*x**6*sqrt(a + b*x**2)/(231*b) + 4*B*a*x**8*sqrt(a + b*x**2)/33 + B*b*x**10*sqrt(a + b*x**2)/11, Ne(b, 0)),
(a**(3/2)*(A*x**6/6 + B*x**8/8), True))

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