3.6.38 \(\int x^5 (a+b x^2)^{5/2} (A+B x^2) \, dx\) [538]

Optimal. Leaf size=103 \[ \frac {a^2 (A b-a B) \left (a+b x^2\right )^{7/2}}{7 b^4}-\frac {a (2 A b-3 a B) \left (a+b x^2\right )^{9/2}}{9 b^4}+\frac {(A b-3 a B) \left (a+b x^2\right )^{11/2}}{11 b^4}+\frac {B \left (a+b x^2\right )^{13/2}}{13 b^4} \]

[Out]

1/7*a^2*(A*b-B*a)*(b*x^2+a)^(7/2)/b^4-1/9*a*(2*A*b-3*B*a)*(b*x^2+a)^(9/2)/b^4+1/11*(A*b-3*B*a)*(b*x^2+a)^(11/2
)/b^4+1/13*B*(b*x^2+a)^(13/2)/b^4

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Rubi [A]
time = 0.05, 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 = {457, 78} \begin {gather*} \frac {a^2 \left (a+b x^2\right )^{7/2} (A b-a B)}{7 b^4}+\frac {\left (a+b x^2\right )^{11/2} (A b-3 a B)}{11 b^4}-\frac {a \left (a+b x^2\right )^{9/2} (2 A b-3 a B)}{9 b^4}+\frac {B \left (a+b x^2\right )^{13/2}}{13 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(A*b - a*B)*(a + b*x^2)^(7/2))/(7*b^4) - (a*(2*A*b - 3*a*B)*(a + b*x^2)^(9/2))/(9*b^4) + ((A*b - 3*a*B)*(
a + b*x^2)^(11/2))/(11*b^4) + (B*(a + b*x^2)^(13/2))/(13*b^4)

Rule 78

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 457

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 )^{5/2} \left (A+B x^2\right ) \, dx &=\frac {1}{2} \text {Subst}\left (\int x^2 (a+b x)^{5/2} (A+B x) \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {a^2 (-A b+a B) (a+b x)^{5/2}}{b^3}+\frac {a (-2 A b+3 a B) (a+b x)^{7/2}}{b^3}+\frac {(A b-3 a B) (a+b x)^{9/2}}{b^3}+\frac {B (a+b x)^{11/2}}{b^3}\right ) \, dx,x,x^2\right )\\ &=\frac {a^2 (A b-a B) \left (a+b x^2\right )^{7/2}}{7 b^4}-\frac {a (2 A b-3 a B) \left (a+b x^2\right )^{9/2}}{9 b^4}+\frac {(A b-3 a B) \left (a+b x^2\right )^{11/2}}{11 b^4}+\frac {B \left (a+b x^2\right )^{13/2}}{13 b^4}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 80, normalized size = 0.78 \begin {gather*} \frac {\left (a+b x^2\right )^{7/2} \left (104 a^2 A b-48 a^3 B-364 a A b^2 x^2+168 a^2 b B x^2+819 A b^3 x^4-378 a b^2 B x^4+693 b^3 B x^6\right )}{9009 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x^2)^(7/2)*(104*a^2*A*b - 48*a^3*B - 364*a*A*b^2*x^2 + 168*a^2*b*B*x^2 + 819*A*b^3*x^4 - 378*a*b^2*B*x
^4 + 693*b^3*B*x^6))/(9009*b^4)

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Maple [A]
time = 0.08, size = 144, normalized size = 1.40

method	result	size
gosper	\(\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (693 B \,x^{6} b^{3}+819 A \,b^{3} x^{4}-378 B a \,b^{2} x^{4}-364 A a \,b^{2} x^{2}+168 B \,a^{2} b \,x^{2}+104 A \,a^{2} b -48 B \,a^{3}\right )}{9009 b^{4}}\)	\(77\)
default	\(B \left (\frac {x^{6} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{13 b}-\frac {6 a \left (\frac {x^{4} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{11 b}-\frac {4 a \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 b^{2}}\right )}{11 b}\right )}{13 b}\right )+A \left (\frac {x^{4} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{11 b}-\frac {4 a \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 b^{2}}\right )}{11 b}\right )\)	\(144\)
trager	\(\frac {\left (693 B \,b^{6} x^{12}+819 A \,b^{6} x^{10}+1701 B a \,b^{5} x^{10}+2093 A a \,b^{5} x^{8}+1113 B \,a^{2} b^{4} x^{8}+1469 A \,a^{2} b^{4} x^{6}+15 B \,a^{3} b^{3} x^{6}+39 A \,a^{3} b^{3} x^{4}-18 B \,a^{4} b^{2} x^{4}-52 A \,a^{4} b^{2} x^{2}+24 B \,a^{5} b \,x^{2}+104 A \,a^{5} b -48 B \,a^{6}\right ) \sqrt {b \,x^{2}+a}}{9009 b^{4}}\)	\(149\)
risch	\(\frac {\left (693 B \,b^{6} x^{12}+819 A \,b^{6} x^{10}+1701 B a \,b^{5} x^{10}+2093 A a \,b^{5} x^{8}+1113 B \,a^{2} b^{4} x^{8}+1469 A \,a^{2} b^{4} x^{6}+15 B \,a^{3} b^{3} x^{6}+39 A \,a^{3} b^{3} x^{4}-18 B \,a^{4} b^{2} x^{4}-52 A \,a^{4} b^{2} x^{2}+24 B \,a^{5} b \,x^{2}+104 A \,a^{5} b -48 B \,a^{6}\right ) \sqrt {b \,x^{2}+a}}{9009 b^{4}}\)	\(149\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(b*x^2+a)^(5/2)*(B*x^2+A),x,method=_RETURNVERBOSE)

[Out]

B*(1/13*x^6*(b*x^2+a)^(7/2)/b-6/13*a/b*(1/11*x^4*(b*x^2+a)^(7/2)/b-4/11*a/b*(1/9*x^2*(b*x^2+a)^(7/2)/b-2/63*a/
b^2*(b*x^2+a)^(7/2))))+A*(1/11*x^4*(b*x^2+a)^(7/2)/b-4/11*a/b*(1/9*x^2*(b*x^2+a)^(7/2)/b-2/63*a/b^2*(b*x^2+a)^
(7/2)))

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Maxima [A]
time = 0.28, size = 132, normalized size = 1.28 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B x^{6}}{13 \, b} - \frac {6 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a x^{4}}{143 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A x^{4}}{11 \, b} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a^{2} x^{2}}{429 \, b^{3}} - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A a x^{2}}{99 \, b^{2}} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a^{3}}{3003 \, b^{4}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A a^{2}}{693 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*(b*x^2 + a)^(7/2)*B*x^6/b - 6/143*(b*x^2 + a)^(7/2)*B*a*x^4/b^2 + 1/11*(b*x^2 + a)^(7/2)*A*x^4/b + 8/429*
(b*x^2 + a)^(7/2)*B*a^2*x^2/b^3 - 4/99*(b*x^2 + a)^(7/2)*A*a*x^2/b^2 - 16/3003*(b*x^2 + a)^(7/2)*B*a^3/b^4 + 8
/693*(b*x^2 + a)^(7/2)*A*a^2/b^3

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Fricas [A]
time = 1.19, size = 147, normalized size = 1.43 \begin {gather*} \frac {{\left (693 \, B b^{6} x^{12} + 63 \, {\left (27 \, B a b^{5} + 13 \, A b^{6}\right )} x^{10} + 7 \, {\left (159 \, B a^{2} b^{4} + 299 \, A a b^{5}\right )} x^{8} - 48 \, B a^{6} + 104 \, A a^{5} b + {\left (15 \, B a^{3} b^{3} + 1469 \, A a^{2} b^{4}\right )} x^{6} - 3 \, {\left (6 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{4} + 4 \, {\left (6 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{9009 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9009*(693*B*b^6*x^12 + 63*(27*B*a*b^5 + 13*A*b^6)*x^10 + 7*(159*B*a^2*b^4 + 299*A*a*b^5)*x^8 - 48*B*a^6 + 10
4*A*a^5*b + (15*B*a^3*b^3 + 1469*A*a^2*b^4)*x^6 - 3*(6*B*a^4*b^2 - 13*A*a^3*b^3)*x^4 + 4*(6*B*a^5*b - 13*A*a^4
*b^2)*x^2)*sqrt(b*x^2 + a)/b^4

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (94) = 188\).
time = 0.72, size = 313, normalized size = 3.04 \begin {gather*} \begin {cases} \frac {8 A a^{5} \sqrt {a + b x^{2}}}{693 b^{3}} - \frac {4 A a^{4} x^{2} \sqrt {a + b x^{2}}}{693 b^{2}} + \frac {A a^{3} x^{4} \sqrt {a + b x^{2}}}{231 b} + \frac {113 A a^{2} x^{6} \sqrt {a + b x^{2}}}{693} + \frac {23 A a b x^{8} \sqrt {a + b x^{2}}}{99} + \frac {A b^{2} x^{10} \sqrt {a + b x^{2}}}{11} - \frac {16 B a^{6} \sqrt {a + b x^{2}}}{3003 b^{4}} + \frac {8 B a^{5} x^{2} \sqrt {a + b x^{2}}}{3003 b^{3}} - \frac {2 B a^{4} x^{4} \sqrt {a + b x^{2}}}{1001 b^{2}} + \frac {5 B a^{3} x^{6} \sqrt {a + b x^{2}}}{3003 b} + \frac {53 B a^{2} x^{8} \sqrt {a + b x^{2}}}{429} + \frac {27 B a b x^{10} \sqrt {a + b x^{2}}}{143} + \frac {B b^{2} x^{12} \sqrt {a + b x^{2}}}{13} & \text {for}\: b \neq 0 \\a^{\frac {5}{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)**(5/2)*(B*x**2+A),x)

[Out]

Piecewise((8*A*a**5*sqrt(a + b*x**2)/(693*b**3) - 4*A*a**4*x**2*sqrt(a + b*x**2)/(693*b**2) + A*a**3*x**4*sqrt
(a + b*x**2)/(231*b) + 113*A*a**2*x**6*sqrt(a + b*x**2)/693 + 23*A*a*b*x**8*sqrt(a + b*x**2)/99 + A*b**2*x**10
*sqrt(a + b*x**2)/11 - 16*B*a**6*sqrt(a + b*x**2)/(3003*b**4) + 8*B*a**5*x**2*sqrt(a + b*x**2)/(3003*b**3) - 2
*B*a**4*x**4*sqrt(a + b*x**2)/(1001*b**2) + 5*B*a**3*x**6*sqrt(a + b*x**2)/(3003*b) + 53*B*a**2*x**8*sqrt(a +
b*x**2)/429 + 27*B*a*b*x**10*sqrt(a + b*x**2)/143 + B*b**2*x**12*sqrt(a + b*x**2)/13, Ne(b, 0)), (a**(5/2)*(A*
x**6/6 + B*x**8/8), True))

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Giac [A]
time = 0.63, size = 104, normalized size = 1.01 \begin {gather*} \frac {693 \, {\left (b x^{2} + a\right )}^{\frac {13}{2}} B - 2457 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} B a + 3003 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} B a^{2} - 1287 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a^{3} + 819 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} A b - 2002 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} A a b + 1287 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A a^{2} b}{9009 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9009*(693*(b*x^2 + a)^(13/2)*B - 2457*(b*x^2 + a)^(11/2)*B*a + 3003*(b*x^2 + a)^(9/2)*B*a^2 - 1287*(b*x^2 +
a)^(7/2)*B*a^3 + 819*(b*x^2 + a)^(11/2)*A*b - 2002*(b*x^2 + a)^(9/2)*A*a*b + 1287*(b*x^2 + a)^(7/2)*A*a^2*b)/b
^4

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Mupad [B]
time = 0.38, size = 136, normalized size = 1.32 \begin {gather*} \sqrt {b\,x^2+a}\,\left (\frac {B\,b^2\,x^{12}}{13}-\frac {48\,B\,a^6-104\,A\,a^5\,b}{9009\,b^4}+\frac {x^{10}\,\left (819\,A\,b^6+1701\,B\,a\,b^5\right )}{9009\,b^4}+\frac {a\,x^8\,\left (299\,A\,b+159\,B\,a\right )}{1287}+\frac {a^3\,x^4\,\left (13\,A\,b-6\,B\,a\right )}{3003\,b^2}-\frac {4\,a^4\,x^2\,\left (13\,A\,b-6\,B\,a\right )}{9009\,b^3}+\frac {a^2\,x^6\,\left (1469\,A\,b+15\,B\,a\right )}{9009\,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)^(5/2),x)

[Out]

(a + b*x^2)^(1/2)*((B*b^2*x^12)/13 - (48*B*a^6 - 104*A*a^5*b)/(9009*b^4) + (x^10*(819*A*b^6 + 1701*B*a*b^5))/(
9009*b^4) + (a*x^8*(299*A*b + 159*B*a))/1287 + (a^3*x^4*(13*A*b - 6*B*a))/(3003*b^2) - (4*a^4*x^2*(13*A*b - 6*
B*a))/(9009*b^3) + (a^2*x^6*(1469*A*b + 15*B*a))/(9009*b))

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