3.1.29 \(\int x^3 (a+b x^2)^5 (A+B x^2) \, dx\) [29]

Optimal. Leaf size=67 \[ -\frac {a (A b-a B) \left (a+b x^2\right )^6}{12 b^3}+\frac {(A b-2 a B) \left (a+b x^2\right )^7}{14 b^3}+\frac {B \left (a+b x^2\right )^8}{16 b^3} \]

[Out]

-1/12*a*(A*b-B*a)*(b*x^2+a)^6/b^3+1/14*(A*b-2*B*a)*(b*x^2+a)^7/b^3+1/16*B*(b*x^2+a)^8/b^3

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Rubi [A]
time = 0.10, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {457, 77} \begin {gather*} \frac {\left (a+b x^2\right )^7 (A b-2 a B)}{14 b^3}-\frac {a \left (a+b x^2\right )^6 (A b-a B)}{12 b^3}+\frac {B \left (a+b x^2\right )^8}{16 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^2)^5*(A + B*x^2),x]

[Out]

-1/12*(a*(A*b - a*B)*(a + b*x^2)^6)/b^3 + ((A*b - 2*a*B)*(a + b*x^2)^7)/(14*b^3) + (B*(a + b*x^2)^8)/(16*b^3)

Rule 77

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

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^3 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx &=\frac {1}{2} \text {Subst}\left (\int x (a+b x)^5 (A+B x) \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {a (-A b+a B) (a+b x)^5}{b^2}+\frac {(A b-2 a B) (a+b x)^6}{b^2}+\frac {B (a+b x)^7}{b^2}\right ) \, dx,x,x^2\right )\\ &=-\frac {a (A b-a B) \left (a+b x^2\right )^6}{12 b^3}+\frac {(A b-2 a B) \left (a+b x^2\right )^7}{14 b^3}+\frac {B \left (a+b x^2\right )^8}{16 b^3}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 114, normalized size = 1.70 \begin {gather*} \frac {1}{4} a^5 A x^4+\frac {1}{6} a^4 (5 A b+a B) x^6+\frac {5}{8} a^3 b (2 A b+a B) x^8+a^2 b^2 (A b+a B) x^{10}+\frac {5}{12} a b^3 (A b+2 a B) x^{12}+\frac {1}{14} b^4 (A b+5 a B) x^{14}+\frac {1}{16} b^5 B x^{16} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^5*A*x^4)/4 + (a^4*(5*A*b + a*B)*x^6)/6 + (5*a^3*b*(2*A*b + a*B)*x^8)/8 + a^2*b^2*(A*b + a*B)*x^10 + (5*a*b^
3*(A*b + 2*a*B)*x^12)/12 + (b^4*(A*b + 5*a*B)*x^14)/14 + (b^5*B*x^16)/16

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(123\) vs. \(2(61)=122\).
time = 0.08, size = 124, normalized size = 1.85

method	result	size
norman	\(\frac {a^{5} A \,x^{4}}{4}+\left (\frac {5}{6} a^{4} b A +\frac {1}{6} a^{5} B \right ) x^{6}+\left (\frac {5}{4} a^{3} b^{2} A +\frac {5}{8} a^{4} b B \right ) x^{8}+\left (a^{2} b^{3} A +a^{3} b^{2} B \right ) x^{10}+\left (\frac {5}{12} a \,b^{4} A +\frac {5}{6} a^{2} b^{3} B \right ) x^{12}+\left (\frac {1}{14} b^{5} A +\frac {5}{14} a \,b^{4} B \right ) x^{14}+\frac {b^{5} B \,x^{16}}{16}\)	\(119\)
gosper	\(\frac {1}{4} a^{5} A \,x^{4}+\frac {5}{6} x^{6} a^{4} b A +\frac {1}{6} x^{6} a^{5} B +\frac {5}{4} x^{8} a^{3} b^{2} A +\frac {5}{8} x^{8} a^{4} b B +A \,a^{2} b^{3} x^{10}+B \,a^{3} b^{2} x^{10}+\frac {5}{12} x^{12} a \,b^{4} A +\frac {5}{6} x^{12} a^{2} b^{3} B +\frac {1}{14} x^{14} b^{5} A +\frac {5}{14} x^{14} a \,b^{4} B +\frac {1}{16} b^{5} B \,x^{16}\)	\(124\)
default	\(\frac {b^{5} B \,x^{16}}{16}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{14}}{14}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{12}}{12}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{10}}{10}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{8}}{8}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{6}}{6}+\frac {a^{5} A \,x^{4}}{4}\)	\(124\)
risch	\(\frac {1}{4} a^{5} A \,x^{4}+\frac {5}{6} x^{6} a^{4} b A +\frac {1}{6} x^{6} a^{5} B +\frac {5}{4} x^{8} a^{3} b^{2} A +\frac {5}{8} x^{8} a^{4} b B +A \,a^{2} b^{3} x^{10}+B \,a^{3} b^{2} x^{10}+\frac {5}{12} x^{12} a \,b^{4} A +\frac {5}{6} x^{12} a^{2} b^{3} B +\frac {1}{14} x^{14} b^{5} A +\frac {5}{14} x^{14} a \,b^{4} B +\frac {1}{16} b^{5} B \,x^{16}\)	\(124\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x^2+a)^5*(B*x^2+A),x,method=_RETURNVERBOSE)

[Out]

1/16*b^5*B*x^16+1/14*(A*b^5+5*B*a*b^4)*x^14+1/12*(5*A*a*b^4+10*B*a^2*b^3)*x^12+1/10*(10*A*a^2*b^3+10*B*a^3*b^2
)*x^10+1/8*(10*A*a^3*b^2+5*B*a^4*b)*x^8+1/6*(5*A*a^4*b+B*a^5)*x^6+1/4*a^5*A*x^4

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Maxima [A]
time = 0.28, size = 118, normalized size = 1.76 \begin {gather*} \frac {1}{16} \, B b^{5} x^{16} + \frac {1}{14} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{14} + \frac {5}{12} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{10} + \frac {1}{4} \, A a^{5} x^{4} + \frac {5}{8} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{8} + \frac {1}{6} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*B*b^5*x^16 + 1/14*(5*B*a*b^4 + A*b^5)*x^14 + 5/12*(2*B*a^2*b^3 + A*a*b^4)*x^12 + (B*a^3*b^2 + A*a^2*b^3)*
x^10 + 1/4*A*a^5*x^4 + 5/8*(B*a^4*b + 2*A*a^3*b^2)*x^8 + 1/6*(B*a^5 + 5*A*a^4*b)*x^6

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Fricas [A]
time = 0.72, size = 118, normalized size = 1.76 \begin {gather*} \frac {1}{16} \, B b^{5} x^{16} + \frac {1}{14} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{14} + \frac {5}{12} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{10} + \frac {1}{4} \, A a^{5} x^{4} + \frac {5}{8} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{8} + \frac {1}{6} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*B*b^5*x^16 + 1/14*(5*B*a*b^4 + A*b^5)*x^14 + 5/12*(2*B*a^2*b^3 + A*a*b^4)*x^12 + (B*a^3*b^2 + A*a^2*b^3)*
x^10 + 1/4*A*a^5*x^4 + 5/8*(B*a^4*b + 2*A*a^3*b^2)*x^8 + 1/6*(B*a^5 + 5*A*a^4*b)*x^6

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (58) = 116\).
time = 0.02, size = 131, normalized size = 1.96 \begin {gather*} \frac {A a^{5} x^{4}}{4} + \frac {B b^{5} x^{16}}{16} + x^{14} \left (\frac {A b^{5}}{14} + \frac {5 B a b^{4}}{14}\right ) + x^{12} \cdot \left (\frac {5 A a b^{4}}{12} + \frac {5 B a^{2} b^{3}}{6}\right ) + x^{10} \left (A a^{2} b^{3} + B a^{3} b^{2}\right ) + x^{8} \cdot \left (\frac {5 A a^{3} b^{2}}{4} + \frac {5 B a^{4} b}{8}\right ) + x^{6} \cdot \left (\frac {5 A a^{4} b}{6} + \frac {B a^{5}}{6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**5*x**4/4 + B*b**5*x**16/16 + x**14*(A*b**5/14 + 5*B*a*b**4/14) + x**12*(5*A*a*b**4/12 + 5*B*a**2*b**3/6)
+ x**10*(A*a**2*b**3 + B*a**3*b**2) + x**8*(5*A*a**3*b**2/4 + 5*B*a**4*b/8) + x**6*(5*A*a**4*b/6 + B*a**5/6)

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Giac [A]
time = 0.65, size = 123, normalized size = 1.84 \begin {gather*} \frac {1}{16} \, B b^{5} x^{16} + \frac {5}{14} \, B a b^{4} x^{14} + \frac {1}{14} \, A b^{5} x^{14} + \frac {5}{6} \, B a^{2} b^{3} x^{12} + \frac {5}{12} \, A a b^{4} x^{12} + B a^{3} b^{2} x^{10} + A a^{2} b^{3} x^{10} + \frac {5}{8} \, B a^{4} b x^{8} + \frac {5}{4} \, A a^{3} b^{2} x^{8} + \frac {1}{6} \, B a^{5} x^{6} + \frac {5}{6} \, A a^{4} b x^{6} + \frac {1}{4} \, A a^{5} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*B*b^5*x^16 + 5/14*B*a*b^4*x^14 + 1/14*A*b^5*x^14 + 5/6*B*a^2*b^3*x^12 + 5/12*A*a*b^4*x^12 + B*a^3*b^2*x^1
0 + A*a^2*b^3*x^10 + 5/8*B*a^4*b*x^8 + 5/4*A*a^3*b^2*x^8 + 1/6*B*a^5*x^6 + 5/6*A*a^4*b*x^6 + 1/4*A*a^5*x^4

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Mupad [B]
time = 0.02, size = 106, normalized size = 1.58 \begin {gather*} x^6\,\left (\frac {B\,a^5}{6}+\frac {5\,A\,b\,a^4}{6}\right )+x^{14}\,\left (\frac {A\,b^5}{14}+\frac {5\,B\,a\,b^4}{14}\right )+\frac {A\,a^5\,x^4}{4}+\frac {B\,b^5\,x^{16}}{16}+a^2\,b^2\,x^{10}\,\left (A\,b+B\,a\right )+\frac {5\,a^3\,b\,x^8\,\left (2\,A\,b+B\,a\right )}{8}+\frac {5\,a\,b^3\,x^{12}\,\left (A\,b+2\,B\,a\right )}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(A + B*x^2)*(a + b*x^2)^5,x)

[Out]

x^6*((B*a^5)/6 + (5*A*a^4*b)/6) + x^14*((A*b^5)/14 + (5*B*a*b^4)/14) + (A*a^5*x^4)/4 + (B*b^5*x^16)/16 + a^2*b
^2*x^10*(A*b + B*a) + (5*a^3*b*x^8*(2*A*b + B*a))/8 + (5*a*b^3*x^12*(A*b + 2*B*a))/12

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