3.1.100 \(\int \frac {(A+B x^2) (a+b x^2+c x^4)^3}{x^2} \, dx\) [100]

Optimal. Leaf size=156 \[ -\frac {a^3 A}{x}+a^2 (3 A b+a B) x+a \left (a b B+A \left (b^2+a c\right )\right ) x^3+\frac {1}{5} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^5+\frac {1}{7} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^7+\frac {1}{3} c \left (b^2 B+A b c+a B c\right ) x^9+\frac {1}{11} c^2 (3 b B+A c) x^{11}+\frac {1}{13} B c^3 x^{13} \]

[Out]

-a^3*A/x+a^2*(3*A*b+B*a)*x+a*(a*b*B+A*(a*c+b^2))*x^3+1/5*(3*a*B*(a*c+b^2)+A*(6*a*b*c+b^3))*x^5+1/7*(3*A*a*c^2+
3*A*b^2*c+6*B*a*b*c+B*b^3)*x^7+1/3*c*(A*b*c+B*a*c+B*b^2)*x^9+1/11*c^2*(A*c+3*B*b)*x^11+1/13*B*c^3*x^13

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Rubi [A]
time = 0.07, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {1275} \begin {gather*} -\frac {a^3 A}{x}+a^2 x (a B+3 A b)+\frac {1}{3} c x^9 \left (a B c+A b c+b^2 B\right )+a x^3 \left (A \left (a c+b^2\right )+a b B\right )+\frac {1}{7} x^7 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac {1}{5} x^5 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {1}{11} c^2 x^{11} (A c+3 b B)+\frac {1}{13} B c^3 x^{13} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x^2)*(a + b*x^2 + c*x^4)^3)/x^2,x]

[Out]

-((a^3*A)/x) + a^2*(3*A*b + a*B)*x + a*(a*b*B + A*(b^2 + a*c))*x^3 + ((3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*
x^5)/5 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^7)/7 + (c*(b^2*B + A*b*c + a*B*c)*x^9)/3 + (c^2*(3*b*B
+ A*c)*x^11)/11 + (B*c^3*x^13)/13

Rule 1275

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] &&
NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

\begin {align*} \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^3}{x^2} \, dx &=\int \left (a^2 (3 A b+a B)+\frac {a^3 A}{x^2}+3 a \left (a b B+A \left (b^2+a c\right )\right ) x^2+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^4+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^6+3 c \left (b^2 B+A b c+a B c\right ) x^8+c^2 (3 b B+A c) x^{10}+B c^3 x^{12}\right ) \, dx\\ &=-\frac {a^3 A}{x}+a^2 (3 A b+a B) x+a \left (a b B+A \left (b^2+a c\right )\right ) x^3+\frac {1}{5} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^5+\frac {1}{7} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^7+\frac {1}{3} c \left (b^2 B+A b c+a B c\right ) x^9+\frac {1}{11} c^2 (3 b B+A c) x^{11}+\frac {1}{13} B c^3 x^{13}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 156, normalized size = 1.00 \begin {gather*} -\frac {a^3 A}{x}+a^2 (3 A b+a B) x+a \left (a b B+A \left (b^2+a c\right )\right ) x^3+\frac {1}{5} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^5+\frac {1}{7} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^7+\frac {1}{3} c \left (b^2 B+A b c+a B c\right ) x^9+\frac {1}{11} c^2 (3 b B+A c) x^{11}+\frac {1}{13} B c^3 x^{13} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x^2)*(a + b*x^2 + c*x^4)^3)/x^2,x]

[Out]

-((a^3*A)/x) + a^2*(3*A*b + a*B)*x + a*(a*b*B + A*(b^2 + a*c))*x^3 + ((3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*
x^5)/5 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^7)/7 + (c*(b^2*B + A*b*c + a*B*c)*x^9)/3 + (c^2*(3*b*B
+ A*c)*x^11)/11 + (B*c^3*x^13)/13

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Maple [A]
time = 0.02, size = 186, normalized size = 1.19

method	result	size
norman	\(\frac {-a^{3} A +\left (3 a^{2} b A +a^{3} B \right ) x^{2}+\left (a^{2} c A +A a \,b^{2}+B \,a^{2} b \right ) x^{4}+\left (\frac {6}{5} A a b c +\frac {1}{5} A \,b^{3}+\frac {3}{5} a^{2} c B +\frac {3}{5} B a \,b^{2}\right ) x^{6}+\left (\frac {3}{7} c^{2} a A +\frac {3}{7} A \,b^{2} c +\frac {6}{7} a b B c +\frac {1}{7} b^{3} B \right ) x^{8}+\left (\frac {1}{3} b \,c^{2} A +\frac {1}{3} c^{2} a B +\frac {1}{3} B \,b^{2} c \right ) x^{10}+\left (\frac {1}{11} c^{3} A +\frac {3}{11} B b \,c^{2}\right ) x^{12}+\frac {B \,c^{3} x^{14}}{13}}{x}\)	\(167\)
default	\(\frac {B \,c^{3} x^{13}}{13}+\frac {A \,c^{3} x^{11}}{11}+\frac {3 B b \,c^{2} x^{11}}{11}+\frac {A b \,c^{2} x^{9}}{3}+\frac {B a \,c^{2} x^{9}}{3}+\frac {B \,b^{2} c \,x^{9}}{3}+\frac {3 A a \,c^{2} x^{7}}{7}+\frac {3 A \,b^{2} c \,x^{7}}{7}+\frac {6 B a b c \,x^{7}}{7}+\frac {B \,b^{3} x^{7}}{7}+\frac {6 A a b c \,x^{5}}{5}+\frac {A \,b^{3} x^{5}}{5}+\frac {3 a^{2} c B \,x^{5}}{5}+\frac {3 B a \,b^{2} x^{5}}{5}+x^{3} a^{2} c A +A a \,b^{2} x^{3}+B \,a^{2} b \,x^{3}+3 a^{2} b A x +a^{3} B x -\frac {a^{3} A}{x}\)	\(186\)
risch	\(\frac {B \,c^{3} x^{13}}{13}+\frac {A \,c^{3} x^{11}}{11}+\frac {3 B b \,c^{2} x^{11}}{11}+\frac {A b \,c^{2} x^{9}}{3}+\frac {B a \,c^{2} x^{9}}{3}+\frac {B \,b^{2} c \,x^{9}}{3}+\frac {3 A a \,c^{2} x^{7}}{7}+\frac {3 A \,b^{2} c \,x^{7}}{7}+\frac {6 B a b c \,x^{7}}{7}+\frac {B \,b^{3} x^{7}}{7}+\frac {6 A a b c \,x^{5}}{5}+\frac {A \,b^{3} x^{5}}{5}+\frac {3 a^{2} c B \,x^{5}}{5}+\frac {3 B a \,b^{2} x^{5}}{5}+x^{3} a^{2} c A +A a \,b^{2} x^{3}+B \,a^{2} b \,x^{3}+3 a^{2} b A x +a^{3} B x -\frac {a^{3} A}{x}\)	\(186\)
gosper	\(-\frac {-1155 B \,c^{3} x^{14}-1365 A \,c^{3} x^{12}-4095 B b \,c^{2} x^{12}-5005 A b \,c^{2} x^{10}-5005 B a \,c^{2} x^{10}-5005 B \,b^{2} c \,x^{10}-6435 A a \,c^{2} x^{8}-6435 A \,b^{2} c \,x^{8}-12870 B a b c \,x^{8}-2145 B \,b^{3} x^{8}-18018 A a b c \,x^{6}-3003 A \,b^{3} x^{6}-9009 B \,a^{2} c \,x^{6}-9009 B a \,b^{2} x^{6}-15015 x^{4} a^{2} c A -15015 A a \,b^{2} x^{4}-15015 B \,a^{2} b \,x^{4}-45045 a^{2} A b \,x^{2}-15015 a^{3} B \,x^{2}+15015 a^{3} A}{15015 x}\)	\(196\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)*(c*x^4+b*x^2+a)^3/x^2,x,method=_RETURNVERBOSE)

[Out]

1/13*B*c^3*x^13+1/11*A*c^3*x^11+3/11*B*b*c^2*x^11+1/3*A*b*c^2*x^9+1/3*B*a*c^2*x^9+1/3*B*b^2*c*x^9+3/7*A*a*c^2*
x^7+3/7*A*b^2*c*x^7+6/7*B*a*b*c*x^7+1/7*B*b^3*x^7+6/5*A*a*b*c*x^5+1/5*A*b^3*x^5+3/5*a^2*c*B*x^5+3/5*B*a*b^2*x^
5+x^3*a^2*c*A+A*a*b^2*x^3+B*a^2*b*x^3+3*a^2*b*A*x+a^3*B*x-a^3*A/x

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Maxima [A]
time = 0.28, size = 162, normalized size = 1.04 \begin {gather*} \frac {1}{13} \, B c^{3} x^{13} + \frac {1}{11} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{11} + \frac {1}{3} \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{9} + \frac {1}{7} \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{7} + \frac {1}{5} \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{5} + {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{3} - \frac {A a^{3}}{x} + {\left (B a^{3} + 3 \, A a^{2} b\right )} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)*(c*x^4+b*x^2+a)^3/x^2,x, algorithm="maxima")

[Out]

1/13*B*c^3*x^13 + 1/11*(3*B*b*c^2 + A*c^3)*x^11 + 1/3*(B*b^2*c + (B*a + A*b)*c^2)*x^9 + 1/7*(B*b^3 + 3*A*a*c^2
+ 3*(2*B*a*b + A*b^2)*c)*x^7 + 1/5*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^5 + (B*a^2*b + A*a*b^2 + A*a
^2*c)*x^3 - A*a^3/x + (B*a^3 + 3*A*a^2*b)*x

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Fricas [A]
time = 0.34, size = 168, normalized size = 1.08 \begin {gather*} \frac {1155 \, B c^{3} x^{14} + 1365 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{12} + 5005 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{10} + 2145 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{8} + 3003 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{6} + 15015 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{4} - 15015 \, A a^{3} + 15015 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}}{15015 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)*(c*x^4+b*x^2+a)^3/x^2,x, algorithm="fricas")

[Out]

1/15015*(1155*B*c^3*x^14 + 1365*(3*B*b*c^2 + A*c^3)*x^12 + 5005*(B*b^2*c + (B*a + A*b)*c^2)*x^10 + 2145*(B*b^3
+ 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^8 + 3003*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^6 + 15015*(B*a^
2*b + A*a*b^2 + A*a^2*c)*x^4 - 15015*A*a^3 + 15015*(B*a^3 + 3*A*a^2*b)*x^2)/x

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Sympy [A]
time = 0.12, size = 185, normalized size = 1.19 \begin {gather*} - \frac {A a^{3}}{x} + \frac {B c^{3} x^{13}}{13} + x^{11} \left (\frac {A c^{3}}{11} + \frac {3 B b c^{2}}{11}\right ) + x^{9} \left (\frac {A b c^{2}}{3} + \frac {B a c^{2}}{3} + \frac {B b^{2} c}{3}\right ) + x^{7} \cdot \left (\frac {3 A a c^{2}}{7} + \frac {3 A b^{2} c}{7} + \frac {6 B a b c}{7} + \frac {B b^{3}}{7}\right ) + x^{5} \cdot \left (\frac {6 A a b c}{5} + \frac {A b^{3}}{5} + \frac {3 B a^{2} c}{5} + \frac {3 B a b^{2}}{5}\right ) + x^{3} \left (A a^{2} c + A a b^{2} + B a^{2} b\right ) + x \left (3 A a^{2} b + B a^{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)*(c*x**4+b*x**2+a)**3/x**2,x)

[Out]

-A*a**3/x + B*c**3*x**13/13 + x**11*(A*c**3/11 + 3*B*b*c**2/11) + x**9*(A*b*c**2/3 + B*a*c**2/3 + B*b**2*c/3)
+ x**7*(3*A*a*c**2/7 + 3*A*b**2*c/7 + 6*B*a*b*c/7 + B*b**3/7) + x**5*(6*A*a*b*c/5 + A*b**3/5 + 3*B*a**2*c/5 +
3*B*a*b**2/5) + x**3*(A*a**2*c + A*a*b**2 + B*a**2*b) + x*(3*A*a**2*b + B*a**3)

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Giac [A]
time = 4.84, size = 185, normalized size = 1.19 \begin {gather*} \frac {1}{13} \, B c^{3} x^{13} + \frac {3}{11} \, B b c^{2} x^{11} + \frac {1}{11} \, A c^{3} x^{11} + \frac {1}{3} \, B b^{2} c x^{9} + \frac {1}{3} \, B a c^{2} x^{9} + \frac {1}{3} \, A b c^{2} x^{9} + \frac {1}{7} \, B b^{3} x^{7} + \frac {6}{7} \, B a b c x^{7} + \frac {3}{7} \, A b^{2} c x^{7} + \frac {3}{7} \, A a c^{2} x^{7} + \frac {3}{5} \, B a b^{2} x^{5} + \frac {1}{5} \, A b^{3} x^{5} + \frac {3}{5} \, B a^{2} c x^{5} + \frac {6}{5} \, A a b c x^{5} + B a^{2} b x^{3} + A a b^{2} x^{3} + A a^{2} c x^{3} + B a^{3} x + 3 \, A a^{2} b x - \frac {A a^{3}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)*(c*x^4+b*x^2+a)^3/x^2,x, algorithm="giac")

[Out]

1/13*B*c^3*x^13 + 3/11*B*b*c^2*x^11 + 1/11*A*c^3*x^11 + 1/3*B*b^2*c*x^9 + 1/3*B*a*c^2*x^9 + 1/3*A*b*c^2*x^9 +
1/7*B*b^3*x^7 + 6/7*B*a*b*c*x^7 + 3/7*A*b^2*c*x^7 + 3/7*A*a*c^2*x^7 + 3/5*B*a*b^2*x^5 + 1/5*A*b^3*x^5 + 3/5*B*
a^2*c*x^5 + 6/5*A*a*b*c*x^5 + B*a^2*b*x^3 + A*a*b^2*x^3 + A*a^2*c*x^3 + B*a^3*x + 3*A*a^2*b*x - A*a^3/x

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Mupad [B]
time = 0.05, size = 163, normalized size = 1.04 \begin {gather*} x^5\,\left (\frac {3\,B\,c\,a^2}{5}+\frac {3\,B\,a\,b^2}{5}+\frac {6\,A\,c\,a\,b}{5}+\frac {A\,b^3}{5}\right )+x^7\,\left (\frac {B\,b^3}{7}+\frac {3\,A\,b^2\,c}{7}+\frac {6\,B\,a\,b\,c}{7}+\frac {3\,A\,a\,c^2}{7}\right )+x\,\left (B\,a^3+3\,A\,b\,a^2\right )+x^{11}\,\left (\frac {A\,c^3}{11}+\frac {3\,B\,b\,c^2}{11}\right )+x^3\,\left (B\,a^2\,b+A\,c\,a^2+A\,a\,b^2\right )+x^9\,\left (\frac {B\,b^2\,c}{3}+\frac {A\,b\,c^2}{3}+\frac {B\,a\,c^2}{3}\right )-\frac {A\,a^3}{x}+\frac {B\,c^3\,x^{13}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(a + b*x^2 + c*x^4)^3)/x^2,x)

[Out]

x^5*((A*b^3)/5 + (3*B*a*b^2)/5 + (3*B*a^2*c)/5 + (6*A*a*b*c)/5) + x^7*((B*b^3)/7 + (3*A*a*c^2)/7 + (3*A*b^2*c)
/7 + (6*B*a*b*c)/7) + x*(B*a^3 + 3*A*a^2*b) + x^11*((A*c^3)/11 + (3*B*b*c^2)/11) + x^3*(A*a*b^2 + A*a^2*c + B*
a^2*b) + x^9*((A*b*c^2)/3 + (B*a*c^2)/3 + (B*b^2*c)/3) - (A*a^3)/x + (B*c^3*x^13)/13

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