3.4.20 \(\int x^m (a+b x^2)^2 (A+B x^2) \, dx\) [320]

Optimal. Leaf size=71 \[ \frac {a^2 A x^{1+m}}{1+m}+\frac {a (2 A b+a B) x^{3+m}}{3+m}+\frac {b (A b+2 a B) x^{5+m}}{5+m}+\frac {b^2 B x^{7+m}}{7+m} \]

[Out]

a^2*A*x^(1+m)/(1+m)+a*(2*A*b+B*a)*x^(3+m)/(3+m)+b*(A*b+2*B*a)*x^(5+m)/(5+m)+b^2*B*x^(7+m)/(7+m)

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Rubi [A]
time = 0.03, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {459} \begin {gather*} \frac {a^2 A x^{m+1}}{m+1}+\frac {a x^{m+3} (a B+2 A b)}{m+3}+\frac {b x^{m+5} (2 a B+A b)}{m+5}+\frac {b^2 B x^{m+7}}{m+7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*x^2)^2*(A + B*x^2),x]

[Out]

(a^2*A*x^(1 + m))/(1 + m) + (a*(2*A*b + a*B)*x^(3 + m))/(3 + m) + (b*(A*b + 2*a*B)*x^(5 + m))/(5 + m) + (b^2*B
*x^(7 + m))/(7 + m)

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin {align*} \int x^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx &=\int \left (a^2 A x^m+a (2 A b+a B) x^{2+m}+b (A b+2 a B) x^{4+m}+b^2 B x^{6+m}\right ) \, dx\\ &=\frac {a^2 A x^{1+m}}{1+m}+\frac {a (2 A b+a B) x^{3+m}}{3+m}+\frac {b (A b+2 a B) x^{5+m}}{5+m}+\frac {b^2 B x^{7+m}}{7+m}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 66, normalized size = 0.93 \begin {gather*} x^{1+m} \left (\frac {a^2 A}{1+m}+\frac {a (2 A b+a B) x^2}{3+m}+\frac {b (A b+2 a B) x^4}{5+m}+\frac {b^2 B x^6}{7+m}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x^(1 + m)*((a^2*A)/(1 + m) + (a*(2*A*b + a*B)*x^2)/(3 + m) + (b*(A*b + 2*a*B)*x^4)/(5 + m) + (b^2*B*x^6)/(7 +
m))

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Maple [A]
time = 0.08, size = 82, normalized size = 1.15

method	result	size
norman	\(\frac {a \left (2 A b +B a \right ) x^{3} {\mathrm e}^{m \ln \left (x \right )}}{3+m}+\frac {a^{2} A x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {b \left (A b +2 B a \right ) x^{5} {\mathrm e}^{m \ln \left (x \right )}}{5+m}+\frac {b^{2} B \,x^{7} {\mathrm e}^{m \ln \left (x \right )}}{7+m}\)	\(82\)
risch	\(\frac {x \left (B \,b^{2} m^{3} x^{6}+9 B \,b^{2} m^{2} x^{6}+A \,b^{2} m^{3} x^{4}+2 B a b \,m^{3} x^{4}+23 m \,x^{6} b^{2} B +11 A \,b^{2} m^{2} x^{4}+22 B a b \,m^{2} x^{4}+15 b^{2} B \,x^{6}+2 A a b \,m^{3} x^{2}+31 A \,b^{2} x^{4} m +B \,a^{2} m^{3} x^{2}+62 B a b \,x^{4} m +26 A a b \,m^{2} x^{2}+21 A \,b^{2} x^{4}+13 B \,a^{2} m^{2} x^{2}+42 B a b \,x^{4}+A \,a^{2} m^{3}+94 a A b \,x^{2} m +47 B \,a^{2} x^{2} m +15 A \,a^{2} m^{2}+70 a A b \,x^{2}+35 B \,a^{2} x^{2}+71 a^{2} A m +105 a^{2} A \right ) x^{m}}{\left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\)	\(261\)
gosper	\(\frac {x^{1+m} \left (B \,b^{2} m^{3} x^{6}+9 B \,b^{2} m^{2} x^{6}+A \,b^{2} m^{3} x^{4}+2 B a b \,m^{3} x^{4}+23 m \,x^{6} b^{2} B +11 A \,b^{2} m^{2} x^{4}+22 B a b \,m^{2} x^{4}+15 b^{2} B \,x^{6}+2 A a b \,m^{3} x^{2}+31 A \,b^{2} x^{4} m +B \,a^{2} m^{3} x^{2}+62 B a b \,x^{4} m +26 A a b \,m^{2} x^{2}+21 A \,b^{2} x^{4}+13 B \,a^{2} m^{2} x^{2}+42 B a b \,x^{4}+A \,a^{2} m^{3}+94 a A b \,x^{2} m +47 B \,a^{2} x^{2} m +15 A \,a^{2} m^{2}+70 a A b \,x^{2}+35 B \,a^{2} x^{2}+71 a^{2} A m +105 a^{2} A \right )}{\left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\)	\(262\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*(2*A*b+B*a)/(3+m)*x^3*exp(m*ln(x))+a^2*A/(1+m)*x*exp(m*ln(x))+b*(A*b+2*B*a)/(5+m)*x^5*exp(m*ln(x))+b^2*B/(7+
m)*x^7*exp(m*ln(x))

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Maxima [A]
time = 0.29, size = 91, normalized size = 1.28 \begin {gather*} \frac {B b^{2} x^{m + 7}}{m + 7} + \frac {2 \, B a b x^{m + 5}}{m + 5} + \frac {A b^{2} x^{m + 5}}{m + 5} + \frac {B a^{2} x^{m + 3}}{m + 3} + \frac {2 \, A a b x^{m + 3}}{m + 3} + \frac {A a^{2} x^{m + 1}}{m + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*b^2*x^(m + 7)/(m + 7) + 2*B*a*b*x^(m + 5)/(m + 5) + A*b^2*x^(m + 5)/(m + 5) + B*a^2*x^(m + 3)/(m + 3) + 2*A*
a*b*x^(m + 3)/(m + 3) + A*a^2*x^(m + 1)/(m + 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (71) = 142\).
time = 1.18, size = 215, normalized size = 3.03 \begin {gather*} \frac {{\left ({\left (B b^{2} m^{3} + 9 \, B b^{2} m^{2} + 23 \, B b^{2} m + 15 \, B b^{2}\right )} x^{7} + {\left ({\left (2 \, B a b + A b^{2}\right )} m^{3} + 42 \, B a b + 21 \, A b^{2} + 11 \, {\left (2 \, B a b + A b^{2}\right )} m^{2} + 31 \, {\left (2 \, B a b + A b^{2}\right )} m\right )} x^{5} + {\left ({\left (B a^{2} + 2 \, A a b\right )} m^{3} + 35 \, B a^{2} + 70 \, A a b + 13 \, {\left (B a^{2} + 2 \, A a b\right )} m^{2} + 47 \, {\left (B a^{2} + 2 \, A a b\right )} m\right )} x^{3} + {\left (A a^{2} m^{3} + 15 \, A a^{2} m^{2} + 71 \, A a^{2} m + 105 \, A a^{2}\right )} x\right )} x^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((B*b^2*m^3 + 9*B*b^2*m^2 + 23*B*b^2*m + 15*B*b^2)*x^7 + ((2*B*a*b + A*b^2)*m^3 + 42*B*a*b + 21*A*b^2 + 11*(2*
B*a*b + A*b^2)*m^2 + 31*(2*B*a*b + A*b^2)*m)*x^5 + ((B*a^2 + 2*A*a*b)*m^3 + 35*B*a^2 + 70*A*a*b + 13*(B*a^2 +
2*A*a*b)*m^2 + 47*(B*a^2 + 2*A*a*b)*m)*x^3 + (A*a^2*m^3 + 15*A*a^2*m^2 + 71*A*a^2*m + 105*A*a^2)*x)*x^m/(m^4 +
16*m^3 + 86*m^2 + 176*m + 105)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1044 vs. \(2 (63) = 126\).
time = 0.40, size = 1044, normalized size = 14.70 \begin {gather*} \begin {cases} - \frac {A a^{2}}{6 x^{6}} - \frac {A a b}{2 x^{4}} - \frac {A b^{2}}{2 x^{2}} - \frac {B a^{2}}{4 x^{4}} - \frac {B a b}{x^{2}} + B b^{2} \log {\left (x \right )} & \text {for}\: m = -7 \\- \frac {A a^{2}}{4 x^{4}} - \frac {A a b}{x^{2}} + A b^{2} \log {\left (x \right )} - \frac {B a^{2}}{2 x^{2}} + 2 B a b \log {\left (x \right )} + \frac {B b^{2} x^{2}}{2} & \text {for}\: m = -5 \\- \frac {A a^{2}}{2 x^{2}} + 2 A a b \log {\left (x \right )} + \frac {A b^{2} x^{2}}{2} + B a^{2} \log {\left (x \right )} + B a b x^{2} + \frac {B b^{2} x^{4}}{4} & \text {for}\: m = -3 \\A a^{2} \log {\left (x \right )} + A a b x^{2} + \frac {A b^{2} x^{4}}{4} + \frac {B a^{2} x^{2}}{2} + \frac {B a b x^{4}}{2} + \frac {B b^{2} x^{6}}{6} & \text {for}\: m = -1 \\\frac {A a^{2} m^{3} x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {15 A a^{2} m^{2} x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {71 A a^{2} m x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {105 A a^{2} x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {2 A a b m^{3} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {26 A a b m^{2} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {94 A a b m x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {70 A a b x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {A b^{2} m^{3} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {11 A b^{2} m^{2} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {31 A b^{2} m x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {21 A b^{2} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {B a^{2} m^{3} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {13 B a^{2} m^{2} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {47 B a^{2} m x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {35 B a^{2} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {2 B a b m^{3} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {22 B a b m^{2} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {62 B a b m x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {42 B a b x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {B b^{2} m^{3} x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {9 B b^{2} m^{2} x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {23 B b^{2} m x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {15 B b^{2} x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-A*a**2/(6*x**6) - A*a*b/(2*x**4) - A*b**2/(2*x**2) - B*a**2/(4*x**4) - B*a*b/x**2 + B*b**2*log(x),
Eq(m, -7)), (-A*a**2/(4*x**4) - A*a*b/x**2 + A*b**2*log(x) - B*a**2/(2*x**2) + 2*B*a*b*log(x) + B*b**2*x**2/2
, Eq(m, -5)), (-A*a**2/(2*x**2) + 2*A*a*b*log(x) + A*b**2*x**2/2 + B*a**2*log(x) + B*a*b*x**2 + B*b**2*x**4/4,
Eq(m, -3)), (A*a**2*log(x) + A*a*b*x**2 + A*b**2*x**4/4 + B*a**2*x**2/2 + B*a*b*x**4/2 + B*b**2*x**6/6, Eq(m,
-1)), (A*a**2*m**3*x*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 15*A*a**2*m**2*x*x**m/(m**4 + 16*m**3 +
86*m**2 + 176*m + 105) + 71*A*a**2*m*x*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 105*A*a**2*x*x**m/(m**4
+ 16*m**3 + 86*m**2 + 176*m + 105) + 2*A*a*b*m**3*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 26*A*a
*b*m**2*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 94*A*a*b*m*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 +
176*m + 105) + 70*A*a*b*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + A*b**2*m**3*x**5*x**m/(m**4 + 16*
m**3 + 86*m**2 + 176*m + 105) + 11*A*b**2*m**2*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 31*A*b**2*
m*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 21*A*b**2*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m +
105) + B*a**2*m**3*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 13*B*a**2*m**2*x**3*x**m/(m**4 + 16*m
**3 + 86*m**2 + 176*m + 105) + 47*B*a**2*m*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 35*B*a**2*x**3
*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 2*B*a*b*m**3*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 10
5) + 22*B*a*b*m**2*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 62*B*a*b*m*x**5*x**m/(m**4 + 16*m**3 +
86*m**2 + 176*m + 105) + 42*B*a*b*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + B*b**2*m**3*x**7*x**m/
(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 9*B*b**2*m**2*x**7*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) +
23*B*b**2*m*x**7*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 15*B*b**2*x**7*x**m/(m**4 + 16*m**3 + 86*m**2
+ 176*m + 105), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (71) = 142\).
time = 0.68, size = 332, normalized size = 4.68 \begin {gather*} \frac {B b^{2} m^{3} x^{7} x^{m} + 9 \, B b^{2} m^{2} x^{7} x^{m} + 2 \, B a b m^{3} x^{5} x^{m} + A b^{2} m^{3} x^{5} x^{m} + 23 \, B b^{2} m x^{7} x^{m} + 22 \, B a b m^{2} x^{5} x^{m} + 11 \, A b^{2} m^{2} x^{5} x^{m} + 15 \, B b^{2} x^{7} x^{m} + B a^{2} m^{3} x^{3} x^{m} + 2 \, A a b m^{3} x^{3} x^{m} + 62 \, B a b m x^{5} x^{m} + 31 \, A b^{2} m x^{5} x^{m} + 13 \, B a^{2} m^{2} x^{3} x^{m} + 26 \, A a b m^{2} x^{3} x^{m} + 42 \, B a b x^{5} x^{m} + 21 \, A b^{2} x^{5} x^{m} + A a^{2} m^{3} x x^{m} + 47 \, B a^{2} m x^{3} x^{m} + 94 \, A a b m x^{3} x^{m} + 15 \, A a^{2} m^{2} x x^{m} + 35 \, B a^{2} x^{3} x^{m} + 70 \, A a b x^{3} x^{m} + 71 \, A a^{2} m x x^{m} + 105 \, A a^{2} x x^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*b^2*m^3*x^7*x^m + 9*B*b^2*m^2*x^7*x^m + 2*B*a*b*m^3*x^5*x^m + A*b^2*m^3*x^5*x^m + 23*B*b^2*m*x^7*x^m + 22*B
*a*b*m^2*x^5*x^m + 11*A*b^2*m^2*x^5*x^m + 15*B*b^2*x^7*x^m + B*a^2*m^3*x^3*x^m + 2*A*a*b*m^3*x^3*x^m + 62*B*a*
b*m*x^5*x^m + 31*A*b^2*m*x^5*x^m + 13*B*a^2*m^2*x^3*x^m + 26*A*a*b*m^2*x^3*x^m + 42*B*a*b*x^5*x^m + 21*A*b^2*x
^5*x^m + A*a^2*m^3*x*x^m + 47*B*a^2*m*x^3*x^m + 94*A*a*b*m*x^3*x^m + 15*A*a^2*m^2*x*x^m + 35*B*a^2*x^3*x^m + 7
0*A*a*b*x^3*x^m + 71*A*a^2*m*x*x^m + 105*A*a^2*x*x^m)/(m^4 + 16*m^3 + 86*m^2 + 176*m + 105)

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Mupad [B]
time = 0.18, size = 177, normalized size = 2.49 \begin {gather*} x^m\,\left (\frac {B\,b^2\,x^7\,\left (m^3+9\,m^2+23\,m+15\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {A\,a^2\,x\,\left (m^3+15\,m^2+71\,m+105\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {a\,x^3\,\left (2\,A\,b+B\,a\right )\,\left (m^3+13\,m^2+47\,m+35\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {b\,x^5\,\left (A\,b+2\,B\,a\right )\,\left (m^3+11\,m^2+31\,m+21\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(A + B*x^2)*(a + b*x^2)^2,x)

[Out]

x^m*((B*b^2*x^7*(23*m + 9*m^2 + m^3 + 15))/(176*m + 86*m^2 + 16*m^3 + m^4 + 105) + (A*a^2*x*(71*m + 15*m^2 + m
^3 + 105))/(176*m + 86*m^2 + 16*m^3 + m^4 + 105) + (a*x^3*(2*A*b + B*a)*(47*m + 13*m^2 + m^3 + 35))/(176*m + 8
6*m^2 + 16*m^3 + m^4 + 105) + (b*x^5*(A*b + 2*B*a)*(31*m + 11*m^2 + m^3 + 21))/(176*m + 86*m^2 + 16*m^3 + m^4
+ 105))

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