3.321 \(\int x^m \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx\)

Optimal. Leaf size=45 \[ \frac{x^{m+3} (a B+A b)}{m+3}+\frac{a A x^{m+1}}{m+1}+\frac{b B x^{m+5}}{m+5} \]

[Out]

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

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Rubi [A]  time = 0.0614579, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{x^{m+3} (a B+A b)}{m+3}+\frac{a A x^{m+1}}{m+1}+\frac{b B x^{m+5}}{m+5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.42324, size = 37, normalized size = 0.82 \[ \frac{A a x^{m + 1}}{m + 1} + \frac{B b x^{m + 5}}{m + 5} + \frac{x^{m + 3} \left (A b + B a\right )}{m + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(b*x**2+a)*(B*x**2+A),x)

[Out]

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

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Mathematica [A]  time = 0.04053, size = 41, normalized size = 0.91 \[ x^m \left (\frac{x^3 (a B+A b)}{m+3}+\frac{a A x}{m+1}+\frac{b B x^5}{m+5}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.005, size = 110, normalized size = 2.4 \[{\frac{{x}^{1+m} \left ( Bb{m}^{2}{x}^{4}+4\,Bbm{x}^{4}+Ab{m}^{2}{x}^{2}+Ba{m}^{2}{x}^{2}+3\,bB{x}^{4}+6\,Abm{x}^{2}+6\,Bam{x}^{2}+Aa{m}^{2}+5\,A{x}^{2}b+5\,B{x}^{2}a+8\,Aam+15\,Aa \right ) }{ \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x^2+a)*(B*x^2+A),x)

[Out]

x^(1+m)*(B*b*m^2*x^4+4*B*b*m*x^4+A*b*m^2*x^2+B*a*m^2*x^2+3*B*b*x^4+6*A*b*m*x^2+6
*B*a*m*x^2+A*a*m^2+5*A*b*x^2+5*B*a*x^2+8*A*a*m+15*A*a)/(5+m)/(3+m)/(1+m)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234854, size = 124, normalized size = 2.76 \[ \frac{{\left ({\left (B b m^{2} + 4 \, B b m + 3 \, B b\right )} x^{5} +{\left ({\left (B a + A b\right )} m^{2} + 5 \, B a + 5 \, A b + 6 \,{\left (B a + A b\right )} m\right )} x^{3} +{\left (A a m^{2} + 8 \, A a m + 15 \, A a\right )} x\right )} x^{m}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((B*b*m^2 + 4*B*b*m + 3*B*b)*x^5 + ((B*a + A*b)*m^2 + 5*B*a + 5*A*b + 6*(B*a + A
*b)*m)*x^3 + (A*a*m^2 + 8*A*a*m + 15*A*a)*x)*x^m/(m^3 + 9*m^2 + 23*m + 15)

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Sympy [A]  time = 2.86559, size = 410, normalized size = 9.11 \[ \begin{cases} - \frac{A a}{4 x^{4}} - \frac{A b}{2 x^{2}} - \frac{B a}{2 x^{2}} + B b \log{\left (x \right )} & \text{for}\: m = -5 \\- \frac{A a}{2 x^{2}} + A b \log{\left (x \right )} + B a \log{\left (x \right )} + \frac{B b x^{2}}{2} & \text{for}\: m = -3 \\A a \log{\left (x \right )} + \frac{A b x^{2}}{2} + \frac{B a x^{2}}{2} + \frac{B b x^{4}}{4} & \text{for}\: m = -1 \\\frac{A a m^{2} x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{8 A a m x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{15 A a x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{A b m^{2} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{6 A b m x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{5 A b x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{B a m^{2} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{6 B a m x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{5 B a x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{B b m^{2} x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{4 B b m x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{3 B b x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-A*a/(4*x**4) - A*b/(2*x**2) - B*a/(2*x**2) + B*b*log(x), Eq(m, -5)),
(-A*a/(2*x**2) + A*b*log(x) + B*a*log(x) + B*b*x**2/2, Eq(m, -3)), (A*a*log(x)
+ A*b*x**2/2 + B*a*x**2/2 + B*b*x**4/4, Eq(m, -1)), (A*a*m**2*x*x**m/(m**3 + 9*m
**2 + 23*m + 15) + 8*A*a*m*x*x**m/(m**3 + 9*m**2 + 23*m + 15) + 15*A*a*x*x**m/(m
**3 + 9*m**2 + 23*m + 15) + A*b*m**2*x**3*x**m/(m**3 + 9*m**2 + 23*m + 15) + 6*A
*b*m*x**3*x**m/(m**3 + 9*m**2 + 23*m + 15) + 5*A*b*x**3*x**m/(m**3 + 9*m**2 + 23
*m + 15) + B*a*m**2*x**3*x**m/(m**3 + 9*m**2 + 23*m + 15) + 6*B*a*m*x**3*x**m/(m
**3 + 9*m**2 + 23*m + 15) + 5*B*a*x**3*x**m/(m**3 + 9*m**2 + 23*m + 15) + B*b*m*
*2*x**5*x**m/(m**3 + 9*m**2 + 23*m + 15) + 4*B*b*m*x**5*x**m/(m**3 + 9*m**2 + 23
*m + 15) + 3*B*b*x**5*x**m/(m**3 + 9*m**2 + 23*m + 15), True))

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GIAC/XCAS [A]  time = 0.307916, size = 225, normalized size = 5. \[ \frac{B b m^{2} x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 4 \, B b m x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + B a m^{2} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + A b m^{2} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 3 \, B b x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 6 \, B a m x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 6 \, A b m x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + A a m^{2} x e^{\left (m{\rm ln}\left (x\right )\right )} + 5 \, B a x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 5 \, A b x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 8 \, A a m x e^{\left (m{\rm ln}\left (x\right )\right )} + 15 \, A a x e^{\left (m{\rm ln}\left (x\right )\right )}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(B*b*m^2*x^5*e^(m*ln(x)) + 4*B*b*m*x^5*e^(m*ln(x)) + B*a*m^2*x^3*e^(m*ln(x)) + A
*b*m^2*x^3*e^(m*ln(x)) + 3*B*b*x^5*e^(m*ln(x)) + 6*B*a*m*x^3*e^(m*ln(x)) + 6*A*b
*m*x^3*e^(m*ln(x)) + A*a*m^2*x*e^(m*ln(x)) + 5*B*a*x^3*e^(m*ln(x)) + 5*A*b*x^3*e
^(m*ln(x)) + 8*A*a*m*x*e^(m*ln(x)) + 15*A*a*x*e^(m*ln(x)))/(m^3 + 9*m^2 + 23*m +
15)