3.1040 \(\int x^5 \left (a+b x^2\right )^p \, dx\)

Optimal. Leaf size=72 \[ \frac{a^2 \left (a+b x^2\right )^{p+1}}{2 b^3 (p+1)}-\frac{a \left (a+b x^2\right )^{p+2}}{b^3 (p+2)}+\frac{\left (a+b x^2\right )^{p+3}}{2 b^3 (p+3)} \]

[Out]

(a^2*(a + b*x^2)^(1 + p))/(2*b^3*(1 + p)) - (a*(a + b*x^2)^(2 + p))/(b^3*(2 + p)
) + (a + b*x^2)^(3 + p)/(2*b^3*(3 + p))

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Rubi [A]  time = 0.0926792, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{a^2 \left (a+b x^2\right )^{p+1}}{2 b^3 (p+1)}-\frac{a \left (a+b x^2\right )^{p+2}}{b^3 (p+2)}+\frac{\left (a+b x^2\right )^{p+3}}{2 b^3 (p+3)} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x^2)^p,x]

[Out]

(a^2*(a + b*x^2)^(1 + p))/(2*b^3*(1 + p)) - (a*(a + b*x^2)^(2 + p))/(b^3*(2 + p)
) + (a + b*x^2)^(3 + p)/(2*b^3*(3 + p))

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Rubi in Sympy [A]  time = 17.6882, size = 58, normalized size = 0.81 \[ \frac{a^{2} \left (a + b x^{2}\right )^{p + 1}}{2 b^{3} \left (p + 1\right )} - \frac{a \left (a + b x^{2}\right )^{p + 2}}{b^{3} \left (p + 2\right )} + \frac{\left (a + b x^{2}\right )^{p + 3}}{2 b^{3} \left (p + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(b*x**2+a)**p,x)

[Out]

a**2*(a + b*x**2)**(p + 1)/(2*b**3*(p + 1)) - a*(a + b*x**2)**(p + 2)/(b**3*(p +
2)) + (a + b*x**2)**(p + 3)/(2*b**3*(p + 3))

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Mathematica [A]  time = 0.0435254, size = 64, normalized size = 0.89 \[ \frac{\left (a+b x^2\right )^{p+1} \left (2 a^2-2 a b (p+1) x^2+b^2 \left (p^2+3 p+2\right ) x^4\right )}{2 b^3 (p+1) (p+2) (p+3)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(a + b*x^2)^p,x]

[Out]

((a + b*x^2)^(1 + p)*(2*a^2 - 2*a*b*(1 + p)*x^2 + b^2*(2 + 3*p + p^2)*x^4))/(2*b
^3*(1 + p)*(2 + p)*(3 + p))

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Maple [A]  time = 0.008, size = 80, normalized size = 1.1 \[{\frac{ \left ( b{x}^{2}+a \right ) ^{1+p} \left ({b}^{2}{p}^{2}{x}^{4}+3\,{b}^{2}p{x}^{4}+2\,{b}^{2}{x}^{4}-2\,abp{x}^{2}-2\,ab{x}^{2}+2\,{a}^{2} \right ) }{2\,{b}^{3} \left ({p}^{3}+6\,{p}^{2}+11\,p+6 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x^2+a)^p,x)

[Out]

1/2*(b*x^2+a)^(1+p)*(b^2*p^2*x^4+3*b^2*p*x^4+2*b^2*x^4-2*a*b*p*x^2-2*a*b*x^2+2*a
^2)/b^3/(p^3+6*p^2+11*p+6)

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Maxima [A]  time = 1.36643, size = 99, normalized size = 1.38 \[ \frac{{\left ({\left (p^{2} + 3 \, p + 2\right )} b^{3} x^{6} +{\left (p^{2} + p\right )} a b^{2} x^{4} - 2 \, a^{2} b p x^{2} + 2 \, a^{3}\right )}{\left (b x^{2} + a\right )}^{p}}{2 \,{\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*x^5,x, algorithm="maxima")

[Out]

1/2*((p^2 + 3*p + 2)*b^3*x^6 + (p^2 + p)*a*b^2*x^4 - 2*a^2*b*p*x^2 + 2*a^3)*(b*x
^2 + a)^p/((p^3 + 6*p^2 + 11*p + 6)*b^3)

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Fricas [A]  time = 0.219338, size = 132, normalized size = 1.83 \[ \frac{{\left ({\left (b^{3} p^{2} + 3 \, b^{3} p + 2 \, b^{3}\right )} x^{6} - 2 \, a^{2} b p x^{2} +{\left (a b^{2} p^{2} + a b^{2} p\right )} x^{4} + 2 \, a^{3}\right )}{\left (b x^{2} + a\right )}^{p}}{2 \,{\left (b^{3} p^{3} + 6 \, b^{3} p^{2} + 11 \, b^{3} p + 6 \, b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*x^5,x, algorithm="fricas")

[Out]

1/2*((b^3*p^2 + 3*b^3*p + 2*b^3)*x^6 - 2*a^2*b*p*x^2 + (a*b^2*p^2 + a*b^2*p)*x^4
+ 2*a^3)*(b*x^2 + a)^p/(b^3*p^3 + 6*b^3*p^2 + 11*b^3*p + 6*b^3)

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Sympy [A]  time = 15.8674, size = 981, normalized size = 13.62 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(b*x**2+a)**p,x)

[Out]

Piecewise((a**p*x**6/6, Eq(b, 0)), (2*a**2*log(-I*sqrt(a)*sqrt(1/b) + x)/(4*a**2
*b**3 + 8*a*b**4*x**2 + 4*b**5*x**4) + 2*a**2*log(I*sqrt(a)*sqrt(1/b) + x)/(4*a*
*2*b**3 + 8*a*b**4*x**2 + 4*b**5*x**4) + 3*a**2/(4*a**2*b**3 + 8*a*b**4*x**2 + 4
*b**5*x**4) + 4*a*b*x**2*log(-I*sqrt(a)*sqrt(1/b) + x)/(4*a**2*b**3 + 8*a*b**4*x
**2 + 4*b**5*x**4) + 4*a*b*x**2*log(I*sqrt(a)*sqrt(1/b) + x)/(4*a**2*b**3 + 8*a*
b**4*x**2 + 4*b**5*x**4) + 4*a*b*x**2/(4*a**2*b**3 + 8*a*b**4*x**2 + 4*b**5*x**4
) + 2*b**2*x**4*log(-I*sqrt(a)*sqrt(1/b) + x)/(4*a**2*b**3 + 8*a*b**4*x**2 + 4*b
**5*x**4) + 2*b**2*x**4*log(I*sqrt(a)*sqrt(1/b) + x)/(4*a**2*b**3 + 8*a*b**4*x**
2 + 4*b**5*x**4), Eq(p, -3)), (-2*a**2*log(-I*sqrt(a)*sqrt(1/b) + x)/(2*a*b**3 +
2*b**4*x**2) - 2*a**2*log(I*sqrt(a)*sqrt(1/b) + x)/(2*a*b**3 + 2*b**4*x**2) - 2
*a**2/(2*a*b**3 + 2*b**4*x**2) - 2*a*b*x**2*log(-I*sqrt(a)*sqrt(1/b) + x)/(2*a*b
**3 + 2*b**4*x**2) - 2*a*b*x**2*log(I*sqrt(a)*sqrt(1/b) + x)/(2*a*b**3 + 2*b**4*
x**2) + b**2*x**4/(2*a*b**3 + 2*b**4*x**2), Eq(p, -2)), (a**2*log(-I*sqrt(a)*sqr
t(1/b) + x)/(2*b**3) + a**2*log(I*sqrt(a)*sqrt(1/b) + x)/(2*b**3) - a*x**2/(2*b*
*2) + x**4/(4*b), Eq(p, -1)), (2*a**3*(a + b*x**2)**p/(2*b**3*p**3 + 12*b**3*p**
2 + 22*b**3*p + 12*b**3) - 2*a**2*b*p*x**2*(a + b*x**2)**p/(2*b**3*p**3 + 12*b**
3*p**2 + 22*b**3*p + 12*b**3) + a*b**2*p**2*x**4*(a + b*x**2)**p/(2*b**3*p**3 +
12*b**3*p**2 + 22*b**3*p + 12*b**3) + a*b**2*p*x**4*(a + b*x**2)**p/(2*b**3*p**3
+ 12*b**3*p**2 + 22*b**3*p + 12*b**3) + b**3*p**2*x**6*(a + b*x**2)**p/(2*b**3*
p**3 + 12*b**3*p**2 + 22*b**3*p + 12*b**3) + 3*b**3*p*x**6*(a + b*x**2)**p/(2*b*
*3*p**3 + 12*b**3*p**2 + 22*b**3*p + 12*b**3) + 2*b**3*x**6*(a + b*x**2)**p/(2*b
**3*p**3 + 12*b**3*p**2 + 22*b**3*p + 12*b**3), True))

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GIAC/XCAS [A]  time = 0.248286, size = 336, normalized size = 4.67 \[ \frac{{\left (b x^{2} + a\right )}^{3} p^{2} e^{\left (p{\rm ln}\left (b x^{2} + a\right )\right )} - 2 \,{\left (b x^{2} + a\right )}^{2} a p^{2} e^{\left (p{\rm ln}\left (b x^{2} + a\right )\right )} +{\left (b x^{2} + a\right )} a^{2} p^{2} e^{\left (p{\rm ln}\left (b x^{2} + a\right )\right )} + 3 \,{\left (b x^{2} + a\right )}^{3} p e^{\left (p{\rm ln}\left (b x^{2} + a\right )\right )} - 8 \,{\left (b x^{2} + a\right )}^{2} a p e^{\left (p{\rm ln}\left (b x^{2} + a\right )\right )} + 5 \,{\left (b x^{2} + a\right )} a^{2} p e^{\left (p{\rm ln}\left (b x^{2} + a\right )\right )} + 2 \,{\left (b x^{2} + a\right )}^{3} e^{\left (p{\rm ln}\left (b x^{2} + a\right )\right )} - 6 \,{\left (b x^{2} + a\right )}^{2} a e^{\left (p{\rm ln}\left (b x^{2} + a\right )\right )} + 6 \,{\left (b x^{2} + a\right )} a^{2} e^{\left (p{\rm ln}\left (b x^{2} + a\right )\right )}}{2 \,{\left (b^{2} p^{3} + 6 \, b^{2} p^{2} + 11 \, b^{2} p + 6 \, b^{2}\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*x^5,x, algorithm="giac")

[Out]

1/2*((b*x^2 + a)^3*p^2*e^(p*ln(b*x^2 + a)) - 2*(b*x^2 + a)^2*a*p^2*e^(p*ln(b*x^2
+ a)) + (b*x^2 + a)*a^2*p^2*e^(p*ln(b*x^2 + a)) + 3*(b*x^2 + a)^3*p*e^(p*ln(b*x
^2 + a)) - 8*(b*x^2 + a)^2*a*p*e^(p*ln(b*x^2 + a)) + 5*(b*x^2 + a)*a^2*p*e^(p*ln
(b*x^2 + a)) + 2*(b*x^2 + a)^3*e^(p*ln(b*x^2 + a)) - 6*(b*x^2 + a)^2*a*e^(p*ln(b
*x^2 + a)) + 6*(b*x^2 + a)*a^2*e^(p*ln(b*x^2 + a)))/((b^2*p^3 + 6*b^2*p^2 + 11*b
^2*p + 6*b^2)*b)