3.1041 \(\int x^3 \left (a+b x^2\right )^p \, dx\)

Optimal. Leaf size=48 \[ \frac{\left (a+b x^2\right )^{p+2}}{2 b^2 (p+2)}-\frac{a \left (a+b x^2\right )^{p+1}}{2 b^2 (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0629979, antiderivative size = 48, 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 \[ \frac{\left (a+b x^2\right )^{p+2}}{2 b^2 (p+2)}-\frac{a \left (a+b x^2\right )^{p+1}}{2 b^2 (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x^2)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.053, size = 37, normalized size = 0.77 \[ - \frac{a \left (a + b x^{2}\right )^{p + 1}}{2 b^{2} \left (p + 1\right )} + \frac{\left (a + b x^{2}\right )^{p + 2}}{2 b^{2} \left (p + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x**2+a)**p,x)

[Out]

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

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Mathematica [A]  time = 0.0291495, size = 40, normalized size = 0.83 \[ \frac{\left (a+b x^2\right )^{p+1} \left (b (p+1) x^2-a\right )}{2 b^2 (p+1) (p+2)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(a + b*x^2)^p,x]

[Out]

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

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Maple [A]  time = 0.007, size = 42, normalized size = 0.9 \[ -{\frac{ \left ( b{x}^{2}+a \right ) ^{1+p} \left ( -{x}^{2}pb-b{x}^{2}+a \right ) }{2\,{b}^{2} \left ({p}^{2}+3\,p+2 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^2+a)^p,x)

[Out]

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

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Maxima [A]  time = 1.36255, size = 63, normalized size = 1.31 \[ \frac{{\left (b^{2}{\left (p + 1\right )} x^{4} + a b p x^{2} - a^{2}\right )}{\left (b x^{2} + a\right )}^{p}}{2 \,{\left (p^{2} + 3 \, p + 2\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.219296, size = 78, normalized size = 1.62 \[ \frac{{\left (a b p x^{2} +{\left (b^{2} p + b^{2}\right )} x^{4} - a^{2}\right )}{\left (b x^{2} + a\right )}^{p}}{2 \,{\left (b^{2} p^{2} + 3 \, b^{2} p + 2 \, b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*x^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.83407, size = 364, normalized size = 7.58 \[ \begin{cases} \frac{a^{p} x^{4}}{4} & \text{for}\: b = 0 \\\frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{a}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{b x^{2} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{b x^{2} \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 a b^{2} + 2 b^{3} x^{2}} & \text{for}\: p = -2 \\- \frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 b^{2}} - \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 b^{2}} + \frac{x^{2}}{2 b} & \text{for}\: p = -1 \\- \frac{a^{2} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} + \frac{a b p x^{2} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} + \frac{b^{2} p x^{4} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} + \frac{b^{2} x^{4} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x**2+a)**p,x)

[Out]

Piecewise((a**p*x**4/4, Eq(b, 0)), (a*log(-I*sqrt(a)*sqrt(1/b) + x)/(2*a*b**2 +
2*b**3*x**2) + a*log(I*sqrt(a)*sqrt(1/b) + x)/(2*a*b**2 + 2*b**3*x**2) + a/(2*a*
b**2 + 2*b**3*x**2) + b*x**2*log(-I*sqrt(a)*sqrt(1/b) + x)/(2*a*b**2 + 2*b**3*x*
*2) + b*x**2*log(I*sqrt(a)*sqrt(1/b) + x)/(2*a*b**2 + 2*b**3*x**2), Eq(p, -2)),
(-a*log(-I*sqrt(a)*sqrt(1/b) + x)/(2*b**2) - a*log(I*sqrt(a)*sqrt(1/b) + x)/(2*b
**2) + x**2/(2*b), Eq(p, -1)), (-a**2*(a + b*x**2)**p/(2*b**2*p**2 + 6*b**2*p +
4*b**2) + a*b*p*x**2*(a + b*x**2)**p/(2*b**2*p**2 + 6*b**2*p + 4*b**2) + b**2*p*
x**4*(a + b*x**2)**p/(2*b**2*p**2 + 6*b**2*p + 4*b**2) + b**2*x**4*(a + b*x**2)*
*p/(2*b**2*p**2 + 6*b**2*p + 4*b**2), True))

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GIAC/XCAS [A]  time = 0.240953, size = 138, normalized size = 2.88 \[ \frac{{\left (b x^{2} + a\right )}^{2} p e^{\left (p{\rm ln}\left (b x^{2} + a\right )\right )} -{\left (b x^{2} + a\right )} a p e^{\left (p{\rm ln}\left (b x^{2} + a\right )\right )} +{\left (b x^{2} + a\right )}^{2} e^{\left (p{\rm ln}\left (b x^{2} + a\right )\right )} - 2 \,{\left (b x^{2} + a\right )} a e^{\left (p{\rm ln}\left (b x^{2} + a\right )\right )}}{2 \,{\left (p^{2} + 3 \, p + 2\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*x^3,x, algorithm="giac")

[Out]

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