3.93 \(\int x^3 \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx\)

Optimal. Leaf size=86 \[ \frac{\left (a+b x^2\right )^3 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{2 b^2 (2 p+3)}-\frac{a \left (a+b x^2\right )^2 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{4 b^2 (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.211069, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ \frac{\left (a+b x^2\right )^3 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{2 b^2 (2 p+3)}-\frac{a \left (a+b x^2\right )^2 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{4 b^2 (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 27.9409, size = 70, normalized size = 0.81 \[ - \frac{a \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p + 1}}{4 b^{2} \left (p + 1\right ) \left (2 p + 3\right )} + \frac{x^{2} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p + 1}}{2 b \left (2 p + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x**2+a)*(b**2*x**4+2*a*b*x**2+a**2)**p,x)

[Out]

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

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Mathematica [A]  time = 0.0348295, size = 45, normalized size = 0.52 \[ \frac{\left (\left (a+b x^2\right )^2\right )^{p+1} \left (2 b (p+1) x^2-a\right )}{4 b^2 (p+1) (2 p+3)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^p,x]

[Out]

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

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Maple [A]  time = 0.007, size = 62, normalized size = 0.7 \[ -{\frac{ \left ({b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2} \right ) ^{p} \left ( -2\,{x}^{2}pb-2\,b{x}^{2}+a \right ) \left ( b{x}^{2}+a \right ) ^{2}}{4\,{b}^{2} \left ( 2\,{p}^{2}+5\,p+3 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^2+a)*(b^2*x^4+2*a*b*x^2+a^2)^p,x)

[Out]

-1/4*(b^2*x^4+2*a*b*x^2+a^2)^p*(-2*b*p*x^2-2*b*x^2+a)*(b*x^2+a)^2/b^2/(2*p^2+5*p
+3)

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Maxima [A]  time = 0.721337, size = 182, normalized size = 2.12 \[ \frac{{\left (b^{2}{\left (2 \, p + 1\right )} x^{4} + 2 \, a b p x^{2} - a^{2}\right )}{\left (b x^{2} + a\right )}^{2 \, p} a}{4 \,{\left (2 \, p^{2} + 3 \, p + 1\right )} b^{2}} + \frac{{\left ({\left (2 \, p^{2} + 3 \, p + 1\right )} b^{3} x^{6} +{\left (2 \, p^{2} + p\right )} a b^{2} x^{4} - 2 \, a^{2} b p x^{2} + a^{3}\right )}{\left (b x^{2} + a\right )}^{2 \, p}}{2 \,{\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*(b^2*x^4 + 2*a*b*x^2 + a^2)^p*x^3,x, algorithm="maxima")

[Out]

1/4*(b^2*(2*p + 1)*x^4 + 2*a*b*p*x^2 - a^2)*(b*x^2 + a)^(2*p)*a/((2*p^2 + 3*p +
1)*b^2) + 1/2*((2*p^2 + 3*p + 1)*b^3*x^6 + (2*p^2 + p)*a*b^2*x^4 - 2*a^2*b*p*x^2
+ a^3)*(b*x^2 + a)^(2*p)/((4*p^3 + 12*p^2 + 11*p + 3)*b^2)

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Fricas [A]  time = 0.284246, size = 124, normalized size = 1.44 \[ \frac{{\left (2 \,{\left (b^{3} p + b^{3}\right )} x^{6} + 2 \, a^{2} b p x^{2} +{\left (4 \, a b^{2} p + 3 \, a b^{2}\right )} x^{4} - a^{3}\right )}{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{4 \,{\left (2 \, b^{2} p^{2} + 5 \, b^{2} p + 3 \, b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*(b^2*x^4 + 2*a*b*x^2 + a^2)^p*x^3,x, algorithm="fricas")

[Out]

1/4*(2*(b^3*p + b^3)*x^6 + 2*a^2*b*p*x^2 + (4*a*b^2*p + 3*a*b^2)*x^4 - a^3)*(b^2
*x^4 + 2*a*b*x^2 + a^2)^p/(2*b^2*p^2 + 5*b^2*p + 3*b^2)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x**2+a)*(b**2*x**4+2*a*b*x**2+a**2)**p,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.270793, size = 281, normalized size = 3.27 \[ \frac{2 \, b^{3} p x^{6} e^{\left (p{\rm ln}\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )\right )} + 2 \, b^{3} x^{6} e^{\left (p{\rm ln}\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )\right )} + 4 \, a b^{2} p x^{4} e^{\left (p{\rm ln}\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )\right )} + 3 \, a b^{2} x^{4} e^{\left (p{\rm ln}\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )\right )} + 2 \, a^{2} b p x^{2} e^{\left (p{\rm ln}\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )\right )} - a^{3} e^{\left (p{\rm ln}\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )\right )}}{4 \,{\left (2 \, b^{2} p^{2} + 5 \, b^{2} p + 3 \, b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*(b^2*x^4 + 2*a*b*x^2 + a^2)^p*x^3,x, algorithm="giac")

[Out]

1/4*(2*b^3*p*x^6*e^(p*ln(b^2*x^4 + 2*a*b*x^2 + a^2)) + 2*b^3*x^6*e^(p*ln(b^2*x^4
+ 2*a*b*x^2 + a^2)) + 4*a*b^2*p*x^4*e^(p*ln(b^2*x^4 + 2*a*b*x^2 + a^2)) + 3*a*b
^2*x^4*e^(p*ln(b^2*x^4 + 2*a*b*x^2 + a^2)) + 2*a^2*b*p*x^2*e^(p*ln(b^2*x^4 + 2*a
*b*x^2 + a^2)) - a^3*e^(p*ln(b^2*x^4 + 2*a*b*x^2 + a^2)))/(2*b^2*p^2 + 5*b^2*p +
3*b^2)