3.1263 \(\int x^3 \left (a+b x^4\right )^p \, dx\)

Optimal. Leaf size=23 \[ \frac{\left (a+b x^4\right )^{p+1}}{4 b (p+1)} \]

[Out]

(a + b*x^4)^(1 + p)/(4*b*(1 + p))

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Rubi [A]  time = 0.016378, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{\left (a+b x^4\right )^{p+1}}{4 b (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x^4)^p,x]

[Out]

(a + b*x^4)^(1 + p)/(4*b*(1 + p))

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Rubi in Sympy [A]  time = 2.74449, size = 15, normalized size = 0.65 \[ \frac{\left (a + b x^{4}\right )^{p + 1}}{4 b \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x**4)**(p + 1)/(4*b*(p + 1))

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Mathematica [A]  time = 0.0114634, size = 22, normalized size = 0.96 \[ \frac{\left (a+b x^4\right )^{p+1}}{4 b p+4 b} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(a + b*x^4)^p,x]

[Out]

(a + b*x^4)^(1 + p)/(4*b + 4*b*p)

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Maple [A]  time = 0.005, size = 22, normalized size = 1. \[{\frac{ \left ( b{x}^{4}+a \right ) ^{1+p}}{4\,b \left ( 1+p \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^4+a)^p,x)

[Out]

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

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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^4 + a)^p*x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234595, size = 34, normalized size = 1.48 \[ \frac{{\left (b x^{4} + a\right )}{\left (b x^{4} + a\right )}^{p}}{4 \,{\left (b p + b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.00935, size = 129, normalized size = 5.61 \[ \begin{cases} \frac{x^{4}}{4 a} & \text{for}\: b = 0 \wedge p = -1 \\\frac{a^{p} x^{4}}{4} & \text{for}\: b = 0 \\\frac{\log{\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + x \right )}}{4 b} + \frac{\log{\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + x \right )}}{4 b} + \frac{\log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + x^{2} \right )}}{4 b} & \text{for}\: p = -1 \\\frac{a \left (a + b x^{4}\right )^{p}}{4 b p + 4 b} + \frac{b x^{4} \left (a + b x^{4}\right )^{p}}{4 b p + 4 b} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x**4+a)**p,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.21908, size = 28, normalized size = 1.22 \[ \frac{{\left (b x^{4} + a\right )}^{p + 1}}{4 \, b{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(b*x^4 + a)^(p + 1)/(b*(p + 1))