3.1262 \(\int x^7 \left (a+b x^4\right )^p \, dx\)

Optimal. Leaf size=48 \[ \frac{\left (a+b x^4\right )^{p+2}}{4 b^2 (p+2)}-\frac{a \left (a+b x^4\right )^{p+1}}{4 b^2 (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0659229, 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 \[ \frac{\left (a+b x^4\right )^{p+2}}{4 b^2 (p+2)}-\frac{a \left (a+b x^4\right )^{p+1}}{4 b^2 (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^7*(a + b*x^4)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.4618, size = 37, normalized size = 0.77 \[ - \frac{a \left (a + b x^{4}\right )^{p + 1}}{4 b^{2} \left (p + 1\right )} + \frac{\left (a + b x^{4}\right )^{p + 2}}{4 b^{2} \left (p + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(b*x**4+a)**p,x)

[Out]

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

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Mathematica [A]  time = 0.0346798, size = 40, normalized size = 0.83 \[ \frac{\left (a+b x^4\right )^{p+1} \left (b (p+1) x^4-a\right )}{4 b^2 (p+1) (p+2)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7*(a + b*x^4)^p,x]

[Out]

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

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Maple [A]  time = 0.008, size = 42, normalized size = 0.9 \[ -{\frac{ \left ( b{x}^{4}+a \right ) ^{1+p} \left ( -{x}^{4}pb-b{x}^{4}+a \right ) }{4\,{b}^{2} \left ({p}^{2}+3\,p+2 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(b*x^4+a)^p,x)

[Out]

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

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Maxima [A]  time = 1.45027, size = 63, normalized size = 1.31 \[ \frac{{\left (b^{2}{\left (p + 1\right )} x^{8} + a b p x^{4} - a^{2}\right )}{\left (b x^{4} + a\right )}^{p}}{4 \,{\left (p^{2} + 3 \, p + 2\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^p*x^7,x, algorithm="maxima")

[Out]

1/4*(b^2*(p + 1)*x^8 + a*b*p*x^4 - a^2)*(b*x^4 + a)^p/((p^2 + 3*p + 2)*b^2)

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Fricas [A]  time = 0.234624, size = 78, normalized size = 1.62 \[ \frac{{\left ({\left (b^{2} p + b^{2}\right )} x^{8} + a b p x^{4} - a^{2}\right )}{\left (b x^{4} + a\right )}^{p}}{4 \,{\left (b^{2} p^{2} + 3 \, b^{2} p + 2 \, b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^p*x^7,x, algorithm="fricas")

[Out]

1/4*((b^2*p + b^2)*x^8 + a*b*p*x^4 - a^2)*(b*x^4 + a)^p/(b^2*p^2 + 3*b^2*p + 2*b
^2)

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Sympy [A]  time = 28.7692, size = 495, normalized size = 10.31 \[ \begin{cases} \frac{a^{p} x^{8}}{8} & \text{for}\: b = 0 \\\frac{a \log{\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + x \right )}}{4 a b^{2} + 4 b^{3} x^{4}} + \frac{a \log{\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + x \right )}}{4 a b^{2} + 4 b^{3} x^{4}} + \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + x^{2} \right )}}{4 a b^{2} + 4 b^{3} x^{4}} + \frac{a}{4 a b^{2} + 4 b^{3} x^{4}} + \frac{b x^{4} \log{\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + x \right )}}{4 a b^{2} + 4 b^{3} x^{4}} + \frac{b x^{4} \log{\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + x \right )}}{4 a b^{2} + 4 b^{3} x^{4}} + \frac{b x^{4} \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + x^{2} \right )}}{4 a b^{2} + 4 b^{3} x^{4}} & \text{for}\: p = -2 \\- \frac{a \log{\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + x \right )}}{4 b^{2}} - \frac{a \log{\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + x \right )}}{4 b^{2}} - \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + x^{2} \right )}}{4 b^{2}} + \frac{x^{4}}{4 b} & \text{for}\: p = -1 \\- \frac{a^{2} \left (a + b x^{4}\right )^{p}}{4 b^{2} p^{2} + 12 b^{2} p + 8 b^{2}} + \frac{a b p x^{4} \left (a + b x^{4}\right )^{p}}{4 b^{2} p^{2} + 12 b^{2} p + 8 b^{2}} + \frac{b^{2} p x^{8} \left (a + b x^{4}\right )^{p}}{4 b^{2} p^{2} + 12 b^{2} p + 8 b^{2}} + \frac{b^{2} x^{8} \left (a + b x^{4}\right )^{p}}{4 b^{2} p^{2} + 12 b^{2} p + 8 b^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(b*x**4+a)**p,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.220958, size = 138, normalized size = 2.88 \[ \frac{{\left (b x^{4} + a\right )}^{2} p e^{\left (p{\rm ln}\left (b x^{4} + a\right )\right )} -{\left (b x^{4} + a\right )} a p e^{\left (p{\rm ln}\left (b x^{4} + a\right )\right )} +{\left (b x^{4} + a\right )}^{2} e^{\left (p{\rm ln}\left (b x^{4} + a\right )\right )} - 2 \,{\left (b x^{4} + a\right )} a e^{\left (p{\rm ln}\left (b x^{4} + a\right )\right )}}{4 \,{\left (p^{2} + 3 \, p + 2\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^p*x^7,x, algorithm="giac")

[Out]

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