3.1264 \(\int x^7 (a+b x^4)^p \, dx\)

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 = 0.0265116, 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, Rules used = {266, 43} \[ \frac{\left (a+b x^4\right )^{p+2}}{4 b^2 (p+2)}-\frac{a \left (a+b x^4\right )^{p+1}}{4 b^2 (p+1)} \]

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))

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^7 \left (a+b x^4\right )^p \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int x (a+b x)^p \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^p}{b}+\frac{(a+b x)^{1+p}}{b}\right ) \, dx,x,x^4\right )\\ &=-\frac{a \left (a+b x^4\right )^{1+p}}{4 b^2 (1+p)}+\frac{\left (a+b x^4\right )^{2+p}}{4 b^2 (2+p)}\\ \end{align*}

Mathematica [A]  time = 0.0200436, 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 = 0.004, size = 42, normalized size = 0.9 \begin{align*} -{\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 ) }} \end{align*}

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 = 1.29999, size = 63, normalized size = 1.31 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^4+a)^p,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 = 1.82391, size = 115, normalized size = 2.4 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^4+a)^p,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 = 11.4989, size = 495, normalized size = 10.31 \begin{align*} \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} \end{align*}

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*s
qrt(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 [B]  time = 1.14254, size = 127, normalized size = 2.65 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{2}{\left (b x^{4} + a\right )}^{p} p -{\left (b x^{4} + a\right )}{\left (b x^{4} + a\right )}^{p} a p +{\left (b x^{4} + a\right )}^{2}{\left (b x^{4} + a\right )}^{p} - 2 \,{\left (b x^{4} + a\right )}{\left (b x^{4} + a\right )}^{p} a}{4 \,{\left (p^{2} + 3 \, p + 2\right )} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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