3.185 \(\int x^n (b+d x^2)^n (b+3 d x^2) \, dx\)

Optimal. Leaf size=22 \[ \frac{x^{n+1} \left (b+d x^2\right )^{n+1}}{n+1} \]

[Out]

(x^(1 + n)*(b + d*x^2)^(1 + n))/(1 + n)

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Rubi [A]  time = 0.0080456, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {449} \[ \frac{x^{n+1} \left (b+d x^2\right )^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

Int[x^n*(b + d*x^2)^n*(b + 3*d*x^2),x]

[Out]

(x^(1 + n)*(b + d*x^2)^(1 + n))/(1 + n)

Rule 449

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[
a*d*(m + 1) - b*c*(m + n*(p + 1) + 1), 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int x^n \left (b+d x^2\right )^n \left (b+3 d x^2\right ) \, dx &=\frac{x^{1+n} \left (b+d x^2\right )^{1+n}}{1+n}\\ \end{align*}

Mathematica [C]  time = 0.0354555, size = 108, normalized size = 4.91 \[ \frac{x^{n+1} \left (b+d x^2\right )^n \left (\frac{d x^2}{b}+1\right )^{-n} \left (3 d (n+1) x^2 \, _2F_1\left (-n,\frac{n+3}{2};\frac{n+5}{2};-\frac{d x^2}{b}\right )+b (n+3) \, _2F_1\left (-n,\frac{n+1}{2};\frac{n+3}{2};-\frac{d x^2}{b}\right )\right )}{(n+1) (n+3)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^n*(b + d*x^2)^n*(b + 3*d*x^2),x]

[Out]

(x^(1 + n)*(b + d*x^2)^n*(b*(3 + n)*Hypergeometric2F1[-n, (1 + n)/2, (3 + n)/2, -((d*x^2)/b)] + 3*d*(1 + n)*x^
2*Hypergeometric2F1[-n, (3 + n)/2, (5 + n)/2, -((d*x^2)/b)]))/((1 + n)*(3 + n)*(1 + (d*x^2)/b)^n)

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Maple [A]  time = 0.003, size = 23, normalized size = 1.1 \begin{align*}{\frac{{x}^{1+n} \left ( d{x}^{2}+b \right ) ^{1+n}}{1+n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^n*(d*x^2+b)^n*(3*d*x^2+b),x)

[Out]

x^(1+n)*(d*x^2+b)^(1+n)/(1+n)

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Maxima [A]  time = 1.19996, size = 42, normalized size = 1.91 \begin{align*} \frac{{\left (d x^{3} + b x\right )} e^{\left (n \log \left (d x^{2} + b\right ) + n \log \left (x\right )\right )}}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^n*(d*x^2+b)^n*(3*d*x^2+b),x, algorithm="maxima")

[Out]

(d*x^3 + b*x)*e^(n*log(d*x^2 + b) + n*log(x))/(n + 1)

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Fricas [A]  time = 1.6317, size = 55, normalized size = 2.5 \begin{align*} \frac{{\left (d x^{3} + b x\right )}{\left (d x^{2} + b\right )}^{n} x^{n}}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^n*(d*x^2+b)^n*(3*d*x^2+b),x, algorithm="fricas")

[Out]

(d*x^3 + b*x)*(d*x^2 + b)^n*x^n/(n + 1)

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Sympy [B]  time = 58.6106, size = 76, normalized size = 3.45 \begin{align*} \begin{cases} \frac{b x x^{n} \left (b + d x^{2}\right )^{n}}{n + 1} + \frac{d x^{3} x^{n} \left (b + d x^{2}\right )^{n}}{n + 1} & \text{for}\: n \neq -1 \\\log{\left (x \right )} + \log{\left (- i \sqrt{b} \sqrt{\frac{1}{d}} + x \right )} + \log{\left (i \sqrt{b} \sqrt{\frac{1}{d}} + x \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**n*(d*x**2+b)**n*(3*d*x**2+b),x)

[Out]

Piecewise((b*x*x**n*(b + d*x**2)**n/(n + 1) + d*x**3*x**n*(b + d*x**2)**n/(n + 1), Ne(n, -1)), (log(x) + log(-
I*sqrt(b)*sqrt(1/d) + x) + log(I*sqrt(b)*sqrt(1/d) + x), True))

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Giac [A]  time = 1.15311, size = 53, normalized size = 2.41 \begin{align*} \frac{{\left (d x^{2} + b\right )}^{n} d x^{3} x^{n} +{\left (d x^{2} + b\right )}^{n} b x x^{n}}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^n*(d*x^2+b)^n*(3*d*x^2+b),x, algorithm="giac")

[Out]

((d*x^2 + b)^n*d*x^3*x^n + (d*x^2 + b)^n*b*x*x^n)/(n + 1)