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

Optimal. Leaf size=19 \[ \frac{\left (b x+d x^3\right )^{n+1}}{n+1} \]

[Out]

(b*x + d*x^3)^(1 + n)/(1 + n)

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Rubi [A]  time = 0.0089349, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {1588} \[ \frac{\left (b x+d x^3\right )^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

Int[(b + 3*d*x^2)*(b*x + d*x^3)^n,x]

[Out]

(b*x + d*x^3)^(1 + n)/(1 + n)

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [C]  time = 0.0645414, size = 106, normalized size = 5.58 \[ \frac{x \left (x \left (b+d x^2\right )\right )^n \left (\frac{d x^2}{b}+1\right )^{-n} \left (3 d (n+1) x^2 \, _2F_1\left (-n,\frac{n+3}{2};\frac{n+5}{2};-\frac{d x^2}{b}\right )+b (n+3) \, _2F_1\left (-n,\frac{n+1}{2};\frac{n+3}{2};-\frac{d x^2}{b}\right )\right )}{(n+1) (n+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(x*(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*Hy
pergeometric2F1[-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 = 26, normalized size = 1.4 \begin{align*}{\frac{x \left ( d{x}^{2}+b \right ) \left ( d{x}^{3}+bx \right ) ^{n}}{1+n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.64072, size = 53, normalized size = 2.79 \begin{align*} \frac{{\left (d x^{3} + b x\right )}{\left (d x^{3} + b x\right )}^{n}}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x^3 + b*x)*(d*x^3 + b*x)^n/(n + 1)

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Sympy [B]  time = 12.4504, size = 73, normalized size = 3.84 \begin{align*} \begin{cases} \frac{b x \left (b x + d x^{3}\right )^{n}}{n + 1} + \frac{d x^{3} \left (b x + d x^{3}\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((3*d*x**2+b)*(d*x**3+b*x)**n,x)

[Out]

Piecewise((b*x*(b*x + d*x**3)**n/(n + 1) + d*x**3*(b*x + d*x**3)**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 = 1.20395, size = 50, normalized size = 2.63 \begin{align*} \frac{{\left (d x^{3} + b x\right )}^{n} d x^{3} +{\left (d x^{3} + b x\right )}^{n} b x}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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