3.2.84 \(\int (b+3 d x^2) (b x+d x^3)^n \, dx\) [184]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1602} \begin {gather*} \frac {\left (b x+d x^3\right )^{n+1}}{n+1} \end {gather*}

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 1602

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*}

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Mathematica [A]
time = 0.06, size = 19, normalized size = 1.00 \begin {gather*} \frac {\left (x \left (b+d x^2\right )\right )^{1+n}}{1+n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(b + d*x^2))^(1 + n)/(1 + n)

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Maple [A]
time = 0.16, size = 20, normalized size = 1.05

method	result	size
derivativedivides	\(\frac {\left (d \,x^{3}+b x \right )^{1+n}}{1+n}\)	\(20\)
default	\(\frac {\left (d \,x^{3}+b x \right )^{1+n}}{1+n}\)	\(20\)
gosper	\(\frac {x \left (d \,x^{2}+b \right ) \left (d \,x^{3}+b x \right )^{n}}{1+n}\)	\(26\)
risch	\(\frac {x \left (d \,x^{2}+b \right ) \left (x \left (d \,x^{2}+b \right )\right )^{n}}{1+n}\)	\(26\)
norman	\(\frac {b x \,{\mathrm e}^{n \ln \left (d \,x^{3}+b x \right )}}{1+n}+\frac {d \,x^{3} {\mathrm e}^{n \ln \left (d \,x^{3}+b x \right )}}{1+n}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 19, normalized size = 1.00 \begin {gather*} \frac {{\left (d x^{3} + b x\right )}^{n + 1}}{n + 1} \end {gather*}

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]

(d*x^3 + b*x)^(n + 1)/(n + 1)

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Fricas [A]
time = 0.40, size = 26, normalized size = 1.37 \begin {gather*} \frac {{\left (d x^{3} + b x\right )} {\left (d x^{3} + b x\right )}^{n}}{n + 1} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (14) = 28\).
time = 5.35, size = 63, normalized size = 3.32 \begin {gather*} \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 (x - \sqrt {- \frac {b}{d}} \right )} + \log {\left (x + \sqrt {- \frac {b}{d}} \right )} & \text {otherwise} \end {cases} \end {gather*}

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(x - sqr
t(-b/d)) + log(x + sqrt(-b/d)), True))

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Giac [A]
time = 3.95, size = 19, normalized size = 1.00 \begin {gather*} \frac {{\left (d x^{3} + b x\right )}^{n + 1}}{n + 1} \end {gather*}

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 + 1)/(n + 1)

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Mupad [B]
time = 2.13, size = 25, normalized size = 1.32 \begin {gather*} \frac {x\,{\left (d\,x^3+b\,x\right )}^n\,\left (d\,x^2+b\right )}{n+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + d*x^3)^n*(b + 3*d*x^2),x)

[Out]

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

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