3.598 \(\int (d x)^m (a+b x^n+c x^{2 n}) \, dx\)

Optimal. Leaf size=58 \[ \frac{a (d x)^{m+1}}{d (m+1)}+\frac{b x^{n+1} (d x)^m}{m+n+1}+\frac{c x^{2 n+1} (d x)^m}{m+2 n+1} \]

[Out]

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

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Rubi [A]  time = 0.0242398, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {14, 20, 30} \[ \frac{a (d x)^{m+1}}{d (m+1)}+\frac{b x^{n+1} (d x)^m}{m+n+1}+\frac{c x^{2 n+1} (d x)^m}{m+2 n+1} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*(a + b*x^n + c*x^(2*n)),x]

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0615337, size = 41, normalized size = 0.71 \[ x (d x)^m \left (\frac{a}{m+1}+x^n \left (\frac{b}{m+n+1}+\frac{c x^n}{m+2 n+1}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*(a + b*x^n + c*x^(2*n)),x]

[Out]

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

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Maple [C]  time = 0.042, size = 205, normalized size = 3.5 \begin{align*}{\frac{x \left ( c{m}^{2} \left ({x}^{n} \right ) ^{2}+cmn \left ({x}^{n} \right ) ^{2}+b{m}^{2}{x}^{n}+2\,bmn{x}^{n}+2\,mc \left ({x}^{n} \right ) ^{2}+c \left ({x}^{n} \right ) ^{2}n+a{m}^{2}+3\,amn+2\,a{n}^{2}+2\,mb{x}^{n}+2\,b{x}^{n}n+c \left ({x}^{n} \right ) ^{2}+2\,am+3\,an+b{x}^{n}+a \right ) }{ \left ( 1+m \right ) \left ( 1+m+n \right ) \left ( 1+m+2\,n \right ) }{{\rm e}^{{\frac{m \left ( -i \left ({\it csgn} \left ( idx \right ) \right ) ^{3}\pi +i \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( id \right ) \pi +i \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) \pi -i{\it csgn} \left ( idx \right ){\it csgn} \left ( id \right ){\it csgn} \left ( ix \right ) \pi +2\,\ln \left ( x \right ) +2\,\ln \left ( d \right ) \right ) }{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a+b*x^n+c*x^(2*n)),x)

[Out]

x*(c*m^2*(x^n)^2+c*m*n*(x^n)^2+b*m^2*x^n+2*b*m*n*x^n+2*m*c*(x^n)^2+c*(x^n)^2*n+a*m^2+3*a*m*n+2*a*n^2+2*m*b*x^n
+2*b*x^n*n+c*(x^n)^2+2*a*m+3*a*n+b*x^n+a)/(1+m)/(1+m+n)/(1+m+2*n)*exp(1/2*m*(-I*csgn(I*d*x)^3*Pi+I*csgn(I*d*x)
^2*csgn(I*d)*Pi+I*csgn(I*d*x)^2*csgn(I*x)*Pi-I*csgn(I*d*x)*csgn(I*d)*csgn(I*x)*Pi+2*ln(x)+2*ln(d)))

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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((d*x)^m*(a+b*x^n+c*x^(2*n)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.89088, size = 371, normalized size = 6.4 \begin{align*} \frac{{\left (c m^{2} + 2 \, c m +{\left (c m + c\right )} n + c\right )} x x^{2 \, n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} +{\left (b m^{2} + 2 \, b m + 2 \,{\left (b m + b\right )} n + b\right )} x x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} +{\left (a m^{2} + 2 \, a n^{2} + 2 \, a m + 3 \,{\left (a m + a\right )} n + a\right )} x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )}}{m^{3} + 2 \,{\left (m + 1\right )} n^{2} + 3 \, m^{2} + 3 \,{\left (m^{2} + 2 \, m + 1\right )} n + 3 \, m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(a+b*x**n+c*x**(2*n)),x)

[Out]

Exception raised: TypeError

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Giac [B]  time = 1.10858, size = 752, normalized size = 12.97 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*m^2*x*x^(2*n)*e^(m*log(d) + m*log(x)) + c*m*n*x*x^(2*n)*e^(m*log(d) + m*log(x)) + b*m^2*x*x^n*e^(m*log(d) +
m*log(x)) + c*m^2*x*x^n*e^(m*log(d) + m*log(x)) + 2*b*m*n*x*x^n*e^(m*log(d) + m*log(x)) + c*m*n*x*x^n*e^(m*lo
g(d) + m*log(x)) + a*m^2*x*e^(m*log(d) + m*log(x)) + b*m^2*x*e^(m*log(d) + m*log(x)) + c*m^2*x*e^(m*log(d) + m
*log(x)) + 3*a*m*n*x*e^(m*log(d) + m*log(x)) + 2*b*m*n*x*e^(m*log(d) + m*log(x)) + c*m*n*x*e^(m*log(d) + m*log
(x)) + 2*a*n^2*x*e^(m*log(d) + m*log(x)) + 2*c*m*x*x^(2*n)*e^(m*log(d) + m*log(x)) + c*n*x*x^(2*n)*e^(m*log(d)
+ m*log(x)) + 2*b*m*x*x^n*e^(m*log(d) + m*log(x)) + 2*c*m*x*x^n*e^(m*log(d) + m*log(x)) + 2*b*n*x*x^n*e^(m*lo
g(d) + m*log(x)) + c*n*x*x^n*e^(m*log(d) + m*log(x)) + 2*a*m*x*e^(m*log(d) + m*log(x)) + 2*b*m*x*e^(m*log(d) +
m*log(x)) + 2*c*m*x*e^(m*log(d) + m*log(x)) + 3*a*n*x*e^(m*log(d) + m*log(x)) + 2*b*n*x*e^(m*log(d) + m*log(x
)) + c*n*x*e^(m*log(d) + m*log(x)) + c*x*x^(2*n)*e^(m*log(d) + m*log(x)) + b*x*x^n*e^(m*log(d) + m*log(x)) + c
*x*x^n*e^(m*log(d) + m*log(x)) + a*x*e^(m*log(d) + m*log(x)) + b*x*e^(m*log(d) + m*log(x)) + c*x*e^(m*log(d) +
m*log(x)))/(m^3 + 3*m^2*n + 2*m*n^2 + 3*m^2 + 6*m*n + 2*n^2 + 3*m + 3*n + 1)