3.723 \(\int (d+e x)^m (a+c x^2) \, dx\)

Optimal. Leaf size=70 \[ \frac{\left (a e^2+c d^2\right ) (d+e x)^{m+1}}{e^3 (m+1)}-\frac{2 c d (d+e x)^{m+2}}{e^3 (m+2)}+\frac{c (d+e x)^{m+3}}{e^3 (m+3)} \]

[Out]

((c*d^2 + a*e^2)*(d + e*x)^(1 + m))/(e^3*(1 + m)) - (2*c*d*(d + e*x)^(2 + m))/(e^3*(2 + m)) + (c*(d + e*x)^(3
+ m))/(e^3*(3 + m))

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Rubi [A]  time = 0.0295774, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {697} \[ \frac{\left (a e^2+c d^2\right ) (d+e x)^{m+1}}{e^3 (m+1)}-\frac{2 c d (d+e x)^{m+2}}{e^3 (m+2)}+\frac{c (d+e x)^{m+3}}{e^3 (m+3)} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^m*(a + c*x^2),x]

[Out]

((c*d^2 + a*e^2)*(d + e*x)^(1 + m))/(e^3*(1 + m)) - (2*c*d*(d + e*x)^(2 + m))/(e^3*(2 + m)) + (c*(d + e*x)^(3
+ m))/(e^3*(3 + m))

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int (d+e x)^m \left (a+c x^2\right ) \, dx &=\int \left (\frac{\left (c d^2+a e^2\right ) (d+e x)^m}{e^2}-\frac{2 c d (d+e x)^{1+m}}{e^2}+\frac{c (d+e x)^{2+m}}{e^2}\right ) \, dx\\ &=\frac{\left (c d^2+a e^2\right ) (d+e x)^{1+m}}{e^3 (1+m)}-\frac{2 c d (d+e x)^{2+m}}{e^3 (2+m)}+\frac{c (d+e x)^{3+m}}{e^3 (3+m)}\\ \end{align*}

Mathematica [A]  time = 0.0402471, size = 59, normalized size = 0.84 \[ \frac{(d+e x)^{m+1} \left (\frac{a e^2+c d^2}{m+1}+\frac{c (d+e x)^2}{m+3}-\frac{2 c d (d+e x)}{m+2}\right )}{e^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^m*(a + c*x^2),x]

[Out]

((d + e*x)^(1 + m)*((c*d^2 + a*e^2)/(1 + m) - (2*c*d*(d + e*x))/(2 + m) + (c*(d + e*x)^2)/(3 + m)))/e^3

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Maple [A]  time = 0.043, size = 100, normalized size = 1.4 \begin{align*}{\frac{ \left ( ex+d \right ) ^{1+m} \left ( c{e}^{2}{m}^{2}{x}^{2}+3\,c{e}^{2}m{x}^{2}+a{e}^{2}{m}^{2}-2\,cdemx+2\,c{e}^{2}{x}^{2}+5\,a{e}^{2}m-2\,cdex+6\,a{e}^{2}+2\,c{d}^{2} \right ) }{{e}^{3} \left ({m}^{3}+6\,{m}^{2}+11\,m+6 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e*x+d)^(1+m)*(c*e^2*m^2*x^2+3*c*e^2*m*x^2+a*e^2*m^2-2*c*d*e*m*x+2*c*e^2*x^2+5*a*e^2*m-2*c*d*e*x+6*a*e^2+2*c*d
^2)/e^3/(m^3+6*m^2+11*m+6)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.0741, size = 308, normalized size = 4.4 \begin{align*} \frac{{\left (a d e^{2} m^{2} + 5 \, a d e^{2} m + 2 \, c d^{3} + 6 \, a d e^{2} +{\left (c e^{3} m^{2} + 3 \, c e^{3} m + 2 \, c e^{3}\right )} x^{3} +{\left (c d e^{2} m^{2} + c d e^{2} m\right )} x^{2} +{\left (a e^{3} m^{2} + 6 \, a e^{3} -{\left (2 \, c d^{2} e - 5 \, a e^{3}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{3} m^{3} + 6 \, e^{3} m^{2} + 11 \, e^{3} m + 6 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*d*e^2*m^2 + 5*a*d*e^2*m + 2*c*d^3 + 6*a*d*e^2 + (c*e^3*m^2 + 3*c*e^3*m + 2*c*e^3)*x^3 + (c*d*e^2*m^2 + c*d*
e^2*m)*x^2 + (a*e^3*m^2 + 6*a*e^3 - (2*c*d^2*e - 5*a*e^3)*m)*x)*(e*x + d)^m/(e^3*m^3 + 6*e^3*m^2 + 11*e^3*m +
6*e^3)

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Sympy [A]  time = 1.4262, size = 952, normalized size = 13.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((d**m*(a*x + c*x**3/3), Eq(e, 0)), (-a*e**2/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) + 2*c*d**2*log(
d/e + x)/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) + 3*c*d**2/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) + 4*c*d*
e*x*log(d/e + x)/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) + 4*c*d*e*x/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2)
+ 2*c*e**2*x**2*log(d/e + x)/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2), Eq(m, -3)), (-a*e**2/(d*e**3 + e**4*x)
- 2*c*d**2*log(d/e + x)/(d*e**3 + e**4*x) - 2*c*d**2/(d*e**3 + e**4*x) - 2*c*d*e*x*log(d/e + x)/(d*e**3 + e**
4*x) + c*e**2*x**2/(d*e**3 + e**4*x), Eq(m, -2)), (a*log(d/e + x)/e + c*d**2*log(d/e + x)/e**3 - c*d*x/e**2 +
c*x**2/(2*e), Eq(m, -1)), (a*d*e**2*m**2*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 5*a*d*e
**2*m*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 6*a*d*e**2*(d + e*x)**m/(e**3*m**3 + 6*e**
3*m**2 + 11*e**3*m + 6*e**3) + a*e**3*m**2*x*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 5*a
*e**3*m*x*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 6*a*e**3*x*(d + e*x)**m/(e**3*m**3 + 6
*e**3*m**2 + 11*e**3*m + 6*e**3) + 2*c*d**3*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) - 2*c*
d**2*e*m*x*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + c*d*e**2*m**2*x**2*(d + e*x)**m/(e**3
*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + c*d*e**2*m*x**2*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m
+ 6*e**3) + c*e**3*m**2*x**3*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 3*c*e**3*m*x**3*(d
+ e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 2*c*e**3*x**3*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2
+ 11*e**3*m + 6*e**3), True))

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Giac [B]  time = 1.12598, size = 319, normalized size = 4.56 \begin{align*} \frac{{\left (x e + d\right )}^{m} c m^{2} x^{3} e^{3} +{\left (x e + d\right )}^{m} c d m^{2} x^{2} e^{2} + 3 \,{\left (x e + d\right )}^{m} c m x^{3} e^{3} +{\left (x e + d\right )}^{m} c d m x^{2} e^{2} - 2 \,{\left (x e + d\right )}^{m} c d^{2} m x e +{\left (x e + d\right )}^{m} a m^{2} x e^{3} + 2 \,{\left (x e + d\right )}^{m} c x^{3} e^{3} +{\left (x e + d\right )}^{m} a d m^{2} e^{2} + 2 \,{\left (x e + d\right )}^{m} c d^{3} + 5 \,{\left (x e + d\right )}^{m} a m x e^{3} + 5 \,{\left (x e + d\right )}^{m} a d m e^{2} + 6 \,{\left (x e + d\right )}^{m} a x e^{3} + 6 \,{\left (x e + d\right )}^{m} a d e^{2}}{m^{3} e^{3} + 6 \, m^{2} e^{3} + 11 \, m e^{3} + 6 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*e + d)^m*c*m^2*x^3*e^3 + (x*e + d)^m*c*d*m^2*x^2*e^2 + 3*(x*e + d)^m*c*m*x^3*e^3 + (x*e + d)^m*c*d*m*x^2*e
^2 - 2*(x*e + d)^m*c*d^2*m*x*e + (x*e + d)^m*a*m^2*x*e^3 + 2*(x*e + d)^m*c*x^3*e^3 + (x*e + d)^m*a*d*m^2*e^2 +
2*(x*e + d)^m*c*d^3 + 5*(x*e + d)^m*a*m*x*e^3 + 5*(x*e + d)^m*a*d*m*e^2 + 6*(x*e + d)^m*a*x*e^3 + 6*(x*e + d)
^m*a*d*e^2)/(m^3*e^3 + 6*m^2*e^3 + 11*m*e^3 + 6*e^3)