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

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

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Rubi [A]  time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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)} \end {gather*}

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*}

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Mathematica [A]  time = 0.04, size = 59, normalized size = 0.84 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.04, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^m \left (a+c x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^m*(a + c*x^2),x]

[Out]

Defer[IntegrateAlgebraic][(d + e*x)^m*(a + c*x^2), x]

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fricas [B]  time = 0.42, size = 149, normalized size = 2.13 \begin {gather*} \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 {gather*}

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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giac [B]  time = 0.17, size = 236, normalized size = 3.37 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.05, size = 100, normalized size = 1.43 \begin {gather*} \frac {\left (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}\right ) \left (e x +d \right )^{m +1}}{\left (m^{3}+6 m^{2}+11 m +6\right ) e^{3}} \end {gather*}

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)^(m+1)*(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 [A]  time = 1.44, size = 89, normalized size = 1.27 \begin {gather*} \frac {{\left (e x + d\right )}^{m + 1} a}{e {\left (m + 1\right )}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} c}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.48, size = 163, normalized size = 2.33 \begin {gather*} {\left (d+e\,x\right )}^m\,\left (\frac {c\,x^3\,\left (m^2+3\,m+2\right )}{m^3+6\,m^2+11\,m+6}+\frac {x\,\left (-2\,c\,d^2\,e\,m+a\,e^3\,m^2+5\,a\,e^3\,m+6\,a\,e^3\right )}{e^3\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {d\,\left (2\,c\,d^2+a\,e^2\,m^2+5\,a\,e^2\,m+6\,a\,e^2\right )}{e^3\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {c\,d\,m\,x^2\,\left (m+1\right )}{e\,\left (m^3+6\,m^2+11\,m+6\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)*(d + e*x)^m,x)

[Out]

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

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sympy [A]  time = 1.73, size = 952, normalized size = 13.60 \begin {gather*} \begin {cases} d^{m} \left (a x + \frac {c x^{3}}{3}\right ) & \text {for}\: e = 0 \\- \frac {a e^{2}}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} + \frac {2 c d^{2} \log {\left (\frac {d}{e} + x \right )}}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} + \frac {3 c d^{2}}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} + \frac {4 c d e x \log {\left (\frac {d}{e} + x \right )}}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} + \frac {4 c d e x}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} + \frac {2 c e^{2} x^{2} \log {\left (\frac {d}{e} + x \right )}}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} & \text {for}\: m = -3 \\- \frac {a e^{2}}{d e^{3} + e^{4} x} - \frac {2 c d^{2} \log {\left (\frac {d}{e} + x \right )}}{d e^{3} + e^{4} x} - \frac {2 c d^{2}}{d e^{3} + e^{4} x} - \frac {2 c d e x \log {\left (\frac {d}{e} + x \right )}}{d e^{3} + e^{4} x} + \frac {c e^{2} x^{2}}{d e^{3} + e^{4} x} & \text {for}\: m = -2 \\\frac {a \log {\left (\frac {d}{e} + x \right )}}{e} + \frac {c d^{2} \log {\left (\frac {d}{e} + x \right )}}{e^{3}} - \frac {c d x}{e^{2}} + \frac {c x^{2}}{2 e} & \text {for}\: m = -1 \\\frac {a d e^{2} m^{2} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {5 a d e^{2} m \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {6 a d e^{2} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {a e^{3} m^{2} x \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {5 a e^{3} m x \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {6 a e^{3} x \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {2 c d^{3} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} - \frac {2 c d^{2} e m x \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {c d e^{2} m^{2} x^{2} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {c d e^{2} m x^{2} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {c e^{3} m^{2} x^{3} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {3 c e^{3} m x^{3} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {2 c e^{3} x^{3} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} & \text {otherwise} \end {cases} \end {gather*}

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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