3.5.36 \(\int (d+e x)^m (c d x+c e x^2) \, dx\) [436]

Optimal. Leaf size=41 \[ -\frac {c d (d+e x)^{2+m}}{e^2 (2+m)}+\frac {c (d+e x)^{3+m}}{e^2 (3+m)} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {640, 12, 45} \begin {gather*} \frac {c (d+e x)^{m+3}}{e^2 (m+3)}-\frac {c d (d+e x)^{m+2}}{e^2 (m+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^m*(c*d*x + c*e*x^2),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 34, normalized size = 0.83 \begin {gather*} \frac {c (d+e x)^{2+m} (-d+e (2+m) x)}{e^2 (2+m) (3+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.42, size = 37, normalized size = 0.90

method	result	size
gosper	\(-\frac {c \left (e x +d \right )^{2+m} \left (-m e x -2 e x +d \right )}{e^{2} \left (m^{2}+5 m +6\right )}\)	\(37\)
risch	\(-\frac {c \left (-e^{3} m \,x^{3}-2 d \,e^{2} m \,x^{2}-2 x^{3} e^{3}-d^{2} e m x -3 d \,x^{2} e^{2}+d^{3}\right ) \left (e x +d \right )^{m}}{\left (2+m \right ) \left (3+m \right ) e^{2}}\)	\(72\)
norman	\(\frac {c e \,x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{3+m}+\frac {c d \left (3+2 m \right ) x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{m^{2}+5 m +6}+\frac {m c \,d^{2} x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+5 m +6\right )}-\frac {c \,d^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{2}+5 m +6\right )}\)	\(109\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m*(c*e*x^2+c*d*x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (41) = 82\).
time = 0.30, size = 117, normalized size = 2.85 \begin {gather*} \frac {{\left ({\left (m + 1\right )} x^{2} e^{2} + d m x e - d^{2}\right )} c d e^{\left (m \log \left (x e + d\right ) - 2\right )}}{m^{2} + 3 \, m + 2} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} x^{3} e^{3} + {\left (m^{2} + m\right )} d x^{2} e^{2} - 2 \, d^{2} m x e + 2 \, d^{3}\right )} c e^{\left (m \log \left (x e + d\right ) - 2\right )}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.57, size = 66, normalized size = 1.61 \begin {gather*} \frac {{\left (c d^{2} m x e + {\left (c m + 2 \, c\right )} x^{3} e^{3} - c d^{3} + {\left (2 \, c d m + 3 \, c d\right )} x^{2} e^{2}\right )} {\left (x e + d\right )}^{m} e^{\left (-2\right )}}{m^{2} + 5 \, m + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (34) = 68\).
time = 0.34, size = 299, normalized size = 7.29 \begin {gather*} \begin {cases} \frac {c d d^{m} x^{2}}{2} & \text {for}\: e = 0 \\\frac {c d \log {\left (\frac {d}{e} + x \right )}}{d e^{2} + e^{3} x} + \frac {c d}{d e^{2} + e^{3} x} + \frac {c e x \log {\left (\frac {d}{e} + x \right )}}{d e^{2} + e^{3} x} & \text {for}\: m = -3 \\- \frac {c d \log {\left (\frac {d}{e} + x \right )}}{e^{2}} + \frac {c x}{e} & \text {for}\: m = -2 \\- \frac {c d^{3} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {c d^{2} e m x \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {2 c d e^{2} m x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {3 c d e^{2} x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {c e^{3} m x^{3} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {2 c e^{3} x^{3} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (41) = 82\).
time = 1.31, size = 118, normalized size = 2.88 \begin {gather*} \frac {{\left (x e + d\right )}^{m} c m x^{3} e^{3} + 2 \, {\left (x e + d\right )}^{m} c d m x^{2} e^{2} + {\left (x e + d\right )}^{m} c d^{2} m x e + 2 \, {\left (x e + d\right )}^{m} c x^{3} e^{3} + 3 \, {\left (x e + d\right )}^{m} c d x^{2} e^{2} - {\left (x e + d\right )}^{m} c d^{3}}{m^{2} e^{2} + 5 \, m e^{2} + 6 \, e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.33, size = 88, normalized size = 2.15 \begin {gather*} {\left (d+e\,x\right )}^m\,\left (\frac {c\,e\,x^3\,\left (m+2\right )}{m^2+5\,m+6}-\frac {c\,d^3}{e^2\,\left (m^2+5\,m+6\right )}+\frac {c\,d\,x^2\,\left (2\,m+3\right )}{m^2+5\,m+6}+\frac {c\,d^2\,m\,x}{e\,\left (m^2+5\,m+6\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*x + c*e*x^2)*(d + e*x)^m,x)

[Out]

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

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