3.5.53 \(\int (d+e x)^2 (a+c x^2) \, dx\) [453]

Optimal. Leaf size=57 \[ \frac {\left (c d^2+a e^2\right ) (d+e x)^3}{3 e^3}-\frac {c d (d+e x)^4}{2 e^3}+\frac {c (d+e x)^5}{5 e^3} \]

[Out]

1/3*(a*e^2+c*d^2)*(e*x+d)^3/e^3-1/2*c*d*(e*x+d)^4/e^3+1/5*c*(e*x+d)^5/e^3

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Rubi [A]
time = 0.02, antiderivative size = 57, 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 = {711} \begin {gather*} \frac {(d+e x)^3 \left (a e^2+c d^2\right )}{3 e^3}+\frac {c (d+e x)^5}{5 e^3}-\frac {c d (d+e x)^4}{2 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*(a + c*x^2),x]

[Out]

((c*d^2 + a*e^2)*(d + e*x)^3)/(3*e^3) - (c*d*(d + e*x)^4)/(2*e^3) + (c*(d + e*x)^5)/(5*e^3)

Rule 711

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)^2 \left (a+c x^2\right ) \, dx &=\int \left (\frac {\left (c d^2+a e^2\right ) (d+e x)^2}{e^2}-\frac {2 c d (d+e x)^3}{e^2}+\frac {c (d+e x)^4}{e^2}\right ) \, dx\\ &=\frac {\left (c d^2+a e^2\right ) (d+e x)^3}{3 e^3}-\frac {c d (d+e x)^4}{2 e^3}+\frac {c (d+e x)^5}{5 e^3}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 53, normalized size = 0.93 \begin {gather*} a d^2 x+a d e x^2+\frac {1}{3} \left (c d^2+a e^2\right ) x^3+\frac {1}{2} c d e x^4+\frac {1}{5} c e^2 x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^2*(a + c*x^2),x]

[Out]

a*d^2*x + a*d*e*x^2 + ((c*d^2 + a*e^2)*x^3)/3 + (c*d*e*x^4)/2 + (c*e^2*x^5)/5

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Maple [A]
time = 0.40, size = 48, normalized size = 0.84

method	result	size
default	\(\frac {c \,e^{2} x^{5}}{5}+\frac {c d e \,x^{4}}{2}+\frac {\left (e^{2} a +c \,d^{2}\right ) x^{3}}{3}+a d e \,x^{2}+a x \,d^{2}\)	\(48\)
norman	\(\frac {c \,e^{2} x^{5}}{5}+\frac {c d e \,x^{4}}{2}+\left (\frac {e^{2} a}{3}+\frac {c \,d^{2}}{3}\right ) x^{3}+a d e \,x^{2}+a x \,d^{2}\)	\(49\)
gosper	\(\frac {1}{5} c \,e^{2} x^{5}+\frac {1}{2} c d e \,x^{4}+\frac {1}{3} x^{3} e^{2} a +\frac {1}{3} x^{3} c \,d^{2}+a d e \,x^{2}+a x \,d^{2}\)	\(50\)
risch	\(\frac {1}{5} c \,e^{2} x^{5}+\frac {1}{2} c d e \,x^{4}+\frac {1}{3} x^{3} e^{2} a +\frac {1}{3} x^{3} c \,d^{2}+a d e \,x^{2}+a x \,d^{2}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c*e^2*x^5+1/2*c*d*e*x^4+1/3*(a*e^2+c*d^2)*x^3+a*d*e*x^2+a*x*d^2

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Maxima [A]
time = 0.55, size = 47, normalized size = 0.82 \begin {gather*} \frac {1}{5} \, c x^{5} e^{2} + \frac {1}{2} \, c d x^{4} e + a d x^{2} e + a d^{2} x + \frac {1}{3} \, {\left (c d^{2} + a e^{2}\right )} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c*x^5*e^2 + 1/2*c*d*x^4*e + a*d*x^2*e + a*d^2*x + 1/3*(c*d^2 + a*e^2)*x^3

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Fricas [A]
time = 1.23, size = 51, normalized size = 0.89 \begin {gather*} \frac {1}{3} \, c d^{2} x^{3} + a d^{2} x + \frac {1}{15} \, {\left (3 \, c x^{5} + 5 \, a x^{3}\right )} e^{2} + \frac {1}{2} \, {\left (c d x^{4} + 2 \, a d x^{2}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*c*d^2*x^3 + a*d^2*x + 1/15*(3*c*x^5 + 5*a*x^3)*e^2 + 1/2*(c*d*x^4 + 2*a*d*x^2)*e

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Sympy [A]
time = 0.01, size = 51, normalized size = 0.89 \begin {gather*} a d^{2} x + a d e x^{2} + \frac {c d e x^{4}}{2} + \frac {c e^{2} x^{5}}{5} + x^{3} \left (\frac {a e^{2}}{3} + \frac {c d^{2}}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d**2*x + a*d*e*x**2 + c*d*e*x**4/2 + c*e**2*x**5/5 + x**3*(a*e**2/3 + c*d**2/3)

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Giac [A]
time = 1.39, size = 49, normalized size = 0.86 \begin {gather*} \frac {1}{5} \, c x^{5} e^{2} + \frac {1}{2} \, c d x^{4} e + \frac {1}{3} \, c d^{2} x^{3} + \frac {1}{3} \, a x^{3} e^{2} + a d x^{2} e + a d^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c*x^5*e^2 + 1/2*c*d*x^4*e + 1/3*c*d^2*x^3 + 1/3*a*x^3*e^2 + a*d*x^2*e + a*d^2*x

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Mupad [B]
time = 0.02, size = 48, normalized size = 0.84 \begin {gather*} x^3\,\left (\frac {c\,d^2}{3}+\frac {a\,e^2}{3}\right )+\frac {c\,e^2\,x^5}{5}+a\,d^2\,x+a\,d\,e\,x^2+\frac {c\,d\,e\,x^4}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)*(d + e*x)^2,x)

[Out]

x^3*((a*e^2)/3 + (c*d^2)/3) + (c*e^2*x^5)/5 + a*d^2*x + a*d*e*x^2 + (c*d*e*x^4)/2

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