3.1030 \(\int (d+e x)^2 \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx\)

Optimal. Leaf size=39 \[ \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 c e} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {642, 609} \[ \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 c e} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]

[Out]

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

Rule 609

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p + 1
)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && NeQ[p, -2^(-1)]

Rule 642

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 28, normalized size = 0.72 \[ \frac {(d+e x) \left (c (d+e x)^2\right )^{3/2}}{4 c e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.12, size = 63, normalized size = 1.62 \[ \frac {{\left (e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x\right )} \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{4 \, {\left (e x + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x, algorithm="fricas")

[Out]

1/4*(e^3*x^4 + 4*d*e^2*x^3 + 6*d^2*e*x^2 + 4*d^3*x)*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)/(e*x + d)

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giac [A]  time = 0.30, size = 51, normalized size = 1.31 \[ \frac {1}{4} \, \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} {\left (d^{3} e^{\left (-1\right )} + {\left (3 \, d^{2} + {\left (x e^{2} + 3 \, d e\right )} x\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 62, normalized size = 1.59 \[ \frac {\left (e^{3} x^{3}+4 e^{2} x^{2} d +6 d^{2} x e +4 d^{3}\right ) \sqrt {c \,e^{2} x^{2}+2 c d e x +c \,d^{2}}\, x}{4 e x +4 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x)

[Out]

1/4*x*(e^3*x^3+4*d*e^2*x^2+6*d^2*e*x+4*d^3)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2)/(e*x+d)

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maxima [A]  time = 1.37, size = 60, normalized size = 1.54 \[ \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} x}{4 \, c} + \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} d}{4 \, c e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.70, size = 76, normalized size = 1.95 \[ \frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}\,\left (c\,d^3+e\,x\,\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )+2\,c\,d^2\,e\,x+c\,d\,e^2\,x^2\right )}{4\,c\,e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^2*(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(1/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {c \left (d + e x\right )^{2}} \left (d + e x\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2*(c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)

[Out]

Integral(sqrt(c*(d + e*x)**2)*(d + e*x)**2, x)

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