3.9.75 \(\int \frac {(d+e x)^3}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx\)

Optimal. Leaf size=34 \[ \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 c^2 e} \]

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Rubi [A]  time = 0.02, antiderivative size = 34, 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 = {643, 629} \begin {gather*} \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 c^2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]

[Out]

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

Rule 629

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

Rule 643

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[e^(m - 1)/c^((m - 1)/2
), Int[(d + e*x)*(a + b*x + c*x^2)^(p + (m - 1)/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 - 1)/2]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 27, normalized size = 0.79 \begin {gather*} \frac {(d+e x)^4}{3 e \sqrt {c (d+e x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]

[Out]

(d + e*x)^4/(3*e*Sqrt[c*(d + e*x)^2])

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IntegrateAlgebraic [A]  time = 0.04, size = 23, normalized size = 0.68 \begin {gather*} \frac {\left (c (d+e x)^2\right )^{3/2}}{3 c^2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^3/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]

[Out]

(c*(d + e*x)^2)^(3/2)/(3*c^2*e)

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fricas [A]  time = 0.39, size = 55, normalized size = 1.62 \begin {gather*} \frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} {\left (e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x\right )}}{3 \, {\left (c e x + c d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.47, size = 50, normalized size = 1.47 \begin {gather*} \frac {1}{3} \, \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} {\left (x {\left (\frac {x e}{c} + \frac {2 \, d}{c}\right )} + \frac {d^{2} e^{\left (-1\right )}}{c}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 49, normalized size = 1.44 \begin {gather*} \frac {\left (e^{2} x^{2}+3 d x e +3 d^{2}\right ) \left (e x +d \right ) x}{3 \sqrt {c \,e^{2} x^{2}+2 c d e x +c \,d^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.49, size = 84, normalized size = 2.47 \begin {gather*} \frac {2 \, d e x^{2}}{3 \, \sqrt {c}} - \frac {4 \, d^{2} x}{3 \, \sqrt {c}} + \frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} e x^{2}}{3 \, c} + \frac {7 \, \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} d^{2}}{3 \, c e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x, algorithm="maxima")

[Out]

2/3*d*e*x^2/sqrt(c) - 4/3*d^2*x/sqrt(c) + 1/3*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*e*x^2/c + 7/3*sqrt(c*e^2*x^2
+ 2*c*d*e*x + c*d^2)*d^2/(c*e)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^3}{\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.19, size = 114, normalized size = 3.35 \begin {gather*} \begin {cases} \frac {d^{2} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3 c e} + \frac {2 d x \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3 c} + \frac {e x^{2} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3 c} & \text {for}\: e \neq 0 \\\frac {d^{3} x}{\sqrt {c d^{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)

[Out]

Piecewise((d**2*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(3*c*e) + 2*d*x*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/
(3*c) + e*x**2*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(3*c), Ne(e, 0)), (d**3*x/sqrt(c*d**2), True))

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