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

Optimal. Leaf size=31 \[ \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 e} \]

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Rubi [A]  time = 0.02, antiderivative size = 31, 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 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.00, size = 20, normalized size = 0.65 \begin {gather*} \frac {\left (c (d+e x)^2\right )^{3/2}}{3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(d + e*x)^2)^(3/2)/(3*e)

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IntegrateAlgebraic [A]  time = 0.04, size = 20, normalized size = 0.65 \begin {gather*} \frac {\left (c (d+e x)^2\right )^{3/2}}{3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(d + e*x)^2)^(3/2)/(3*e)

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fricas [B]  time = 0.38, size = 55, normalized size = 1.77 \begin {gather*} \frac {{\left (c e^{2} x^{3} + 3 \, c d e x^{2} + 3 \, c d^{2} x\right )} \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{3 \, {\left (e x + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 2*((16*c*exp(1)^4*1/96/exp(1)^3*x+32*c*e
xp(1)^3*d*1/96/exp(1)^3)*x+16*c*exp(1)^2*d^2*1/96/exp(1)^3)*sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))+2*(c^2*d^4
*exp(1)^4-2*c^2*d^4*exp(1)^2*exp(2)+c^2*d^4*exp(2)^2)*2/2/exp(1)^4/d/sqrt(c*exp(1)^2-c*exp(2))*atan((-d*sqrt(c
*exp(2))+(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*exp(1))/d/sqrt(c*exp(1)^2-c*exp(2)))

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maple [A]  time = 0.05, size = 51, normalized size = 1.65 \begin {gather*} \frac {\left (e^{2} x^{2}+3 d x e +3 d^{2}\right ) \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}} x}{3 \left (e x +d \right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.34, size = 27, normalized size = 0.87 \begin {gather*} \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}}}{3 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.48, size = 16, normalized size = 0.52 \begin {gather*} \frac {{\left (c\,{\left (d+e\,x\right )}^2\right )}^{3/2}}{3\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*(d + e*x)^2)^(3/2)/(3*e)

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sympy [A]  time = 3.89, size = 39, normalized size = 1.26 \begin {gather*} \begin {cases} \frac {\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac {3}{2}}}{3 e} & \text {for}\: e \neq 0 \\\frac {x \left (c d^{2}\right )^{\frac {3}{2}}}{d} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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