3.11.45 \(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^{3/2}}{(d+e x)^3} \, dx\) [1045]

Optimal. Leaf size=29 \[ \frac {c \sqrt {c d^2+2 c d e x+c e^2 x^2}}{e} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 29, 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 = {657, 643} \begin {gather*} \frac {c \sqrt {c d^2+2 c d e x+c e^2 x^2}}{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)^3,x]

[Out]

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

Rule 643

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 657

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

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Mathematica [A]
time = 0.01, size = 22, normalized size = 0.76 \begin {gather*} \frac {x \left (c (d+e x)^2\right )^{3/2}}{(d+e x)^3} \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)^3,x]

[Out]

(x*(c*(d + e*x)^2)^(3/2))/(d + e*x)^3

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Maple [A]
time = 0.57, size = 32, normalized size = 1.10

method	result	size
risch	\(\frac {c \sqrt {\left (e x +d \right )^{2} c}\, x}{e x +d}\)	\(22\)
default	\(\frac {x \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}}{\left (e x +d \right )^{3}}\)	\(32\)
trager	\(\frac {c x \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{e x +d}\)	\(33\)

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)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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)^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [A]
time = 3.37, size = 33, normalized size = 1.14 \begin {gather*} \frac {\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} c x}{x e + d} \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)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 1.63, size = 39, normalized size = 1.34 \begin {gather*} c \left (\begin {cases} \frac {x \sqrt {c d^{2}}}{d} & \text {for}\: e = 0 \\\frac {\sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{e} & \text {otherwise} \end {cases}\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)**3,x)

[Out]

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

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Giac [A]
time = 0.74, size = 27, normalized size = 0.93 \begin {gather*} {\left (c d e^{\left (-1\right )} \mathrm {sgn}\left (x e + d\right ) + c x \mathrm {sgn}\left (x e + d\right )\right )} \sqrt {c} \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)^3,x, algorithm="giac")

[Out]

(c*d*e^(-1)*sgn(x*e + d) + c*x*sgn(x*e + d))*sqrt(c)

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

[Out]

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

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