3.1045 \(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^{3/2}}{(d+e x)^3} \, dx\)

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 = {643, 629} \[ \frac {c \sqrt {c d^2+2 c d e x+c e^2 x^2}}{e} \]

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 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)^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 \[ \frac {x \left (c (d+e x)^2\right )^{3/2}}{(d+e x)^3} \]

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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fricas [A]  time = 0.87, size = 32, normalized size = 1.10 \[ \frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c x}{e x + d} \]

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

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

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]

Exception raised: NotImplementedError >> Unable to parse Giac output: 4*c*1/4/exp(1)*sqrt(c*d^2+2*c*d*x*exp(1)
+c*x^2*exp(2))+2*(-(5*c^2*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*d^2*exp(1)^5-11*c^2*exp
(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*d^2*exp(1)^3+6*c^2*exp(2)^2*(sqrt(c*d^2+2*c*d
*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*d^2*exp(1)-7*c^2*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*e
xp(2))-sqrt(c*exp(2))*x)^2*d^3*exp(1)^4+17*c^2*exp(2)*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-
sqrt(c*exp(2))*x)^2*d^3*exp(1)^2-10*c^2*exp(2)^2*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(
c*exp(2))*x)^2*d^3+3*c^3*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*d^4*exp(1)^5-9*c^3*exp(2)*
(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*d^4*exp(1)^3+6*c^3*exp(2)^2*(sqrt(c*d^2+2*c*d*x*exp
(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*d^4*exp(1)+c^3*sqrt(c*exp(2))*d^5*exp(1)^4-c^3*exp(2)*sqrt(c*exp(2))*d^5*e
xp(1)^2)/2/exp(1)^4/(-(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^2*exp(1)+2*sqrt(c*exp(2))*(sq
rt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*d-c*d^2*exp(1))^2+(3*c^2*d^2*exp(1)^4-9*c^2*exp(2)*d^2
*exp(1)^2+6*c^2*exp(2)^2*d^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 = 0.04, size = 32, normalized size = 1.10 \[ \frac {\left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}} x}{\left (e x +d \right )^{3}} \]

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)

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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

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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sympy [A]  time = 4.55, size = 39, normalized size = 1.34 \[ c \left (\begin {cases} \frac {x \sqrt {c d^{2}}}{d} & \text {for}\: e = 0 \\\frac {\sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{e} & \text {otherwise} \end {cases}\right ) \]

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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