3.9.59 \(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^{3/2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=32 \[ -\frac {c^2}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

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Rubi [A]  time = 0.02, antiderivative size = 32, 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 {c^2}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \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)^5,x]

[Out]

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

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)^5} \, dx &=c^3 \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {c^2}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 0.78 \begin {gather*} -\frac {\left (c (d+e x)^2\right )^{3/2}}{e (d+e x)^4} \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)^5,x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.05, size = 26, normalized size = 0.81 \begin {gather*} -\frac {c \sqrt {c (d+e x)^2}}{e (d+e x)^2} \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)^5,x]

[Out]

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

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fricas [A]  time = 0.41, size = 47, normalized size = 1.47 \begin {gather*} -\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c}{e^{3} x^{2} + 2 \, d e^{2} x + d^{2} 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)^5,x, algorithm="fricas")

[Out]

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

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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)^5,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: exp(1)^2*(2*(-(exp(1)*x+d)^-1/exp(1)*(-C
_1/exp(1)^2/c+1/2*c^2*sign((exp(1)*x+d)^-1)/exp(1)^3/c)+C_0)*sqrt(c*exp(2)-c*d^2*(-(exp(1)*x+d)^-1/exp(1))^2*e
xp(1)^4+2*c*d*(exp(1)*x+d)^-1/exp(1)*exp(1)^3-2*c*d*(exp(1)*x+d)^-1/exp(1)*exp(1)*exp(2)+c*d^2*(-(exp(1)*x+d)^
-1/exp(1))^2*exp(1)^2*exp(2))+2*C_1*sqrt(-c*exp(1)^2+c*exp(2))*ln(abs(-d*sqrt(-c*exp(1)^2+c*exp(2))+((exp(1)*x
+d)^-1/exp(1)*sqrt(-c*d^2*exp(1)^4+c*d^2*exp(1)^2*exp(2))+sqrt(c*exp(2)-c*d^2*(-(exp(1)*x+d)^-1/exp(1))^2*exp(
1)^4+2*c*d*(exp(1)*x+d)^-1/exp(1)*exp(1)^3-2*c*d*(exp(1)*x+d)^-1/exp(1)*exp(1)*exp(2)+c*d^2*(-(exp(1)*x+d)^-1/
exp(1))^2*exp(1)^2*exp(2)))*abs(d)))/(c*exp(1)^3-c*exp(1)*exp(2))/abs(d))

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maple [A]  time = 0.04, size = 35, normalized size = 1.09 \begin {gather*} -\frac {\left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}}{\left (e x +d \right )^{4} e} \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)^5,x)

[Out]

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

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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)^5,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 [B]  time = 0.43, size = 35, normalized size = 1.09 \begin {gather*} -\frac {c\,\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{e\,{\left (d+e\,x\right )}^2} \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)^5,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c \left (d + e x\right )^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{5}}\, dx \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)**5,x)

[Out]

Integral((c*(d + e*x)**2)**(3/2)/(d + e*x)**5, x)

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