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

Optimal. Leaf size=31 \[ \frac {c^2 \sqrt {c d^2+2 c d e x+c e^2 x^2}}{e} \]

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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 {c^2 \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)^(5/2)/(d + e*x)^5,x]

[Out]

(c^2*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 )^{5/2}}{(d+e x)^5} \, dx &=c^3 \int \frac {d+e x}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx\\ &=\frac {c^2 \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 = 23, normalized size = 0.74 \begin {gather*} \frac {c^3 x (d+e x)}{\sqrt {c (d+e x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^5,x]

[Out]

(c^3*x*(d + e*x))/Sqrt[c*(d + e*x)^2]

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

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^5,x]

[Out]

(c^2*Sqrt[c*(d + e*x)^2])/e

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fricas [A]  time = 0.40, size = 34, normalized size = 1.10 \begin {gather*} \frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c^{2} x}{e x + 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)^(5/2)/(e*x+d)^5,x, algorithm="fricas")

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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)^(5/2)/(e*x+d)^5,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep)]Evaluation time: 0.87Unable to divide, perhaps due to rounding error%%%{%%%{1,[0,12,0]%%%}+%%%{-2,
[0,10,1]%%%}+%%%{1,[0,8,2]%%%},[4,0,0,4,0]%%%}+%%%{%%%{2,[0,10,1]%%%}+%%%{-2,[0,8,2]%%%},[4,0,0,2,2]%%%}+%%%{%
%%{1,[0,8,2]%%%},[4,0,0,0,4]%%%}+%%%{%%%{-2,[1,12,1]%%%}+%%%{4,[1,10,2]%%%}+%%%{-2,[1,8,3]%%%},[2,0,0,4,0]%%%}
+%%%{%%%{-4,[1,10,2]%%%}+%%%{4,[1,8,3]%%%},[2,0,0,2,2]%%%}+%%%{%%%{-2,[1,8,3]%%%},[2,0,0,0,4]%%%}+%%%{%%%{1,[2
,12,2]%%%}+%%%{-2,[2,10,3]%%%}+%%%{1,[2,8,4]%%%},[0,0,0,4,0]%%%}+%%%{%%%{2,[2,10,3]%%%}+%%%{-2,[2,8,4]%%%},[0,
0,0,2,2]%%%}+%%%{%%%{1,[2,8,4]%%%},[0,0,0,0,4]%%%} / %%%{%%%{-1,[1,2,0]%%%}+%%%{1,[1,0,1]%%%},[4,0,0,0,0]%%%}+
%%%{%%%{2,[2,2,1]%%%}+%%%{-2,[2,0,2]%%%},[2,0,0,0,0]%%%}+%%%{%%%{-1,[3,2,2]%%%}+%%%{1,[3,0,3]%%%},[0,0,0,0,0]%
%%} Error: Bad Argument Value

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maple [A]  time = 0.05, size = 32, normalized size = 1.03 \begin {gather*} \frac {\left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}} x}{\left (e x +d \right )^{5}} \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)^(5/2)/(e*x+d)^5,x)

[Out]

(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^5*x

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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)^(5/2)/(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 [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 )}^{5/2}}{{\left (d+e\,x\right )}^5} \,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)^(5/2)/(d + e*x)^5,x)

[Out]

int((c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(5/2)/(d + e*x)^5, x)

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sympy [A]  time = 7.74, size = 41, normalized size = 1.32 \begin {gather*} c^{2} \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)**(5/2)/(e*x+d)**5,x)

[Out]

c**2*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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