3.9.72 \(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^{5/2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=32 \[ -\frac {c^3}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

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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^3}{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)^(5/2)/(d + e*x)^7,x]

[Out]

-(c^3/(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 )^{5/2}}{(d+e x)^7} \, dx &=c^4 \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^3}{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 )^{5/2}}{e (d+e x)^6} \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)^7,x]

[Out]

-((c*(d + e*x)^2)^(5/2)/(e*(d + e*x)^6))

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IntegrateAlgebraic [A]  time = 0.05, size = 28, normalized size = 0.88 \begin {gather*} -\frac {c^2 \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)^(5/2)/(d + e*x)^7,x]

[Out]

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

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fricas [A]  time = 0.39, size = 49, normalized size = 1.53 \begin {gather*} -\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c^{2}}{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)^(5/2)/(e*x+d)^7,x, algorithm="fricas")

[Out]

-sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*c^2/(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)^(5/2)/(e*x+d)^7,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 2*(-(-15*c^3*(sqrt(c*d^2+2*c*d*x*exp(1)+
c*x^2*exp(2))-sqrt(c*exp(2))*x)^11*exp(1)^11+48*c^3*exp(2)^3*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*e
xp(2))*x)^11*exp(1)^5+165*c^3*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^10*d*e
xp(1)^10-288*c^3*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)^10*d*exp(1
)^6-240*c^3*exp(2)^3*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^10*d*exp(1)^4-8
5*c^4*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^9*d^2*exp(1)^11-740*c^4*exp(2)*(sqrt(c*d^2+2*
c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^9*d^2*exp(1)^9+960*c^4*exp(2)^2*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*
exp(2))-sqrt(c*exp(2))*x)^9*d^2*exp(1)^7+1040*c^4*exp(2)^3*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp
(2))*x)^9*d^2*exp(1)^5+640*c^4*exp(2)^4*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^9*d^2*exp(1
)^3+765*c^4*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^8*d^3*exp(1)^10-210*c^4*
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)^8*d^3*exp(1)^8-2880*c^4*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)^8*d^3*exp(1)^6-2160*c^4*exp(2)^3
*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^8*d^3*exp(1)^4-960*c^4*exp(2)^4*sqr
t(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^8*d^3*exp(1)^2-198*c^5*(sqrt(c*d^2+2*c*
d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^7*d^4*exp(1)^11-360*c^5*exp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp
(2))-sqrt(c*exp(2))*x)^7*d^4*exp(1)^9+3480*c^5*exp(2)^2*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2)
)*x)^7*d^4*exp(1)^7+4320*c^5*exp(2)^3*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^7*d^4*exp(1)^
5+2880*c^5*exp(2)^4*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^7*d^4*exp(1)^3+768*c^5*exp(2)^5
*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^7*d^4*exp(1)-150*c^5*sqrt(c*exp(2))*(sqrt(c*d^2+2*
c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^6*d^5*exp(1)^10-2800*c^5*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)^6*d^5*exp(1)^8-5480*c^5*exp(2)^2*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*e
xp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^6*d^5*exp(1)^6-4320*c^5*exp(2)^3*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(
1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^6*d^5*exp(1)^4-2240*c^5*exp(2)^4*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+
c*x^2*exp(2))-sqrt(c*exp(2))*x)^6*d^5*exp(1)^2-256*c^5*exp(2)^5*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^
2*exp(2))-sqrt(c*exp(2))*x)^6*d^5+198*c^6*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^5*d^6*exp
(1)^11+2160*c^6*exp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^5*d^6*exp(1)^9+4920*c^6*exp(
2)^2*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^5*d^6*exp(1)^7+4320*c^6*exp(2)^3*(sqrt(c*d^2+2
*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^5*d^6*exp(1)^5+2880*c^6*exp(2)^4*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^
2*exp(2))-sqrt(c*exp(2))*x)^5*d^6*exp(1)^3+768*c^6*exp(2)^5*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*ex
p(2))*x)^5*d^6*exp(1)-990*c^6*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^4*d^7*
exp(1)^10-3540*c^6*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)^4*d^7*exp(
1)^8-3240*c^6*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)^4*d^7*exp(1)^
6-2160*c^6*exp(2)^3*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^4*d^7*exp(1)^4-9
60*c^6*exp(2)^4*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^4*d^7*exp(1)^2+85*c^
7*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*d^8*exp(1)^11+1640*c^7*exp(2)*(sqrt(c*d^2+2*c*d
*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*d^8*exp(1)^9+2040*c^7*exp(2)^2*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*ex
p(2))-sqrt(c*exp(2))*x)^3*d^8*exp(1)^7+1040*c^7*exp(2)^3*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2
))*x)^3*d^8*exp(1)^5+640*c^7*exp(2)^4*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*d^8*exp(1)^
3-255*c^7*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^9*exp(1)^10-960*c^7*ex
p(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^9*exp(1)^8-360*c^7*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^9*exp(1)^6-240*c^7*exp(2)^3*sqr
t(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^9*exp(1)^4+15*c^8*(sqrt(c*d^2+2*c*d
*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*d^10*exp(1)^11+180*c^8*exp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2
))-sqrt(c*exp(2))*x)*d^10*exp(1)^9+120*c^8*exp(2)^2*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)
*d^10*exp(1)^7+48*c^8*exp(2)^3*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*d^10*exp(1)^5-15*c^8
*sqrt(c*exp(2))*d^11*exp(1)^10-10*c^8*exp(2)*sqrt(c*exp(2))*d^11*exp(1)^8-8*c^8*exp(2)^2*sqrt(c*exp(2))*d^11*e
xp(1)^6)/48/exp(1)^6/(-(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))*(s
qrt(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))^6+5*c^3/16/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 = 35, normalized size = 1.09 \begin {gather*} -\frac {\left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}}}{\left (e x +d \right )^{6} 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)^(5/2)/(e*x+d)^7,x)

[Out]

-1/(e*x+d)^6/e*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)

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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)^(5/2)/(e*x+d)^7,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 = 37, normalized size = 1.16 \begin {gather*} -\frac {c^2\,\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)^(5/2)/(d + e*x)^7,x)

[Out]

-(c^2*(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 {5}{2}}}{\left (d + e x\right )^{7}}\, 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)**(5/2)/(e*x+d)**7,x)

[Out]

Integral((c*(d + e*x)**2)**(5/2)/(d + e*x)**7, x)

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