3.812 \(\int \frac{(d^2-e^2 x^2)^{7/2}}{(d+e x)^{10}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{\left (d^2-e^2 x^2\right )^{9/2}}{99 d^2 e (d+e x)^9}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{11 d e (d+e x)^{10}} \]

[Out]

-(d^2 - e^2*x^2)^(9/2)/(11*d*e*(d + e*x)^10) - (d^2 - e^2*x^2)^(9/2)/(99*d^2*e*(d + e*x)^9)

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Rubi [A]  time = 0.021408, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {659, 651} \[ -\frac{\left (d^2-e^2 x^2\right )^{9/2}}{99 d^2 e (d+e x)^9}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{11 d e (d+e x)^{10}} \]

Antiderivative was successfully verified.

[In]

Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^10,x]

[Out]

-(d^2 - e^2*x^2)^(9/2)/(11*d*e*(d + e*x)^10) - (d^2 - e^2*x^2)^(9/2)/(99*d^2*e*(d + e*x)^9)

Rule 659

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1)
)/(2*c*d*(m + p + 1)), x] + Dist[Simplify[m + 2*p + 2]/(2*d*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p,
x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2
], 0]

Rule 651

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1))
/(2*c*d*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rubi steps

\begin{align*} \int \frac{\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{10}} \, dx &=-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{11 d e (d+e x)^{10}}+\frac{\int \frac{\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx}{11 d}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{11 d e (d+e x)^{10}}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{99 d^2 e (d+e x)^9}\\ \end{align*}

Mathematica [A]  time = 0.0671445, size = 48, normalized size = 0.72 \[ -\frac{(d-e x)^4 (10 d+e x) \sqrt{d^2-e^2 x^2}}{99 d^2 e (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^10,x]

[Out]

-((d - e*x)^4*(10*d + e*x)*Sqrt[d^2 - e^2*x^2])/(99*d^2*e*(d + e*x)^6)

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Maple [A]  time = 0.043, size = 43, normalized size = 0.6 \begin{align*} -{\frac{ \left ( ex+10\,d \right ) \left ( -ex+d \right ) }{99\, \left ( ex+d \right ) ^{9}{d}^{2}e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^10,x)

[Out]

-1/99*(-e*x+d)*(e*x+10*d)*(-e^2*x^2+d^2)^(7/2)/(e*x+d)^9/d^2/e

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.80534, size = 428, normalized size = 6.39 \begin{align*} -\frac{10 \, e^{6} x^{6} + 60 \, d e^{5} x^{5} + 150 \, d^{2} e^{4} x^{4} + 200 \, d^{3} e^{3} x^{3} + 150 \, d^{4} e^{2} x^{2} + 60 \, d^{5} e x + 10 \, d^{6} +{\left (e^{5} x^{5} + 6 \, d e^{4} x^{4} - 34 \, d^{2} e^{3} x^{3} + 56 \, d^{3} e^{2} x^{2} - 39 \, d^{4} e x + 10 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{99 \,{\left (d^{2} e^{7} x^{6} + 6 \, d^{3} e^{6} x^{5} + 15 \, d^{4} e^{5} x^{4} + 20 \, d^{5} e^{4} x^{3} + 15 \, d^{6} e^{3} x^{2} + 6 \, d^{7} e^{2} x + d^{8} e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^10,x, algorithm="fricas")

[Out]

-1/99*(10*e^6*x^6 + 60*d*e^5*x^5 + 150*d^2*e^4*x^4 + 200*d^3*e^3*x^3 + 150*d^4*e^2*x^2 + 60*d^5*e*x + 10*d^6 +
(e^5*x^5 + 6*d*e^4*x^4 - 34*d^2*e^3*x^3 + 56*d^3*e^2*x^2 - 39*d^4*e*x + 10*d^5)*sqrt(-e^2*x^2 + d^2))/(d^2*e^
7*x^6 + 6*d^3*e^6*x^5 + 15*d^4*e^5*x^4 + 20*d^5*e^4*x^3 + 15*d^6*e^3*x^2 + 6*d^7*e^2*x + d^8*e)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-e**2*x**2+d**2)**(7/2)/(e*x+d)**10,x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^10,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError