3.813 \(\int \frac{(d^2-e^2 x^2)^{7/2}}{(d+e x)^{11}} \, dx\)

Optimal. Leaf size=100 \[ -\frac{2 \left (d^2-e^2 x^2\right )^{9/2}}{1287 d^3 e (d+e x)^9}-\frac{2 \left (d^2-e^2 x^2\right )^{9/2}}{143 d^2 e (d+e x)^{10}}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{13 d e (d+e x)^{11}} \]

[Out]

-(d^2 - e^2*x^2)^(9/2)/(13*d*e*(d + e*x)^11) - (2*(d^2 - e^2*x^2)^(9/2))/(143*d^2*e*(d + e*x)^10) - (2*(d^2 -
e^2*x^2)^(9/2))/(1287*d^3*e*(d + e*x)^9)

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Rubi [A]  time = 0.0359951, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {659, 651} \[ -\frac{2 \left (d^2-e^2 x^2\right )^{9/2}}{1287 d^3 e (d+e x)^9}-\frac{2 \left (d^2-e^2 x^2\right )^{9/2}}{143 d^2 e (d+e x)^{10}}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{13 d e (d+e x)^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^11,x]

[Out]

-(d^2 - e^2*x^2)^(9/2)/(13*d*e*(d + e*x)^11) - (2*(d^2 - e^2*x^2)^(9/2))/(143*d^2*e*(d + e*x)^10) - (2*(d^2 -
e^2*x^2)^(9/2))/(1287*d^3*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)^{11}} \, dx &=-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{13 d e (d+e x)^{11}}+\frac{2 \int \frac{\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{10}} \, dx}{13 d}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{13 d e (d+e x)^{11}}-\frac{2 \left (d^2-e^2 x^2\right )^{9/2}}{143 d^2 e (d+e x)^{10}}+\frac{2 \int \frac{\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx}{143 d^2}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{13 d e (d+e x)^{11}}-\frac{2 \left (d^2-e^2 x^2\right )^{9/2}}{143 d^2 e (d+e x)^{10}}-\frac{2 \left (d^2-e^2 x^2\right )^{9/2}}{1287 d^3 e (d+e x)^9}\\ \end{align*}

Mathematica [A]  time = 0.0715973, size = 60, normalized size = 0.6 \[ -\frac{(d-e x)^4 \sqrt{d^2-e^2 x^2} \left (119 d^2+22 d e x+2 e^2 x^2\right )}{1287 d^3 e (d+e x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^11,x]

[Out]

-((d - e*x)^4*Sqrt[d^2 - e^2*x^2]*(119*d^2 + 22*d*e*x + 2*e^2*x^2))/(1287*d^3*e*(d + e*x)^7)

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Maple [A]  time = 0.044, size = 55, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,{e}^{2}{x}^{2}+22\,dex+119\,{d}^{2} \right ) \left ( -ex+d \right ) }{1287\, \left ( ex+d \right ) ^{10}{d}^{3}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)^11,x)

[Out]

-1/1287*(-e*x+d)*(2*e^2*x^2+22*d*e*x+119*d^2)*(-e^2*x^2+d^2)^(7/2)/(e*x+d)^10/d^3/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)^11,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.66569, size = 522, normalized size = 5.22 \begin{align*} -\frac{119 \, e^{7} x^{7} + 833 \, d e^{6} x^{6} + 2499 \, d^{2} e^{5} x^{5} + 4165 \, d^{3} e^{4} x^{4} + 4165 \, d^{4} e^{3} x^{3} + 2499 \, d^{5} e^{2} x^{2} + 833 \, d^{6} e x + 119 \, d^{7} +{\left (2 \, e^{6} x^{6} + 14 \, d e^{5} x^{5} + 43 \, d^{2} e^{4} x^{4} - 352 \, d^{3} e^{3} x^{3} + 628 \, d^{4} e^{2} x^{2} - 454 \, d^{5} e x + 119 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{1287 \,{\left (d^{3} e^{8} x^{7} + 7 \, d^{4} e^{7} x^{6} + 21 \, d^{5} e^{6} x^{5} + 35 \, d^{6} e^{5} x^{4} + 35 \, d^{7} e^{4} x^{3} + 21 \, d^{8} e^{3} x^{2} + 7 \, d^{9} e^{2} x + d^{10} 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)^11,x, algorithm="fricas")

[Out]

-1/1287*(119*e^7*x^7 + 833*d*e^6*x^6 + 2499*d^2*e^5*x^5 + 4165*d^3*e^4*x^4 + 4165*d^4*e^3*x^3 + 2499*d^5*e^2*x
^2 + 833*d^6*e*x + 119*d^7 + (2*e^6*x^6 + 14*d*e^5*x^5 + 43*d^2*e^4*x^4 - 352*d^3*e^3*x^3 + 628*d^4*e^2*x^2 -
454*d^5*e*x + 119*d^6)*sqrt(-e^2*x^2 + d^2))/(d^3*e^8*x^7 + 7*d^4*e^7*x^6 + 21*d^5*e^6*x^5 + 35*d^6*e^5*x^4 +
35*d^7*e^4*x^3 + 21*d^8*e^3*x^2 + 7*d^9*e^2*x + d^10*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)**11,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)^11,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError