3.814 \(\int \frac{(d^2-e^2 x^2)^{7/2}}{(d+e x)^{12}} \, dx\)

Optimal. Leaf size=133 \[ -\frac{2 \left (d^2-e^2 x^2\right )^{9/2}}{6435 d^4 e (d+e x)^9}-\frac{2 \left (d^2-e^2 x^2\right )^{9/2}}{715 d^3 e (d+e x)^{10}}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{65 d^2 e (d+e x)^{11}}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{15 d e (d+e x)^{12}} \]

[Out]

-(d^2 - e^2*x^2)^(9/2)/(15*d*e*(d + e*x)^12) - (d^2 - e^2*x^2)^(9/2)/(65*d^2*e*(d + e*x)^11) - (2*(d^2 - e^2*x
^2)^(9/2))/(715*d^3*e*(d + e*x)^10) - (2*(d^2 - e^2*x^2)^(9/2))/(6435*d^4*e*(d + e*x)^9)

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Rubi [A]  time = 0.0559151, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {659, 651} \[ -\frac{2 \left (d^2-e^2 x^2\right )^{9/2}}{6435 d^4 e (d+e x)^9}-\frac{2 \left (d^2-e^2 x^2\right )^{9/2}}{715 d^3 e (d+e x)^{10}}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{65 d^2 e (d+e x)^{11}}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{15 d e (d+e x)^{12}} \]

Antiderivative was successfully verified.

[In]

Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^12,x]

[Out]

-(d^2 - e^2*x^2)^(9/2)/(15*d*e*(d + e*x)^12) - (d^2 - e^2*x^2)^(9/2)/(65*d^2*e*(d + e*x)^11) - (2*(d^2 - e^2*x
^2)^(9/2))/(715*d^3*e*(d + e*x)^10) - (2*(d^2 - e^2*x^2)^(9/2))/(6435*d^4*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)^{12}} \, dx &=-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{15 d e (d+e x)^{12}}+\frac{\int \frac{\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx}{5 d}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{15 d e (d+e x)^{12}}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{65 d^2 e (d+e x)^{11}}+\frac{2 \int \frac{\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{10}} \, dx}{65 d^2}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{15 d e (d+e x)^{12}}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{65 d^2 e (d+e x)^{11}}-\frac{2 \left (d^2-e^2 x^2\right )^{9/2}}{715 d^3 e (d+e x)^{10}}+\frac{2 \int \frac{\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx}{715 d^3}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{15 d e (d+e x)^{12}}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{65 d^2 e (d+e x)^{11}}-\frac{2 \left (d^2-e^2 x^2\right )^{9/2}}{715 d^3 e (d+e x)^{10}}-\frac{2 \left (d^2-e^2 x^2\right )^{9/2}}{6435 d^4 e (d+e x)^9}\\ \end{align*}

Mathematica [A]  time = 0.0742261, size = 71, normalized size = 0.53 \[ -\frac{(d-e x)^4 \sqrt{d^2-e^2 x^2} \left (141 d^2 e x+548 d^3+24 d e^2 x^2+2 e^3 x^3\right )}{6435 d^4 e (d+e x)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^12,x]

[Out]

-((d - e*x)^4*Sqrt[d^2 - e^2*x^2]*(548*d^3 + 141*d^2*e*x + 24*d*e^2*x^2 + 2*e^3*x^3))/(6435*d^4*e*(d + e*x)^8)

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Maple [A]  time = 0.045, size = 66, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,{e}^{3}{x}^{3}+24\,{e}^{2}{x}^{2}d+141\,x{d}^{2}e+548\,{d}^{3} \right ) \left ( -ex+d \right ) }{6435\, \left ( ex+d \right ) ^{11}{d}^{4}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)^12,x)

[Out]

-1/6435*(-e*x+d)*(2*e^3*x^3+24*d*e^2*x^2+141*d^2*e*x+548*d^3)*(-e^2*x^2+d^2)^(7/2)/(e*x+d)^11/d^4/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)^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 5.07142, size = 612, normalized size = 4.6 \begin{align*} -\frac{548 \, e^{8} x^{8} + 4384 \, d e^{7} x^{7} + 15344 \, d^{2} e^{6} x^{6} + 30688 \, d^{3} e^{5} x^{5} + 38360 \, d^{4} e^{4} x^{4} + 30688 \, d^{5} e^{3} x^{3} + 15344 \, d^{6} e^{2} x^{2} + 4384 \, d^{7} e x + 548 \, d^{8} +{\left (2 \, e^{7} x^{7} + 16 \, d e^{6} x^{6} + 57 \, d^{2} e^{5} x^{5} + 120 \, d^{3} e^{4} x^{4} - 1440 \, d^{4} e^{3} x^{3} + 2748 \, d^{5} e^{2} x^{2} - 2051 \, d^{6} e x + 548 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6435 \,{\left (d^{4} e^{9} x^{8} + 8 \, d^{5} e^{8} x^{7} + 28 \, d^{6} e^{7} x^{6} + 56 \, d^{7} e^{6} x^{5} + 70 \, d^{8} e^{5} x^{4} + 56 \, d^{9} e^{4} x^{3} + 28 \, d^{10} e^{3} x^{2} + 8 \, d^{11} e^{2} x + d^{12} 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)^12,x, algorithm="fricas")

[Out]

-1/6435*(548*e^8*x^8 + 4384*d*e^7*x^7 + 15344*d^2*e^6*x^6 + 30688*d^3*e^5*x^5 + 38360*d^4*e^4*x^4 + 30688*d^5*
e^3*x^3 + 15344*d^6*e^2*x^2 + 4384*d^7*e*x + 548*d^8 + (2*e^7*x^7 + 16*d*e^6*x^6 + 57*d^2*e^5*x^5 + 120*d^3*e^
4*x^4 - 1440*d^4*e^3*x^3 + 2748*d^5*e^2*x^2 - 2051*d^6*e*x + 548*d^7)*sqrt(-e^2*x^2 + d^2))/(d^4*e^9*x^8 + 8*d
^5*e^8*x^7 + 28*d^6*e^7*x^6 + 56*d^7*e^6*x^5 + 70*d^8*e^5*x^4 + 56*d^9*e^4*x^3 + 28*d^10*e^3*x^2 + 8*d^11*e^2*
x + d^12*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)**12,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)^12,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError