3.815 \(\int \frac{(d^2-e^2 x^2)^{7/2}}{(d+e x)^{13}} \, dx\)

Optimal. Leaf size=166 \[ -\frac{8 \left (d^2-e^2 x^2\right )^{9/2}}{109395 d^5 e (d+e x)^9}-\frac{8 \left (d^2-e^2 x^2\right )^{9/2}}{12155 d^4 e (d+e x)^{10}}-\frac{4 \left (d^2-e^2 x^2\right )^{9/2}}{1105 d^3 e (d+e x)^{11}}-\frac{4 \left (d^2-e^2 x^2\right )^{9/2}}{255 d^2 e (d+e x)^{12}}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{17 d e (d+e x)^{13}} \]

[Out]

-(d^2 - e^2*x^2)^(9/2)/(17*d*e*(d + e*x)^13) - (4*(d^2 - e^2*x^2)^(9/2))/(255*d^2*e*(d + e*x)^12) - (4*(d^2 -
e^2*x^2)^(9/2))/(1105*d^3*e*(d + e*x)^11) - (8*(d^2 - e^2*x^2)^(9/2))/(12155*d^4*e*(d + e*x)^10) - (8*(d^2 - e
^2*x^2)^(9/2))/(109395*d^5*e*(d + e*x)^9)

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Rubi [A]  time = 0.0758125, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {659, 651} \[ -\frac{8 \left (d^2-e^2 x^2\right )^{9/2}}{109395 d^5 e (d+e x)^9}-\frac{8 \left (d^2-e^2 x^2\right )^{9/2}}{12155 d^4 e (d+e x)^{10}}-\frac{4 \left (d^2-e^2 x^2\right )^{9/2}}{1105 d^3 e (d+e x)^{11}}-\frac{4 \left (d^2-e^2 x^2\right )^{9/2}}{255 d^2 e (d+e x)^{12}}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{17 d e (d+e x)^{13}} \]

Antiderivative was successfully verified.

[In]

Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^13,x]

[Out]

-(d^2 - e^2*x^2)^(9/2)/(17*d*e*(d + e*x)^13) - (4*(d^2 - e^2*x^2)^(9/2))/(255*d^2*e*(d + e*x)^12) - (4*(d^2 -
e^2*x^2)^(9/2))/(1105*d^3*e*(d + e*x)^11) - (8*(d^2 - e^2*x^2)^(9/2))/(12155*d^4*e*(d + e*x)^10) - (8*(d^2 - e
^2*x^2)^(9/2))/(109395*d^5*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)^{13}} \, dx &=-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{17 d e (d+e x)^{13}}+\frac{4 \int \frac{\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{12}} \, dx}{17 d}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{17 d e (d+e x)^{13}}-\frac{4 \left (d^2-e^2 x^2\right )^{9/2}}{255 d^2 e (d+e x)^{12}}+\frac{4 \int \frac{\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx}{85 d^2}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{17 d e (d+e x)^{13}}-\frac{4 \left (d^2-e^2 x^2\right )^{9/2}}{255 d^2 e (d+e x)^{12}}-\frac{4 \left (d^2-e^2 x^2\right )^{9/2}}{1105 d^3 e (d+e x)^{11}}+\frac{8 \int \frac{\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{10}} \, dx}{1105 d^3}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{17 d e (d+e x)^{13}}-\frac{4 \left (d^2-e^2 x^2\right )^{9/2}}{255 d^2 e (d+e x)^{12}}-\frac{4 \left (d^2-e^2 x^2\right )^{9/2}}{1105 d^3 e (d+e x)^{11}}-\frac{8 \left (d^2-e^2 x^2\right )^{9/2}}{12155 d^4 e (d+e x)^{10}}+\frac{8 \int \frac{\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx}{12155 d^4}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{17 d e (d+e x)^{13}}-\frac{4 \left (d^2-e^2 x^2\right )^{9/2}}{255 d^2 e (d+e x)^{12}}-\frac{4 \left (d^2-e^2 x^2\right )^{9/2}}{1105 d^3 e (d+e x)^{11}}-\frac{8 \left (d^2-e^2 x^2\right )^{9/2}}{12155 d^4 e (d+e x)^{10}}-\frac{8 \left (d^2-e^2 x^2\right )^{9/2}}{109395 d^5 e (d+e x)^9}\\ \end{align*}

Mathematica [A]  time = 0.0816001, size = 82, normalized size = 0.49 \[ -\frac{(d-e x)^4 \sqrt{d^2-e^2 x^2} \left (660 d^2 e^2 x^2+2756 d^3 e x+8627 d^4+104 d e^3 x^3+8 e^4 x^4\right )}{109395 d^5 e (d+e x)^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^13,x]

[Out]

-((d - e*x)^4*Sqrt[d^2 - e^2*x^2]*(8627*d^4 + 2756*d^3*e*x + 660*d^2*e^2*x^2 + 104*d*e^3*x^3 + 8*e^4*x^4))/(10
9395*d^5*e*(d + e*x)^9)

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Maple [A]  time = 0.045, size = 77, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 8\,{e}^{4}{x}^{4}+104\,{e}^{3}{x}^{3}d+660\,{e}^{2}{x}^{2}{d}^{2}+2756\,x{d}^{3}e+8627\,{d}^{4} \right ) \left ( -ex+d \right ) }{109395\, \left ( ex+d \right ) ^{12}{d}^{5}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)^13,x)

[Out]

-1/109395*(-e*x+d)*(8*e^4*x^4+104*d*e^3*x^3+660*d^2*e^2*x^2+2756*d^3*e*x+8627*d^4)*(-e^2*x^2+d^2)^(7/2)/(e*x+d
)^12/d^5/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)^13,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 7.09905, size = 720, normalized size = 4.34 \begin{align*} -\frac{8627 \, e^{9} x^{9} + 77643 \, d e^{8} x^{8} + 310572 \, d^{2} e^{7} x^{7} + 724668 \, d^{3} e^{6} x^{6} + 1087002 \, d^{4} e^{5} x^{5} + 1087002 \, d^{5} e^{4} x^{4} + 724668 \, d^{6} e^{3} x^{3} + 310572 \, d^{7} e^{2} x^{2} + 77643 \, d^{8} e x + 8627 \, d^{9} +{\left (8 \, e^{8} x^{8} + 72 \, d e^{7} x^{7} + 292 \, d^{2} e^{6} x^{6} + 708 \, d^{3} e^{5} x^{5} + 1155 \, d^{4} e^{4} x^{4} - 20508 \, d^{5} e^{3} x^{3} + 41398 \, d^{6} e^{2} x^{2} - 31752 \, d^{7} e x + 8627 \, d^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{109395 \,{\left (d^{5} e^{10} x^{9} + 9 \, d^{6} e^{9} x^{8} + 36 \, d^{7} e^{8} x^{7} + 84 \, d^{8} e^{7} x^{6} + 126 \, d^{9} e^{6} x^{5} + 126 \, d^{10} e^{5} x^{4} + 84 \, d^{11} e^{4} x^{3} + 36 \, d^{12} e^{3} x^{2} + 9 \, d^{13} e^{2} x + d^{14} 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)^13,x, algorithm="fricas")

[Out]

-1/109395*(8627*e^9*x^9 + 77643*d*e^8*x^8 + 310572*d^2*e^7*x^7 + 724668*d^3*e^6*x^6 + 1087002*d^4*e^5*x^5 + 10
87002*d^5*e^4*x^4 + 724668*d^6*e^3*x^3 + 310572*d^7*e^2*x^2 + 77643*d^8*e*x + 8627*d^9 + (8*e^8*x^8 + 72*d*e^7
*x^7 + 292*d^2*e^6*x^6 + 708*d^3*e^5*x^5 + 1155*d^4*e^4*x^4 - 20508*d^5*e^3*x^3 + 41398*d^6*e^2*x^2 - 31752*d^
7*e*x + 8627*d^8)*sqrt(-e^2*x^2 + d^2))/(d^5*e^10*x^9 + 9*d^6*e^9*x^8 + 36*d^7*e^8*x^7 + 84*d^8*e^7*x^6 + 126*
d^9*e^6*x^5 + 126*d^10*e^5*x^4 + 84*d^11*e^4*x^3 + 36*d^12*e^3*x^2 + 9*d^13*e^2*x + d^14*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)**13,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)^13,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError