### 3.811 $$\int \frac{(d^2-e^2 x^2)^{7/2}}{(d+e x)^9} \, dx$$

Optimal. Leaf size=33 $-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 d e (d+e x)^9}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0090743, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.042, Rules used = {651} $-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 d e (d+e x)^9}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^9,x]

[Out]

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

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

Mathematica [A]  time = 0.0626643, size = 41, normalized size = 1.24 $-\frac{(d-e x)^4 \sqrt{d^2-e^2 x^2}}{9 d e (d+e x)^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^9,x]

[Out]

-((d - e*x)^4*Sqrt[d^2 - e^2*x^2])/(9*d*e*(d + e*x)^5)

________________________________________________________________________________________

Maple [A]  time = 0.042, size = 36, normalized size = 1.1 \begin{align*} -{\frac{-ex+d}{9\, \left ( ex+d \right ) ^{8}de} \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)^9,x)

[Out]

-1/9/(e*x+d)^8*(-e*x+d)/d/e*(-e^2*x^2+d^2)^(7/2)

________________________________________________________________________________________

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)^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.37059, size = 333, normalized size = 10.09 \begin{align*} -\frac{e^{5} x^{5} + 5 \, d e^{4} x^{4} + 10 \, d^{2} e^{3} x^{3} + 10 \, d^{3} e^{2} x^{2} + 5 \, d^{4} e x + d^{5} +{\left (e^{4} x^{4} - 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} - 4 \, d^{3} e x + d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{9 \,{\left (d e^{6} x^{5} + 5 \, d^{2} e^{5} x^{4} + 10 \, d^{3} e^{4} x^{3} + 10 \, d^{4} e^{3} x^{2} + 5 \, d^{5} e^{2} x + d^{6} 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)^9,x, algorithm="fricas")

[Out]

-1/9*(e^5*x^5 + 5*d*e^4*x^4 + 10*d^2*e^3*x^3 + 10*d^3*e^2*x^2 + 5*d^4*e*x + d^5 + (e^4*x^4 - 4*d*e^3*x^3 + 6*d
^2*e^2*x^2 - 4*d^3*e*x + d^4)*sqrt(-e^2*x^2 + d^2))/(d*e^6*x^5 + 5*d^2*e^5*x^4 + 10*d^3*e^4*x^3 + 10*d^4*e^3*x
^2 + 5*d^5*e^2*x + d^6*e)

________________________________________________________________________________________

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)**9,x)

[Out]

Timed out

________________________________________________________________________________________

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)^9,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError