3.850 \(\int \frac{(d+e x)^5}{(d^2-e^2 x^2)^{7/2}} \, dx\)

Optimal. Leaf size=33 \[ \frac{(d+e x)^5}{5 d e \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

(d + e*x)^5/(5*d*e*(d^2 - e^2*x^2)^(5/2))

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Rubi [A]  time = 0.0094206, 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} \[ \frac{(d+e x)^5}{5 d e \left (d^2-e^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^5/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(d + e*x)^5/(5*d*e*(d^2 - e^2*x^2)^(5/2))

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{(d+e x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{(d+e x)^5}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0944502, size = 41, normalized size = 1.24 \[ \frac{(d+e x)^3}{5 d e (d-e x)^2 \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^5/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(d + e*x)^3/(5*d*e*(d - e*x)^2*Sqrt[d^2 - e^2*x^2])

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Maple [A]  time = 0.043, size = 36, normalized size = 1.1 \begin{align*}{\frac{ \left ( ex+d \right ) ^{6} \left ( -ex+d \right ) }{5\,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*x+d)^5/(-e^2*x^2+d^2)^(7/2),x)

[Out]

1/5*(e*x+d)^6*(-e*x+d)/d/e/(-e^2*x^2+d^2)^(7/2)

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Maxima [B]  time = 1.17891, size = 200, normalized size = 6.06 \begin{align*} \frac{e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{5 \, d e^{2} x^{3}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{2 \, d^{2} e x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{7 \, d^{3} x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{d^{4}}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{d x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{x}{5 \, \sqrt{-e^{2} x^{2} + d^{2}} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

e^3*x^4/(-e^2*x^2 + d^2)^(5/2) + 5/2*d*e^2*x^3/(-e^2*x^2 + d^2)^(5/2) + 2*d^2*e*x^2/(-e^2*x^2 + d^2)^(5/2) + 7
/10*d^3*x/(-e^2*x^2 + d^2)^(5/2) + 1/5*d^4/((-e^2*x^2 + d^2)^(5/2)*e) + 1/10*d*x/(-e^2*x^2 + d^2)^(3/2) + 1/5*
x/(sqrt(-e^2*x^2 + d^2)*d)

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Fricas [B]  time = 1.965, size = 197, normalized size = 5.97 \begin{align*} \frac{e^{3} x^{3} - 3 \, d e^{2} x^{2} + 3 \, d^{2} e x - d^{3} -{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{5 \,{\left (d e^{4} x^{3} - 3 \, d^{2} e^{3} x^{2} + 3 \, d^{3} e^{2} x - d^{4} e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{5}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral((d + e*x)**5/(-(-d + e*x)*(d + e*x))**(7/2), x)

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Giac [B]  time = 1.30536, size = 103, normalized size = 3.12 \begin{align*} -\frac{{\left (d^{4} e^{\left (-1\right )} +{\left (5 \, d^{3} +{\left (10 \, d^{2} e +{\left (x{\left (\frac{x e^{4}}{d} + 5 \, e^{3}\right )} + 10 \, d e^{2}\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{5 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

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