3.7.80 \(\int \frac {(d+e x)^3}{(d^2-e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=33 \[ \frac {(d+e x)^3}{3 d e \left (d^2-e^2 x^2\right )^{3/2}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {(d+e x)^3}{3 d e \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d + e*x)^3/(3*d*e*(d^2 - e^2*x^2)^(3/2))

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)^3}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\frac {(d+e x)^3}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 41, normalized size = 1.24 \begin {gather*} \frac {(d+e x)^2}{3 d e (d-e x) \sqrt {d^2-e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.38, size = 40, normalized size = 1.21 \begin {gather*} \frac {(d+e x) \sqrt {d^2-e^2 x^2}}{3 d e (e x-d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.40, size = 63, normalized size = 1.91 \begin {gather*} \frac {e^{2} x^{2} - 2 \, d e x + d^{2} + \sqrt {-e^{2} x^{2} + d^{2}} {\left (e x + d\right )}}{3 \, {\left (d e^{3} x^{2} - 2 \, d^{2} e^{2} x + d^{3} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(-e^2*x^2+d^2)^(5/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.28, size = 56, normalized size = 1.70 \begin {gather*} \frac {\sqrt {-x^{2} e^{2} + d^{2}} {\left (d^{2} e^{\left (-1\right )} + {\left (x {\left (\frac {x e^{2}}{d} + 3 \, e\right )} + 3 \, d\right )} x\right )}}{3 \, {\left (x^{2} e^{2} - d^{2}\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 36, normalized size = 1.09 \begin {gather*} \frac {\left (e x +d \right )^{4} \left (-e x +d \right )}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3/(-e^2*x^2+d^2)^(5/2),x)

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 1.40, size = 80, normalized size = 2.42 \begin {gather*} \frac {e x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {4 \, d x}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {d^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e} - \frac {x}{3 \, \sqrt {-e^{2} x^{2} + d^{2}} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(-e^2*x^2+d^2)^(5/2),x, algorithm="maxima")

[Out]

e*x^2/(-e^2*x^2 + d^2)^(3/2) + 4/3*d*x/(-e^2*x^2 + d^2)^(3/2) + 1/3*d^2/((-e^2*x^2 + d^2)^(3/2)*e) - 1/3*x/(sq
rt(-e^2*x^2 + d^2)*d)

________________________________________________________________________________________

mupad [B]  time = 0.57, size = 35, normalized size = 1.06 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (d+e\,x\right )}{3\,d\,e\,{\left (d-e\,x\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^3/(d^2 - e^2*x^2)^(5/2),x)

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3/(-e**2*x**2+d**2)**(5/2),x)

[Out]

Integral((d + e*x)**3/(-(-d + e*x)*(d + e*x))**(5/2), x)

________________________________________________________________________________________