3.9.44 \(\int \frac {1}{(d+e x)^3 (d^2-e^2 x^2)^{5/2}} \, dx\) [844]

Optimal. Leaf size=148 \[ \frac {8 x}{63 d^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{9 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{21 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{21 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 x}{63 d^7 \sqrt {d^2-e^2 x^2}} \]

[Out]

8/63*x/d^5/(-e^2*x^2+d^2)^(3/2)-1/9/d/e/(e*x+d)^3/(-e^2*x^2+d^2)^(3/2)-2/21/d^2/e/(e*x+d)^2/(-e^2*x^2+d^2)^(3/
2)-2/21/d^3/e/(e*x+d)/(-e^2*x^2+d^2)^(3/2)+16/63*x/d^7/(-e^2*x^2+d^2)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {673, 198, 197} \begin {gather*} -\frac {2}{21 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{9 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 x}{63 d^7 \sqrt {d^2-e^2 x^2}}+\frac {8 x}{63 d^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{21 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^3*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

(8*x)/(63*d^5*(d^2 - e^2*x^2)^(3/2)) - 1/(9*d*e*(d + e*x)^3*(d^2 - e^2*x^2)^(3/2)) - 2/(21*d^2*e*(d + e*x)^2*(
d^2 - e^2*x^2)^(3/2)) - 2/(21*d^3*e*(d + e*x)*(d^2 - e^2*x^2)^(3/2)) + (16*x)/(63*d^7*Sqrt[d^2 - e^2*x^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

Rule 673

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]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=-\frac {1}{9 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{3 d}\\ &=-\frac {1}{9 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{21 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{21 d^2}\\ &=-\frac {1}{9 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{21 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{21 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{21 d^3}\\ &=\frac {8 x}{63 d^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{9 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{21 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{21 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{63 d^5}\\ &=\frac {8 x}{63 d^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{9 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{21 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{21 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 x}{63 d^7 \sqrt {d^2-e^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.47, size = 104, normalized size = 0.70 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-19 d^6+6 d^5 e x+66 d^4 e^2 x^2+56 d^3 e^3 x^3-24 d^2 e^4 x^4-48 d e^5 x^5-16 e^6 x^6\right )}{63 d^7 e (d-e x)^2 (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^3*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-19*d^6 + 6*d^5*e*x + 66*d^4*e^2*x^2 + 56*d^3*e^3*x^3 - 24*d^2*e^4*x^4 - 48*d*e^5*x^5 -
16*e^6*x^6))/(63*d^7*e*(d - e*x)^2*(d + e*x)^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(267\) vs. \(2(128)=256\).
time = 0.56, size = 268, normalized size = 1.81

method	result	size
gosper	\(-\frac {\left (-e x +d \right ) \left (16 e^{6} x^{6}+48 d \,e^{5} x^{5}+24 e^{4} x^{4} d^{2}-56 d^{3} e^{3} x^{3}-66 d^{4} e^{2} x^{2}-6 d^{5} e x +19 d^{6}\right )}{63 \left (e x +d \right )^{2} d^{7} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\)	\(99\)
trager	\(-\frac {\left (16 e^{6} x^{6}+48 d \,e^{5} x^{5}+24 e^{4} x^{4} d^{2}-56 d^{3} e^{3} x^{3}-66 d^{4} e^{2} x^{2}-6 d^{5} e x +19 d^{6}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{63 d^{7} \left (e x +d \right )^{5} \left (-e x +d \right )^{2} e}\)	\(101\)
default	\(\frac {-\frac {1}{9 d e \left (x +\frac {d}{e}\right )^{3} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {2 e \left (-\frac {1}{7 d e \left (x +\frac {d}{e}\right )^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {5 e \left (-\frac {1}{5 d e \left (x +\frac {d}{e}\right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {4 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{6 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e^{2} d^{4} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}\right )}{7 d}\right )}{3 d}}{e^{3}}\)	\(268\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^3/(-e^2*x^2+d^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/e^3*(-1/9/d/e/(x+d/e)^3/(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(3/2)+2/3*e/d*(-1/7/d/e/(x+d/e)^2/(-e^2*(x+d/e)^2+2*d
*e*(x+d/e))^(3/2)+5/7*e/d*(-1/5/d/e/(x+d/e)/(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(3/2)+4/5*e/d*(-1/6*(-2*e^2*(x+d/e)
+2*d*e)/d^2/e^2/(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(3/2)-1/3/e^2/d^4*(-2*e^2*(x+d/e)+2*d*e)/(-e^2*(x+d/e)^2+2*d*e*
(x+d/e))^(1/2)))))

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Maxima [A]
time = 0.29, size = 238, normalized size = 1.61 \begin {gather*} -\frac {1}{9 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d x^{3} e^{4} + 3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} x^{2} e^{3} + 3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} e\right )}} - \frac {2}{21 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} x^{2} e^{3} + 2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} e\right )}} - \frac {2}{21 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} e\right )}} + \frac {8 \, x}{63 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5}} + \frac {16 \, x}{63 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(-e^2*x^2+d^2)^(5/2),x, algorithm="maxima")

[Out]

-1/9/((-x^2*e^2 + d^2)^(3/2)*d*x^3*e^4 + 3*(-x^2*e^2 + d^2)^(3/2)*d^2*x^2*e^3 + 3*(-x^2*e^2 + d^2)^(3/2)*d^3*x
*e^2 + (-x^2*e^2 + d^2)^(3/2)*d^4*e) - 2/21/((-x^2*e^2 + d^2)^(3/2)*d^2*x^2*e^3 + 2*(-x^2*e^2 + d^2)^(3/2)*d^3
*x*e^2 + (-x^2*e^2 + d^2)^(3/2)*d^4*e) - 2/21/((-x^2*e^2 + d^2)^(3/2)*d^3*x*e^2 + (-x^2*e^2 + d^2)^(3/2)*d^4*e
) + 8/63*x/((-x^2*e^2 + d^2)^(3/2)*d^5) + 16/63*x/(sqrt(-x^2*e^2 + d^2)*d^7)

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Fricas [A]
time = 2.94, size = 218, normalized size = 1.47 \begin {gather*} -\frac {19 \, x^{7} e^{7} + 57 \, d x^{6} e^{6} + 19 \, d^{2} x^{5} e^{5} - 95 \, d^{3} x^{4} e^{4} - 95 \, d^{4} x^{3} e^{3} + 19 \, d^{5} x^{2} e^{2} + 57 \, d^{6} x e + 19 \, d^{7} + {\left (16 \, x^{6} e^{6} + 48 \, d x^{5} e^{5} + 24 \, d^{2} x^{4} e^{4} - 56 \, d^{3} x^{3} e^{3} - 66 \, d^{4} x^{2} e^{2} - 6 \, d^{5} x e + 19 \, d^{6}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{63 \, {\left (d^{7} x^{7} e^{8} + 3 \, d^{8} x^{6} e^{7} + d^{9} x^{5} e^{6} - 5 \, d^{10} x^{4} e^{5} - 5 \, d^{11} x^{3} e^{4} + d^{12} x^{2} e^{3} + 3 \, d^{13} x e^{2} + d^{14} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(-e^2*x^2+d^2)^(5/2),x, algorithm="fricas")

[Out]

-1/63*(19*x^7*e^7 + 57*d*x^6*e^6 + 19*d^2*x^5*e^5 - 95*d^3*x^4*e^4 - 95*d^4*x^3*e^3 + 19*d^5*x^2*e^2 + 57*d^6*
x*e + 19*d^7 + (16*x^6*e^6 + 48*d*x^5*e^5 + 24*d^2*x^4*e^4 - 56*d^3*x^3*e^3 - 66*d^4*x^2*e^2 - 6*d^5*x*e + 19*
d^6)*sqrt(-x^2*e^2 + d^2))/(d^7*x^7*e^8 + 3*d^8*x^6*e^7 + d^9*x^5*e^6 - 5*d^10*x^4*e^5 - 5*d^11*x^3*e^4 + d^12
*x^2*e^3 + 3*d^13*x*e^2 + d^14*e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**3/(-e**2*x**2+d**2)**(5/2),x)

[Out]

Integral(1/((-(-d + e*x)*(d + e*x))**(5/2)*(d + e*x)**3), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.71, size = 168, normalized size = 1.14 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {197\,x}{1008\,d^5}-\frac {155}{1008\,d^4\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}}{36\,d^3\,e\,{\left (d+e\,x\right )}^5}-\frac {13\,\sqrt {d^2-e^2\,x^2}}{252\,d^4\,e\,{\left (d+e\,x\right )}^4}-\frac {23\,\sqrt {d^2-e^2\,x^2}}{336\,d^5\,e\,{\left (d+e\,x\right )}^3}+\frac {16\,x\,\sqrt {d^2-e^2\,x^2}}{63\,d^7\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d^2 - e^2*x^2)^(1/2)*((197*x)/(1008*d^5) - 155/(1008*d^4*e)))/((d + e*x)^2*(d - e*x)^2) - (d^2 - e^2*x^2)^(1
/2)/(36*d^3*e*(d + e*x)^5) - (13*(d^2 - e^2*x^2)^(1/2))/(252*d^4*e*(d + e*x)^4) - (23*(d^2 - e^2*x^2)^(1/2))/(
336*d^5*e*(d + e*x)^3) + (16*x*(d^2 - e^2*x^2)^(1/2))/(63*d^7*(d + e*x)*(d - e*x))

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