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3.215 \(\int \frac{1}{x^3 (1+x)^{3/2}} \, dx\)

Optimal. Leaf size=52 \[ -\frac{1}{2 x^2 \sqrt{x+1}}+\frac{5}{4 x \sqrt{x+1}}+\frac{15}{4 \sqrt{x+1}}-\frac{15}{4} \tanh ^{-1}\left (\sqrt{x+1}\right ) \]

[Out]

15/(4*Sqrt[1 + x]) - 1/(2*x^2*Sqrt[1 + x]) + 5/(4*x*Sqrt[1 + x]) - (15*ArcTanh[Sqrt[1 + x]])/4

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Rubi [A]  time = 0.010101, antiderivative size = 53, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {51, 63, 207} \[ -\frac{5 \sqrt{x+1}}{2 x^2}+\frac{2}{x^2 \sqrt{x+1}}+\frac{15 \sqrt{x+1}}{4 x}-\frac{15}{4} \tanh ^{-1}\left (\sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(1 + x)^(3/2)),x]

[Out]

2/(x^2*Sqrt[1 + x]) - (5*Sqrt[1 + x])/(2*x^2) + (15*Sqrt[1 + x])/(4*x) - (15*ArcTanh[Sqrt[1 + x]])/4

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
 FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^3 (1+x)^{3/2}} \, dx &=\frac{2}{x^2 \sqrt{1+x}}+5 \int \frac{1}{x^3 \sqrt{1+x}} \, dx\\ &=\frac{2}{x^2 \sqrt{1+x}}-\frac{5 \sqrt{1+x}}{2 x^2}-\frac{15}{4} \int \frac{1}{x^2 \sqrt{1+x}} \, dx\\ &=\frac{2}{x^2 \sqrt{1+x}}-\frac{5 \sqrt{1+x}}{2 x^2}+\frac{15 \sqrt{1+x}}{4 x}+\frac{15}{8} \int \frac{1}{x \sqrt{1+x}} \, dx\\ &=\frac{2}{x^2 \sqrt{1+x}}-\frac{5 \sqrt{1+x}}{2 x^2}+\frac{15 \sqrt{1+x}}{4 x}+\frac{15}{4} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{1+x}\right )\\ &=\frac{2}{x^2 \sqrt{1+x}}-\frac{5 \sqrt{1+x}}{2 x^2}+\frac{15 \sqrt{1+x}}{4 x}-\frac{15}{4} \tanh ^{-1}\left (\sqrt{1+x}\right )\\ \end{align*}

Mathematica [C]  time = 0.0042073, size = 20, normalized size = 0.38 \[ \frac{2 \, _2F_1\left (-\frac{1}{2},3;\frac{1}{2};x+1\right )}{\sqrt{x+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(1 + x)^(3/2)),x]

[Out]

(2*Hypergeometric2F1[-1/2, 3, 1/2, 1 + x])/Sqrt[1 + x]

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Maple [A]  time = 0.012, size = 73, normalized size = 1.4 \begin{align*} 2\,{\frac{1}{\sqrt{1+x}}}+{\frac{1}{8} \left ( 1+\sqrt{1+x} \right ) ^{-2}}+{\frac{7}{8} \left ( 1+\sqrt{1+x} \right ) ^{-1}}-{\frac{15}{8}\ln \left ( 1+\sqrt{1+x} \right ) }-{\frac{1}{8} \left ( -1+\sqrt{1+x} \right ) ^{-2}}+{\frac{7}{8} \left ( -1+\sqrt{1+x} \right ) ^{-1}}+{\frac{15}{8}\ln \left ( -1+\sqrt{1+x} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(1+x)^(3/2),x)

[Out]

2/(1+x)^(1/2)+1/8/(1+(1+x)^(1/2))^2+7/8/(1+(1+x)^(1/2))-15/8*ln(1+(1+x)^(1/2))-1/8/(-1+(1+x)^(1/2))^2+7/8/(-1+
(1+x)^(1/2))+15/8*ln(-1+(1+x)^(1/2))

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Maxima [A]  time = 0.936766, size = 74, normalized size = 1.42 \begin{align*} \frac{15 \,{\left (x + 1\right )}^{2} - 25 \, x - 17}{4 \,{\left ({\left (x + 1\right )}^{\frac{5}{2}} - 2 \,{\left (x + 1\right )}^{\frac{3}{2}} + \sqrt{x + 1}\right )}} - \frac{15}{8} \, \log \left (\sqrt{x + 1} + 1\right ) + \frac{15}{8} \, \log \left (\sqrt{x + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(1+x)^(3/2),x, algorithm="maxima")

[Out]

1/4*(15*(x + 1)^2 - 25*x - 17)/((x + 1)^(5/2) - 2*(x + 1)^(3/2) + sqrt(x + 1)) - 15/8*log(sqrt(x + 1) + 1) + 1
5/8*log(sqrt(x + 1) - 1)

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Fricas [A]  time = 1.78244, size = 174, normalized size = 3.35 \begin{align*} -\frac{15 \,{\left (x^{3} + x^{2}\right )} \log \left (\sqrt{x + 1} + 1\right ) - 15 \,{\left (x^{3} + x^{2}\right )} \log \left (\sqrt{x + 1} - 1\right ) - 2 \,{\left (15 \, x^{2} + 5 \, x - 2\right )} \sqrt{x + 1}}{8 \,{\left (x^{3} + x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(1+x)^(3/2),x, algorithm="fricas")

[Out]

-1/8*(15*(x^3 + x^2)*log(sqrt(x + 1) + 1) - 15*(x^3 + x^2)*log(sqrt(x + 1) - 1) - 2*(15*x^2 + 5*x - 2)*sqrt(x
+ 1))/(x^3 + x^2)

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Sympy [B]  time = 2.69473, size = 3966, normalized size = 76.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(1+x)**(3/2),x)

[Out]

Piecewise((-30*(x + 1)**(17/2)*acoth(sqrt(x + 1))/(8*(x + 1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x + 1)**(13/2
) - 448*(x + 1)**(11/2) + 560*(x + 1)**(9/2) - 448*(x + 1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)**(3/2) + 8
*sqrt(x + 1)) - 15*I*pi*(x + 1)**(17/2)/(8*(x + 1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x + 1)**(13/2) - 448*(x
 + 1)**(11/2) + 560*(x + 1)**(9/2) - 448*(x + 1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqrt(x +
1)) + 240*(x + 1)**(15/2)*acoth(sqrt(x + 1))/(8*(x + 1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x + 1)**(13/2) - 4
48*(x + 1)**(11/2) + 560*(x + 1)**(9/2) - 448*(x + 1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqrt
(x + 1)) + 120*I*pi*(x + 1)**(15/2)/(8*(x + 1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x + 1)**(13/2) - 448*(x + 1
)**(11/2) + 560*(x + 1)**(9/2) - 448*(x + 1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqrt(x + 1))
- 840*(x + 1)**(13/2)*acoth(sqrt(x + 1))/(8*(x + 1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x + 1)**(13/2) - 448*(
x + 1)**(11/2) + 560*(x + 1)**(9/2) - 448*(x + 1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqrt(x +
 1)) - 420*I*pi*(x + 1)**(13/2)/(8*(x + 1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x + 1)**(13/2) - 448*(x + 1)**(
11/2) + 560*(x + 1)**(9/2) - 448*(x + 1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqrt(x + 1)) + 16
80*(x + 1)**(11/2)*acoth(sqrt(x + 1))/(8*(x + 1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x + 1)**(13/2) - 448*(x +
 1)**(11/2) + 560*(x + 1)**(9/2) - 448*(x + 1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqrt(x + 1)
) + 840*I*pi*(x + 1)**(11/2)/(8*(x + 1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x + 1)**(13/2) - 448*(x + 1)**(11/
2) + 560*(x + 1)**(9/2) - 448*(x + 1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqrt(x + 1)) - 2100*
(x + 1)**(9/2)*acoth(sqrt(x + 1))/(8*(x + 1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x + 1)**(13/2) - 448*(x + 1)*
*(11/2) + 560*(x + 1)**(9/2) - 448*(x + 1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqrt(x + 1)) -
1050*I*pi*(x + 1)**(9/2)/(8*(x + 1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x + 1)**(13/2) - 448*(x + 1)**(11/2) +
 560*(x + 1)**(9/2) - 448*(x + 1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqrt(x + 1)) + 1680*(x +
 1)**(7/2)*acoth(sqrt(x + 1))/(8*(x + 1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x + 1)**(13/2) - 448*(x + 1)**(11
/2) + 560*(x + 1)**(9/2) - 448*(x + 1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqrt(x + 1)) + 840*
I*pi*(x + 1)**(7/2)/(8*(x + 1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x + 1)**(13/2) - 448*(x + 1)**(11/2) + 560*
(x + 1)**(9/2) - 448*(x + 1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqrt(x + 1)) - 840*(x + 1)**(
5/2)*acoth(sqrt(x + 1))/(8*(x + 1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x + 1)**(13/2) - 448*(x + 1)**(11/2) +
560*(x + 1)**(9/2) - 448*(x + 1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqrt(x + 1)) - 420*I*pi*(
x + 1)**(5/2)/(8*(x + 1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x + 1)**(13/2) - 448*(x + 1)**(11/2) + 560*(x + 1
)**(9/2) - 448*(x + 1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqrt(x + 1)) + 240*(x + 1)**(3/2)*a
coth(sqrt(x + 1))/(8*(x + 1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x + 1)**(13/2) - 448*(x + 1)**(11/2) + 560*(x
 + 1)**(9/2) - 448*(x + 1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqrt(x + 1)) + 120*I*pi*(x + 1)
**(3/2)/(8*(x + 1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x + 1)**(13/2) - 448*(x + 1)**(11/2) + 560*(x + 1)**(9/
2) - 448*(x + 1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqrt(x + 1)) - 30*sqrt(x + 1)*acoth(sqrt(
x + 1))/(8*(x + 1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x + 1)**(13/2) - 448*(x + 1)**(11/2) + 560*(x + 1)**(9/
2) - 448*(x + 1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqrt(x + 1)) - 15*I*pi*sqrt(x + 1)/(8*(x
+ 1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x + 1)**(13/2) - 448*(x + 1)**(11/2) + 560*(x + 1)**(9/2) - 448*(x +
1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqrt(x + 1)) + 30*(x + 1)**8/(8*(x + 1)**(17/2) - 64*(x
 + 1)**(15/2) + 224*(x + 1)**(13/2) - 448*(x + 1)**(11/2) + 560*(x + 1)**(9/2) - 448*(x + 1)**(7/2) + 224*(x +
 1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqrt(x + 1)) - 230*(x + 1)**7/(8*(x + 1)**(17/2) - 64*(x + 1)**(15/2) + 224
*(x + 1)**(13/2) - 448*(x + 1)**(11/2) + 560*(x + 1)**(9/2) - 448*(x + 1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x
+ 1)**(3/2) + 8*sqrt(x + 1)) + 766*(x + 1)**6/(8*(x + 1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x + 1)**(13/2) -
448*(x + 1)**(11/2) + 560*(x + 1)**(9/2) - 448*(x + 1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqr
t(x + 1)) - 1446*(x + 1)**5/(8*(x + 1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x + 1)**(13/2) - 448*(x + 1)**(11/2
) + 560*(x + 1)**(9/2) - 448*(x + 1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqrt(x + 1)) + 1690*(
x + 1)**4/(8*(x + 1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x + 1)**(13/2) - 448*(x + 1)**(11/2) + 560*(x + 1)**(
9/2) - 448*(x + 1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqrt(x + 1)) - 1250*(x + 1)**3/(8*(x +
1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x + 1)**(13/2) - 448*(x + 1)**(11/2) + 560*(x + 1)**(9/2) - 448*(x + 1)
**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqrt(x + 1)) + 570*(x + 1)**2/(8*(x + 1)**(17/2) - 64*(x
+ 1)**(15/2) + 224*(x + 1)**(13/2) - 448*(x + 1)**(11/2) + 560*(x + 1)**(9/2) - 448*(x + 1)**(7/2) + 224*(x +
1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqrt(x + 1)) - 146*(x + 1)/(8*(x + 1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x
+ 1)**(13/2) - 448*(x + 1)**(11/2) + 560*(x + 1)**(9/2) - 448*(x + 1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)
**(3/2) + 8*sqrt(x + 1)) + 16/(8*(x + 1)**(17/2) - 64*(x + 1)**(15/2) + 224*(x + 1)**(13/2) - 448*(x + 1)**(11
/2) + 560*(x + 1)**(9/2) - 448*(x + 1)**(7/2) + 224*(x + 1)**(5/2) - 64*(x + 1)**(3/2) + 8*sqrt(x + 1)), Abs(x
 + 1) > 1), (-15*(x + 1)**(17/2)*atanh(sqrt(x + 1))/(4*(x + 1)**(17/2) - 32*(x + 1)**(15/2) + 112*(x + 1)**(13
/2) - 224*(x + 1)**(11/2) + 280*(x + 1)**(9/2) - 224*(x + 1)**(7/2) + 112*(x + 1)**(5/2) - 32*(x + 1)**(3/2) +
 4*sqrt(x + 1)) + 120*(x + 1)**(15/2)*atanh(sqrt(x + 1))/(4*(x + 1)**(17/2) - 32*(x + 1)**(15/2) + 112*(x + 1)
**(13/2) - 224*(x + 1)**(11/2) + 280*(x + 1)**(9/2) - 224*(x + 1)**(7/2) + 112*(x + 1)**(5/2) - 32*(x + 1)**(3
/2) + 4*sqrt(x + 1)) - 420*(x + 1)**(13/2)*atanh(sqrt(x + 1))/(4*(x + 1)**(17/2) - 32*(x + 1)**(15/2) + 112*(x
 + 1)**(13/2) - 224*(x + 1)**(11/2) + 280*(x + 1)**(9/2) - 224*(x + 1)**(7/2) + 112*(x + 1)**(5/2) - 32*(x + 1
)**(3/2) + 4*sqrt(x + 1)) + 840*(x + 1)**(11/2)*atanh(sqrt(x + 1))/(4*(x + 1)**(17/2) - 32*(x + 1)**(15/2) + 1
12*(x + 1)**(13/2) - 224*(x + 1)**(11/2) + 280*(x + 1)**(9/2) - 224*(x + 1)**(7/2) + 112*(x + 1)**(5/2) - 32*(
x + 1)**(3/2) + 4*sqrt(x + 1)) - 1050*(x + 1)**(9/2)*atanh(sqrt(x + 1))/(4*(x + 1)**(17/2) - 32*(x + 1)**(15/2
) + 112*(x + 1)**(13/2) - 224*(x + 1)**(11/2) + 280*(x + 1)**(9/2) - 224*(x + 1)**(7/2) + 112*(x + 1)**(5/2) -
 32*(x + 1)**(3/2) + 4*sqrt(x + 1)) + 840*(x + 1)**(7/2)*atanh(sqrt(x + 1))/(4*(x + 1)**(17/2) - 32*(x + 1)**(
15/2) + 112*(x + 1)**(13/2) - 224*(x + 1)**(11/2) + 280*(x + 1)**(9/2) - 224*(x + 1)**(7/2) + 112*(x + 1)**(5/
2) - 32*(x + 1)**(3/2) + 4*sqrt(x + 1)) - 420*(x + 1)**(5/2)*atanh(sqrt(x + 1))/(4*(x + 1)**(17/2) - 32*(x + 1
)**(15/2) + 112*(x + 1)**(13/2) - 224*(x + 1)**(11/2) + 280*(x + 1)**(9/2) - 224*(x + 1)**(7/2) + 112*(x + 1)*
*(5/2) - 32*(x + 1)**(3/2) + 4*sqrt(x + 1)) + 120*(x + 1)**(3/2)*atanh(sqrt(x + 1))/(4*(x + 1)**(17/2) - 32*(x
 + 1)**(15/2) + 112*(x + 1)**(13/2) - 224*(x + 1)**(11/2) + 280*(x + 1)**(9/2) - 224*(x + 1)**(7/2) + 112*(x +
 1)**(5/2) - 32*(x + 1)**(3/2) + 4*sqrt(x + 1)) - 15*sqrt(x + 1)*atanh(sqrt(x + 1))/(4*(x + 1)**(17/2) - 32*(x
 + 1)**(15/2) + 112*(x + 1)**(13/2) - 224*(x + 1)**(11/2) + 280*(x + 1)**(9/2) - 224*(x + 1)**(7/2) + 112*(x +
 1)**(5/2) - 32*(x + 1)**(3/2) + 4*sqrt(x + 1)) + 15*(x + 1)**8/(4*(x + 1)**(17/2) - 32*(x + 1)**(15/2) + 112*
(x + 1)**(13/2) - 224*(x + 1)**(11/2) + 280*(x + 1)**(9/2) - 224*(x + 1)**(7/2) + 112*(x + 1)**(5/2) - 32*(x +
 1)**(3/2) + 4*sqrt(x + 1)) - 115*(x + 1)**7/(4*(x + 1)**(17/2) - 32*(x + 1)**(15/2) + 112*(x + 1)**(13/2) - 2
24*(x + 1)**(11/2) + 280*(x + 1)**(9/2) - 224*(x + 1)**(7/2) + 112*(x + 1)**(5/2) - 32*(x + 1)**(3/2) + 4*sqrt
(x + 1)) + 383*(x + 1)**6/(4*(x + 1)**(17/2) - 32*(x + 1)**(15/2) + 112*(x + 1)**(13/2) - 224*(x + 1)**(11/2)
+ 280*(x + 1)**(9/2) - 224*(x + 1)**(7/2) + 112*(x + 1)**(5/2) - 32*(x + 1)**(3/2) + 4*sqrt(x + 1)) - 723*(x +
 1)**5/(4*(x + 1)**(17/2) - 32*(x + 1)**(15/2) + 112*(x + 1)**(13/2) - 224*(x + 1)**(11/2) + 280*(x + 1)**(9/2
) - 224*(x + 1)**(7/2) + 112*(x + 1)**(5/2) - 32*(x + 1)**(3/2) + 4*sqrt(x + 1)) + 845*(x + 1)**4/(4*(x + 1)**
(17/2) - 32*(x + 1)**(15/2) + 112*(x + 1)**(13/2) - 224*(x + 1)**(11/2) + 280*(x + 1)**(9/2) - 224*(x + 1)**(7
/2) + 112*(x + 1)**(5/2) - 32*(x + 1)**(3/2) + 4*sqrt(x + 1)) - 625*(x + 1)**3/(4*(x + 1)**(17/2) - 32*(x + 1)
**(15/2) + 112*(x + 1)**(13/2) - 224*(x + 1)**(11/2) + 280*(x + 1)**(9/2) - 224*(x + 1)**(7/2) + 112*(x + 1)**
(5/2) - 32*(x + 1)**(3/2) + 4*sqrt(x + 1)) + 285*(x + 1)**2/(4*(x + 1)**(17/2) - 32*(x + 1)**(15/2) + 112*(x +
 1)**(13/2) - 224*(x + 1)**(11/2) + 280*(x + 1)**(9/2) - 224*(x + 1)**(7/2) + 112*(x + 1)**(5/2) - 32*(x + 1)*
*(3/2) + 4*sqrt(x + 1)) - 73*(x + 1)/(4*(x + 1)**(17/2) - 32*(x + 1)**(15/2) + 112*(x + 1)**(13/2) - 224*(x +
1)**(11/2) + 280*(x + 1)**(9/2) - 224*(x + 1)**(7/2) + 112*(x + 1)**(5/2) - 32*(x + 1)**(3/2) + 4*sqrt(x + 1))
 + 8/(4*(x + 1)**(17/2) - 32*(x + 1)**(15/2) + 112*(x + 1)**(13/2) - 224*(x + 1)**(11/2) + 280*(x + 1)**(9/2)
- 224*(x + 1)**(7/2) + 112*(x + 1)**(5/2) - 32*(x + 1)**(3/2) + 4*sqrt(x + 1)), True))

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Giac [A]  time = 1.07459, size = 66, normalized size = 1.27 \begin{align*} \frac{2}{\sqrt{x + 1}} + \frac{7 \,{\left (x + 1\right )}^{\frac{3}{2}} - 9 \, \sqrt{x + 1}}{4 \, x^{2}} - \frac{15}{8} \, \log \left (\sqrt{x + 1} + 1\right ) + \frac{15}{8} \, \log \left ({\left | \sqrt{x + 1} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(1+x)^(3/2),x, algorithm="giac")

[Out]

2/sqrt(x + 1) + 1/4*(7*(x + 1)^(3/2) - 9*sqrt(x + 1))/x^2 - 15/8*log(sqrt(x + 1) + 1) + 15/8*log(abs(sqrt(x +
1) - 1))

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