3.2.17 \(\int \frac {1}{\sqrt {x} (a+b x)^3} \, dx\) [117]

Optimal. Leaf size=57 \[ \sqrt {x} \left (\frac {1}{2 a (a+b x)^2}+\frac {1}{4 a^2 (a+b x)}\right )+\frac {3 \arctan \left (\frac {b x}{a}\right )}{4 a^2 \sqrt {a b}} \]

[Out]

(1/2/a/(b*x+a)^2+1/4/a^2/(b*x+a))*x^(1/2)+3/4/a^2/(a*b)^(1/2)*arctan(b*x/a)

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Rubi [A]
time = 0.01, antiderivative size = 70, normalized size of antiderivative = 1.23, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {44, 65, 211} \begin {gather*} \frac {3 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b}}+\frac {3 \sqrt {x}}{4 a^2 (a+b x)}+\frac {\sqrt {x}}{2 a (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(a + b*x)^3),x]

[Out]

Sqrt[x]/(2*a*(a + b*x)^2) + (3*Sqrt[x])/(4*a^2*(a + b*x)) + (3*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(5/2)*S
qrt[b])

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

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 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\frac {\sqrt {x}}{2 a (a+b x)^2}+\frac {3 \int \frac {1}{\sqrt {x} (a+b x)^2} \, dx}{4 a}\\ &=\frac {\sqrt {x}}{2 a (a+b x)^2}+\frac {3 \sqrt {x}}{4 a^2 (a+b x)}+\frac {3 \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{8 a^2}\\ &=\frac {\sqrt {x}}{2 a (a+b x)^2}+\frac {3 \sqrt {x}}{4 a^2 (a+b x)}+\frac {3 \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{4 a^2}\\ &=\frac {\sqrt {x}}{2 a (a+b x)^2}+\frac {3 \sqrt {x}}{4 a^2 (a+b x)}+\frac {3 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b}}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.07, size = 59, normalized size = 1.04 \begin {gather*} \frac {\sqrt {x} (5 a+3 b x)}{4 a^2 (a+b x)^2}+\frac {3 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*(a + b*x)^3),x]

[Out]

(Sqrt[x]*(5*a + 3*b*x))/(4*a^2*(a + b*x)^2) + (3*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(5/2)*Sqrt[b])

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Maple [A]
time = 0.03, size = 59, normalized size = 1.04

method	result	size
derivativedivides	\(\frac {\sqrt {x}}{2 a \left (b x +a \right )^{2}}+\frac {\frac {3 \sqrt {x}}{4 a \left (b x +a \right )}+\frac {3 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 a \sqrt {a b}}}{a}\)	\(59\)
default	\(\frac {\sqrt {x}}{2 a \left (b x +a \right )^{2}}+\frac {\frac {3 \sqrt {x}}{4 a \left (b x +a \right )}+\frac {3 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 a \sqrt {a b}}}{a}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^3/x^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*x^(1/2)/a/(b*x+a)^2+3/2/a*(1/2*x^(1/2)/a/(b*x+a)+1/2/a/(a*b)^(1/2)*arctan(b*x^(1/2)/(a*b)^(1/2)))

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Maxima [A]
time = 0.40, size = 60, normalized size = 1.05 \begin {gather*} \frac {3 \, b x^{\frac {3}{2}} + 5 \, a \sqrt {x}}{4 \, {\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )}} + \frac {3 \, \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(3*b*x^(3/2) + 5*a*sqrt(x))/(a^2*b^2*x^2 + 2*a^3*b*x + a^4) + 3/4*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a
^2)

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Fricas [A]
time = 0.59, size = 186, normalized size = 3.26 \begin {gather*} \left [-\frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) - 2 \, {\left (3 \, a b^{2} x + 5 \, a^{2} b\right )} \sqrt {x}}{8 \, {\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}}, -\frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) - {\left (3 \, a b^{2} x + 5 \, a^{2} b\right )} \sqrt {x}}{4 \, {\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/8*(3*(b^2*x^2 + 2*a*b*x + a^2)*sqrt(-a*b)*log((b*x - a - 2*sqrt(-a*b)*sqrt(x))/(b*x + a)) - 2*(3*a*b^2*x +
5*a^2*b)*sqrt(x))/(a^3*b^3*x^2 + 2*a^4*b^2*x + a^5*b), -1/4*(3*(b^2*x^2 + 2*a*b*x + a^2)*sqrt(a*b)*arctan(sqr
t(a*b)/(b*sqrt(x))) - (3*a*b^2*x + 5*a^2*b)*sqrt(x))/(a^3*b^3*x^2 + 2*a^4*b^2*x + a^5*b)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 632 vs. \(2 (48) = 96\).
time = 7.83, size = 632, normalized size = 11.09 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 \sqrt {x}}{a^{3}} & \text {for}\: b = 0 \\- \frac {2}{5 b^{3} x^{\frac {5}{2}}} & \text {for}\: a = 0 \\\frac {3 a^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{4} b \sqrt {- \frac {a}{b}} + 16 a^{3} b^{2} x \sqrt {- \frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {- \frac {a}{b}}} - \frac {3 a^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{4} b \sqrt {- \frac {a}{b}} + 16 a^{3} b^{2} x \sqrt {- \frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {- \frac {a}{b}}} + \frac {10 a b \sqrt {x} \sqrt {- \frac {a}{b}}}{8 a^{4} b \sqrt {- \frac {a}{b}} + 16 a^{3} b^{2} x \sqrt {- \frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {- \frac {a}{b}}} + \frac {6 a b x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{4} b \sqrt {- \frac {a}{b}} + 16 a^{3} b^{2} x \sqrt {- \frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {- \frac {a}{b}}} - \frac {6 a b x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{4} b \sqrt {- \frac {a}{b}} + 16 a^{3} b^{2} x \sqrt {- \frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {- \frac {a}{b}}} + \frac {6 b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{8 a^{4} b \sqrt {- \frac {a}{b}} + 16 a^{3} b^{2} x \sqrt {- \frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {- \frac {a}{b}}} + \frac {3 b^{2} x^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{4} b \sqrt {- \frac {a}{b}} + 16 a^{3} b^{2} x \sqrt {- \frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {- \frac {a}{b}}} - \frac {3 b^{2} x^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{4} b \sqrt {- \frac {a}{b}} + 16 a^{3} b^{2} x \sqrt {- \frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/x**(5/2), Eq(a, 0) & Eq(b, 0)), (2*sqrt(x)/a**3, Eq(b, 0)), (-2/(5*b**3*x**(5/2)), Eq(a, 0)), (
3*a**2*log(sqrt(x) - sqrt(-a/b))/(8*a**4*b*sqrt(-a/b) + 16*a**3*b**2*x*sqrt(-a/b) + 8*a**2*b**3*x**2*sqrt(-a/b
)) - 3*a**2*log(sqrt(x) + sqrt(-a/b))/(8*a**4*b*sqrt(-a/b) + 16*a**3*b**2*x*sqrt(-a/b) + 8*a**2*b**3*x**2*sqrt
(-a/b)) + 10*a*b*sqrt(x)*sqrt(-a/b)/(8*a**4*b*sqrt(-a/b) + 16*a**3*b**2*x*sqrt(-a/b) + 8*a**2*b**3*x**2*sqrt(-
a/b)) + 6*a*b*x*log(sqrt(x) - sqrt(-a/b))/(8*a**4*b*sqrt(-a/b) + 16*a**3*b**2*x*sqrt(-a/b) + 8*a**2*b**3*x**2*
sqrt(-a/b)) - 6*a*b*x*log(sqrt(x) + sqrt(-a/b))/(8*a**4*b*sqrt(-a/b) + 16*a**3*b**2*x*sqrt(-a/b) + 8*a**2*b**3
*x**2*sqrt(-a/b)) + 6*b**2*x**(3/2)*sqrt(-a/b)/(8*a**4*b*sqrt(-a/b) + 16*a**3*b**2*x*sqrt(-a/b) + 8*a**2*b**3*
x**2*sqrt(-a/b)) + 3*b**2*x**2*log(sqrt(x) - sqrt(-a/b))/(8*a**4*b*sqrt(-a/b) + 16*a**3*b**2*x*sqrt(-a/b) + 8*
a**2*b**3*x**2*sqrt(-a/b)) - 3*b**2*x**2*log(sqrt(x) + sqrt(-a/b))/(8*a**4*b*sqrt(-a/b) + 16*a**3*b**2*x*sqrt(
-a/b) + 8*a**2*b**3*x**2*sqrt(-a/b)), True))

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Giac [A]
time = 0.46, size = 47, normalized size = 0.82 \begin {gather*} \frac {3 \, \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{2}} + \frac {3 \, b x^{\frac {3}{2}} + 5 \, a \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^2) + 1/4*(3*b*x^(3/2) + 5*a*sqrt(x))/((b*x + a)^2*a^2)

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Mupad [B]
time = 0.12, size = 57, normalized size = 1.00 \begin {gather*} \frac {\frac {5\,\sqrt {x}}{4\,a}+\frac {3\,b\,x^{3/2}}{4\,a^2}}{a^2+2\,a\,b\,x+b^2\,x^2}+\frac {3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,a^{5/2}\,\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(a + b*x)^3),x)

[Out]

((5*x^(1/2))/(4*a) + (3*b*x^(3/2))/(4*a^2))/(a^2 + b^2*x^2 + 2*a*b*x) + (3*atan((b^(1/2)*x^(1/2))/a^(1/2)))/(4
*a^(5/2)*b^(1/2))

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