3.5.86 \(\int \frac {1}{\sqrt {x} (-a+b x)^3} \, dx\) [486]

Optimal. Leaf size=72 \[ -\frac {\sqrt {x}}{2 a (a-b x)^2}-\frac {3 \sqrt {x}}{4 a^2 (a-b x)}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {44, 65, 214} \begin {gather*} -\frac {3 \tanh ^{-1}\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]

-1/2*Sqrt[x]/(a*(a - b*x)^2) - (3*Sqrt[x])/(4*a^2*(a - b*x)) - (3*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(5/
2)*Sqrt[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 214

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

Rubi steps

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

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Mathematica [A]
time = 0.08, size = 60, normalized size = 0.83 \begin {gather*} \frac {\sqrt {x} (-5 a+3 b x)}{4 a^2 (a-b x)^2}-\frac {3 \tanh ^{-1}\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*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(5/2)*Sqrt[b])

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Maple [A]
time = 0.12, size = 61, normalized size = 0.85

method	result	size
derivativedivides	\(-\frac {\sqrt {x}}{2 a \left (-b x +a \right )^{2}}-\frac {3 \left (\frac {\sqrt {x}}{2 a \left (-b x +a \right )}+\frac {\arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}\right )}{2 a}\)	\(61\)
default	\(-\frac {\sqrt {x}}{2 a \left (-b x +a \right )^{2}}-\frac {3 \left (\frac {\sqrt {x}}{2 a \left (-b x +a \right )}+\frac {\arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}\right )}{2 a}\)	\(61\)

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)*arctanh(b*x^(1/2)/(a*b)^(1/2)))

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Maxima [A]
time = 0.51, size = 77, normalized size = 1.07 \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 \, \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{8 \, \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) + 3/8*log((b*sqrt(x) - sqrt(a*b))/(b*sqrt(x) +
sqrt(a*b)))/(sqrt(a*b)*a^2)

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Fricas [A]
time = 0.44, size = 185, normalized size = 2.57 \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(sqrt(-
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. 580 vs. \(2 (63) = 126\).
time = 12.33, size = 580, normalized size = 8.06 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{5 b^{3} x^{\frac {5}{2}}} & \text {for}\: a = 0 \\- \frac {2 \sqrt {x}}{a^{3}} & \text {for}\: b = 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/(5*b**3*x**(5/2)), Eq(a, 0)), (-2*sqrt(x)/a**3, Eq(b, 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 = 0.96, size = 51, normalized size = 0.71 \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 = 0.13, size = 58, normalized size = 0.81 \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 {atanh}\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*atanh((b^(1/2)*x^(1/2))/a^(1/2)))
/(4*a^(5/2)*b^(1/2))

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