3.5.85 \(\int \frac {\sqrt {x}}{(-a+b x)^3} \, dx\) [485]

Optimal. Leaf size=75 \[ -\frac {\sqrt {x}}{2 b (a-b x)^2}+\frac {\sqrt {x}}{4 a b (a-b x)}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {43, 44, 65, 214} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}}+\frac {\sqrt {x}}{4 a b (a-b x)}-\frac {\sqrt {x}}{2 b (a-b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(-a + b*x)^3,x]

[Out]

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

Rule 43

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

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

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Mathematica [A]
time = 0.08, size = 60, normalized size = 0.80 \begin {gather*} -\frac {\sqrt {x} (a+b x)}{4 a b (a-b x)^2}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(-a + b*x)^3,x]

[Out]

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

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 8.41, size = 580, normalized size = 7.73 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\text {DirectedInfinity}\left [\frac {1}{x^{\frac {3}{2}}}\right ],a\text {==}0\text {\&\&}b\text {==}0\right \},\left \{\frac {-2}{3 b^3 x^{\frac {3}{2}}},a\text {==}0\right \},\left \{\frac {-2 x^{\frac {3}{2}}}{3 a^3},b\text {==}0\right \}\right \},-\frac {a^2 \text {Log}\left [\sqrt {x}-\sqrt {\frac {a}{b}}\right ]}{8 a^3 b^2 \sqrt {\frac {a}{b}}-16 a^2 b^3 x \sqrt {\frac {a}{b}}+8 a b^4 x^2 \sqrt {\frac {a}{b}}}+\frac {a^2 \text {Log}\left [\sqrt {x}+\sqrt {\frac {a}{b}}\right ]}{8 a^3 b^2 \sqrt {\frac {a}{b}}-16 a^2 b^3 x \sqrt {\frac {a}{b}}+8 a b^4 x^2 \sqrt {\frac {a}{b}}}-\frac {2 a b \sqrt {x} \sqrt {\frac {a}{b}}}{8 a^3 b^2 \sqrt {\frac {a}{b}}-16 a^2 b^3 x \sqrt {\frac {a}{b}}+8 a b^4 x^2 \sqrt {\frac {a}{b}}}-\frac {2 a b x \text {Log}\left [\sqrt {x}+\sqrt {\frac {a}{b}}\right ]}{8 a^3 b^2 \sqrt {\frac {a}{b}}-16 a^2 b^3 x \sqrt {\frac {a}{b}}+8 a b^4 x^2 \sqrt {\frac {a}{b}}}+\frac {2 a b x \text {Log}\left [\sqrt {x}-\sqrt {\frac {a}{b}}\right ]}{8 a^3 b^2 \sqrt {\frac {a}{b}}-16 a^2 b^3 x \sqrt {\frac {a}{b}}+8 a b^4 x^2 \sqrt {\frac {a}{b}}}-\frac {2 b^2 x^{\frac {3}{2}} \sqrt {\frac {a}{b}}}{8 a^3 b^2 \sqrt {\frac {a}{b}}-16 a^2 b^3 x \sqrt {\frac {a}{b}}+8 a b^4 x^2 \sqrt {\frac {a}{b}}}-\frac {b^2 x^2 \text {Log}\left [\sqrt {x}-\sqrt {\frac {a}{b}}\right ]}{8 a^3 b^2 \sqrt {\frac {a}{b}}-16 a^2 b^3 x \sqrt {\frac {a}{b}}+8 a b^4 x^2 \sqrt {\frac {a}{b}}}+\frac {b^2 x^2 \text {Log}\left [\sqrt {x}+\sqrt {\frac {a}{b}}\right ]}{8 a^3 b^2 \sqrt {\frac {a}{b}}-16 a^2 b^3 x \sqrt {\frac {a}{b}}+8 a b^4 x^2 \sqrt {\frac {a}{b}}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[Sqrt[x]/(-a + b*x)^3,x]')

[Out]

Piecewise[{{DirectedInfinity[1 / x ^ (3 / 2)], a == 0 && b == 0}, {-2 / (3 b ^ 3 x ^ (3 / 2)), a == 0}, {-2 x
^ (3 / 2) / (3 a ^ 3), b == 0}}, -a ^ 2 Log[Sqrt[x] - Sqrt[a / b]] / (8 a ^ 3 b ^ 2 Sqrt[a / b] - 16 a ^ 2 b ^
3 x Sqrt[a / b] + 8 a b ^ 4 x ^ 2 Sqrt[a / b]) + a ^ 2 Log[Sqrt[x] + Sqrt[a / b]] / (8 a ^ 3 b ^ 2 Sqrt[a / b
] - 16 a ^ 2 b ^ 3 x Sqrt[a / b] + 8 a b ^ 4 x ^ 2 Sqrt[a / b]) - 2 a b Sqrt[x] Sqrt[a / b] / (8 a ^ 3 b ^ 2 S
qrt[a / b] - 16 a ^ 2 b ^ 3 x Sqrt[a / b] + 8 a b ^ 4 x ^ 2 Sqrt[a / b]) - 2 a b x Log[Sqrt[x] + Sqrt[a / b]]
/ (8 a ^ 3 b ^ 2 Sqrt[a / b] - 16 a ^ 2 b ^ 3 x Sqrt[a / b] + 8 a b ^ 4 x ^ 2 Sqrt[a / b]) + 2 a b x Log[Sqrt[
x] - Sqrt[a / b]] / (8 a ^ 3 b ^ 2 Sqrt[a / b] - 16 a ^ 2 b ^ 3 x Sqrt[a / b] + 8 a b ^ 4 x ^ 2 Sqrt[a / b]) -
2 b ^ 2 x ^ (3 / 2) Sqrt[a / b] / (8 a ^ 3 b ^ 2 Sqrt[a / b] - 16 a ^ 2 b ^ 3 x Sqrt[a / b] + 8 a b ^ 4 x ^ 2
Sqrt[a / b]) - b ^ 2 x ^ 2 Log[Sqrt[x] - Sqrt[a / b]] / (8 a ^ 3 b ^ 2 Sqrt[a / b] - 16 a ^ 2 b ^ 3 x Sqrt[a
/ b] + 8 a b ^ 4 x ^ 2 Sqrt[a / b]) + b ^ 2 x ^ 2 Log[Sqrt[x] + Sqrt[a / b]] / (8 a ^ 3 b ^ 2 Sqrt[a / b] - 16
a ^ 2 b ^ 3 x Sqrt[a / b] + 8 a b ^ 4 x ^ 2 Sqrt[a / b])]

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Maple [A]
time = 0.11, size = 53, normalized size = 0.71

method	result	size
derivativedivides	\(-\frac {2 \left (\frac {x^{\frac {3}{2}}}{8 a}+\frac {\sqrt {x}}{8 b}\right )}{\left (-b x +a \right )^{2}}+\frac {\arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 a b \sqrt {a b}}\)	\(53\)
default	\(-\frac {2 \left (\frac {x^{\frac {3}{2}}}{8 a}+\frac {\sqrt {x}}{8 b}\right )}{\left (-b x +a \right )^{2}}+\frac {\arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 a b \sqrt {a b}}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(1/8/a*x^(3/2)+1/8*x^(1/2)/b)/(-b*x+a)^2+1/4/a/b/(a*b)^(1/2)*arctanh(b*x^(1/2)/(a*b)^(1/2))

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Maxima [A]
time = 0.34, size = 80, normalized size = 1.07 \begin {gather*} -\frac {b x^{\frac {3}{2}} + a \sqrt {x}}{4 \, {\left (a b^{3} x^{2} - 2 \, a^{2} b^{2} x + a^{3} b\right )}} - \frac {\log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{8 \, \sqrt {a b} a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(b*x^(3/2) + a*sqrt(x))/(a*b^3*x^2 - 2*a^2*b^2*x + a^3*b) - 1/8*log((b*sqrt(x) - sqrt(a*b))/(b*sqrt(x) +
sqrt(a*b)))/(sqrt(a*b)*a*b)

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Fricas [A]
time = 0.32, size = 183, normalized size = 2.44 \begin {gather*} \left [\frac {{\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 (a b^{2} x + a^{2} b\right )} \sqrt {x}}{8 \, {\left (a^{2} b^{4} x^{2} - 2 \, a^{3} b^{3} x + a^{4} b^{2}\right )}}, -\frac {{\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 (a b^{2} x + a^{2} b\right )} \sqrt {x}}{4 \, {\left (a^{2} b^{4} x^{2} - 2 \, a^{3} b^{3} x + a^{4} b^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 7.10, size = 575, normalized size = 7.67 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {3}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{3 b^{3} x^{\frac {3}{2}}} & \text {for}\: a = 0 \\- \frac {2 x^{\frac {3}{2}}}{3 a^{3}} & \text {for}\: b = 0 \\- \frac {a^{2} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {\frac {a}{b}} - 16 a^{2} b^{3} x \sqrt {\frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {\frac {a}{b}}} + \frac {a^{2} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {\frac {a}{b}} - 16 a^{2} b^{3} x \sqrt {\frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {\frac {a}{b}}} - \frac {2 a b \sqrt {x} \sqrt {\frac {a}{b}}}{8 a^{3} b^{2} \sqrt {\frac {a}{b}} - 16 a^{2} b^{3} x \sqrt {\frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {\frac {a}{b}}} + \frac {2 a b x \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {\frac {a}{b}} - 16 a^{2} b^{3} x \sqrt {\frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {\frac {a}{b}}} - \frac {2 a b x \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {\frac {a}{b}} - 16 a^{2} b^{3} x \sqrt {\frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {\frac {a}{b}}} - \frac {2 b^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b}}}{8 a^{3} b^{2} \sqrt {\frac {a}{b}} - 16 a^{2} b^{3} x \sqrt {\frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {\frac {a}{b}}} - \frac {b^{2} x^{2} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {\frac {a}{b}} - 16 a^{2} b^{3} x \sqrt {\frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {\frac {a}{b}}} + \frac {b^{2} x^{2} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {\frac {a}{b}} - 16 a^{2} b^{3} x \sqrt {\frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {\frac {a}{b}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 74, normalized size = 0.99 \begin {gather*} -2 \left (-\frac {-\sqrt {x} x b-\sqrt {x} a}{8 b a \left (x b-a\right )^{2}}+\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{4 b a\cdot 2 \sqrt {-a b}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.14, size = 57, normalized size = 0.76 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,a^{3/2}\,b^{3/2}}-\frac {\frac {x^{3/2}}{4\,a}+\frac {\sqrt {x}}{4\,b}}{a^2-2\,a\,b\,x+b^2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x^(1/2)/(a - b*x)^3,x)

[Out]

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

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